Radiation in the atmosphere (International Geophysics, 12)

RADIATION IN THE ATMOSPHERE lnternutiond Geophysics Series Edited by J. VAN MIEGHEM Royal Belgian Meteorological Ins...

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RADIATION IN THE ATMOSPHERE

lnternutiond Geophysics Series Edited by

J. VAN MIEGHEM Royal Belgian Meteorological Institute Uccle, Belgium Volume 1 BENOGUTENBERG. Physics of the Earth's Interior. 1959 Volume 2 JOSEPHW. CHAMBERLAIN. Physics of the Aurora and Airglow. 1961

S. K. RUNCORN (ed.). Continental Drift. 1962 C. E. JUNGE.Air Chemistry and Radioactivity. 1963 ROBERTC. FLEAGLE AND JOOSTA. BUSINGER. An Introduction to Atmospheric Physics. 1963 Volume 6 L. DUFOUR AND R. DEFAY.Thermodynamics of Clouds. 1963 Volume 7 H. U. ROLL.Physics of the Marine Atmosphere. 1965 Volume 8 RICHARD A. CRAIG.The Upper Atmosphere: Meteorology and Physics. 1965 Volume 9 WILLISL. WEBB.Structure of the Stratosphere and Mesosphere. 1966 Volume 10 MICHELECAPUTO.The Gravity Field of the Earth from Classical and Modern Methods. 1967 Volume 11 S. MATSUSHITA AND WALLACE H. CAMPBELL(eds.). Physics of Geomagnetic Phenomena. (In two volumes.) 1967 Volume 12 K. YA.KONDRATYEV. Radiation in the Atmosphere. 1969. Volume 3 Volume 4 Volume5

In preparation ERIKH. PALMEN A N D CHESTER W. NEWTON. Atmosphere Circulation Systems: Their Structure and Physical Interpretation. A N D OWENK. GARRIOTT. Introduction to IonoHENRYRISHBETH spheric Physics.

RADIATION IN THE ATMOSPHERE K. YA. K O N D R A T Y E V DEPARTMENT OF ATMOSPHERIC LENINGRAD UNIVERSITY LENINGRAD, U.S.S.R.

@

PHYSICS

1969

ACADEMIC PRESS

New York and London

COPYRIGHT@ 1969,

BY

ACADEMICPRESS,INC.

ALL RIGHTS RESERVED. NO PART OF THIS BOOK MAY BE REPRODUCED I N ANY FORM, BY PHOTOSTAT, MICROFILM, OR ANY OTHER MEANS, WITHOUT WRITTEN PERMISSION FROM THE PUBLISHERS.

ACADEMIC PRESS, INC. 111 Fifth Avenue, New York, New York 10003

United Kingdom Edition published by ACADEMIC PRESS, INC. (LONDON) LTD. Berkeley Square House, London W.l

LIBRARY OF CONGRESS CATALOG CARDNUMBER: 67-23163

PRINTED IN THE UNITED STATES OF AMERICA

To the memory of my mother and sister

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EDITOR’S PREFACE

I wish to thank Professor K. Ya. Kondratyev, rector of the University of Leningrad, for having kindly accepted the invitation to prepare in English the original manuscript for this volume. I agreed with the publishers that the author’s style should be preserved rather than risk the introduction of possible inaccuracies through excessive editing of this large and detailed work. The substance of such a work is what is of primary importance; the aim of the International Geophysics Series is to make known the important work in geophysics carried out everywhere in the world. Thanks to this excellent volume, we have not only a comprehensive view of this very important subject but an opportunity to become acquainted with the work achieved in this field by the U.S.S.R., to which the author has made a major contributions. J. VAN MIEGHAM January, 1969

vii

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PREFACE

The investigation of radiative processes in the earth’s atmosphere is becoming more and more factual. Solar and terrestrial radiation, the only known sources of energy determining atmospheric circulation, have always been of great importance to meteorology and to atmospheric physics. The progress achieved in this field during recent years has been chiefly due to the success of meteorological satellites in studying the regularities of the global distribution of outgoing radiation in various spectral regions and of the planetary heat reservoir. Although the detailed mechanism of energy transformation is not yet known (which limits the value of the radiation factors in formulating an exact theory of general circulation and climate), the prospect of obtaining the first data about the earth’s heat reservoir on a planetary scale is highly attractive. No less important to atmospheric physics is the fact that the recent radiation studies employed (along with classical descriptive and observational methods) modern physical and mathematical techniques. As a result, the experimental approaches have been greatly enhanced. Among the instrumental techniques, the use of the new high resolution spectrometers (interferometric spectrometers in their number) and of the lasers has contributed a major means of evaluation. Many theoretical investigations, in their turn, have been made possible only by the application of large computers with extensive memory banks. The correct representation of radiative transfer taxes the capabilities of even the most advanced computers. For example, the consideration of the radiation factors in the numerical experiments on general atmospheric circulation, performed by J. Smagorinski and collaborators, consumed about three-quarters of the entire computing time. Radiation studies have always had important applications in biology. ix

X

Preface

Today these studies are of special interest in connection with the problem of photosynthesis. Transportation, building, and architecture also reserve an interest in the various aspects of radiative transfer in the atmosphere, and recently space navigation and orientation have become involved. The demands of the latter have been satisfied, for example, in the solution of the problem of the orientation of satellites relative to the earth, beyond the infrared spectrum. This book is written from the standpoint of atmospheric physics and meteorology. Its purpose is to coordinate an accurate physical approach and the descriptive empirical results that are important in meteorological practice. At the same time an attempt is made to present the material in a form that will be sufficiently universal to be used with any of the mentioned subjects. It is natural that the large volume and variability of the material considered make the presentation of certain aspects purely informative. “Radiation in the Atmosphere” has been preceded by several monographs devoted to various aspects of radiative transfer in the atmosphere, of which “Actinometry” (Gidrometeoizdat, Leningrad, 1965) was the first presentation of the problems considered in this publication. For the details of certain problems that are not fully treated within this monograph, the author refers the interested reader to Professor R. M. Goody’s excellent work on the theory of radiative transfer (“Atmospheric Radiation,” Oxford Univ. Press, London and New York, 1964). Those who wish to become acquainted with the technique of standard actinometric measurements may look for Y. D. Yanishevsky’s “Actinometric Instruments and Methods of Observation” (Gidrometeoizdat, Leningrad, 1957) and for “Rayonnement Solaire et Gchanges Radiatifs Naturels. Mtthodes Actinometriques,” by Ch. Perrin de Brichambaut (Gauthier-Villars, Paris, 1963). The monograph of K. Ya. Kondratyev, 0. A. Avaste, M. P. Fedorova, and K. E. Yakushevskaya, “The Radiative Field of the Earth as a Planet” (Gidrometeoizdat, Leningrad, 1967), contains a special review of corresponding information. The author expresses his gratitude to his colleagues for their contributions to the present volume: Y. K. Ross, S. V. Ashcheulov, and V. V. Mikhailov, Chapter 2, E. M. Feugelson, Chapter 6, Section 3.6 Z. F. Mironova, Chapter 7, M. P. Fedorova, Chapter 8, Section 4, S. V. Ashcheulov and D. B. Styro, Chapter 9, Section 8, Y. M. Timofeyev, Chapter 9, Section 9,

Preface

xi

to 0. P. Filipovich whose supervision was important in the preparation of the manuscript, and to Larissa Kondratyeva for translating the manuscript from the Russian. The collaboration with Professor J. Van Mieghem, who acts as a scientific editor of this series of monographs, has always been most pleasant and fruitful. The author realizes that his work is in many aspects incomplete. The reader’s remarks and wishes can have an important role in the further betterment of the book. Toward this end, the author would appreciate receiving criticisms and comments. These may be directed to him at The University, Leningrad, U.S.S.R. K. YA. KONDRATYEV Leningrad January, 1969

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CONTENTS

Editor's Preface

vii

Preface

ix

1. Radiant Energy. The Main Concepts and Definitions

1

1.1. The Sun as the Source of Radiation. The Concept of Stellar Temperatures 1.2. General Information on Radiant Fluxes in the Atmosphere 1.3. Main Quantitative Characteristics of a Radiation Field 1.4. Fundamental Laws of Thermal Radiation

1.5. Thermal Emission of Real Bodies 1.6. The Equation of Radiative Transfer for a Stationary Radiation Field References

2. Methods of Actinometric Measurements 2.1. General Characteristic of the Methods for Measurement of Radiant Energy 2.2. Instruments for Measuring Direct Solar Radiation 2.3. Instruments for Measuring Global and Diffuse Radiation and Albedo 2.4. Instruments for Measuring Brightness and Illumination 2.5. Instruments for Measuring Radiation Balance and Effective Radiation 2.6. Main Types of Instruments for Spectral Measurements 2.7. Instruments for Measuring Atmospheric Thermal Emission 2.8. Instruments for Measuring Shortwave Radiation Fluxes References xiii

1 6 9 22 35 43 47

49

49 52 60 65 66 72 77 79 83

xiv

Contents

3. Radiation Absorption in the Atmosphere 3.1. 3.2. 3.3. 3.4. 3.5.

General Principles of Selective Radiation Absorption The Absorption Spectrum of Water and Water Vapor The Abs,orption Spectrum of CO, The Absorption Spectra of Ozone and Oxygen General Characteristic of Minor Radiation-Absorbing Components of the Atmosphere 3.6. The Integral Transmission Function of the Atmosphere for Thermal Radiation 3.7. Absorption Spectroscopy of the Atmosphere as a Method for Investigation of the Atmospheric Composition References

4. Scattering of Radiation in the Atmosphere 4.1. 4.2. 4.3. 4.4. 4.5. 4.6. 4.7.

5.2. 5.3. 5.4. 5.5.

5.6. 5.7. 5.8. 5.9. 5.10. 5.11.

86 107 123 132 139 141 144 151

161

The Solar Ray Path in the Atmosphere 161 Scattering of Radiation (General Considerations) 169 171 Rayleigh Scattering 180 Scattering of Radiation on Large Particles Computation of the Attenuation in the Atmosphere due to Scattering 194 Elementary Theory of Radiative Transfer, Including Multiple Scattering 200 209 Radiation Scattering and the Structure of Atmospheric Aerosol 212 References

5. Direct Solar Radiation 5.1.

85

Distribution of Energy in the Solar Spectrum at the Earth’s Surface Level Spectral Atmospheric Transparency Energy Distribution in the Solar Spectrum outside the Atmosphere The Solar Constant Total Attenuation of Solar Radiation in an Ideal Atmosphere Quantitative Characteristics of the Real Atmospheric Transparency Some Data of Observations on the Variation of the Atmospheric Transparency State Attenuation of Solar Radiation by Clouds Theoretical Calculations of Irradiation of the Earth’s Surface by the Sun Temporal and Spatial Variability of Fluxes and Totals of Solar Radiation Income of Solar Radiation on Slant Surfaces References

217

217 234 245 252 260 263 283 300 304 317 342 355

Contents

6. Diffuse Radiation of the Atmosphere

xv 363

6.1. Energy Distribution in the Spectrum of Diffuse Radiation 363 6.2. Angular Distribution of Diffuse Radiation Intensity 368 376 6.3. Fluxes of Diffuse Radiation 6.4. The Main Observed Regularities of the Variability of Diffuse Radiation 400 Totals References 408

7. Albedo of the Underlying Surface and Clouds 7.1. Spectral Albedo of Natural Underlying Surfaces 7.2. Albedo of Various Continental Underlying Surfaces 7.3. Albedo of Water Basins 7.4. Albedo of Clouds 7.5. Geographical Distribution of Albedo References

8. Global Radiation

41 1 41 1 422 431 440 444 449

453

8.1. Energy Distribution in the Spectrum of Global Radiation 453 451 8.2.Fluxes of Global Radiation 8.3. The Main Observed Regularities in the Variability of Global Radiation 410 Totals 485 8.4. Incoming Shortwave Radiation on Oriented Slant Surfaces 8.5. Income of Global Radiation Under Vegetative Covers 502 517 8.6. Penetration of Radiant Energy into Water, Ice, and Snow 531 8.7. Illumination References 532

9. Thermal Radiation of the Atmosphere

538

9.1. The Phenomenological Theory of the Transfer of Thermal Radiation in the Atmosphere

538

9.2. Approximate Transfer Equations and Their Use for Calculating Thermal Radiative Transfer in the Atmosphere 9.3. Radiation Charts 9.4. Radiative Heat Transfer in Clouds 9.5. Effective Radiation of the Underlying Surface and the Downward Atmospheric Radiation 9.6. Angular Distribution of Intensity of Effective Radiation and Atmospheric Emission over the Celestial Sphere

546 551 558 559 584

xvi

Contents 9.7. Some Practical Applications of Data on the Angular Distribution of the

Intensities of Effective Radiation and Atmospheric Emission

599

9.8. Distribution of Energy in the Spectrum of Effective Radiation and Down-

ward Atmospheric Radiation

617

9.9. Emission Spectroscopy as a Means of Investigating the Structure and

Composition of the Atmosphere References

10. Net Radiation

624 646

655

10.1. Observed Regularities in Variation of Net Radiation of the Underlying

Surface Results of Calculations of Net Radiation at the Underlying Surface The Net Radiation of Slopes Net Radiation and Its Components in a Free Atmosphere Climatology of Net Radiation of the Earth 10.6. Investigations of the Earth’s Net Radiation by Means of Satellites 10.7. Statistical Features of the Net Radiation of the Earth-Atmosphere System References 10.2. 10.3. 10.4. 10.5.

657 672 680 686 728 750 765 782

11. Temperature Variation in the Atmosphere Due to Radiative Heat Exchange 79 1 11.1. Equation for the Heat Inflow

79 1

11.2. Methods for Calculating Radiative Heat Inflow 11.3. Results of Calculations of Radiative Flux Divergence

795 804

11.4. Radiation Factors in the Thermal Regime of the Stratosphere and

Mesosphere

814

11.5. Relation between Radiative and Turbulent Heat Exchange in the Surface

Layer of the Atmosphere References

832

ADDITIONAL BIBLIOGRAPHY

837

APPENDIXES

860

Author Index

889

Subject Index

907

829

RADIANT ENERGY. THE MAIN CONCEPTS AND DEFINITIONS

1.1. The Sun as the Source of Radiation. The Concept of Stellar Temperatures The radiant energy of the sun is practically the only source of energy that influences atmospheric motions and the many various processes in the atmosphere and surface layers of the Earth’s crust. Table 1.1, made up from Gerson’s [20] data, gives a spectacular presentation of the degree to which solar radiation exceeds all other sources of energy. TABLE 1.1 Sources of Energy for the Atmosphere (the Values of Energy Are Determined for the Earth as a Whole). After Gerson [20]

Energy Source of Energy erg/sec

1.76 x 3.09 x 1.60 x 2.61 x 2.53 x 1.63 x 1.44 x 1.12 x

Sun Moon (the full moon) Lightnings Stellar light Bright Aurora Borealis Cosmic rays Meteors Luminosity of the night sky

1

1024

1019 1019 1017 1017 1017 1017 1017

Relative to Sun

1 1.76 x 9.09 x 1.48 x 10-7 1.44 x 10-7 9.26 x 8.18 x 10-8 6.37 x

2

Radiant Energy. The Main Concepts and Definitions

As seen from the table even the maximum (in comparison with all other) radiant energy from the moon constitutes less than 0.002 percent in relation to solar radiation. Detailed treatment of the physics of the sun can be found in numerous publications on astronomy and astrophysics. A brief description follows : The sun is the star nearest to us. It is a huge gaseous globe whose temperature at the radiating surface is about 6000’K. The internal parts of the sun are at still higher temperatures, reaching 40,000,000°. The following data characterize the dimensions of the sun : 139.14 x lo4 km

Diameter Surface area

6.093 x 10l2km2

Volume

1.412 x 10l8km3

The geometrical dimensions, mass, and density of the sun, in relation to the corresponding values for the earth, are characterized by the following figures : Diameter Surface Volume Mass Density

109.05 11,918 1,301,000 332,488 0.255

The distance from the earth to the sun varies within a year, averaging 1.4953 x lo* km. This mean earth to sun distance corresponds to the mean visible solar angular diameter, equal to 31‘59.3”. The variation of the earth to sun distance can be followed with the help of Table 1.2. TABLE 1.2 Variation of the Earth to Sun Distance ~

Season

Distance, km

Jan. 1

147,001,000 (min.)

Apr. 1

149,501,000

July 1

152,003,000 (max.)

Oct. 1

149,501, OOO

1.1. The Sun as the Source of Radiation

3

As seen from this table, the variation of the earth to sun distance within a year is 5,000,000 km. Annual fluctuations of the income of solar radiation on the earth’s surface, constituting f3.5 percent of the value of solar radiation averaged over a year, are the sequel of the earth to sun distance variation. The composition of the sun is very complicated. The outer layer directly accessible for observation is called the photosphere. The photosphere is enveloped by a luminous but almost transparent solar atmosphere consisting of much rarefied gases. The sun’s atmosphere is composed of two layers. The inner layer is called chromosphere and extends about 12.000 to 14,000 km. A greatly extended corona constitutes the outer part of the solar atmosphere. In the direction from the photosphere to the corona the temperature of the solar atmosphere rapidly increases. Modern optical observations, as well as observations of radio-emission, show that the temperature of the corona is l,OOO,OOOo, and this high temperature extends at least the distance of two solar radii from the surface of the sun. The existence of the extended solar corona, possessing high temperatures, has enabled Chapman to suggest a hypothesis about the outer part of the earth’s atmosphere (the so-called exosphere) as being heated by the high-temperature coronal gas. According to this hypothesis, the earth is enveloped not by cold space but by a strongly rarefied (though “incandescent”) coronal gas, the temperature of which in the vicinity of several terrestrial radii is about 200,000°. Quantitative evaluation show that the flux of energy consumed in heating the outer part of the earth’s atmosphere is from several tenths to several ergs per square centimeters per second. Both at the surface of the sun and in its atmosphere, stormy motions of gaseous masses take place so that the solar surface is not homogeneous. To particularly essential phenomena of the solar surface belong the so-called sunspots, dark formations on the surface of the sun. Measurements of their temperature have shown the sun’s surface to be at a significantly lower temperature than that of the photosphere (about 4500’K). The sunspotforming activity at the surface of the photosphere is very unstable; however, numerous observations have discovered that it possesses a number of cyclic phenomena, of which the best known is the 11-year cycle of variation in the number of sunspots. A number of indices has been proposed for quantitative characteristic of variations in the sunspot-forming activity of the sun. The most widely used is the Wolf number R, determined by the relation

4

Radiant Energy. The Main Concepts and Definitions

where

f

= number

of isolated spots g = number of groups of spots k = empirical coefficient

It has been found from observations that the Wolf number correlates well with many geophysical phenomena, which gives evidence of the existence of the influence of the sunspot-forming activity on atmospheric processes. It was mentioned above that the solar surface temperature is about 6000'K. This value is the so-called effective temperature of the sun. Besides the effective temperature, there exist several other characteristics of the thermal state of stars in general and the sun in particular. All these characteris tics are known as stellar temperatures. Since they find their application in actinometry as well, we give a brief definition of some of them.

The effective temperature T, of a star or any other source of radiant energy is determined by F = 0Te4 (1.2) where

F

= flux

of radiant energy from the object, the effective temperature of which is being determined cr = Stefan-Boltzmann constant

As seen from the equation, the effective temperature serves for comparison of the radiative flux of a given object with that of a perfect blackbody. The color temperature of a source of radiation, T,, is determined as the temperature at which the spectral distribution of the radiant intensity of the blackbody coincides in the best possible way with that of a given object, within a narrow frequency interval (in such determination the concept of color temperature is valid throughout the entire wavelength spectrum). The radiation temperature T, is a third stellar temperature of interest to actinometry. The determination of the radiation temperature TE is given by

Fa B,

= monochromatic

flux of radiation of a given object = monochromatic flux of radiation of the blackbody at the temperature TE

5

1.1. The Sun as the Source of Radiation

We see here that this temperature value is somewhat analogous to the effective temperature, the essential difference being that the latter is determined by comparing the full radiant fluxes of the given object with that of the blackbody, whereas the former is determined from the comparison of monochromatic fluxes. Consider now the numerical values of the mentioned stellar temperatures for the sun. According to the most recent data, the precise value of the effective temperature of the sun is 5784'K. The color temperature of the sun, determined in the above way, has the value T, = 7140'K for the spectral region 0.4 to 0.7 p. For the ultraviolet (A < 0.4 p), it equals 4850'K. The data for solar radiation temperature are given in Table 1.3. For wavelengths shorter than those of Table 1.3, the solar radiation temperature significantly decreases (the sun is much "colder" in the ultraviolet than in the visible region of the spectrum). TABLE 1.3 The Radiation Temperature of the Sun

Radiation Temperature, OK Wavelength, p Center of Solar Disk

Total Radiation Emitted by Solar Disk

0.7

6000

5800

0.55

6300

6100

0.45

6400

6200

Rocket investigations in the ultraviolet spectrum of the sun have produced radiation temperature values of the order of 4000 to 5000'K. The solar radiation temperature in the infrared also has lower values than those in Table 1.3 (as, for example, according to the data of Saiedy and Goody [21], TE = 5036'K at = 11.1 p). As to the radiation temperature for the region of solar radio-emission (centimeter-meter radio waves), it has an order of value of 106'K. In Fig. 1.1 is given a curve of the spectral distribution of the radiation temperature from the author's [22] measurement data and the available published results. The data given here for stellar temperatures present a great variety of numerical values of these quantities, which shows that the sun as a ra-

6

Radiant Energy. The Main Concepts and Definitions

50

x - l

1

1

1

1

+-2

0-7 0-8

FIG. 1.1 The dependence of the solar radiation temperature upon wave Iength. After Kondratyev et al. [22]. (1) Peyturaux (1952); (2) Saiedy (1959); (3) Farmer and Todd (1961); (4) Leningrad State University (1963); ( 5 ) Naumov (1963); (6) Coates (1958); (7) Troitzky, Zelinskaya (1958); (9) Kovington (1954); (10) Talbert, Stratton (1961); (11) Murcray et al. (1964).

diator differs noticeably from a blackbody, for if it did not, the values of all stellar temperatures for the sun would have to be equal. 1.2. General Information on Radiant Fluxes in the Atmosphere The radiant energy of the sun as it passes through the atmosphere undergoes complicated transformations. On the way from the outer boundary of the atmosphere to the earth’s surface, absorption and scattering of radiant energy take place. Because of the scattering of radiant energy, we observe a t the surface level not only direct solar radiation in the form of parallel beams of the sun’s rays, but also dzxuuse radiation, falling from every point of the sky. Direct solar radiation and diffuse radiation comprise global radiation. On reaching the earth, the global radiation is partly reflected by the earth’s surface and a flux of reflected radiation thus ap-

1.2. General Information on Radiant Fluxes in the Atmosphere

7

pears. The unreflected part of direct solar radiation and diffuse radiation is absorbed by the earth’s surface and constitutes the absorbed radiation. The heat released in the absorption of the global radiation then heats the soil. The heated surface of the soil becomes a source of the thermal radiation of the earth’s surface, directed to the atmosphere. The atmosphere is in turn heated in the course of the heat exchange (mainly turbulent) with the earth‘s surface, and then also emits thermal radiation surfaceward (downward atmospheric radiation) and spaceward (upward atmospheric radiation). Since the relative emissivity (and, correspondingly, absorptivity) of the earth’s surface 6 is less than unity, reflection of the downward atmospheric radiation by the earth’s surface takes place, and so it is necessary to take into consideration the presence of the rejected atmospheric radiation. For practical purposes it is most important to determine the value of the radiative heat exchange between the earth’s surface and the atmosphere. This value is characterized by the concept of the effective radiation. By the efective radiation of the earth’s surface is meant the difference between the thermal radiation of the earths’ surface and the part of the downward atmospheric radiation absorbed by the earth’s surface. However, sometimes another way of determining the effective radiation is to take the difference between the upward and downward radiant fluxes at the surface level. We shall see later that both methods are identical. The difference between the absorbed global radiation and the effective radiation is called the net radiation of the underlying surface. Similarly, it is possible to introduce the concepts of the net radiation of the atmosphere and the surface-atmosphere system. These quantities will be determined in Chapter 10. Thus we observe in the atmosphere a whole system of radiant fluxes. These radiant fluxes possess an essential peculiarity in that they have different spectral compositions. It is known that the spectral composition of a radiant flux is characterized by the distribution of radiant energy according to wavelengths. Wavelengths of electromagnetic radiation are usually measured in the following units : 1 ,LA (micron) = cm = mm mm 1 mp (millimicron) = lo-’ cm = 1 A (Angstrom) = 10-8 cm = 10-7 mm

1p

=

10,0008,

=

1,OoOmp

Since wavelengths of the radiations met with in nature vary within very wide limits, the entire electromagnetic spectrum is divided into several regions. The spectral region corresponding to wavelengths il < 0.4,; is called

8

Radiant Energy. The Main Concepts and Definitions

ultraviolet (UV). Often the whole of the ultraviolet is subdivided into the near (0.4 to 0.3 p), f a r (0.3 to 0.2 y ) , and the vacuum (2.10-1 to p) regions of the ultraviolet. The division of the radiation spectrum from 0.4 to 0.75 y is known as the visible spectral region, characterized by the fact that the human eye is sensitive to radiation of any wavelength within this spectral division. The radiation of different wavelengths of the visible spectrum is perceived as different color sensations. In Table 1.4 are given wavelength divisions corresponding to different colors. TABLE 1.4 Wavelengths and the Corresponding Colors

Color

Wavelength Interval,

Typical Wavelength,

mP

mP

390-455 455485 485-505 505-550 55G575 575-585 585-620 620-760

430 470 495 530 560 580 600 640

Violet Dark blue Light blue Green Yellow-green Yellow Orange Red

The radiation of longwaves (A > 0.75 p ) relates to the so-called infrared spectral region. The infrared spectrum is subdivided into the near-infrared (0.75 to 25 p ) and the far-infrared (25 to 1,000 p). It should be noted that for some of the above spectral regions, the designations given in the accompanying tabulation are sometimes introduced.

Spectral Region,

Designation

Spectral Region, ,u

Designation

0.20-0.40

uv

0.52-0.62

0.20-0.28

0.62-0.75

S-C

0.75-24

IR IR-A IR-B IR-C

0.324.40

UV-C UV-B UV-A

0.40-0. 75

S

1.4 -3.0

0.40-0.52

S-A

3.0 -24

0 . 2 8 4 . 32

0.75-1.4

S-B

1.3. Main Quantitative Characteristics of a Radiation Field

9

Special investigations have shown that practically all radiant energy of direct solar, diffuse, and reflected radiation falls in the region of short wavelengths from 0.2 to 5 p, with the major portion of radiation in the visible and the near-infrared spectral regions. So the fluxes listed in this region are known as shortwave. The thermal radiation of the earth’s surface and the atmosphere has, on the contrary, a Zongwave characteristic, since it is localized in the infrared (5 to 100 p), and thus, the flux is called longwave. The longwave radiation of the earth’s surface is often called terrestrial radiation. After summarizing this brief information on the atmospheric radiant fluxes, we proceed to the problem of the main quantitative characteristics of a radiation field.

1.3. Main Quantitative Characteristics of a Radiation Field At the present time there is no adopted uniform system of quantitative characteristics of a radiation field. Depending on what aspect of a radiation field (its energy or luminosity) is studied, either the energy or photometric system of quantitative characteristics is used. Both systems are interrelated; however, for a number of reasons, it is expedient, depending on the circumstances, to choose one or the other. One important reason is that in studying the energy effects of the radiation field, we are usually interested in the quantity of heat released per unit time in full absorption of the radiant flux under consideration. The measurement of the quantity of heat is made with the help of instruments, the receiving surfaces of which are perfectly black (in the technically possible approximation, of course). The measurement of luminous effects is also connected with transfer of a certain amount of radiant energy to the receiver. However, in this case the receiver (the eye, in particular) possesses considerable selectivity as to the radiation falling on it, being sensitive only to a very limited range of light wavelengths. The noted difference between the energy and luminous effects of a radiation field is one of the main justifications for using two systems of measuring quantitative characteristics of a radiation field-the energy and the photometric effects. Further, since we shall consider almost exclusively the energy effects of a radiation field, the main quantitative characteristics of a radiation field will be defined according to the author’s monographs [1,2]. 1. Radiant Intensity. The intensity of radiation, denoted by I, is the main quantitative characteristic of a radiation field. The value of radiant

10

Radiant Energy. The Main Concepts and Definitions

intensity depends on the wavelength of radiation A, the time t , the coordinates x, y , z of the point under consideration, P, and the direction r of the ray. The dependence of the radiant intensity upon all these values is usually denoted as Ia(t, P,r ) Let us turn now to the examination of the physical sense of the concept of radiant intensity. Consider a surface element do at a point P (Fig. 1.2). Through this surface element many parallel beams of variously directed rays are passing. Consider only those beams that are grouped around some n

dU

FIG. 1.2 The determination of radiant intensity.

definite direction and find the quantity of radiation passing in this direction. Let n be the normal to do at the point P,and r the line passing through the point P and constituting the angle 0 with the direction of the normal n. In building an elementary cone with a solid angle dw round the direction r, consider the volume limited by the truncated cone adjacent to the surface do. Denote by dFA the quantity of radiant energy corresponding to the spectral interval (A, A dA) and passing through da during the time dt within the limits of the considered volume. Determine now the value of the intensity ZA(t, P, r ) of the wavelength 1 at the point P in the direction r, using the following relation:

+

dF,

= Za(t,

P,r ) cos 0 do dw dil dt

(1.4)

Thus the radiant intensity In([, P,r ) is a quantity of energy in the unit wavelength interval, and in the unit solid angle per unit time per unit area of the surface perpendicular to the direction r of the beam of rays. The analogous quantity in photometry is called luminosity. In our case it may be called radiance. The distribution of radiant intensity in the spectra of different radiation sources is usually characterized graphically by curves of the spectral energy

11

1.3. Main Quantitative Characteristics of a Radiation Field

distribution. It is particularly convenient for description of the spectral composition of the radiation observed in the atmosphere, since in this case, as a rule, the radiation spectra are continuous. Most often the energy distribution in the spectrum is depicted in the form of the dependence of radiant energy upon the wavelength I. In this case the spectral radiation intensity must be determined for equal wavelength intervals in the entire range being considered. Commonly, however, the spectral composition of radiation is described by basing the dependence of the radiant energy upon the wave number n = 112 (or frequency v = CIA; c is the velocity of light), with the spectral radiation intensity determined for equal wavenumber (or frequency) intervals. It is easily seen that the curves of the radiant intensity distribution in the spectrum, if drawn with dependence on the wavelength I or the wave number (frequency) n, will not coincide. Now consider the wavelength interval ( I , I d I ) and the corresponding wave-number intervals (n, n dn) or frequencies (v, v dv). The validity of the following identities is evident:

+

+

dI,, = IAdI

= dI, = I,

dn

+

= dl, = I ,

dv

Hence I A = I, -; dn dI

I,

I>. = I,,-; dv dI

=

1,-dv

(1.5)

dn

where dn dil

-=---

1 --$; I2

dv dil

C I2

-

v2 . C

’

dv dn

- --

c

From these relations it follows that

Consequently the position of the maximum of the function It, must be displaced toward longer wavelengths in comparison with the maximum of the function IA (the same holds, apparently, for I,). In the case of the solar spectrum, the maximum In falls at the wavelengths I, = 527 m p and the maximum I,, (or I,) is displaced into the infrared and is at I , = 927 mp. Thus the values A, for I]. and I,, differ in the given case by 1.76 times. An identical presentation of the spectral radiation intensity distributions on the scale of wavelengths and frequencies can be realized with the help of the system of coordinates ( I J , In A), (Inn, In n), (12,v,In v). This becomes immediately apparent if we hold the following identities to be valid :

12

Radiant Energy. The Main Concepts and Definitions

dI,

d(ln1)

dil 1

dn n

= I,1 - = dI,, = Inn - = dIv=

dil il

dn n

= - = - -= -

dln n

I,,v

dv

(1.6a)

~

dv

= - -= V

V

dln v

(1.6b)

In the given case the intensity of radiation is determined for equal intervals of nondimensional quantities. The maximum of intensity in the solar spectrum now falls in each case on the wavelength A, = 668 mp. The total (integral) intensity of radiation can be obtained by integrating over all wavelengths and frequencies :

s

W

I=

I, dil = Jw I,,dv

The most often used unit of measurement of the integral radiant intensity in actinometry is 1 cal cm-2 min-l sr-l, which equals 0.6976 x lo6erg cm-2 sec-’ = 0.0697 W cm-2. The value, equal to 1 cal cm-2, is also known as a “langley” and is denoted by ly. 2. Flux of Radiation. The second of the more important radiation field characteristics is flux of radiation, which is understood to be the quantity of radiant energy of the wavelength il per unit area (at a given direction of the normal) per unit time. It should be noted that there are many terminological variants for the quantity. which we here call radiant fiux. Among other names, it is also called irradiance, radiant emittance, and energetical illumination. Taking the explanation given above as our basic definition, we obtain the following expression for the flux of radiant energy of the wavelength il across the surface, whose orientation is characterized by the direction of the normal n :

FA,n =

IA(t,P,I ) cos 0 dco

where the integral refers to all the possible directions r. By substituting the known relation A

cos 0

= cos(n,

A

A

r) = cos(n, x ) cos(r, x )

A

A

A

A

+ cos(n, y ) cos(r, y ) + cos(n, z ) cos(r, z )

it is possible to transform (1.8) in the following manner:

1.3. Main Quantitative Characteristics of a Radiation Field

13

where the integrals A

FA,

=

J IAcos(r, x ) dw, =

I

A

FA,v=

J I, cos(r, y ) dw

A

IAcos(r, z ) dw

are the expressions for the radiant fluxes in the directions of the axes OX, OY, OZ, respectively. As seen from (1.9), FA,nis the projection of a certain vector FA@,+, FA,y,FA,,) on the normal to the surface da. The vector F A is also called radiant flux. All that we have said about the intensity of radiation also holds for the description of the energy distribution in the spectrum of radiant fluxes. The total (integral) radiant flux is determined [analogously to (1.7)], by the following integral :

F

=

jrn FAdA= JwF,.dv

(1.10)

0

Introducing the spherical coordinates 8 and p, and taking into consideration that du, = sin 8 d8 dp, we can use the formula (1.8) to derive the flux from a hemisphere : 2n

FA,,, = Jo dp

12.(t,P,O, p) cos 8 sin 8 d8

(1.11)

The flux of radiant energy of the wavelength il thus can be determined at a given moment and at a given point by integration over 8 and p, given in ( l . l l ) , provided the intensity IA as the function of the coordinates 8 and p is known. It IAdoes not depend on the direction, the radiation field is isotropic. In this case the integration is easily performed, and we obtain the following expression for the flux:

FA= nIA An analogous relation will evidently hold for the total flux:

F = nI

(1.12)

Thus the flux of radiation through an arbitrarily oriented surface in the case of an isotropic field is n times as large as the intensity. This relation is often called Lambert’s law. It is known that only blackbody radiation is strictly isotropic. However, we shall see later that many real emitting and absorbing bodies approximately meet the demand of isotropy.

14

Radiant Energy. The Main Concepts and Definitions

The principal unit of measurement of radiant flux is 1 cal cm-2 min-I (or 1 ly min-I). The definitions of intensity and radiant flux given here refer to the case of diffuse radiation, that is, radiation spreading in various directions. Aside from this consideration, in the case of direct solar radiation, we must remember that radiant intensity is not zero only within a small solid angle do, corresponding to the angular solar diameter; in the directions outside d o the intensity of direct solar radiation equals zero. Taking into account these circumstances, we define the quantity of radiant solar energy as the flux of direct solar radiation per unit area per unit time. Let us proceed now to the determination of quantities, describing the interaction between the radiant energy and the medium inside which the propagation of radiant energy is taking place.

3. The Emission Coeficient. Suppose that a mass element dm is emitting equal quantities of energy in all directions. The quantity of radiant energy dF, emitted by the mass element per unit time within the solid angle d o and in the wavelength interval dA, will then equal dF,

=

ra(t,P ) d o dm dA

(1.13)

The value 7, of this expression is called the mass emission coefficient. As seen from the definition, the mass emission coefficient is numerically equal to the quantity of radiant energy emitted by the unit mass element per unit time within the unit solid angle and in the unit wavelength interval. Integrating (1.13) over all possible directions by assuming the independence of 7, of direction, we obtain the following expression for the entire quantity of radiant energy of the wavelength 1 emitted by the element dm: 4ny,(t, P ) dm dA

(1.14)

For the integral radiation of all wavelengths we obtain the following integral mass emission coefficient:

To characterize the radiation of surfaces, the concept of emissivity is often introduced. The determination of this quantitative characteristic of the radiation field of the surface is, however, identical to the above determination of radiant intensity, so there is no need to give it here.

1.3. Main Quantitative Characteristics of a Radiation Field

15

In some cases it is expedient to introduce the concept of the relative emissivity (or simply emissivity) 6,. The latter is a dimensionless quantity, which is the ratio of the radiant intensity of a given body to that of a blackbody at the same temperature. The concept of emissivity is introduced for both monochromatic and nonmonochromatic radiations. 4. The Absorption Coefficient.

Consider now the attenuation of radiant intensity due to absorption. Assume that the attenuation of radiant intensity Zj.(t, P, r ) due to absorption on the way ds between points P and P' is proportional to the distance ds and the density of medium Q at the point P. The value of the attenuation of radiation on way ds will then be expressed as

the the the the the

The proportionality coefficient k , is called the mass absorption coefficient. As seen from formula (1.15), it has the dimension L2M-l. In actinometry the absorption coefficient is usually expressed in cm2 g-l. In many cases, instead of the mass coefficient, the volume absorption coefficient is used, which can be determined as

It is evident that the volume absorption coefficient has the dimension L-*. To characterize the surface absorption of radiant energy, the concept of absorptivity is introduced. As distinct from the absorption coefficient, the absorptivity is a dimensionless quantity and is determined by the relation dF,' = a,(t, P, r ) Ia(t, P, r ) cos 19do dw d;l dt (1.17) Here dF,' is the quantity of radiant energy absorbed by the surface from the incident radiant energy: dF,

= Z,(t,

P, r ) cos 8 do dw dt

Thus the absorptivity is a value numerically equal to the ratio of the absorbed radiation to the radiation incident on a given surface. In the general case, the absorptivity of a surface depends on the wavelength, the position of the area under consideration, the direction of rays absorbed by the surface, and the time. We shall now consider the effect of the most important of these factors.

16

Radiant Energy. The Main Concepts and Definitions

5. The Scattering Coeflcient. The attenuation of radiant intensity due to scattering can be [analogous to (1.15)] expressed by the formula dI,(t, P, r ) = -a,(t, P ) Z,(t, P,r)e ds

(1.18)

The proportionality coefficient G , in this formula is called the massscattering coefficient. It is convenient also to introduce the volume-scattering coefficient oAe. The dimensions and the measurement units are the same as those of the corresponding absorption coefficients. The value a, characterizes the attenuation of radiation due to scattering in all possible directions. The concept of the scattering function is introduced to describe the angular intensity distribution of scattered light. The scattering function is determined as the function y,(P, r', r ) / 4 n (r is the direction of the falling ray, r' is the direction of scattering), which characterizes the portion of radiation scattered in the direction r' relative to the entire scattered radiation. Also

provided the integration includes all directions. The concept of the absolute scattering function is used to characterize the angular distribution of the absolute intensity values of radiation scattered at a given point. 6. Reflectivity. The property of a body to reflect the incident radiation can be given by the value of reflectivity, R,. The latter is a dimensionless quantity, expressing the ratio of the radiant intensity of the wavelength 1 reflected from the given surface to the intensity of the radiation of the same wavelength incident on the given surface. Along with this concept of reflectivity for monochromatic light, an analogous quantity is used for nonmonochromatic light to characterize the reflective properties of a body in the spectral region of a certain width. It must be noted that the preceding definition of reflectivity is reasonably applicable only in the case of specular reflection. In the presence of diffuse reflection, we use a quantity equal to the ratio of the reflected radiant flux to the flux of incident radiation, which is known as the surface albedo, A,.

7 . The Absorption and Transmission Functions. The quantitative characteristics of the interaction between the radiation field and the medium may be called local, for they relate to definite points (more precisely, to ele-

1.3. Main Quantitative Characteristics of a Radiation Field

17

ments of mass, or surface) in space. Besides these quantitative characteristics, it is also necessary to introduce those that determine the integral properties of absorbing and emitting bodies. Among such integral quantitative characteristics are the functions of absorption and transmission. The absorption function is a relative value of the absorption of radiation by a layer containing the mass of the absorbing substance, equal to w,and is determined in the following way: (1.19)

where Z(O), Z(w) are the intensities of radiation incident on the given layer and passing through it, respectively. Also introduced is the absorption function for radiant fluxes :

(1.20) We may also use the transmission function, determined by the relation P(w) = 1 - A(w)

(1.21)

As in the case of the absorption function, one should distinguish between the transmission functions for directed PI and diffuse PF radiations. These functions can be determined also for monochromatic or nonmonochromatic radiation. If the monochromatic transmission function is determined from laboratory measurement data for the directed radiation, then the integral transmission (absorption) function for diffuse radiation can also be found. Since the problem of the relation of transmission functions (later we ,shall discuss transmission functions in greater detail) for the directed and diffuse monochromatic and nonmonochromatic radiations is of great interest, let us discuss it more fully now. The problem of the relation of transmission functions for diffuse and directed radiation is solved in the simplest way in the case of monochromatic radiation. In this case the law of the attenuation of radiant intensity due to absorption is determined by the following formula (expressing Bouger’s law) IA= e - b w see 0 (1.22) where 8 is the zenith angle that determines the direction of radiation

18

Radiant Energy. The Main Concepts and Definitions

propagation. Thus, for the transmission function of the monochromatic directed radiation, we have

pIa

= e-kAw sec 0

(1.23)

Note that this formula is based on the assumption that it is possible to neglect the dependence of the absorption coefficient k, upon pressure and temperature (on the vertical coordinate 2). In the opposite case, the exponent in the formula (1.23) must be changed for the expression sec 0

k,e dz

where z is the thickness of the radiation absorbing layer and e is the density of the absorbing substance. Using (1.22) and (1.1 I), we obtain for the monochromatic radiation flux:

Fa

=

s:" r'' dp

Io,a e-kaw seC cos 8 sin 0 d0

By introducing a new variable, t = sec 8 and assuming the incident radiation intensity to be independent of direction, it is possible to present this expression in the form c k a w t t-3 dt

FA = Fovn2

where Eo,a = is the monochromatic radiation flux incident on the absorbing layer, and

so 00

E,(x)

=

e-& t-3 dt

(1.25)

is a certain transcendent function. On the basis of (1.24) we can find for the transmission function of monochromatic diffuse radiation :

PFa= 2E3(kw) = e-pPAw

(1.26)

where is a certain coefficient. Figure 1.3, from Kaplan's data [3], gives the curve 1, characterizing the dependence of the coefficient ,8 values, at which the identity (1.26) is taking place, on the transmission 2E3(x) values. As seen, with the variation of the latter from 0 to 1, the coefficient values vary within the range from 1.2 to

19

1.3. Main Quantitative Characteristics of a Radiation Field

2.0. At PFA= 2F3(x) = 4, the value B = 1.66. It is vital that the curve in Fig. 1.3 be practically linear in the region of the transmission function values from 0.2 to 0.8. In this region of PFnvalues the coefficient B varies by not more than 10 percent, so we assume that the identity (1.26) has I .a

0.9

0.0 0.7

--

0.6

0.5

(u

0.4

0.3 0.2 0. I 0 1.

3

FIG. 1.3 The quantity j 9 at diferent values of 2E3(x). After Kaplan [23].

a fairly sufficient degree of accuracy at ,f?= 1.66. The curves 2,3 in Fig. 1.3 characterize the variation of ,4 in its dependence on the functions pxE3(x) and P2x2E3(x).These results are of interest to the theory of radiative heat exchange in the atmosphere. Thus we can write, instead of (1.26), pFA

e-1.66k~w

(1.27)

Comparing (1.27) and (1.23), we see that the transmission of diffuse radiation is equivalent to the transmission of directed radiation at the same w but at sec 8 1.66, which corresponds to 8 N 53”. As will be shown further on, this fact is of great importance for the development of approximate methods of calculation of atmospheric thermal radiation fluxes. The formula (1.26) is valid only in the case where radiant flux incident on the absorbing layer is isotropic. In actual fact, however, the observed atmospheric radiant fluxes are not isotropic. The consideration of this =1

20

Radiant Energy. The Main Concepts and Definitions

fact significantly complicates the problem of calculation of the transmission function for diffuse radiation and this problem in its general form has not yet been solved. Let us now proceed to the derivation of the transmission function for the nonmonochromatic (integral) radiation, making use of the method of radiation absorption selectivity, which was proposed by Ambartzumian [8] and developed in application to the earth’s atmosphere by Lebedinsky [see Kondratyev (2)]. Let there be the following notation:

where integration is performed over all the wavelengths, for which k < k dk. Denote further that

+

< k,

where I,, is the integral incident radiation intensity. It is obvious that Imf(k) dk

=

1

It should be noted that in the case of absorption of thermal radiation, the valuef(k) is a function not only of k but also of the temperature T, since in the variation of temperature a displacement of the distribution of energy in the spectrum of blackbody radiation or of the atmospheric thermal radiation takes place. In the first approximation, though, the influence of the “displacement effect” may be neglected. Now substitute (1.22) by the following relation, derived from the integration of both parts of (1.22) over the preceding combination of wavelengths : = I0 f(k) Ckw Integrating this relation over all k from 0 to 00 (which evidently corresponds to integrating over all wavelengths from 0 to m), we obtain

I

=

10

I

03

Ik dk

=

I,

f(k) e-kw

@

dk

Hence, for the transmission function of the integral directed radiation we have I ( I .28) dk P,(w sec 0) = -= f(k) e-kw 10

I

@

1.3. Main Quantitative Characteristics of a Radiation Field

21

or, taking account of (1.22), this can be written as

As seen, the transmission function of integral radiation is uniquely determined through the transmission function of monochromatic radiation if the function f ( k ) is known. The physical sense of the latter consists in its characterizing the spectral composition of incident radiation (the value f ( k ) determines the portion of the incident radiation intensity falling on the combination of the spectral regions to which correspond infinitesimally differing values of the absorption coefficient). The following formula will then be for the radiant flux: F

= Zo

j m dk 12”dp

y12 f ( k ) e-kw

cos 8 sin 8 d8

Considering (1.23), we find that in this case the transmission function is of the form F m (1.30) PF(w)= -= f(k)&(kw) dk Fo 0

1

It is easily seen that the integral transmission functions for diffuse and directed radiations are related to

PF(w)= 2 [ I 2 P,(w sec 0) cos 8 sin 8 d8

(1.31)

This relation enables calculation of the transmission function for diffuse radiation from the one for directed radiation. Using the approximate formula (1.27) instead of (1.31), we have PF(W) = PICBW)

(1.32)

According to (1.27), the coefficient B is equal to 1.66. In a number of cases this approximate relation proves to be quite satisfactory. For example, the experimental determination of the coefficient B for the case of the integral transmission function of thermal radiation, performed by the author and Yelovskikh [4], has given the value /I= 1.68, which practically coincides with the theoretical evaluation. Strictly speaking, the formulas (1.31) and (1.32) hold only for the case where the values of radiative transmission are unique functions of the radiation absorbent content, whereas transmission (and absorption, cor-

22

Radiant Energy. The Main Concepts and Definitions

respondingly) of radiation depends on a number of other factors. Only in laboratory conditions, when the influence of these other factors is eliminated, does PF = PF(w). In this connection there arises a very important and complicated problem of determining the transmission function of a nonhomogeneous medium from laboratory measurement data obtained for the transmission function of a homogeneous medium with fixed values of pressure and temperature, for which PF = PF(w).This problem has so far been solved only for some particular and comparatively simple cases. Plass [5-71 for example, has solved the problem of the determination of the transmission function in conditions of the real atmosphere by using laboratory measurement data for two finite cases of very slight and very strong absorption.

1.4. Fundamental Laws of Thermal Radiation Only a brief review of the basic laws of thermal radiation is given here, since we are more concerned with the detailed characteristic of the peculiarities of their application to atmospheric conditions.

1. Kirchhofl’s Law. Consider a homogeneous medium in the state of thermodynamic equilibrium. In such a medium the thermal radiation intensity will not depend on the direction. On the other hand, the variation of radiant intensity along the ray path can be determined by the equation (1.33) where

E,

= monochromatic

thermal radiation intensity ds = element of the length in the direction of the ray e =density of substance Since dE,/ds

= 0,

we obtain

This formula expresses the known Kirchhoff’s law: In the state of thermodynamic equilibrium the ratio of the mass coefficients of emission and absorption does not depend on the nature of the absorbing and radiating matter, but is a universal function of wavelength and temperature. The relation (1.34) applies to the mass element (since rAand k , are the

1.4. Fundamental Laws of Thermal Radiation

23

mass coefficient of emission and absorption). Consider now in place of the mass element a surface element on the side of a perfect black enclosure in conditions of thermodynamic equilibrium. Determine the emissivity and absorptivity of the side I>. and A , by analogy with the determination of the mass coefficients of emission and absorption. Then, instead of (1.34), we have

In this way the ratio of the emissivity of a body to its absorptivity in thermodynamic equilibrium equals the intensity of blackbody radiation, that is, a certain universal function ;Iand T. That E,(T) is the blackbody radiation intensity becomes apparent if we take into consideration that if A]. = 1, the value of Ia = Ea. Since we know that there is no thermodynamic equilibrium in the atmosphere, the formulas (1.34) and (1.35) cannot be applied to the atmosphere. Mustel [8] notes that strict thermodynamic equilibrium is disturbed by three main factors. The first is a temperature gradient owing to which the intensity of radiation appears to be dependent on direction. This makes the temperature TE dependent on direction, too, whereas in the conditions of thermodynamic equilibrium, the radiation field is isotropic and the radiation temperature is constant in all directions. The most clearly marked nonisotropy of radiation will evidently take place in the outer (upper) layers of the atmosphere, where a “dilution” effect of radiation has been observed : the intensity of the upward radiation appreciably exceeds that of the downward radiation. The second factor in the breakdown of thermodynamic equilibrium is a non-Planckian character of the distribution of energy in the radiation spectrum. This factor finds expression in the fact that the radiation temperature TE at a given point and for a given direction depends on frequency. In the case of the Planckian energy distribution (discussed further on in this section), the radiation temperature appears to be independent of frequency. Finally, the third factor that violates thermodynamic equilibrium is the difference between the kinetic temperature in the formula of Maxwell’s law of distribution of molecule speeds and the radiation temperature. The inequality Tk f T, in the absence of thermodynamic equilibrium follows because in this case TE is dependent on direction and frequency, whereas Tk must not possess such a dependence if equilibrium is to be obtained.

24

Radiant Energy. The Main Concepts and Definitions

Thus the strict thermodynamic equilibrium cannot be realized in the atmosphere, and consequently Kirchhoff’s law is not fulfilled. In these conditions the solution of the problem of thermal radiation transfer in the atmosphere becomes extremely complicated. The concept of “local thermodynamic equilibrium,” introduced by astrophysicists, offers a way out. Local thermodynamic equilibrium is understood to be that state of the medium in which the emission and absorption of each small portion of the medium at the temperature T is the same as that of a perfectly black enclosure at the temperature T in equilibrium. There appears to be no necessity in the isothermal medium such as that required for the conditions of thermodynamic equilibrium. Temperature may vary from point to point, but each element of the environment behaves as if it were in thermodynamic equilibrium at the temperature of a given point. As was shown by Milne [9], whether the demands for local thermodynamic equilibrium can be met is determined by the role of collisions in establishing the equilibrium state. Local thermodynamic equilibrium takes place when the collision effect controls the excitation and de-excitation of atoms and molecules; that is, when the “scattering” of radiant energy can be neglected. We shall see later that similar conditions are realized in the atmosphere at elevations not exceeding 50 km. Thus, the concept of local thermodynamic equilibrium, cannot be applied to atmospheric layers above 50 km. At high altitudes, transition from the state of local thermodynamic equilibrium t o the state of monochromatic radiative equilibrium takes place. The latter is defined as that state of the medium in which for each definite frequency the same quantity of radiation (and of the same frequency) is emitted and absorbed. The problem of determination of the conditions for the state of local thermodynamic equilibrium and of departures from this state is quite complex. To date this has been solved for only a small number of particular cases. To understand the physical sense of it, let us consider the simplest case of a discrete spectrum in the presence of two energetic levels, investigated by Milne [9]. Taking account of (1.33) and (1.34) and introducing the frequency Y as the index, in place of the wavelength A, we have, in the state of local thermodynamic equilibrium, (1.36)

where Ev is the intensity of atmospheric thermal radiation.

1.4. Fundamental Laws of Thermal Radiation

25

In the case of monochromatic radiative equilibrium, the absorbed (emitted) radiation equals k , J I, d o , where we intergrate over the entire space round the given point to obtain (1.37) Local thermodynamic equilibrium and monochromatic radiative equilibrium are two finite cases of a more general equilibrium. Consider now the way to characterize the more general type of equilibrium, assuming that atoms (or molecules) are in one of the following states; (1) normal, state 1 of the lesser energy; (2) excited, state 2 of the greater energy. Let n, and n2 be the number of atoms in 1 cm3 in state 1 and 2, respectively, at a point P. Let Azl denote the probability of a spontaneous transition of the atom from the normal to the excited state and let B,, and Bzl be the probabilities of induced transitions (under the influence of the radiation field, accompanied by either absorption (B,,) or emission (BZl). Note here that in this case the transition probabilities are determined in relation to the intensity and not to the density of radiation (for example, the value B,,I,,dt is the probability of the atom’s absorbing radiation during a time interval dt in the field of isotropic radiation of the intensity Z,,).It should also be mentioned that in the case of isotropic radiation, the spectral radiation density el. (4n/c)Z,,,where c is the velocity of light. If g, and g, are the observed weights of state 1 and state 2, then the following relation must be fulfilled: = I

(1.38) (1.39) where h is Planck’s constant. Let a, denote the absorption coefficient, calculated for a single atom in the normal state. Now it is possible to show the approximate relation B12hv a,dv = 4n

where the integration covers the range of frequencies v within which the atom can absorb radiation.

26

Radiant Energy. The Main Concepts and Definitions

For simplification of the preceding formula, we introduce the mean absorption coefficient Or, and obtain (1.40) We let the values b,, and aZ1characterize the probability of transition of the atom from the normal state to the excited, and vice versa, under the influence of collisions with other atoms. The transition probabilities A21, B 1 2 ,B,, are the atom constants, independent of temperature, while the values b,, and aZ1appear to be the temperature functions. These values also depend upon the general number and the kind of colliding particles, and at the constant chemical composition of the atmosphere are proportional to the density. It must be added that in the absence of thermodynamic equilibrium, the usual determination of temperature is meaningless. In the case of equilibrium of the general type, the concept of temperature can be explained as follows: Suppose that the distribution of speeds of colliding particles is described by Maxwell’s law and determine the temperature in such a way as to make the Maxwell’s speed distribution identical to the speeds observed. In the state of thermodynamic equilibrium by virtue of the principle of detail equilibrium, n1b12 = n2a21 (1.41) This means that transitions from the normal to the excited state, and vice versa, under the influence of collisions are put in equilibrium. At the same time, according to Boltzmann’s law, (1.42) where k is the Boltzmann constant. So, applying (1.41) and (1.42), we obtain -

021

gz

e-(hvlkT)

(1.43)

g1

This formula is derived for the case of thermodynamic equilibrium. However, it must take place in other conditions as well if the speed distribution is to be considered Maxwellian. We shall therefore assume that the formula ( I .43) is fulfilled in the case of equilibrium of the general type. It also appears that the ratio blz/a21must not depend on the density.

1.4. Fundamental Laws of Thermal Radiation

27

As to the separate dependence of b,, and aZ1on temperature, it can be easily deduced from the general considerations that the greater temperature dependence is due to the coefficient b,, ,and consequently 61, -e-(hv/kT). Consider now, on the basis of the preceding assumptions, the radiative transfer in absorbing and emitting media. The number of emission quanta in the frequency interval from v to v dv, per unit time per unit area in the direction normal to this area and within the limits of a solid angle dw, will equal I.. Av dw hv

+

The number of quanta passing through an analogous surface at a distance ds from the first one along the path of the ray will be

+

(Z, dZ,.) Av dw hv Thus the variation in the quantum number at the distance ds is equal to

dI,,Av dw hv On the other hand, the variation of the quantum number on the path ds results from: (1) Increase in the number of quanta, due to emission, by the value

(2) Decrease in the number of quanta, due to absorption, by the value

dw 4n

n, ds B,,I, -

So reducing by dw, we can write

Making use of the formulas (1.38) to (1.40), this equation can be transformed to 1 dZ,, GV ds

28

Radiant Energy. The Main Concepts and Definitions

Let us now make some transformations of Eq. (1.44). To this effect let us write the condition for a steady state of the medium, that is, specify an equality of the number of transitions from the normal to the excited state, and vice versa, which can be expressed in the form dw

+ b,,]

=

n,[A,,

+ B,, J I,. 4dwn + a,,]

(1.45)

Introducing the designation E = blz/Blz and using (1.38), (1.39), and (1.43), we can rewrite Eq. (1.45) as dw

g, 2hv3 n, - - &[n,- fi & I k T ]

sz

+

c2

=0

(1.46)

g2

Hence

Transforming with the help of Eq. (1.44), we find

From the comparison of (1.44) and (1.36) there follows the relation below, establishing the connection between the mass absorption coefficient and the absorption coefficient a, for the single atom: (1.49) Taking account of (1.49) and introducing the value E

r = r

(1.50)

we finally transform (1.48) in the following manner (1.51) This equation is the general case of transfer of radiant energy for equilibrium of the absorbing and emitting medium. It is easy to see that the conditions of local thermodynamic and monochromatic radiative equilibria are finite cases, following from (1.51). At

1.4. Fundamental Laws of Thermal Radiation

29

7 + 0, Eq. (1.51) becomes identical to (1.37), which corresponds to the conditions of monochromatic equilibrium. If 7 + 00, then (1.51) coincides with (1.36); that is, it represents the condition of local thermodynamic equilibrium. Since 7 E , and E ee-hv/kT,it means that at great density (the lower atmospheric layers) the conditions of thermodynamic equilibrium are fulfilled, and at lesser density (the upper layers), conditions of monochromatic equilibrium exist. From these considerations it is evident that dependence of the character of equilibrium on the density of the medium is determined by the importance of collisions as the cause of absorption and emission of radiation by atoms and molecules. Starting from this conclusion, we shall now investigate the way in which we can obtain a quantitative estimate of the altitude of the upper boundary of the zone of local thermodynamic equilibrium. Table 1.5 gives some data

-

-

TABLE 1.5 Structure of the Atmosphere: The Mean Free Path and Frequency of Collisions between Molecules. Afrer Goody [lo]

Height, km

0 11 32 62 84 100

Temperature, OK

288 218 218 330 200 300

Pressure, mb

1013 230 8.6 0.2 1.2 x 10-2 1.5 x 10-3

Number of Molecules per cm3

2.5 x 7.8 x 2.9 x 4.5 x 4.4 x 3.6 x

10lg 1018

1017 1015 1014 1013

Mean Free Path, cm

6.3 x 2.1 x 10-5 5.6 x 10-4 3.6 x 3.7 x 10-1 4.5

Frequency of Collisions, sec-l

7.3 x loQ 1.9 x 109 7.1 x 107 1.4 x lo6 1.0 x 105 1.0 x 1w

that characterize the structure of the atmosphere and the mean free path and frequency of collisions between molecules a t different heights in the atmosphere, as shown by Goody [lo]. The last two values in the table are obtained on the assumption that air is a mixture of two gases, nitrogen and oxygen. On the other hand, laboratory measurements give the following approximate values of the lifetime of the excited state in vibrational transitions for some basic atmospheric absorption bands (see Table 1.6). As seen from Table 1.6, the rate of de-excitation resulting from spontaneous transitions is about 10 molecules per second. It is also known

30

Radiant Energy. The Main Concepts and Definitions TABLE 1.6

Lifetime of the Excited State in Vibrational Transitions for Some Absorption Bands

Gas

Center of Band, P

6.3

Lifetime in Excited State, sec

6 x

7.8

9 x

15

4 x 10-1

10-2

that the effecticity of de-excitation due to collisions for vibrational transifor an oxygen molecule. Using this figure and the tions is about 5 x data of Table 1.5, we find that at the height of 32 km the loss of molecules in the excited state due to collisions is 355 molecules per second; and at 62 km, 7 molecules per second. On the basis of the data in Table 1.6, this means that at the height of 62 km collisions lose their domination in the deexcitation of molecules and consequently there is no local thermodynamic equilibrium. So we can assume that the conditions of local thermodynamic equilibrium are fulfilled up to the height of about 50 km. This estimate was derived from data on vibrational transitions. In the case of rotational transitions it may somewhat increase because of a longer life time of the excited state and a higher effectivity of de-excitation of molecules due to collisions. The conditions for the fulfillement of local thermodynamic equilibrium have been further considered in the works of Woolley [ ll] , Mustel [8], Curtis and Goody [12], Filipovich [13], and Shved [14]. In the present work, however, we shall confine our discussion to a brief treatment of the problem, omitting detail. 2. PZunck’s Law. The formula (1.34) that expresses Kirchhoff’s law contains the function E, , which characterizes the distribution of radiant intensity in the emission spectrum of a blackbody. The determination of the form of the function E,(T) is the fundamental problem of the theory of emission. One of the first fruitful attempts to solve this problem was undertaken by a Russian physicist V. A. Mihelson. However, only Planck had succeeded in finding a completely right solution by formulating a hypothesis of the quantum nature of the emission process. The investigations of Planck in the distribution of energy in the blackbody emission spectrum initiated the creation of quantum mechanics.

1.4. Fundamental Laws of Thermal Radiation

31

The form of the function E,(T), known as the Planck function, is easily obtained by making use of the results of the preceding section, If we compare (1.44) and (1.36), we see that these equations are identical in the case of the fulfillment of (1.49) and of the following relation: (1.52) Dividing (1.52) by (1.49), we have

But in the presence of local thermodynamic equilibrium, the relation (1.42) holds. Taking this relation into consideration, it is possible to present the preceding relation in the form

E,(T)

2hv3

1

c2 ehvlkT - 1

(1.53)

Now, if we substitute the frequency scale for the wavelength scale, using the relations EAdA= E,,dv, v = c/A, I dA I = (c/v2)dv, we have 2hc2 1 EA(T) = 15 ehelfiT - 1

(1.54)

The last two formulas determine the Planck function. The values of the constants in these formulas are (according to the summary of the world constant values [23]): the Planck constant h = (6.6256 f 0.0005) x lop2' erg sec; the velocity of light, c = (2.99725 f 0.00003)/1010 cm sec-l; the Boltzmann constant, k = (1.38054 0.00018) x erg deg-'. The Planck formula (1.54) is easily transformable if the coordinates

6 -,A F

1,

E'

=En

Ea,nb

(1.55)

are introduced, where A, is the wavelength corresponding to the maximum value of the monochromatic emission intensity EA,,. It can be shown that

A,

C'

= 1,E = -5

T

EA = E Am E ' = c"T5E' where c' and c'' are certain constants.

(1S6a) (1.56b)

32

Radiant Energy. The Main Concepts and Definitions

Taking into account (1.55) and (1.56), we present the formula (1.54) in the following way: (1.57)

where cl' = 2hc2, c, = hc/k. From (1.57) it is apparent that the relative blackbody emission intensity E' is the function of the nondimensional parameter t.This enables construction of the curve I?'([), which may be treated as a universal curve of the energy distribution in the blackbody emission spectrum. Figure 1.4 gives

B C

10 15

20

30

30

45

40 60

50 75

60 90

70 105

A,F

FIG. 1.4

Energy distribution in the blackbody emission spectrum.

a graphic curve of this kind. On the axis of ordinates are plotted the values (B, = nE,) in 10-l6 cal cm-2 sec-l deg-5 p-l; on the axis of abscissas the wavelengths A are given in microns. The upper scale A of the wavelengths corresponds to the source of emission at T = 6000'K; the scales B and C refer to the emission sources at T = 300' and T = 200°, respectively. The maximum energy of the source at T = 6000'K (approximately the sun's temperature) falls on the wavelength near 0.5 p, whereas at atmospheric and terrestrial temperatures the maximum of energy displaces toward the region of wavelengths about 10 to 15 p. Using the nondimensional values of (1.56), one obtains an expression for the portion of the integral blackbody intensity falling on a spectral region in the wavelength interval (A,, A2). Denoting this portion of the integral emission intensity by p, we have

-

1.4. Fundamental Laws of Thermal Radiation

33

By introduction of the designation

we get (1.58)

The function y(5) is presented in Fig. 1.5.

t FIG. 1.5

The function

~(6).

As was mentioned above, the coordinate system (ELI,In A ) is conveniently applicable for characterization of the spectral distribution of the value p because in this case the distributions in the spectrum of the values EAA and p are identical. The same holds if the frequency or wave number is used instead of the wavelength. 3. Stefan-Boltzmann Law. Integrating (1.54) over all wavelengths from 0 to 00, we obtain the total blackbody emission intensity:

The last integral turns out to be equal to n4/15. So finally (1.59)

34

Radiant Energy. The Main Concepts and Definitions

where g = -

2n5k4 15c2h3

Since the black-surface emission intensity is independent of direction (Lambert's law), then we have the following expression, according to the formula (1.12), for the radiant flux of a perfect black surface: B = nE

= aT4

(1.60)

For the constant 0, numerical values are cr = 5.75 x erg cm-2 sec-l deg-4, or (in other units) (T = 0.826 x 10-lo cal cm-2 min-l deg-4. According to current data, (T = (0.56697 f 0.00029) x erg cm-2 sec-' deg-4 is derived, which corresponds to cr = (0.81581 f 0.00014) x 10-lo cal cm-2 deg-4 in other units. Differentiating the Planck function over A and determining the value of the wavelength corresponding to the maximum of the function E,(T), we obtain the following relation:

4. Wien's Displacement Law.

A,T

=a

(1.61)

where a = 0.28978 f 0.00004 cm deg if the wavelength is expressed in centimeters, and EA,,, = c"T5 (1.62) where c" =

(&)

1.301 x lOI5 W/cm2,u deg5

The relation (I .61) is known as Wien's displacement law. It characterizes the displacement of the maximum of the intensity of blackbody emission dependent on the temperature of the blackbody. The formula (1.62) indicates that the maximum intensity of blackbody emission is proportional to one-fifth of the absolute temperature. We must stress here that all these conclusions refer to the case when the spectral composition of emission is described as the dependence of the monochromatic emission intensity upon the wavelength. If, instead of E,, we consider E,, or E, , then 1 , is displaced into the infrared. Neither the earth's surface nor the atmosphere are blackbodies; that is why direct application of the preceding formulas to the computation of terrestrial or atmospheric emission is impossible.

1.5. Thermal Emission of Real Bodies

35

In the following section we shall see how thermal emission of real bodies can be computed.

1.5. Thermal Emission of Real Bodies We have just described the main regularities of thermal blackbody radiation. We know that real bodies are never perfectly black and therefore we must know what factors primarily affect the thermal emission of real bodies. Once these are identified, it will be possible to calculate this emission and also learn how the emission of a given body can be brought as near as possible to that of a blackbody, since the latter is the most important factor in preparing the receiving surfaces of actinometric instruments. Thermal radiation of bodies is influenced most of all by their electric properties (conductivity) and the state of surface (degree of roughness). First let us consider the emission of smooth surfaces of dielectrics and conductors, basing our computations on some fundamental formulas of the electromagnetic theory of light. In reality, of course, the surface layers of a given body rather than their geometric surfaces are sources of emission. We speak of the emission of a surface only in the sense that all the emitted radiant flux passes over the surface.

1. Thermal Emission of Dielectrics. As Shifrin [15] noted, the absorptivity u, (and the emissivity, if Kirchhoff's law applies) of a body can be determined by (1.63) a, = (1 - R1)(1 - e-"AL) where R,

=

reflectivity of a body

a,

=

I

=

volume absorption coefficient (aA= kAe) body thickness in the direction of refracted rays

The value 1 - RA in this formula determines the portion of radiant intensity penetrating the body; the value 1 - e-";1 indicates how much of this portion is absorbed by the body. We speak here of intensity, not of flux, for as will be apparent further on, aa depends on the direction. We shall discuss here only those bodies for which the volume absorption coefficient is sufficiently large. Therefore it is possible to assume that at the surface of a body the following condition is satisfied approximately:

36

Radiant Energy. The Main Concepts and De6nitions

or dA

+ R, = 1

(1.65)

provided Kirchhoff’s law is fulfilled at the surface and a, = 8, (6, denotes the relative emissivity). The majority of dielectrics and semiconductors can be considered “gray” bodies; that is, it is possible to assume that the absorptivity an or relative emissivity 8, is independent of the wavelength. However, this is an approximate assumption. The spectral coefficients of diffuse reflection of infrared radiation from blackened surfaces, investigated by Kozyrev and Vershinin [ 161, show that even in this case there is a considerable selectivity of radiation reflection. Different kinds of soot, which appear equally black to the eye, are far from being very “black” in the infrared region of the spectrum. Figure 1.6 illustrates this by several results of measurements by the authors [16].

80

60

#

IL

40

20 0

PA

FIG. 1.6 Reflection coefficients of bismuth-blackened foil. After Kozyrev and Vershinin [16]. ( l ) g = 0.1, sublimation at the residual pressure p = 1.5; (2) g = 0.2, p = 0.7; (3) g = 0.8, p = 0.1 ; (4) g = 0.8, p = 0.7; (5) g = 0.9,p = 1.0 (g is the mass of coating in mg cm-2, p is pressure in mm Hg).

We rewrite the relation (1.65) for dielectrics: dd,i

+

Rd,i =

1

(1.66)

where the subscript d means that the considered value refers to dielectrics, and the subscript i indicates the dependence of 6 and R on the angle of incidence of radiation, i.

1.5. Thermal Emission of Real Bodies

37

According to the known Fresnel formula, 1

Rd*i =

sin2(i - r )

z [ sinz(i + r )

+

+

"I

(1.67)

tanz(i - r )

where r is the refraction angle. Taking into account (1.66) and (1.67), we have for the relative emissivity: 1

3'''

=

[

sin2(i - r )

- z sinZ(i + r )

tan2(i - r ) r)

+ tanz(i

+

1

(1.68)

The angles i and r are related simply by

n=- sin i sin r

(1.69)

where n is the refraction index. With the help of (1.68) and (1.69) it is possible to calculate the body's relative emissivity in various directions at different values of n. In calculations of this kind it is necessary, strictly speaking, to take into consideration the dependence of n on the wavelength. In many cases one may neglect this dependence in the first approximation. The value of the emissivity in the direction of the normal to the surface (i = 0) can be obtained from

Bd,o

=1

i::3

- -

(1.70)

In Table 1.7 are presented computational results Bd,' for the values i from 0 ' to 90' and n from 1 to 3. The value n = 1 corresponds to the blackbody case. This table shows that the blackbody radiation is isotropic (obeying Lambert's law). The other cases are considerable dapartures from the isotropic distribution of emission. At i > 70°, the emissivity notably decreases, reaching zero at i = 90'. Using (1.66), we can, with the help of Table 1.7, analyze the angular distribution of reflectivity of dielectrics. According to (1.66), the variation of reflectivity with the variation of the angle of incidence is inverse to the variation of the relative emissivity. The maximum reflectivity is present at large angles of incidence (R = 1 at i = 90'), and the minimum is in the direction of the normal to the surface of a body. Laboratory investigation data on the anisotropy of emissivity (and absorptivity, correspondingly) confirm the above computational results. Bolz's

38

Radiant Energy. The Main Concepts and Definitions TABLE 1.7 The Angular Distribution of Emissivity of DieIectrics n i, deg

0 10 20 30 40 50 60 70 80 90

1 .o

1.41

2

3

1.Ooo 1 .Ooo 1 .Ooo 1 .Ooo 1 .Ooo 1.ow

0.970 0.970 0.970 0.969

0.889 0.889 0.889 0.888

0.750 0.750 0.750 0.749

0.966 0.955

0.881 0.869

1.Ooo

0.925

0.839

0.746 0.742 0.728

1 .om 1 .Ooo 1.ooo

0.846 0.628

0.763 0.573

0.690 0.567

0.000

0.000

0.Ooo

results (Table 1.8) illustrate this conclusion. Bolz investigated how the absorptivity of the blackened receiving surface of the vibrational pyrgeometer depended on direction. Measurement data show, in accordance with the theoretical calculations, that the greatest departures from isotropy occur at large angles of incidence. TABLE 1.8 The Angular Distribution of Absorptivity for a Blackened Mica Plate. After Bolz [17]

i, deg

30

50

60

70

80

85

a, percent

98

92

90

80

67

38

In the given case, however, the total value of absorbed radiation differs only by 5 percent from the radiation that could be absorbed by a perfect black surface. Hence it is clear that the receiving surface in question may be approximately considered as possessing an isotropic distribution of absorptivity, if we deal with isotropic (or approximately isotropic) fluxes of radiant energy. Let us now proceed to the calculation of radiant flux. According to

39

1.5. Thermal Emission of Real Bodies

Kirchhoff's law, the emissivity (emission intensity) of a dielectric is related to the intensity of blackbody emission as 1d.i =

(1.71)

dd,iE

From the general formula, (1.12), expressing radiant flux through intensity, we have for the radiant flux of dielectric Fa (1.72) using (1.68) and performing simple integration,

Fd=B--- 1 2

dp, J;I~

~ 2 "

0

[sin 2( I sin2(i

' _

r ) + tan2(i - r ) ]

+ r)

tan2(i

+ r)

sin i cos i di (1.73)

where B = nE = aT4. First, the formula (1.73) confirms an obvious fact, namely, that always Fd < B if the blackbody radiation flux is calculated for the same temperature as that of the given body. The value of the flux Fd can be computed from (1.73) with consideration of (1.69) by graphical o r numerical integration. As the majority of dielectrics can be considered gray bodies, it is natural to present the radiant flux of a dielectric in the form

Fd

=

6dB

=

(1.74)

ddaT4

where dd is the integral relative emissivity and T is the body's temperature. Computing Fa from (1.73), we can also compute 8 d = Fd/B. The results of similar computations for different values of n are presented in Table 1.9. TABLE 1.9 The Integral Relative Emissivity of Dielectrics n

1

1.41

2

3

4

5

6,

1 .oo

0.91

0.83

0.72

0.63

0.56

For the majority of dielectrics, n < 2, which accounts for most dielectrics having a high integral relative emissivity (about 90 percent). Experimental investigations are in conformity with this conclusion. Table 1.10 displays some measurement results for several dielectrics.

40

Radiant Energy. The Main Concepts and Definitions TABLE 1.10 The Integral Relative Emissivity of Some Dielectrics" ~~

Material

~

~~

a,, %

State of the Surface

White frost Ice a t -9.6OC Soot cover (thick) Soot cover (thin) Asbestos plate Glass Marble (light gray)

98.5 96.5 96.5 94.5 96 93.5 93

Smooth

Rough Smooth Polished

For water laboratory measurements, 6 = 96.5%.

2. Thermal Emission of Conductors. The theory of reflection of radiation from metals, owing to its complexity, has not yet been sufficiently worked out. Computations and measurements of the reflectivity of metals show that they are good reflectors of both thermal and visible radiations. Accordingly, the absorptivity (and emissivity) of metals is not high. Table 1.1 1 gives experimental data on the absorptivity of a number of metals for the radiation incident normally to the metallic surfaces at T = 290OK. TABLE 1.11 The Absorptivity of Some Metals (Percent)

Wavelength, ,u

1.0

1.5

2.0

2.5

3.3

5.0

7.0

10.0 13.0

Silver foil, lop

1.32 1.05 0.92 0.92 0.92 0.89 0.90 0.87 0.81

Gold foil, lop

1.74 1.31 1.18 1.13 1.04 1.01 1.03 1.05 0.95

Wavelength

1.06 1.71 3.06 3.96 5.24 6.75 8.02 9.38 10.49 12.03

Aluminum"

26.2 19.2 11.7 8.6 6.2 4.8 3.1 2.6 3.1 2.7 46.0 41.7 31.4 28.3 23.3 19.7 16.8 13.0 13.0 13.1

Tinb

Mechanically polished.

Cast.

NOTES: We see that the absorptivity decreases with the increase of wavelength (especially in the case of metals that are comparatively strongly absorbing). Silver and gold are practically ideal reflectors, whereas aluminum, often used in mirrors, is not. As to the angular dependence of reflectivity of metals, there is an observed increase of reflectivity with the growth of the angle of incidence, the same as in dielectrics.

41

1.5. Thermal Emission of Real Bodies

As already mentioned, along with the electric properties of bodies, the state of their surface considerably affects their thermal emission. This is particularly marked in the case of metals. Much oxydized and very rough metallic surfaces approach the blackbody in their emissive properties. Table 1.12 provides measurement data on the absorptivity of copper and brass in different states of the surface. TABLE 1.12 The Influence of the Surface State on the Absorptivity a, %

Metal

State of the Surface

Copper

Polished Polished, slightly dulled Shaved Oxidized, black

78

Polished Polished, slightly oxidized After rolling Tarnished Oxidized at 600OC

4 4.5 6 22 60

Brass

3

3.5 7

The cause of the increase of the absorptivity (and emissivity, correspondingly) of a rough surface is evident. We know that the best model of the blackbody is an enclosure with a small opening. The radiation leaving through such an opening corresponds most closely to that of a blackbody. It is natural, therefore, that the roughness of a surface (as expressed by the presence of many ‘‘enclosures’’ on the surface) adds to the increase of the absorpitivity (emissivity) of the latter. Theoretically, the influence of the surface roughness on the relative emissivity can be estimated in the following manner. As can be shown, the relative emissivity of the enclosing walls 6 is related to the “apparent” relative emissivity of the opening as follows: (1.75) where and I7, are the areas of the enclosed surface and its opening, respectively.

42

Radiant Energy. The Main Concepts and Definitions

In the finite case 17,/17,+ 0 (an enclosure with an infinitesimal opening), the equality 6, = 1 (blackbody) takes place. If I7, = D,, then 6, = 6. If 6 = 0.9 and IT2/U1 = 1/2, from (1.75) we have the value 6, = 0.95. Thus, even at IT,/IT, = 1/2, there exists a notable difference between 6 and 6,. With this kind of roughness the surface relative emissivity increases by 5.5 percent. After having studied the influence of different factors on thermal radiation of real bodies, we can answer the question about the case in which the emissive (absorptive) properties of a body will maximally approach the corresponding properties of the blackbody. The formula (1.63) indicates that in the case of a smooth surface, the absorptivity is determined by two factors-the reflectivity R, and the absorption function of a given body, A,([) = 1 - ecai2. From the consideration of these two factors, Shifrin [15] has stated the following peculiarities of the thermal radiation of smooth-surface bodies. In the case of metals, a, will, at constant thickness, approach zero at both small and at large values of a,. The maximum value a, will be reached at some intermediate al. The cause of this phenomenon in the case of metals is that large values of a, correspond to large values R, (“metallic” reflection). Consequently, smooth metallic surfaces are poor absorbers and emitters of thermal radiation. Only by making the metallic surface rough and oxidized (that is, by changing both the geometric and electric properties of the metallic surface layer) can one bring its absorptive and emissive properties close to those of a blackbody. This situation does not hold in the case of dielectrics. Even smoothsurface dielectrics have a very high relative emissivity.

3. Thermal Emission of Natural Surfaces. It is perfectly evident that thermal emission of natural underlying surfaces cannot be computed from the above formulas for conductors, dielectrics, or semitransparent bodies. It must be mentioned, however, that natural underlying surfaces are closer in their emissive properties to those of dielectrics and semiconductors. Both of the latter are gray emitters, and the intensity of their thermal emission can be computed from I = 6E (1.76) which is analogous to (1.71). A similar formula is valid for radiant flux, since the emission of natural underlying surfaces may be approximately considered isotropic : F = 6B

( I .77)

43

1.6. The Equation of Radiative Transfer

It is appropriate to mention here that the surfaces in question may be assumed to be gray emitters only in the first approximation. For example, the investigation of the spectral reflectivity of different trees and herbaceous plants in the infrared range from 3 to 25 p has shown the presence of a pronounced selectivity of radiation reflection, which evidently indicates that in the given case the relative emissivity must also depend on wavelength. The value of the relative emissivity has been measured in laboratory and under natural conditions. The most reliable measurements of this kind have been performed by Gayevsky [18], who has measured the relative emissivity of various natural covers in the spectral region 9 to 12 p. His results are as follows: Sand of fine grains, dry

0.949

Sand of fine grains, well wetted Sandy clayey soil, dry

0.962 0.954

Sandy clayey soil, well wetted

0.968

Peat, dry

0.970

Peat, well wetted

0.983

Thick green grass Thin green grass on wet, sandy, clayey soil

0.986 0.975 0.971

Coniferous needles Fresh snow Dirty snow

Kuzmin gives 6

= 0.96

0.986 0.969

for the water surface. Alexandrov and Kurtener

[19] have derived from their laboratory measurements the value 6 = 0.95

as the mean relative emissivity of the earth’s surface. Taking into consideration the existing variability of 6, we can state that on the average the relative emissivity of natural underlying surfaces is within the range 0.90 to 0.98. At present, however, the problem of the measurement of the relative emissivity of natural underlying surfaces is far from being satisfactorily solved. Both improvement of measurement methods and performance of these measurements on a larger scale are needed.

1.6. The Equation of Radiative Transfer for a Stationary Radiation Field In the atmosphere, radiant energy undergoes absorption and scattering. If we consider radiation of long wavelengths (those exceeding several microns), we must also take into consideration the emission of thermal radiation by different atmospheric layers. In the general case the transfer of radiant energy in the atmosphere (or, in general, in any absorbing, emitting,

44

Radiant Energy. The Main Concepts and Definitions

and scattering medium) is described by the so-called equation of radiative transfer. Due to a great complexity of the processes that determine the radiative transfer, the transfer equation is complex. In the theory of atmospheric radiative transfer we usually confine the problem to consideration of equations of radiative transfer for a stationary field of unpolarized radiation, without taking account of refraction. It is a reasonable assumption also to consider the radiation field to be stationary, since the process of radiative transfer in the atmosphere may be considered quasistationary. The influence of the nonstationary state of radiation is determined in the transfer equation by the term (l/c) (aZ/dt), where c is the velocity of light and t is the time. The variations of radiant intensity with time are so slow that this term may be considered as practically zero. A far more serious error would be to ignore the polarization of radiation. If the atmospheric thermal emission were considered unpolarized, the shortwave (scattered) radiation would be always polarized. The problem of determining polarization is so difficult that at present it is solved only for the simplest case of molecular scattering. Hence the neglect of polarization of radiation is unavoidable to a large extent. The influence of refraction on atmospheric radiative transfer is of importance only in some special cases, and therefore it is possible to ignore this influence in the general consideration of the problem. Let us now turn to the derivation of the general transfer equation for a stationary field of unpolarized radiation. Let us take an arbitrarily directed ray ( r = direction) and consider an environmental element in the form of a cylinder of unit section, the axis of which coincides with the direction of the ray. Let the ray cut the bases, perpendicular to the ray, at points P and P'. Denote E' = ds. The intensity of radiation at P and P' will then be

ZA(P,r ) and ZA(P',r ) = ZA(P,r )

+ -!% - ds ds

On the other hand, the variation of intensity in the transition from the point

P to P' is caused by: (a) Attenuation due to absorption of radiant energy, which can be expressed as

-k,(p)Z,(p> r)e(P) ds where e is the density and k , is the mass absorption coefficient at the point P .

45

1.6. The Equation of Radiative Transfer

We have already noted that such expression of attenuation of radiant energy due to absorption is dependent on the assumption that this attenuation is proportional to the initial intensity of radiant energy, the path, and the density. The mass absorption coefficient (that is, calculated per mass unit) is therefore the proportionality coefficient. (b) Attenuation due to scattering of radiant energy by the considered environmental element, which can be presented in the form

where u, is the mass scattering coefficient. This expression evidently takes account of the same assumption as that for the case of absorption. (c) Increase of the intensity of radiation owing to the considered element of the medium emitting in the direction I, which can be written as

where 7, is the mass emission coefficient. (d) Increase of the intensity of radiation resulting from the scattering process, due to which the rays of various directions passing through the elementary cylinder add a part of their energy to the ray of the direction r. If we consider the ray of the direction r‘ passing through an element of the medium, then that part of its energy equal to O,(W,(P,

r’le ds

will be scattered by this element, and a part

of this quantity will go in the direction r. Here the expression (1/4n)y,(P, r’, r ) characterizes the scattering function, and this means that of the total quantity of scattered radiation, the portion uJne ds equal to (1/4n)ya,Z,~ ds departs in the direction r. Integrating the expression (1.78) over all possible directions, we have

*s

4n

I,(P, r’)y,(P, r’, r ) do‘p ds

Taking into consideration all the obtained results and reducing by ds,

%(P,r’)yAP, r‘, r> do’

-

(k.

+

(1.79)

46

Radiant Energy. The Main Concepts and Definitions

As mentioned above, Kirchhoff’s law (1.34) is valid in the presence of local thermodynamic equilibrium. Taking into account also that ds = dz sec 8, where z is the vertical coordinate and 8 is the zenith angle, we can finally rewrite (I .79) in the following form :

In the case where we consider the problem of short-wave radiation transfer, EA is practically equal to zero. Thus, in place of (1.80), we have ‘OS Q

a”dz

-

cA 47c

I

IA(z,r’)yA(z,r‘, r ) dw‘ - (k,

+ oA)IA

(1.81)

If thermal radiation transfer is studied, it is possible to neglect scattering (onis practically zero) except in the case of transfer in clouds, fogs, or smo-

kes. It follows, then, that for thermal radiation, (1.80) will hold: (1.82) As seen from the comparison of (1.82) and (1.80), the problem of computing the monochromatic intensity of thermal radiation in the absence of large scattering particles in the atmosphere can be solved in a simpler way than in the case where such particles are present. In the former case the problem actually reduces to integration of an ordinary differential equation of the first order, whereas in the latter case it is necessary to solve a complex integral differential equation. Instead of one transfer equation (1.82), it is convenient to consider two similar equations, which one can derive in the following way. Let us introduce two new functions:

The function G,(z, 8 ) characterizes the intensity of radiation directed downward from the upper hemisphere; the function UA(z,0 ) is the radiant intensity going upward from the lower hemisphere. For the function UA(z,e), the transfer equation will retain the previous

References

form; for G,(z, result,

e),

we substitute n cos

e

t)

a ~ , ( z e, ) dZ

47

-

0 in place of 6 in (1.82). As the

=

k,[G,(z, 0 ) - E,1

(1.83a) (1.83b)

These equations of transfer of longwave radiation are basic for solving computational problems of atmospheric thermal radiation fluxes. Chapter 2 will show how t o derive the general formulas for thermal radiation fluxes by integrating these equations. Let us note here that (1.83) describe only the case where the atmosphere is assumed t o have the sole component of absorbing and emitting radiation. We shall also see how these equations can be generalized for the presence of several absorbing components. When deriving (1.83), we made an assumption of the state of local thermodynamic equilibrium. This assumption does not restrict the general approach t o investigations of thermal radiant fluxes in the troposphere, but can be of vital importance in other cases; for example, in the consideration of thermal radiative transfer in the upper atmosphere. Various equations of radiative transfer of a more general type can be found in the monographs of S. Chandrasekhar and V. V. Sobolev [23,24].

REFERENCES 1. Kondratyev, K. Ya. (1954) “The Radiant Energy of the Sun.” Gidrometeoizdat, Leningrad. 2. Kondratyev, K. Ya. (1956) “Radiative Heat Exchange in the Atmosphere.” Gidrometeoizdat, Leningrad. 3. Kaplan L. D. (1952) On the calculation of atmospheric transmission functions for the infrared J . Meteorol., 9 No. 2. 4. Kondratyev, K. Ya., and Yelovskikh, M. P. (1955) The distribution of the intensity of effective radiation and counterradiation over the sky. Proc. Acad. Sci. USSR, Geophys. Sect. (English Tratul.) No. 5 . 5. Plass, G . N. (1952). Parallel-beam and diffuse radiation in the atmosphere. J . Meteorof. 9, No. 6. 6. Plass, G. N. (1952). Method for determination of atmospheric transmission functions from laboratory absorption measurements. J. Opt. SOC.Am. 42, No. 9. 7. Plass, G. N. (1963). Spectral band absorptance for atmospheric slant paths. Appl. Opt. 2, No. 5 . 8. Ambartzumian, V. A., Mustel, E. R., Severny, A. B., and Sobolev V. V. (1952). “Theoretical Astrophysics.” Moscow. 9. Milne, E. A,, (1928). Influence of collisions on monochromatic radiative equilibrium. Moritly Notices Roy. Astron. SOC.88, No. 5 .

48

Radiant Energy. The Main Concepts and Definitions

10. Goody, R. M. (1954). “The Physics of the Stratosphere.” Cambridge Univ. Press, London and New York. 11. Woolley, R. (1947). Radiative equilibrium in the ionosphere. Proc. Roy. SOC.A189. 12. Curtis, A. R., and Goody, R. M. (1954). Spectral line shape and its effect on atmospheric transmissions. Quart. J. Roy. Meteorol. SOC.80. 13. Kondratyev, K.Ya., and Filipovich, 0. P. Tech. Transl. (1962). The thermal state of upper atmospheric levels. NASA, Tech. Transl. F103. 14. Shved, G. M. (1965). Method for consideration of the Kirchhoff‘s law deviation in the mesosphere due to radiative transfer in CO, 15 micron band. Proc. Leningrad State Univ., Chem. Phys. 1,No. 4. 15. Shifrin, K. S. (1951). “Light Scattering in a Turbid Medium.” Moscow-Leningrad. 16. Kozyrev, B. P., and Vershinin, 0. Ye. (1959). “Transmissivity of Materials for Wave Lengths 0.8-15.0 micron.” Leningrad. 17. Bolz, H. M. (1947). Applicability of Lambert’s law for the Falckenberg vibration pyrgeometer. Z. Meteorol. 1, No. 11/12. 18. Gayevsky, V. L. (1951). Surface temperature of large territories. Proc. Main Geophys. Obs. No. 26 (88). 19. Alexandrov, B. P., and Kurtener, A. V. (1936). Determination of radiation constants for solid dispersed bodies. Ch. Phys. USSR III, No. 3. 20. Gerson, N. S. (1951). Critical review of ionospheric temperature data. Rept. Progr. Phys. 14, 316. 21. Goody, R. M., and Saiedy, F. (1959). The solar emission intensity at 11 micron. Monthly Notices Roy. Astron. SOC.119,No. 3. 22. Kondratyev, K. Ya. Badinov, I. Y., Ashcheulov, S. V. and Andreyev, S. D. (1965). Atmospheric infrared spectra of absorption and emission. Results of surface measurements. Proc. Acad. Sci. USSR, Phys. Atmosphere Ocean No. 4. 23. Chandrasekhar, S. (1950). “Radiative Transfer.” Oxford Univ. Press, London and New York. 24. Sobolev, V. V. (1950). “Radiative Transfer in Stellar and Planetary Atmospheres.” Leningrad.

2 METHODS OF ACTINOMETRIC MEASUREMENTS?

The intention of this chapter is to give only a brief survey of the available actinometric instruments. A more detailed treatment of the problems concerning actinometric instruments and measurement methods can be found in special literature (see, for example [l, 21).

2.1. General Characteristic of the Methods for Measurement of Radiant Energy The different methods for measurement of radiant energy are based on the use of the effects that radiant energy can produce on various receivers, changing thereby into other kinds of energy. For example, in the absorption of solar radiant energy by a blackened surface, the transition of radiant energy into thermal energy is taking place. By recording the amount of heat released in this process or the increase of the receiving surface temperature, it is possible to measure the magnitude of the solar radiant flux incident on the receiving surface. Similar principles form the basis of the calorimetric method. The phenomenon of photoeffect and the photochemical effects of light have been used in working out the photoelectric and the photographic methods of the radiant energy measurement. Fairly popular are visual measurements in which the human eye-which possesses in the visible spectral region a high sensitivity to light effects-is a receiver of radiation. Table 2.1 summarizes the peculiarities of the above methods. Table 2.1 gives the wavelength range in which a measurement method is sensitive, the sensitivity, linearity, and selectivity of the method. Linearity Y.K. Ross, S. V. Ashcheulov, and V. V. Mikhailov are co-authors of this chapter. 49

50

Methods of Actinometric Measurements TABLE 2.1 General Characteristic of the Methods for Mesurement of Radiant Energy

Method of Measurement

Wavelength Range, 8,

Calorimetric

All wavelengths To 400,000 To 12,000 4000-7500

Photoelectric Photographic Visual

Sensitivity

Linearity

Selectivity

Low

Very good

Absent

High High

Good Bad

High High

High

Very bad

High

is understood to be the proportionality of the receiver’s readings to the measured radiation value ; selectivity characterizes the degree of the dependence of the receiver’s sensitivity upon the wavelength. As seen from Table 2.1, the calorimetric method is characterized by the absence of selectivity (equal sensitivity t o radiation of all wavelengths) and by a very good linearity of readings. These properties account for the wide use of the calorimetric method in actinometric measurements, in spite of its low sensitivity. The most important task of actinometric measurements is the determination of the integral flux values of shortwave and longwave radiation, and in this particular function the calorimetric method fits best. In the investigation of spectral fluxes of shortwave radiation at the present time, the most widely used method is the photoelectric method of recording with the help of photocells, photomultipliers, bolometers, and Golay detectors. For measurement of longwave radiation spectral flux bolometers, Golay detectors, and other sensors are used. Let us now proceed to a closer acquaintance with various ways of radiant energy measurement by the calorimetric method. Consider the following schematic classification of the basic ways of radiant energy measurement as applied in actinometry, as suggested by Courvoisier and Wierzejewski [3]. As the result of the absorption of radiant energy by the instrument’s receiving surface, the transformation of radiant energy into thermal energy occurs, and in the system of the receiving body a heat flux appears. Different ways of calorimetric measurement of radiant energy are distinguished by the method for determination of this heat flux. Two groups of measurement methods are isolated : those without phase transformations and those with them. In the latter case the heat flux value is determined through the quantity of heat lost in a certain phase transformation (for example, ice melts),

51

2.1. Methods for Measurement of Radiant Energy

Radiant energy transforming into heat through absorption

Thermal flux to the receiver’s system

Phase transformations

1 I

Without phase transformations I

Heat flux measurement

I

1 I

I Temperature difference measurement I

I

Relative method

Absolute (compensation) method

the measure of which is the quantity of the matter changed into another phase state (for example, the quantity of melted ice). In the first of the above cases one must distinguish those methods based on direct measurement of the heat flux and those based on the difference in temperature between the receiving surface and the environment (the massive parts of the instrument at a constant temperature equal to the air temperature). The direct determination of heat flux is taken in actinometric instruments from the heating of the water around the receiver. The increase of its temperature is a measure of the heat flux, and of the amount of radiant energy incident on the receiving surface. On this principle are based various types of water pyrheliometers intended for measurement of direct solar radiation flux. More widely used are instruments in which the heat flux (and consequently the flux of radiant energy) is determined through the difference in temperature between the receiving surface and the environment. The temperature

52

Methods of Actinometric Measurements

difference is most often measured thermoelectrically. This is done by taking the value of the electric current rising in the circuit of consecutive thermocouples, the “hot” junctions of which contact the receiving surface and are exposed to irradiation, and the “cold” junctions are at constant temperature. Instruments of this kind which measure radiant energy by the difference between “hot” and “cold” junctions, are relative and need calibration by reference to absolute instruments that establish a standard. Among such relative actinometric instruments for measurement of solar and diffuse radiation fluxes are the much used in the U.S.S.R. Savinov-Yanishevsky actinometer and the Yanishevsky pyranometer, the description of which will be given later. Absolute measurements of radiant energy are made possible by application of the compensation method consisting in heating the “cold” junctions up to the temperature of the “hot” junctions. The quantity of heat used for the heating can serve as the absolute measure of the amount of heat released in the absorption of radiant energy by the receiving surface of the instrument. The Angstrom absolute pyrheliometer is built according to this principle. Certain types of actinometric instruments with a blackened bimetallic lamina for the receiving surface employ the deformation of this lamina as it is irradiated by solar radiation, to measure the amount of heat received by the surface. The Mihelson actinometer is the most successful instrument of this kind. Let us now examine the instruments most widely used at present.

2.2. Instruments for Measuring Direct Solar Radiation For measurement of direct solar radiation flux a large number of instruments such as pyrheliometers and actinometers are used. We shall treat here, however, only those in wider use in Europe.

1. Angstrom Compensation Pyrheliometer. The electrical pyrheliometer of K. Angstrom can be used for the absolute measurement of direct solar radiation flux. Its receiver (Fig. 2.1) consists of two similar manganin strips A and B blackened on the outside. To the lower sides of the strips over the insulating coat are attached the junctions a and b of the thermocouples connected to the sensitive mirror galvanometer GI. The manganin strip (strip B in Fig. 2.1) can be heated by the electric current passing through it from the battery E. The rheostat R regulates the current and thus the amount of heat released when current passes through the strip. The index galvano-

2.2. Instruments for Measuring Direct Solar Radiation

53

FIG. 2.1 The scheme of the ,&gstriim compensation pyrheliomefer.

meter G, serves for measuring current strength. The receiving strips are set at the head of pyrheliometer 1, which is inserted in the lower end of the tube 2 (Fig. 2.2). In the upper end of the tube, stopped by the cover 3, are two aperture slots in opposition to the receiving strips. Each of these apertures can be shaded by a shield capable of half-rotations. The worm screws 4 and 5 serve for orientation of the instrument to the sun. The setting of the instrument on the target (sun) is made with the help of pointing control 6. The operation takes place in the following manner. One of the strips (strip a) is exposed to the sun’s rays and the other (strip b ) is shaded. This arrangement generates electric current in the circuit of the thermocouple because of the difference between the temperatures of the strips a and by the first of which is heated by the absorption of the incident solar radiation, and the second (being shaded) of which has no change its temperature. If now the current passes through the second strip and heats it up to the temperature of the first, it is evident that the quantity of heat released in strip a will equal the quantity of solar radiation absorbed by strip b. The absence of current in the galvanometer circuit GI determines the moment when the temperature of both strips is equal. Thus by recording the strength of current i in the galvanometer G, at the zero reading of the galvanometer G I , we have for the quantity of heat Q (cal/sec) released in strip a, by virtue of Joule-Lenz’s law, Q

= 0.24i2r

(2.1)

54

Methods of Actinometric Measurements

3

FIG. 2.2

The h g s t r i i m pyrheliometer.

where i is the strength of current in amperes and r is the resistance of the strip in ohms. On the other hand, the same quantity of heat (cal/sec), received by strip a in the absorption of solar radiation is determined by the relation

SBlb 60

Q=-

where S is the flux of direct solar radiation in cal cm-2 min-', 6 is the absorptivity of the surface of the strip, and b and I are its width and length in centimeters. On the basis of these relations we find the following expression for direct

2.2. Instruments for Measuring Direct Solar Radiation

55

solar radiation flux in cal cm-2 min-’:

where k is the pyrheliometer’s constant ( k = 14.4 r/61 b). This elementary theory, however, does not produce fully reliable results, for this theory does not take into consideration certain factors that influence the pyrheliometer readings. Therefore each individual pyrheliometer is not an absolute instrument. Since the constants of network pyrheliometers are not determined by computation, their readings must be compared and adjusted to those of a specially prepared and carefully handled standard pyrheliometer. The Angstrom pyrheliometer possesses a systematic error caused by the so-called edge efSect [ 5 ] , caused when the exposed strip is not heated uniformly in the absorption of solar radiation. This occurs because its edges are partly shaded by the instrument’s diaphragm and also because the radiation is absorbed by a thin surface layer of the strip, whereas the shaded strip undergoes quite an even heating by electric current. The equal temperatures of the strips, thus, fixed by the galvanometer G I , are observed factually in the case when Q < S 61 b/60, which leads to the lowering of the pyrheliometer readings. The application of Ansgstrom pyrheliometers is also complicated by the presence in the instrument of rectangular splitting diaphragms, the solid angle of which does not correspond to the solid angle of most widely used actinometers either in form or in magnitude. This makes a difference between the standard pyrheliometer readings and those of actinometers, owing to different amounts of the circumsolar diffuse radiation falling on the aperture of instruments together with the direct solar radiation. In this connection Yanishevsky has improved the Angstrom pyrheliometer by building circular diaphragms identical with the Yanishevsky actinometer diaphragms. In the new pyrheliometer, the thermocouples u and b (Fig. 2. I ) have been replaced by thermopiles, which allows replacement of the mirror galvanometer GI by one with an index. Some other attempts at changing the Angstrom pyrheliometer have also been made. The Angstrom compensation pyrheliometer is the main standard absolute instrument in Europe. In the United States another pyrheliometer, the so-called waterstreurn, is used as the standard. In this type, the direct solar radiation flux value is determined from the rise of the temperature of the water that washes the receiving part of the instrument heated in the radiation absorption. This

56

Methods of Actinometric Measurements

principle of measurement was first suggested by Mihelson ; Shulghin then considerably improved the instrument, showing that the initial model was giving too high readings. An interesting model of pyrheliometer has recently been proposed by Koocherov [4]. The receiving surface is first cooled down to the temperature below the air temperature and then is heated under the influence of the absorption of solar radiation. It can be shown that the direct solar radiation flux value is proportional to the variation of the receiving surface temperature with time at the moment of the equality of the temperature of the receiving lamina and the air temperature. Both standard pyrheliometers possess a rather low (for a standard) degree of accuracy and even have systematic errors. Owing to this, there are two actinometric scales, European and American, based on the use of the two standards, respectively. It has been found in the American scale that the readings exceed by 3.5 percent those in the European. The improvement of the waterstream pyrheliometer has shown the previous reading to be in error of +2.4 percent. On the other hand, the improvement of the observation processing in the Angstrom pyrheliometer gives an error - 1.3 percent on the old European actinometric scale. Taking account of these new data, we see that the precise American scale is increased by 1.1 percent and the precise European by 1.3 percent relative to the old European scale. At the present time, both active scales coincide, the difference between them being only 0.2 percent. In connection with the conduct of the International Geophysical Year, a decision was made to adopt, beginning Jan. 1, 1957, the “International Pyrheliometric Scale 1956.” Measurements made according to the original uncorrected European scale are to be increased by 1.5 percent; measurements made according to the American scale are to be decreased by 2.0 percent. 2. Mihelson Actinometer. The Mihelson actinometer is intended for relative measurements of direct solar radiation flux. Owing to its compactness and simplicity, it is used extensively. Figure 2.3 gives a schematic cross section of the instrument. A sootblackened bimetallic lamina 1 (usually iron-invar), irradiated by the solar beam passing through the rectangular aperture 5, is heated and deflected in the process of heating. The deflection is caused by the fact that iron lengthens in heating while invar does not undergo notable expansion by heat. The shift of the lamina’s edge, caused by deflection, is then transmitted to the attached index 3, made of aluminium or quartz filament. At the point of the index, whose shift is more pronounced that at the lamina

2.2. Instruments for Measuring Direct Solar Radiation

I

FIG. 2.3

57

Ill

A schematic section of the Mihelson actinometer.

edge, is fixed a quartz filament 2 serving as a shift indicator. The filament is observed against the mirror 4,forming a part of the index through a microscope, whose setting 6 is screwed to the case. In the first approximation the deflection of the lamina, and consequently the filament shift, is directly proportional to the quantity of solar radiation absorbed by the lamina; thus it is possible from the shift to judge the solar radiation flux. The actinometer’s calibration factor at this point must be verified by comparison with the angstrom pyrheliometer or some other absolute instrument. A considerable dependence of the calibration factor upon temperature is a peculiarity of the Mihelson actinometer. The Mihelson actinometer has many modifications, as by Marten, Buttner, Kalitin, and others. A model for absolute measurements has been constructed by Krylov. In the United States the so-called silver disk pyrheliometer is widely used as a relative nonthermoelectric instrument. The measure of solar intensity is judged from the temperature, taken with a mercury thermometer, of a blackened silver disk, positioned at the lower end of a receiving tube whose aperture angle is approximately 6 deg. 3. Savinov- Yanishevsky Thermoelectric Actinometer. This actinometer is at the present time the main instrument in the U.S.S.R. for measuring direct solar radiation flux. Its main parts are a receiver with a thermopile, a case with a tube in which the thermopile is inserted, and a support that also serves for setting

58

FIG. 2.4

Methods of Actinometric Measurements

The longitudinal section of the Savinov- Yanishevsky actinometer’s tube.

the tube on the sun. A galvanometer is connected to the thermopile. Figure 2.4 shows a longitudinal section of the tube, whose aperture angle is 10 deg. The solar beam passing through the tube falls on the receiving surface, which is a thin silver disk, 1, blackened on the side of the sun. To the inner side of the disk are stuck “hot” junctions, 2, in a system of zigzagconnected manganin-constantan thermoelements. “Cold” junctions, 3, are fixed t o a copper ring, 4, mounted in the case. The thermopile of the Savinov-Yanishevsky actinometer is usually called thermostar because of the zigzag shape of its junctions. Between the junctions and the silver disk and copper ring is an insulation of tissue paper impregnated with shellac. Under the influence of solar radiation, the ‘‘hot” junctions get warm; thermocurrent passes from the thermobattery circuit through conductors 5 and then through the insulated copper conductors 6 connected to the galvanometer. In the first approximation the difference in temperature between the hot and cold junctions is proportional to the falling flux of radiation. The thermocurrent, in its turn, is proportional to the temperature difference, too. If the galvanometer scale is linear, then the galvanometer deviations must be proportional to the measured flux. The proportionality coefficient, the calibration factor of the Savinov-Yanishevsky actinometer, is obtained by comparing its reading with those of the Angstrom pyrheliometer. The sensitivity of the instrument is 4-7 mV per cal cm-2 min-I, the internal resistance 13 to 20 ohms. The temperature coefficient of the calibration factor is -0.1 percent per OC. The Savinov-Yanishevsky actinometer serves both for single measurements and for continuous recording of direct solar radiation. In the latter case it is mounted on a heliostat, a clock mechanism automatically orienting the actinometer on the sun. The current i n the actinometer circuit is recorded

2.2. Instruments for Measuring Direct Solar Radiation

59

with a galvanograph, of which different types are used, with recent preference for automatic electronic potentiometers. 4. Linke-Feussner Actinometer. This is one of the modern constructions of actinometer and has mainly a European application. It is distinguished by a massive case with thick copper diaphragms (Fig. 2.5), securing a high

II

7

3\

4-

2

I----

FIG. 2.5 A schematic section of the Linke-Feussner actinometer. (1) Moll thermopile; (2) copper diaphragm; (3) screening head; (4) rotating disk; (5) filter.

60

Methods of Actinometric Measurements

stability and good sensitivity of the instrument. A rotating disk 4 with filters 5 is positioned in the upper end of the tube. At the upper extremity of the tube is a special screening head 3, which eliminates unwanted reflection in filter measurements. The actinometer employs a Moll thermopile receiver consisting of thin constantan-manganin strips (of tenths of micron thickness). The hot junctions, blackened with Parsons' optical black lacquer, are situated along the central cross line of the thermobattery; the cold junctions are put at the edges and have a good thermal contact with the massive case through the copper shanks that support the thermobattery. The thermal capacity of the cold junctions exceeds by 10,000 times that of the hot junctions, and, in measuring (or, during measurements), the temperature of the latter increases by 25 to 30' above the cold junction temperatures. The new model employs two sections, each consisting of 20 thermoelements connected in opposition. One section is exposed to the sun's rays and the other is screened from radiation, which enables elimination of the undesired temperature and pressure effects. The design of the Moll thermopile makes it possible to attain high sensitivity and stability of readings and almost absolute zero position, increasing at the same time the dependence of the sensitivity on temperature. The aperture angle is approximately 10 deg, the sensitivity 5 to 10 mV per cal cm-2 min-l, the internal resistance 60 Q, and the temperature sensitivity coefficient -0.2 percent per "C. Certain other models are used in actinometry, among which are the Moll-Gorczynski actinometer and the Eppley pyrheliometer. 2.3. Instruments for Measuring Global and Diffuse Radiation and Albedo There is no agreement on which name for these instruments is the best. The three types of most used pyranometers are called the Yanishevsky pyranometer, the Moll-Gorczynski solarimeter, and the Eppley pyrheliometer, in spite of their common purpose of measuring global, diffuse, and reflected radiation. 1. Yunishevsky Pyvanometev. It is the main instrument for global and diffuse radiation measurement in the U.S.S.R. A schematic presentation of this instrument is given in Fig. 2.6. The receiving surface consists of a system of consecutively connected manganin-constantan thermoelements (Fig. 2.6, B and D, where the manganin strips are cross-hatched). The thermoelements are placed either in the form of rectangular -arrangement

61

FIG. 2.6 The scheme of the Yanishevsky pyranometer.

2.3. Measuring Global and Diffuse Radiation and Albedo

(1) battery; (2) removable plate; (3) case; (4) diaphragm; (5) joint-pin; (6) screw preventing overturning of the diaphragm; (7) ring; (8) glass; (9) cover for determining the zero position; (10) level; (11) lamina; (12) desiccator; (13) screw for setting one certain side of the pyranometer on the sun; (14) nails; (15) setting screws; (16) screen; (17) collapsible tubulated shank to be fixed with a screw; (18, 19) clips guarded against overturning and loss of nuts.

62

Methods of Actinometric Measurements

(Fig. 2.6 B, C ) or radially from the center of the receiving surface. The hot junctions are sooted black, and the cold are whitened with magnesia so that the receiving surface is like a black and white checkerboard. A hemispherical glass cover prevents wind effect (Fig. 2.6, A ) . When irradiated, the cold and hot junctions create a temperature difference, approximately proportional to the measured radiation flux. Since the temperature difference in the thermobattery circuit and the deviation of the attached galvanometer are directly proportional, it is possible to measure the quantity of the incident radiant flux from the galvanometer deviations. In measuring diffuse radiation flux the receiving surface is screened from direct solar radiation (Fig. 2.5 ). The pyranometer is a relative instrument. The determination of the calibration factor of the instrument is made according to the ‘‘sun-shadow’’ method : The difference between the readings of the screened and unscreened instrument, characterizing the direct solar radiation flux value, is compared with the actinometer reading. The processing of pyranometer observational data must consider the dependence of the calibration factor upon the angle of incidence of radiation (the solar zenith distance). The characteristic pyranometer angular dependence of sensitivity, and also a pronounced selectivity of sensitivity for radiation of different wavelengths, necessitates introduction of the so-called angular and spectral corrections to the readings in the measurement of diffuse radiation. These corrections take into consideration the differences between the angular and spectral distributions of diffuse and direct solar radiation intensity. The same types of self-recorder, which are employed in recording the direct solar radiation, provide for continuous recording of the global or diffuse radiations. Pyranometers are also used for measuring the albedo of the underlying surface. When specially modified for such measurements, they are known as albedometers, the main distinction being the presence of a device for alternate rotation of the albedometer to position its receiving surface either upward or downward. The so-called field model secures the maintenance of horizontal receivers by the use of gimbals. At present, pyranometers employ such supports, which enables their use both as pyranometers and as albedometers without any auxiliary devices. In some cases a pair of instruments is used, one receiving the downcoming and one the reflected flux of radiation. This method shortens the time necessary for measuring the albedo, which is very important in view of the possible rapid variation of global (and, accordingly, reflected) radiation with time. Its drawback,

2.3. Measuring Global and Diffuse Radiation and Albedo

63

however, is the need for reliable control over the sensitivity of both pyranometers. For measurements of global and reflected radiation at different elevations, pyranometers are installed on aircraft, balloons, and also on special masts and towers. They are also employed for sea observations on board a ship. Higher sensitivity pyranometers (usually engaging four thermobatteries of a common pyranometer) are used as underwater, undersnow, and under-ice pyranometers for measuring the shortwave radiation penetrating water, snow, and ice. A recent model of the Yanishevsky pyranometer has a checkerboard receiving surface of 30 x 30 mm, a sensitivity of 7 to 10 mV per cal cm-2 min-l, the internal resistance of 28 to 32 Q, an insignificant temperature coefficient of the calibration factor, and a good fulfillment of the cosine law.

2. Moll-Gorczynski Solarimeter. This instrument is widely used in Europe. It employs a Moll thermopile (see Linke-Feussner actinometer) consisting of 14 constantan-manganin thermoelements in the form of strips of the dimensions 10 x 1 x 0.005 mm. In contrast with the Yanishevsky pyranometer, this instrument does not have a white surface; its cold junctions are in good thermal contact with a heavy case. To increase the stability of readings of the 10 x 14 mm thermopile, the latter is covered by two concentric ground and polished optical glass domes, with diameters of 26 and 46 mm. Provision is made for the use of a white metallic guard disk of 300-mm diameter at the receiving surface level, to prevent the direct sun’s rays from heating the case. The instrument is intended for measuring and recording global radiation. The sensitivity is about 7 to 8 mV per cal cm-2 min-l, the resistance 10 Q, the temperature sensitivity coefficient 0.2 percent OC; the cosine law is fulfilled satisfactorily. 3. Eppley Wide-Angle Pyrheliometer. This instrument is used mainly in America. The receiver is a thin silver disk with an attached ring (29 mm in diameter). The central circular part of the disk and the outer ring are coated with magnesium oxide. The innermost ring of the receiver is coated with Parsons’ optical black lacquer. Behind these black and white rings are respectively attached hot and cold junctions of 50 thermoelements of gold-palladium and platinum-rhodium alloys, which measure the temperature difference between the black and white rings. The thermobattery occupies the central position in an almost spherical bulb, of diameter 75 mm, made of optical glass. The lower part of the hermetically sealed bulb, filled with dry air, is fixed to the support.

64

Methods of Actinometric Measurements

The instrument serves for measuring and recording global radiation. Its sensitivity is 7 to 8 mV per cal cm-2 min-l, the resistance 10052, the temperature coefficient -0.5 percent per “C. The cosine law is fulfilled within an error of 10 percent. 4. Other Models of Pyranometers. The Robitzsch bimetallic actinograph for recording global radiation is in widespread use in spite of its various drawbacks (an error not less than 5 to 10 percent). Its construction is similar to that of the Mihelson actinometer. It consists of a receiving plane of three closely set, bimetallic strips (about 8.5 x 1.5 cm). The upper surface of the central strip is blackened; the outermost pair are covered with white paint. The deformation of the central strip, caused by heating in the absorption of global radiation, is transmitted with the help of levers to the recording pen. The deflection of the pen arm is proportional to the difference in temperature between the blackened and white strips, and thus proportional to the measured global radiation flux. The sensitive platform is protected by a glass cover. There are many modifications of Robitzsch pyranometer, but they are all similar in the principle of operation. Another widely used instrument, especially in radiation problems related to biology, is the Bellani spherical pyranometer. It consists of two concentric glass spheres, the innermost one having a metal coating that absorbs shortwave radiation. Thus the outer surface of the innermost sphere is a receiving surface, and the outside protects the arrangement from wind and other atmospheric effects. The interspherical space is exhausted, creating a fairly high vacuum that enables insulation of the innermost sphere from the heat exchange resulting from thermal conductivity. The internal sphere contains very pure ethyl spirit, which is evaporated by the heat released in the absorption of radiation by the metallic surface. The spirit vapor leaves the sphere through a thin pipe, condensing on an attached, vertically pointed pipe. The quantity of spirit condensed in the pipe per day or half-day is a linear function of the quantity of energy reaching the receiving surface. The calibration of Bellani pyranometers is made by the “sun-shadow’’ method. The difference in reading between two pyranometers, one of which is shaded from direct solar radiation, is compared with actinometer readings. If S is a total of direct solar radiation measured by the actinometer per time unit and Ax is a difference in thickness between liquid layers condensed in the shaded and exposed pyranometers, the calibration factor k is determined from k=- S 4 Ax

2.4. Measuring Brightness and Illumination

65

The coefficient 4 is introduced for the purpose of taking into account the fact that the surface of the sphere is four times the area of its crosssection. Once calibration factor is determined, it is possible to use it in the processing of pyranometer readings in the measurement of the global and reflected radiation on the spherical surface. As estimated by Courvoisier and Wierzejewski [3], the Bellani pyranometer produces a daily radiation total measurement with error not exceeding 3 percent. 2.4. Instruments for Measuring Brightness and Illumination

At present there are no commonly used standard instruments for measuring brightness (light intensity of radiation) and illumination (light flux of radiation) in natural conditions. This is partly explained by the fact that measurements of brightness and illumination are conducted on a far smaller scale compared with that of actinometric observations. Two types of photometer, the visual and the photoelectric, are used in these measurements. The latter is more convenient because it records automatically. The same photometer can be employed for measuring both luminosity and the illumination, depending on whether its receiving surface is exposed hemispherically (illumination measurement) or confined to a sufficiently small solid angle (luminosity measurement). In the U.S.S.R., the Weber-Bylov visual photometer is used for measuring natural illumination, used for both total illumination and illumination by diffuse radiation, when the instrument is screened from direct solar influence. Usually a selenium photocell receiver is preferred, owing to the fact that selenium photocells possess a spectral response conforming as closely as possible to that of the human eye. When illumination is being measured, especially with the use of photocells, the so-called photometer sphere (spherical photometer, the Ulbricht sphere) is sometimes used. This is a sphere whitened inside with a paint that reflects radiation in a rather nonselective and orthotropic way (according to Lambert’s law). Magnesium oxide, zinc white, or barium sulfate are commonly used as paint. If there is a horizontal aperture for receiving global radiation in the upper hemisphere and another aperture at an unilluminated spot of the inner spherical surface, through which a beam of light is directed on the receiver (or the aperture is an immediate receiver), the readings of the latter will be directly proportional to the magnitude of illumination of the horizontal surface. However, the aperture radii must be considerably

66

Methods of Actinometric Measurements

smaller than the radius of the sphere. This construction also has wide application in photometry for measuring reflection, transmission, and absorption of radiation by different samples. It should be noted that for approximate characteristic of light regime, global (or diffuse) radiation data may be used, since in the first approximation there is a linear connection between the energetical and luminosity radiation fluxes.

2.5. Instruments for Measuring Radiation Balance and Effective Radiation At present there is no satisfactory solution of the problem of measurement of thermal radiation flux and radiation balance,+although recent years have produced many new models of pyrgeometer and balancemeter. Moreover, while shortwave radiation can be measured with a sufficiently high degree of accuracy, the most reliable instruments for measuring thermal radiation flux reach an accuracy not more than 10 to 15 percent, and even these are not sufficiently accurate for daylight measurements. The main difficulties are the following : (a) When the receiving surfaces are exposed, the readings depend considerably on the turbulent pulsations of the wind speed and the air temperature ; the introduction of corrections takes only a partial account of these effects. If filters (polyethylene, KRS 5, etc.) are used to eliminate the turbulent pulsations of air, the integral sensitivity of the instrument will differ for the direct, diffuse, and reflected radiations as well as for the atmospheric and the earth’s thermal emission. The sensitivity corrections make data processing a complicated affair. (b) There is no sufficient unification of the receiving surface coating at present, and the difference in absorption coefficient between the shortwave and longwave spectral regions have not been fully investigated. (c) There are no well-constructed and unified low-temperature models of the blackbody; neither are there methods of calibration of receivers with reference to thermal emission.

No agreement has been reached as regards a general term to identify instruments in this field. Usually the ones that measure radiation balance are called balance meters or net radiometers, and those that measure effective radiation at the receiver are called pyrgeometers. Net radiation flux.

2.5. Measuring Radiation Balance and Effective Radiation

67

Of these, the most commonly used are the Schulze balance meter, the Funk balance meter, and the Yanishevsky balance meter. 1. Schulze Balance Meter. This instrument is intended for measuring the incoming and outgoing radiant fluxes or their difference (radiation balance). It consists of an aluminium case, 1 (Fig. 2.7) of diameter 80mm

FIG. 2.7

A schematic section of the Schulze balancemeter.

and height 40 mm. To the case are attached two constantan-silver thermopiles, 2 (26 x 5 mm). The hot thermojunctions are coated with Parsons’ matt black lacquer; the cold junctions are in good thermal contact with the case. The receiving platforms are protected from wind by hemispherical polyethylene (Lupolen-H) covers of 60 mm diam and 0.1 mm thickness(3), made by pouring the melted material over a mold. The most recent model provides means for ventilation of the outermost surfaces of the covers, to eliminate overheating and deposition of hydrometeors. The casing temperature is measured with a thermoelectric constantan-copper sensor, one couple of which is situated behind the receiving surface and one is buried at a depth of 1 m in the ground. Nearby is placed a drawing thermometer. There are two metallic disks of 300 mm diam(4) attached to the casing, one facing upward and one downward, to provide for a better heat exchange from the casing airward and to protect the casing from the sun’s rays. The sensitivity of the instrument is 18 to 24 mV per cal cm-2 min-’, and the effective resistance 250 9. There is practically no wind effect. The

68

Methods of Actinometric Measurements

dependence of the calibration factor on temperature does not exceed 0.1 per O C . The balance meter requires a horizontal installation at a minimum I-m elevation. The calibration for shortwave radiation is performed according to the “sunshadow” method for thermal radiation with reference to a low-temperature black emitter.

2. Funk Balance Meter. This type of instrument serves for radiation balance measurement and, as in the Schulze balance meter, employs hemispherical protective polyethylene covers of 0.05-mm thickness. To preserve this sphericity and to exclude the deposition of water vapor, the hemispheres are filled with dry nitrogen of several centimeters of watercolumn pressure. The pressure is kept even with the help of a gas balloon connected to the balancemeter through the reductor. The black receiving surface is striped white with the purpose of decreasing the difference in sensitivity between the shortwave and longwave spectral regions. The receiver is horizontally surrounded with a heating ring of 180-mm diameter, to prevent the setting of dew. The sensitivity is 40 mV per cal cm-2 min-’ and the effective resistance is about 80 9. 3. Yanishevsky Balance Meter. This instrument is in network use in the U.S.S.R. and measures the radiation balance (net radiation). Of the available models this is the simplest. The receiver (Fig. 2.8) consists of two thin copper laminas 1, blackened on the outside. To the inside surface are

FIG. 2.8

The Yanishevsky balancemeter mounted on a binge wirh a shade.

2.5. Measuring Radiation Balance and Effective Radiation

69

attached, through an insulating material, thermocouples of 10 thermopiles of 32 elements each. Copper bars are inserted between the thermojunctions to secure thermal contact of the upper and lower laminas. The laminas are fixed in the square aperture of a nickel-plated brass disk, 2, which is in good thermal contact with the laminas. The receiver is set horizontally with the help of gimbals, 3. The screen, 4, shades it from the sun. The pole 5 and coupling 6 secure the required position of the screen. When not in operation, the balance meter is protected with a cover 7. The instrument working principle is based on the difference in temperature between the upper and lower laminas and is proportional to the measured radiation balance value. The calibration is made by taking the difference in reading between the shaded and exposed balance meter and comparing it with actinometer readings. Since the receiving platforms are open to wind, the calibration factor appears to be dependent on the wind speed. Thus each individual instrument must be evaluated for its own dependence of calibration factor upon wind speed, which can be done both in laboratory and natural conditions. Sometimes in processing balance meter readings, such factors are taken into consideration as the unequality of the sensitivity of the upper and lower sides of the instrument, the difference in sensitivity between the shortwave and longwave radiations and certain others. The balance meter can also be used as pyrgeometer. In this case the lower receiving surface is screened from below, the inner side of the screen being black and the outer one polished. The pyrgeometer measures the net radiation of the instrument’s blackened surface. Sometimes such instruments are called effective pyranometers. The radiation balance of the underlying surface can be found from the difference in reading between the upward and downward directed instrument. 4. Other Models of Pyrgeometers. No longer widely used are several

formerly popular models of pyrgeometer, among which is the SavinovYanishevsky pyrgeometer, based on the same principle as the Yanishevsky construction. Its receiving surface is made of metallic strips, alternately black and shining. The cold junctions contact the black strips and the hot are connected to the shining part. The current in the thermopile circuit, proportional t o the difference in temperature between the black and the shining strips, serves as the measure of the black-strip radiation balance. As in the case of balance meter, the calibration factor is notably dependent on wind speed. The instrument is calibrated by reference to balance meter readings. Since it is mainly used for night measurements, a calibration method by reference t o a blackbody is also used. In this case the source

70

Methods of Actinometric Measurements

of radiation, incident on the pyrgeometer receiving surface, is a blackbody usually represented as an enclosure aperture or in the form of the “snow sky,” a snow-filled hemisphere. As mentioned in Chapter 1, snow possesses a very high relative emissivity. Knowing the blackbody temperature, and thus the radiant flux, incident on the pyrgeometer receiving surface, it is possible to find the calibration factor of the latter. Pyrgeometers as well as balance meters have different sensitivity to shortwave and longwave radiations. Owing to this, the results of the calibrations from reference to the blackbody and to balance meter, calibrated by the “sun-shadow” method, do not coincide as a rule. There are also a number of compensation pyrgeometers, which measure the quantity of effective radiation (during the nighttime only) from the strength of current passing through the blackened strips for leveling the temperatures of the black and the shining strips. This is the principle of the Angstrom compensation pyrgeometer. Other models, as by Falckenberg, Braslavsky, Scheunesson et al., have a totally blackened receiving surface with cold junctions either suspended inside the instrument or positioned in good thermal contact (but insulated electrically) with massive metallic parts of the instrument. An important advantage of this model is a lower dependence of the calibration factor upon wind speed. Certain successful models (as the Braslavsky type) completely eliminate this dependence in the practical sense. The principle of compensation pyrgeometers allows its use as an absolute instrument, which is another advantage. In this case, however, the instrument demands particular care in the determination of its characteristics and the processing of observational data. These requirements limit its use to that of a relative instrument.

5. Other Models of Balance Meters. At present, different countries use about 20 models of balance meters, this great variety being due to the absence of one really reliable instrument. The main difficulty in the measurement of radiation balance lies in the necessity to eliminate the influence of wind on balance-meter readings. There are three ways to overcome this difficulty: first, artificial ventilation of the receiving part, which allows determination of a constant wind correction ; second, wind protection in the form of a light filter; and third, the use of the compensation radiationbalance measurement principle. Some of the balance meters described below employ these methods. The so-called Falckenberg vibration balance meter provides ventilation

2.5. Measuring Radiation Balance and Effective Radiation

71

by rapidly vibrating the receiving surface, which is a mica lamina of 2 x 4 cm2, blackened on both sides. Each side has 12 manganin-constantan thermocouples. The sensing plate is fixed to a steel strip 14-cm long, vibrated by an eccentric that is rotated by an electric motor. Laboratory investigations of the instrument show that at a sufficiently high vibration frequency, the calibration factor is independent of the wind speed if the latter does not exceed 10 m per sec. The Skeib balance meter engages a rapidly rotating (driven by an electric motor) ring for the sensing surface. The thermobattery voltage is measured according to the induction method. Gier and Dunke have built a ventilated model, which directs a blast of air from the blower, through a nozzle in the form of a diffusor having a slot aperture, over the receiving area. At present it is widely used in the United States. A similar instrument has been constructed by Suomi and Fransilla. The Courvoisier type ventilates not only the blackened upper and lower sensing plates but also their interstices. Besides, it provides separate electric heating for both receiving areas. It goes without saying that a ventilated balance meter cannot totally “stabilize” the wind effect, for in many cases this would require too high a speed of ventilation. In this connection other types were often proposed in which the wind effect was eliminated by enveloping the receiver with a light filter, transparent for thermal radiation. In the U.S.S.R., Alexandrov and Kurtener, as early as in 1941, suggested a pyrgeometer with a flat filter of selenium-covered salt rock. Later, Khvoles employed a flat sylvite fllter coated with a thin layer of selenium. In this direction other improvements have been taking place lately. Stern and Schwarzman, for example, offered a KRS-5 hemispherical cover pyrgeometer. (This filter possesses a good transparency for both longwave and (partly) shortwave radiation.) A similar filter is used in the Houghton and Brewer radiometer. A number of balance meters and pyrgeometers (for example, Schulze, Funk, and Georgi) employ polyethylene which is of good transparency in the spectral range from 0.3 to 100 p. Wagner produced a compensation type of radiation balance meter, the receiving body of which resembles that in the Yanishevsky model. The temperatures of both faces are compensated by electrically heating one of them. The compensation-current magnitude serves for the radiation balance measure. The wind effect is supposed to be eliminated in the presence of equal temperatures of both receiving areas.

72

Methods of Actinometric Measurements

A new principle of the radiation balance measurement in the case of its positive value is applied in the Eisenstat “heliocompensation” installation. The principle of heliocompensation consists in shading the upper sensing plate from direct solar radiation with the help of a rotating section disk. The aperture angle of the section disk is so regulated as to set the balancemeter in the zero position. In this case the portion of direct solar radiation that is screened by the disk is equal to the radiation balance. The given method is meant for stable radiation conditions only. Still another type, the so-called Laikhtman-Koocherov differential balance meter (pyrgeometer, essentially), cools the receiving plate below the air temperature. Following the rise of the receiving surface temperature, mainly in the radiation absorption, it is possible to state the moment of equality in temperature between the lamina and the ambient air. Determining the speed of the temperature increase at the moment of equality of the temperature and knowing the lamina’s thermal capacity, it is possible to find the effective radiation of the lamina, since there is no heat exchange between the receiving surface and the air. 2.6. Main Types of Instruments for Spectral Measurements

The use of light filters is the simplest and most feasible method of radiation investigation in different spectral regions. A more complex, but also more accurate way of monochromatization of radiation consists in application of monochromators. In both cases there is a great variety of technical means, owing first of all to the fact that we have no special instruments for actinometric spectral investigations. Later we shall describe briefly a characteristic of the main types of this kind of instrument. 1. Application of Light Filters. These filters are available for different spectral regions, from the ultraviolet to the infrared. In practice, both solid (in the form of plates) and liquid (flat liquid layer) filters are used. For actinometric purposes, however, the solid type is to be preferred. The principal demands that an actinometric filter meets are, first of all, the stability of optical properties in relation to the outside influences (temperature, humidity, precipitation, etc.) and a simple manipulation. This is the case of glass filters and that is why they are so widespread, especially the Schott filters of different types. The International Radiation Commission recommends the following standard Schott filters :

2.6. Main Types of Instruments for Spectral Measurements

Filter Mark

Transmission Region, mp

Thickness, mm

OG 1 RG 2 RG 8

525-2800 630-2800 710-2700

2.4

73

1.5

3.0

We mention here that for colorless glass and quartz, the transmission range is characterized by the following limits, respectively: 350 to 2800 mp (glass) and 250 to 4000 mp (quartz). It is natural that the filter-transmission function value is for any wavelength less than unity. On the other hand, an ideal filter may be considered the one whose transmission equals unity in the investigated spectral region and zero outside its limits. For the comparison of the spectral characteristics of the real and ideal filters is introduced the concept of reduction factor, which is determined as such an independent of the wave coefficient by which the real filter transmission must be multiplied to obtain the ideal filter transmission. Thus the reduction factor is a value inverse to the mean filter transmission P, determined by the formula

Jw SAP,dii P=

O

where S, is the energy distribution in the emission source spectrum and P, is the filter transmission. In connection with the concept of reduction factor and mean filter transmission, it is important to determine the effective wavelength of the filter, which is usually calculated from

JmS,P,J Jeff =

dii

O

Jw SAP,dJ

When using selective radiation receivers the formulas (2.4) and (2.5) are changed for the more general forms:

74

Methods of Actinometric Measurements

Aefi =

”-

where V, is the radiation receiver spectral sensitivity. In Table 2.2 are given some reduction factors for the above filters and also for common glass or quartz as obtained at Davos in different years. TABLE 2.2 The Reduction-Factor Values. (After Measurement Data at Davos from 1930 to 1957)

Reduction Factor

1930-32 1937-39 1949-51 1952-53 1954 1955 1956 1957

Filters OG1

%

1.124f0.3 1.118 f 0 . 3 1.130 f 0 . 5 1.095 f 0 . 5 1.099 f 0.5 1.112 f 0 . 5 1.099 f 1 1.105f0.5

RG2

%

1.145 f 0 . 5 1.118 f 0 . 5 1.135 f 0 . 5 1.130 f 0 . 5 1 . 1 1 1 f0 . 5 1.112 f 0 . 5 1.100 f 0 . 5 1.098f1.0

RG8

%

Glass or Quartz %

1.080 f 0 . 5 1.135 f 3.5 1.070 f 2.0 1.092 f 1.0 1.097 f 0 . 5 (1.13) 1.082f1.5

1.072f0.5

As seen, the reduction factor value somewhat varies from year to year. This means that the filter optical properties are not strictly constant. In particular, it has been stated that the shortwave filter-transmission boundary shifts toward longer wavelengths with the increase in temperature. The data of Table 2.2, strictly speaking, refer to the temperature range 10 to 3.5’. If the filter temperature is found to be outside this range, the reduction factors must consider temperature corrections. The mean values of such corrections are +0.02 percent per O C for OG 1, +0.03 percent per O C for RG 2, and $0.04 percent per O C for RG 8. Table 2.3 gives data on transmission (in percent) for the Schott filters after the reduction of real filters by multiplying by the reduction factor. Since the individual characteristics of filters may somewhat (at times notably) differ, the data in the table should be treated as approximate. We see that although the Schott-filter spectral characteristics are “flat” over wide spectral intervals, nevertheless they differ considerably from the

75

2.6. Main Types of Instruments for Spectral Measurements

TABLE 2.3 The Schott Filters Spectral Transparency (Percent)

500 505 510 515 520 525 530 535 540 545 550 560 580 0 0 1 3 18 56 79 89 95 97 98 99 100 600 605 610 615 620 625 630 635 640 645 650 660 680

101 0

0

1

5

16

35

58

75

86

92

101 96

99 100

690 695 700 705 710 715 720 725 730 740 760 780 800 101 101 101 101 1 3 11 28 50 67 80 88 93 97 99 99 98 900 1000 1200

100 99 99 101 100 99 100 101 102

1500 100 100 100

1750 2000 100 100 101 100 100 99

2200 2400 2600 99 97 96 100 98 97 98 97 86

2700 80 90 50

Ideal filter transmission range 2800 3000 3500 9 42 20 5 50 15 5 26 15

4000 4500 5000

11 9 8

0 1 1

0 0 0

525-2,800 63&2,800 710-2,700

ideal even after reduction. Note that the appearance of the transmission values exceeding 100 percent results from the reduction of real values. An essential peculiarity of the Schott filters (and, as a rule, of glass filters in general) is that they have wide radiation transmission ranges. To narrow the transmission range one may use a combination of measurement results obtained with different filters. Having measurement data on the fluxes of the integral direct solar radiation (S) and the spectral fluxes measuremed by using filters OG 1 (&), RG 2 (S2), and R G 8 (&), and taking into consideration the respective reduction factors R1,R 2 ,R8,it is possible to determine the solar radiation fluxes in the spectral regions mentioned in Table 2.4. From Table 2.4 it is seen that the combination of measurement results, performed with the help of different filters, permits obtaining voluminous information on the radiation spectral composition.

76

Methods of Actinometric Measurements

TABLE 2.4 Spectral Fluxes of Direct Solar Radiation Measured by Means of Schott Filters

Spectral Region

Abbreviation

Blue and violet

s*

Yellow and green Short wavelength

sk

Yellow and red Visible Red and infrared Infrared

s, SD sv

sr

si

Wavelength 525

Method of Determination S-RiSi

525,630 630

RiSi-RvS S-RZSZ

525,710 710 630 710

RiSi-ReSa

S-R& RaSa Rase

To distinguish comparatively narrow transmission bands is also possible by combining measurement results derived with filters of the same type but of different thickness; for example, the use of OG 1 filters having 1and 4-mm thickness makes possible the separation of a rather narrow transmission band with the center at about 530 mp. The glass filters mentioned are intended for investigations of direct, diffuse, and global radiation in the visible and near-infrared regions. It must be mentioned that in the cases where diffuse or global radiation is measured, hemispherical filters are often substituted for the usual pyranometer glass cover. Whenever only flat filters are available, it is possible to use a photometer sphere (the sensing aperture of which is covered with a flat filter) as the radiation receiver. For distinguishing ultraviolet radiation, special glass filters may be used. To this end, thin coatings of silver pulverized over a quartz lamina are often used. A filter of this kind has a narrow transmission band, with the center at about 320 mp, and is employed in instruments for spectroscopic determination of the total ozone content in the atmosphere. Recently much progress has been made in constructing filters for the infrared spectral region. Crystallic germanium filters are very conveniently used to filter infrared radiation. These filters are totally nontransparent for visible radiation and easily transmit infrared radiation in the range from 1.5 to 2 p to 50 to 60 p. Also used for this purpose are KRS-5 filters (a crystal mixture of gallium bromide and gallium iodide), with a transmission range approximately from 0.6 to 30 to 40 p. Also available at present are glass filters that are transparent to infrared radiation in different intervals up to A = 21 p .

2.7. Measuring Atmospheric Thermal Emission

77

The so-called neutral filters of nonselective transmission for both shortwave and longwave radiation present a considerable interest to actinometry. For neutral filters, films of polyethylene, which is transparent to radiation of wavelengths from 0.3 to 50 p (except narrow intervals near 3, 5, 7 and 14 ,u where absorption bands are observed), have been found practical. All these filters are characterized by very wide transmission regions, thus making the monochromatization, performed with their help, very approximate. In this connection there have been attempts to work out types with very narrow transmission bands so as to enable them to compete with spectral instruments. It appears that metallic filters, known as interferential, possess such properties. Recent years have also brought interferential multilayer dielectrical filters. As a rule, the transmission band width of interferential filters is of the order of 10 to 20 A. The maximum transmission varies within 20 to 50 percent. Their use allows comparatively fine spectral investigations.

2. Spectral Instruments? Spectral measurements of radiant fluxes performed for solution of various problems of actinometry and atmospheric optics are specific, owing both to the peculiarities of the object’s properties (temporal, spatial, angular, spectral, polarizational, and energetical) and to the conditions in which these measurements are conducted (field measurements, measurements from different carriers, etc.). This puts certain demands on the instruments in use. Since up to now no special industrial models have been produced, we shall describe a number of instruments developed at the Chair of Atmospheric Physics of Leningrad University. 2.7. Instruments for Measuring Atmospheric Thermal Emission

In spectral photometric measurements of the low-temperature sources of thermal radiation (the atmosphere among them) the so-called differential method is commonly used. This method consists essentially of measuring the difference between the investigated radiation and a certain referent flux of radiation. The method is realized by making use of a double-beam lighting system at the aperture of the spectrophotometer. A mirror chopper sends beams of radiation from both the investigated and the referent emitters, which The general problems related to spectral instruments are treated in several publica-

tions (see [7-101).

78

Methods of Actinometric Measurements

are directed alternately on the aperture slot of the monochromator located behind the modulator. The amplitude of the variable component of the modulated fiux is equal at the aperture to a half-difference of these fluxes. The use of a special referent emission, which may have any degree of stability, is the advantage of this method. It is evident that single-beam schemes, employing the measurement of the considered emission in relation to that of the instrument’s details (in particular, chopper), cannot avoid the influence of instrument temperature variation on the stability of measurements. The schematic of a spectrophotometer for the atmospheric thermal emission measurement according to the differential method is presented in Fig. 2.9.

FIG. 2.9 The scheme of a spectrometer for atmospheric thermal emission measurements.

2.8. Measuring Shortwave Radiation Fluxes

79

The main components of the instrument are presented in Fig. 2.9 and described as follows: (a) A system of pivotable mirrors (1, 2, 3), allowing the investigator to observe any chosen section of the sky. (b) A double-beam lighting system of mirrors (4,5, 6) with a chopper (7), whose position directs the aperture slot either on the sky 8 (beam 2.5O x 0.5') or on the referent emission source 9. (c) The sources of the referent 9 and calibration (10, 12, 11) emission. The latter (calibrated by reference to the black emission) are directed on the aperture slot by a turn of the pivotable mirrors. (d) A prism monochromator (13) of the Littrow scheme for the wavelength range 4 to 40 p. (e) A miniature Golay cell (14) with the noise level 1 : 2 x lO-*O W cycle-1/2, which has proved to be the most convenient for photometric measurement because of the high stability of its characteristics. (f) An amplifier with a phase detector (15) and a self-recording potentiometer (16). The use of the phase detector (the referent signal is taken from chopper 7) allows choice of the transmission band of the receivingrecording system in accordance with the speed of scanning over the spectrum and slot spectral width, which is necessary for minimizing total measurement errors. In determining intensities in the atmospheric emission spectra the photometric instrument is operated according to the replacement method, that is, by substituting the investigated source (sky) with radiation sources of a known intensity (10, 11, 12) and comparing the recorder readings. To work with negative radiant fluxes (the atmospheric temperature is as a rule lower than the cell temperature) is possible only when both referent and calibration source temperatures have been carefully chosen to enable satisfactory calibration. Figure 2.9 gives one such set of temperatures. Recent years have introduced a new method for spectral investigations in the infrared spectra, the so-called Fourier spectroscopy. This method, as compared with the classical spectral method (employing prisms or diffraction gratings) gains hundredfold in energetics, thus permitting rapid measurements or a better resolution.

2.8. Instruments for Measuring Shortwave Radiation Fluxes These instruments differ greaty from those intended for thermal emission measurements for several reasons. First, the shortwave radiant flux values

80

Methods of Actinometric Measurements

are considerably higher than those of the longwave, which makes it possible to use instruments of a lower light power but a greater spectral resolution, especially in measuring direct solar radiation fluxes. Second, in the shortwave spectral range it is possible to use as sensitive elements those photoelectric radiation receivers possessing low inertia, which allows speed-up of spectrum recording when demanded by the problem’s conditions. Third, in the same range and for the same purpose, a photographic plate may be engaged to record simultaneously a fairly wide spectral interval, which is an advantage under certain conditions.

FIG. 2.10

A fast-working spectrophotorneter for measuring spectral shortwave radiation fluxes. (1) semitransparent mirror; (2) chopper; (3) aperture slot; (4) replica; ( 5 ) oscillating mechanism; (6) flat mirror; (7) exit slots; (8) spherical mirror; (9) light filters; (10) preamplified radiation receivers; (1 1) selective amplifiers; (12) recording device; (13) wavelength marker; (14) photometric; (15) feeding block of the photometric standard.

As an example of a fast-working spectrophotometer, Fig. 2.10 is the schematic of the instrument for measuring spectral fluxes of atmospheric shortwave radiation. The main parts of the instrument are: (a) (b) (c) (d)

A A A A

modulator of the light flux mirror monochromator with receiving-recording part (10, source of standard radiation

(2). diffraction grating (3, 4, 6, 7, 8). 11, 12). (14, 15).

The modulator is a perforated disk interrupting the radiant flux with a

2.8. Measuring Shortwave Radiation Fluxes

81

frequency of 800 cycles. The monochromator follows the vertically symmetrical scheme. It consists of an aperture slot 8, two concave spherical mirrors (one is missing in Fig. 2.10) with a common center of incurvation, a replica with 600 cross hatches per millimeter, and three outlet slits 7. The working range extends from 260 to 1000 mp. The inverse linear dispersion in the first order equals 33 8, per millimeter. Periodical scanning of the spectrum is performed by the replica’s being oscillated with the help of a cam gear ( 5 ) . Guiding marks on the wavelength scale are made by an optical mechanical marker (13) whose diaphragm is stiffly connected with the cam. Thus the scale-mark reproduction is very high. Photoelectric multipliers (10) are used as radiation detectors. Behind one of the aperture slots is located an antimony-cesium cathode photomultiplier; behind the second, a multialkaline cathode; and the third slot faces the oxygen-cesium one. The first multiplier helps to record the ultraviolet spectral division; the second, the visible; and the third, the infrared region. To prevent the undesired spectra from striking the cathodes, corresponding filters (9) are placed before the photomultipliers. The detector photocurrents are preintensified by the devices assembled together with the multipliers and are brought through a screened cable to the selective RC amplifiers. After the main amplification, the alternating signals are fed to the diode detectors and recorded at the multichannel train oscillograph (12). An incandescent lamp (14) fed from a stabilized voltage source (15) serves as a photometric standard. The standard radiation recording is performed immediately after the investigated radiation spectrum has been registered. The instrument is mounted on a turret, which enables it to be aligned to any chosen point of the sky. Figure 2.11 is a schematic presentation of an instrument based on the serial spectrophotometer’s monochromator. It has a high spectral resolution and serves for measuring spectral fluxes of global and diffuse radiation within the 0.3 to 1.1 p range. The monochromator is built according to the autocollimator scheme. An off-axis parabolical mirror with the focal distance of about 500 mm is used as the projecting element; a diffraction grating (3) with 600 cross hatches per millimeter serves for dispersion. The relative aperture equals 1 : 10. The inverse linear dispersion in the first order is 32 8, per millimeter. The filters (6) over the aperture (2) eliminate the unwanted spectra. An Ulbricht photometer of 200-mm diameter is placed before the aperture, whose diameter is 30 mm. The inside of the photometer is coated with

82

Methods of Actinometric Measurements

FIG. 2.11

A spectrophotometer for measuring spectral fluxes of global and difuse radiation. (1) Ulbricht sphere; (2) entrance slot; (3) grating; (4) parabolical mirror; (5) exit slot; (6) light filters; (7) photomultipliers; (8) amplifier; (9) recording instrument; (10) photometric standard.

barium sulfate. Its purpose is to direct on to the aperture slot the radiant fluxes f from practically the entire hemisphere of the sky. In the case of measuring diffuse radiation, the photometer aperture is shaded from the direct sun’s ray by a special shield. The photometer has a photometric standard, a low-voltage incandescent lamp (10). Two photomultipliers (7) are radiation detectors, one with an antimony-cesium and one with an oxygen-cesium cathode. A parallelbalanced, continuous-current amplifier (8) is used to amplify the photocurrents. The amplified signal is received by an electronic self-recording potentiometer (9). The photographic method used for recording radiation spectra in the instrument is based on a serial three-prism spectrograph with glass optics. Figure 2.12 shows the setup of this instrument as applied in measuring the sky’s spectral brightness. A nozzle (1) with a condenser is placed at the aperture of the spectrograph to narrow its solid angle. Before the condenser is a revolving head (4) with a lock. On the ray’s path, thus, are alternately placed either an aperture covered with two sheets of tracing paper (depolarizer), or three combinations (each consisting of three polaroids) with different reciprocal orientation of axes in measurement of the degree and angle of inclination of diffuse radiation polarization. At the aperture is located a cassette (3) with a film store for 20 spectrograms. A turning device ensures the setting of the instrument on a chosen section of the sky.

References

83

FIG. 2.12 An instrument for measuring spectral brightness of the sky. (1) nozzle; (2) spectrograph; (3) cassette; (4) revolving head.

The scale of ordinates of the instruments for measuring shortwave radiation is calibrated with the help of a secondary standard of radiant energy, consisting of a band incandescent lamp calibrated in its turn after a blackbody. REFERENCES 1. Yanishevsky, Y. D. (1957). “Actinometric Instruments and Methods for Observations.’’ Gidrometeoizdat, Leningrad. 2. “The International Geophysical Year Instruction Manual.” Part. VI. Pergamon Press, Oxford. 3. Courvoisier, P., and Wierzejewski, I. (1948). Beitrage zur Stahlungsmethodik. I. Die physikalischen Grundlagen der kalorischen Stahlungsmermethoden. Arch. Meteorol., Geophys. Bioklimatol. B1, No. 1. 4. Koocherov, N. V. (1957) An instrument for pyrheliometric measurements. Proc. Main Geophys. Obs. No. 69. 5. Angstrom, A. (1958). On pyrheliometric measurements. Tellus 10, No. 3. 6. Foitzik, L., and Hinzpeter, H. (1958) “Sonnenstrahlung und Lufttriibung.” Leipzig. 7. Toporetz, A. S. (1955). “Monochromators.” GITTL. Moscow. 8. Shishlovsky, A. A. (1961). “Applied Physical Optics.” Phizmatgiz, Moscow. 9. Prokofjev, V. K. (195 1). “Photographic Methods in the Quantative Spectral Analysis of Metals and Alloys.” GITTL, Leningrad. 10. Smith, R. A., Jones, F. E., and Chasmar, R. P. (1957). “The Detection and Measurement of Infrared Radiation.” Oxford Univ. Press (Clarendon). London and New York.

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RADIATION ABSORPTION IN THE ATMOSPHERE

The atmosphere is known to consist mainly of nitrogen and oxygen, the vclume concentration of which is about 99 percent. However, neither nitrogen nor oxygen are of prime importance in the radiation absorption by the atmosphere, the main contribution to this absorption being made by such quantitatively insignificant atmospheric components as water vapor, carbon dioxide, ozone, oxides of nitrogen, and some others, the aerosol component in their number. The absorption spectrum of the atmosphere extends over a wide range from the X-ray region to the ultrashort radio waves, which makes the physical nature of absorption highly varied and greatly complicates the structure of the spectrum. The problem of our fundamental concern is the determination of the function of absorption (transmission, correspondingly) for different spectral intervals and also for wide spectral regions embracing the shortwave and longwave radiation. Before proceeding to the experimental determination of the absorption function, we must clarify what factors are its basic determinants. In all cases the main factor is the content of the radiation absorbent. In a number of cases, however, the pressure and temperature in the absorbing medium are also of importance. The correct analytical presentation of the absorption function demands consideration of fine structure of the absorption spectrum. Since the latter is extremely intricate, we have to model the real absorption lines and bands after simple schematic presentations. This is particularly important in the investigation of the infrared atmospheric absorption spectrum, which possesses the most complex structure and thus presents a special problem as regards the general regularities of selective radiation absorption. In this connection it is appropriate to mention that the radiation absorption in the near-infrared 85

86

Radiation Absorption in the Atmosphere

is related to vibrational-rotational molecular transitions, and in the farinfrared, to purely rotational transitions. 3.1. General Principles of Selective Radiation Absorption

A spectral line is the basic “elementary cell” of the absorption spectrum. Therefore we shall start with the study of the regularities of radiation absorption for the case of an isolated spectral line. Since the absorption caused by a spectral line depends considerably on the line shape, let us first briefly discuss this problem. 1. The Shape of a Spectral Line. As known, monochromatic emission is practically never observed. Due to external influences on atoms and molecules and also due to the loss of energy in emission, energy levels are “washed out,” and the radiation emitted in energy transitions turns out to be nonmonochromatic. That is why, in the investigation of atomic and molecular emission, we observe spectral lines of finite width and absorption bands of finite width. In the quantum theory the frequency of radiation I,, emitted in energy transitions is determined by the relation

where En is the energy level of the atom (or molecule) from which transition takes place, Em is the energy level at which it occurs, and h is the Planck constant. We shall omit the details of the theory of broadening of spectral lines (see [I-61) and only mention its main results. It has been stated that in the absence of strong outside electromagnetic fields, the finite width of spectral lines is explained by the following causes: (a) Damping of vibrations of oscillators, due to the loss of energy in emission (by the finite lifetime of the excited levels). The broadening of lines due to this factor, is considered normal. (b) Perturbations due to reciprocal collisions of the absorbing molecules. (c) Perturbations due to collisions of the absorbing molecules with those of nonabsorbing gases. (d) Doppler effect resulting from the difference in thermal velocities of atoms and molecules. The effects of natural broadening of lines are small in comparison with those caused by factors 2, 3, and 4, and may practically be neglected. The broadening due to collisions and the Doppler effect in the atmosphere are of far greater importance.

87

3.1. General Principles of Selective Radiation Absorption

Consider how the shape of an absorption line can be calculated on the basis of the classic Lorentz’s presentation, taking into account the effect of interatomic collisions. (The term coZZision in the given case should be understood as a kind of interaction between atoms occurring during comparatively short time intervals.) Interactions (collisions) of this sort perturb the atom to such a degree that its vibrations are interrupted and their phases are changed. If the duration of an unperturbed vibration with a circular frequency wo equals t, then the vibration of the dipole moment of the atom p in time can be determined by p

in the interval (0, t )

= poe-(yt/2)eiwot

p

=0

in the interval (t, 00)

(3.2a) (3.2b)

where the multiplier e-(ytl2)is introduced to account for the natural damping due to emission: y is the damping coefficient. If we denote t as the time interval between two successive collisions, the probability of the atom’s staying undisturbed during the time interval from f to t dt equals 1 e-(t/T)dt -

+

t

The probable number of undisturbed atoms per unit volume will be dN

N

= -e-(t/T)dt

t

where N is the number of atoms per 1 cm3. Since not all atoms are in the excited state, only a part of them will be emitting. Let us denote by N* the number of excited atoms per 1 cm3. Introducing a quantity q = N * / N , we obtain for the flux of energy due to the emission by undisturbed atoms, (3.3) where S, is the flux of energy per atom. Assuming the emission by atoms to be dipole, the value S, can be determined from the known relation (3.4) where c is the velocity of light.

88

Radiation Absorption in the Atmosphere

Integrating (3.3) from t flux F: F

=0

=

to t

@ ! z

= 00,

1: S,

we have for the entire energy

e-Wd dt

(3.5)

Taking the function

{ [

f(t) = pe--(t/2T) = po exp -

+2(1'yt}cos

mot

we present it, with the help of the Fourier integral, in the form

Here y(o) is determined by the integral

1 ~ ( o=)2n / r f ( t ) e-id dt

(3.7)

The formula (3.5) can now be rewritten as

Since, according to (3.7),

, rearwhere q * ( w ) is the complex conjugate function related to ~ ( w ) we range (3.8) as

The calculation of complex amplitudes of electromagnetic waves v(w) and v*(w), with the help of (3.9) and under the condition that y < o o (for the infrared region in the range of 10 p, the ratio is y / o o 21 5 x lo-'), leads to the following result: (3.10) where y'

=y

+ (l/t).

89

3.1. General Principles of Selective Radiation Absorption

Substituting this expression in (3.9), we have

F=

qNw04p02 6nc3t

dw

Io

(3.11)

The subintegral function in this expression characterizes the dependence of the radiant intensity Z on the circular frequency w . Introducing v = w/2n, we have the following distribution of the lineemission intensity depending on Y : Z(Y) =

4n2qNvo4p02

(3.12)

At v = vo the function has its maximum. For the maximum intensity value we obtain (3.13) ,z = 64n4qNvo4p02 3c3ty12 Taking account of (3.13), the formula (3.12) can be transformed to Z(Y) =

Y'2L

(3.14)

This kind of distribution of the line emission intensity is called the Lorentz shape and the formdla (3.14), is often called the Lorentz formula. The distance from the center of the line to the frequency at which the intensity decreases. by twice (Z(v) = ZJ2) is called a half-width of the line. Thus we have for the half-width,

Y' Y a=-=-+4n 4n

1 4nz

(3.15)

The first expression in this formula determines the natural half-width and the second determines the half-width caused by the collision effect. Calculations show that at the pressure of about 1000 mb and temperature OOC, the natural half-width is negligibly small as compared with the other half-width and can be ignored. Only at very low' pressure when the time interval between two successive collisions by far exceeds the duration of the atom's excited state, that is, in the upper atmospheric layers, does the consideration of natural broadening become vital.

90

Radiation Absorption in the Atmosphere

In still another case when “collisions” are impacts of emitting atoms with atoms of foreign gases (impacts of the second kind), in which the emitting atoms lose all their energy of excitation, the average time of the mean free path is determined by 1 - = ne2NZB (3.16) t

where Q is the sum of the effective radii of colliding particles, N2 is the number of atoms of the foreign gas per 1 cm3, and (3.17)

+

where k = Boltzmann constant and M = m,m2/(m, mz) is a reduced mass of colliding atoms (ml and m2 are masses of the colliding atoms). The formula (3.16) is valid only when the number of atoms of the basic gas iVl is noticeably smaller than that of the foreign gas N 2 . According to the kinetic theory of gases,

where p is the pressure of the foreign gas. Thus, for the half-width, we have a=-- 1

4nz

-

1/

4&TM

“’

(3.18)

If we designate by a,,the half-width corresponding to the normal pressure po and temperature To, the formula (3.18) can be presented in the form (3.19) As seen from this formula, the half-width is directly proportional to pressure and inversely proportional to the square root of temperature. In atmospheric conditions the number of emitting and absorbing gas molecules (water vapor above all) is much smaller than the number of foreign gas molecules (in the lower layers approximately by 100 times). It may be assumed, then, that the formula (3.19) meets the demand of atmospheric conditions ; consequently the pressure and temperature in this formula are to be taken as the pressure and temperature at a certain at-

91

3.1. General Principles of Selective Radiation Absorption

mospheric level for which the determination of the half-width is being performed. The Lorentz formula (3.14) characterizes the emission intensity distribution. Bearing in mind Kirchhoff's law, it can be understood that the distribution of the absorption coefficient for an isolated line must be completely analogous, that is, the shapes of lines of emission and absorption must be similar. It should be stressed that this conclusion holds only in the case where the shape of an emission line is computed without considering its transformation by the nonhomogeneous thickness of the absorbent, which may lead to deformation of the shape and sometimes causes inversion of emission lines (the frequency of absorption maximum corresponds to emission minimum; see [7]). The Lorentz formula for an absorption line is usually written in the following way: Sa 1 (3.20) k(v) = 7c (v - y o ) , +a2 where the value S=

I

m

k(v) dv

--cQ

is called the line intensity, and k(v) is the absorption coefficient corresponding to the frequency v. A number of theoretical and experimental investigations [4,8-10,231 have shown that the formula (3.20) is approximate. In reality the line profile is symmetrical, as it follows from (3.20). It also appears that as the total pressure increases, the center of the line displaces toward the side of lower frequencies. All these specifications, however, are of little importance. Benedict et al. [l 1] have found that for a number of gases (CO, in particular), the formula (3.20) gives exaggerated absorption values in the wings of lines. For this reason, a suggestion has been made to use the Lorentz shape in such cases only for central parts of lines (at I v - vo < d where d = 2.5 cm-l) and to introduce into (3.20) an empirical correction factor A [exp(- a I v - vo Ib)] for the wings area (I v - vo I 2 d). Here a = 0.0675 and b = 0.7. The constant A is determined from the condition of continuity of transition of one spectral line contour to another at 1 v - vo I = d = 2.5 cm-'. For some other gases (for example, HCI) the formula (3.20), on the contrary, gives lowered values of the absorption coefficient. In the case of self-broadening of HC1 (see [lo]) the absorption coefficient in the wings of rotation lines varies proportionally to [ v - vo

I

92

Radiation Absorption in the Atmosphere

The results developed above enable characterization of the profile of lines in the case when their broadening is caused by collisions. It is perfectly natural that the collision effect must be of prime importance in the lower layers of the atmosphere where density and pressure are high. There are reasons to suppose that the Lorentz profile takes place in each case for the conditions of the troposphere and lower stratosphere. In the upper layers of the atmosphere, however, where the time interval between two successive collisions increases significantly, the influence of collisions on line broadening decreases. In such conditions the line broadening with respect to the Doppler effect is of decided importance. As we know, in the movement of atoms or molecules relative to the observer, the frequency of the emitted radiation varies according to the law v

= vo( 1

+

u)

(3.21)

where vo is the emission frequency of the atom, immobile as to the observer; u is the projection of the speed of the emitting (or absorbing) atom on the observer's direction. The probable number of atoms, dN, of the speed in the limits from u to u du can be determined by the relation

+

where m, is the atom mass. Consequently, for the flux dF of emission by atoms of this kind, we have (3.22)

Here the value of the emission flux from a single atom S, is determined by the expression s, = 32n4v04p02 3ca where n is the number of excitations per second. By using (3.21), it is easy to transform (3.22) in the following manner: dF

=

qS Nc

vz

{exp( -

&)(-)'}

dv

(3.23)

3.1. General Principles of Selective Radiation Absorption

93

Hence for the distribution of the line emission intensity, Z(Y) = dF =

dv

{ [

(

I( Y o ) exp - - ;o"~)"]}

(3.24)

From the condition a = Y - yo for I(Y) = 4Z(v0), we find for the halfwidth, (3.25) Substituting rn, for M (the weight of a gram atom or gram molecule) and introducing a universal gas constant R, we obtain (3.26) Numerical evaluations show that the Doppler broadening by far exceeds natural broadening, especially in the case of light atoms. However, in comparison with broadening due to collisions, Doppler broadening may dominate only in the upper atmosphere. The preceding conclusion may be illustrated by the results of computations of the Doppler half-width (ad)and the half-width due to the collision effect (a,) for the main absorption bands in the atmosphere as given in Table 3.1, where the last two columns show the values of pressure p and altitude h at which a d = a,. It appears that these values of h are to be considered as somewhat lowered. As shown by Elsasser [20], Doppler broadening begins t o be of notable importance only beyond 40 km. Recent measurements by Bastin et al. [19] of the rotation water vapor line I-, - l1 (the wavelength 0.538 mm) have given the following results: Gas Mixture Water-water Water-air Water-ni trogen Water-oxygen

Lorentz Half-Wldth, cm-1 atm-l 0.5 f 0.2 0.05 f 0.01

0.06 f 0.02 0.03 f 0.01

As shown by Saiedy [21], the half-width of the rotation water vapor line 914.03 cm-l (R3,-3) at normal pressure in the conditions of the real atmosphere is 0.047 f 0.002 cm-l.

94

Radiation Absorption in the Atmosphere

TABLE 3.1 The Lorentz and Doppler Line Hay- Width for the Main Absorption Bands in the Atmosphere

Band Center, ad at 30O0K, cm-l x lo3 c1

Gas

HzO

6.4 2.8 0.88

0.11

P,

-

h, km

17 23 32 37

0.15

13 4

26 34

1.5

0.16

7

30

4.7 9.6 14.1

2.3 1.1 0.76

0.16

11 5 4

27 33 34

3.3 7.7

5.6 2.4

0.18

24 10

22 27

0.44

co,

4.3 15

2.6 0.75

NZO

7.8

CH,

cm-l

19 6 3

2.7 6.3 20 40

0 8

aP at NPT,

44

Vasilevsky and Neporent [25, 321 obtained a = 0.5 cm-' for the line 4025.4 cm-', at the pressure 1 atm and temperature 2OoC (for the mixture water-water). Table 3.2, borrowed from Vasilevsky [32], characterizes the importance of reciprocal collisions of water vapor molecules and of collisions H,O-x as factors determining the dependence of the half-width upon pressure (these data relate to the lines of the band 2.7 p). TABLE 3.2 Values of the Relative Eficiency aR,O-% and the Absolute Optical Diameters dfl,o-2 of Collisions between Molecules for the System Water-x. After Vasilevsky 1321

X

QH,O-z:

1 0.123 0.165 0.170 0.239

Line 5 , - 6-s in the Band vs, dH,04 X 108,Cm 12.7 4.9 5.5 5.6 8.8

3.1. General Principles of Selective Radiation Absorption

95

The value aH,O-xpzis the relative effectivity of optical collisions between molecules of water and foreign gas (x) as included in the relation 01 = @o@H,O

+

GE,O-zPx)

where a, is the value of u at the partial pressure of the absorbent p H , O = 1 ; px is the partial pressure of foreign gas. Table 3.2 shows that the efficiency of reciprocal water vapor molecule collisions is higher by approximately six times than that of the mixture water-air. If we remember, however, that the partial pressure of water vapor is by two orders of magnitude smaller than the atmospheric pressure, it will be clear that in the conditions of the real atmosphere, Q: = a ~ , ~ - ~ p ~ practically. So, in particular, p in the formula (3.19) may be understood as atmospheric pressure. Burch et al. [142] have performed an investigation similar to Vasilevsky's [32] for nitrous oxide, carbon monoxide, carbon dioxide, methane, and water vapor. Let us now turn to the case of the line for which the influences of both the Doppler and the collision effect (it may be important in the upper stratosphere and higher) have been considered. When both effects are taken into account on the basis of (3.14) and (3.24), we have for the emission intensity distribution near the frequency v for atoms with the speed u (to this speed corresponds the displacement of frequencies dv = (u/c)v) the formula

where by a d is denoted the Doppler half-width as determined in (3.25). The total emission intensity near the frequency v, resulting from the emission by all atoms (or molecules), can be found from Z(v) = Jrn Z'(v, u) du --m

In view of the complexity of the subintegral function, integration here can be performed only approximately. The absorption coefficient distribution, corresponding to (3.27), is of the form (see [12])

96

Radiation Absorption in the Atmosphere

where

As was shown by Plass and Five1 [12], the integral for this formula can be presented in the form of an exponential series. With the use of a series expansion, the formula (3.28) can be transformed as nn] (3.29)

is the nth derivative of the function

F(w) =

1;

e-xe sin 2wx dx

>

If 1 Y - I a p , 1 Y - v0 I > a d , then instead of (3.29), the following asymptotic relation can be used :

(3.30) At ad = 0 (co = co) an analogous asymptotic relation is obtained, corresponding to the Lorentz profile (3.20). Having considered the main results of the problem of line profiles, let us now turn to general regularities of radiation absorption for the case of isolated lines.

2. The Absorption Function in the Presence of an Isolated Spectral Line. From the definition of the absorption function we have (3.31) where Z,,,o is the intensity of the incident radiation; I,, is the intensity of the radiation after passing through a layer of absorbent. The integration here extends over the entire frequency region covered by the given line. Note also, that further we shall talk about the absorption

3.1. General Principles of Selective Radiation Absorption

97

function for radiant intensity, confining our task to the treatment of the case when line broadening is caused by the effect of collisions between molecules of the main (absorbing) and foreign gases only. As stated above, this particular case presents the greatest interest for meteorological applications. The importance of Doppler broadening is discussed, e.g., in Plass’s work 1221. Consider instead of (3.31) the following quantity, characterizing radiation absorption in an isolated line,

Ws,(w)=

1;

(1 - e-k(u)ur) dv

(3.32)

s

where w = Q dl is the content of the absorbent on the ray’s path computed per unit cross section of a beam of rays; e is the density of the absorbent. The value W,, is generally called the equivalent line width, and the dependence Wsz(w)is known as the curve of growth. Substituting the expression for k(v) in (3.32), according to (3.20) we have (3.33) W,, = 2na~e-~[J,(x) J,(x)]

+

where

x=- s w 2na

(3.34)

and J, and J1 are the Bessel functions of the imaginary argument. For small values of x, the formula (3.33) takes form

w,,= 2 n a x

(3.35)

The calculation error here does not exceed m percent if x 5 0.02. As seen from (3.35), at a small value x, the radiation absorption is directly proportional to the mass of the absorbent. For large values of x, instead of (3.33) the following approximate relation may be used:

w,,= 2a-

(3.36)

The calculation error here does not exceed m percent if x 2 12.5(l/m). From (3.36) it follows that at large values of x, the absorption of radiation is proportional to the square root of the absorbing mass. The formula (3.36) is sometimes called the square root law.

98

Radiation Absorption in the Atmosphere

The above formulas refer to the abstract case of a single line in the absorption spectrum. In reality, however, only isolated lines separated by spectral intervals of finite width can be observed. In this case the integration in (3.32) must be performed over frequency intervals of finite width, which significantly complicates the derived results. A general expression for the absorption function of a single line in a finite spectral interval has been obtained by Zuyev and Tvorogov [154, 1551. More complex results appear for lines of the Doppler and mixed Lorentz-Doppler contours. The case most closely approaching the real conditions is the one with the observed overlapping spectral lines of the absorption band. Since the real absorption band structure is quite intricate, it is necessary to consider schematic bands, the simplest model of which kind was suggested by Elsasser [13].

3. The Elsasser Absorption Band. Suppose the absorption band consists of an infinite number of equally spaced lines of equal intensity and halfwidth, each of the Lorentz shape. It can be shown that in this case the absorption function A for the frequency interval d, corresponding to the spacing between the centers of adjacent lines, will be A = I - - J 1 - n exp[2n --n cash p - cos z

(3.37)

where

and sinh and cosh are hyperbolic sine and cosine. Seitz and Lundholm [24] have found a series that represents this integral for all parameter values. The finite relations, corresponding to small or large argument values, are expressed as

Here

is the error probability integral.

3.1. General Principles of Selective Radiation Absorption

99

The formula (3.38a) is known as the strong-line approximation and (3.38b) as the weak-line approximation. Also interesting is the nonoverlapping line approximation. In this case the absorption function takes the form (see [22J) (3.38~) A = Bxe-W0(x) J,(x)l

+

where JoyJ, are the Bessel functions. Table 3.3, compiled by Plass [22J, displays the values of parameters characterizing the range of applicability of the various approximations. Introducing the notation

I = - 2xaS

(3.39)

d2

we can rewrite (3.38a) as A

=

v(E)

(3.40)

4. Random (Statistical) Absorption Band. The model of the schematic

band just presented, consisting of equally spaced lines of equal intensity, is a far-going abstraction. In reality the dislocation of lines in a band is not so regular and the line intensities are far from equal. It is more natural to suppose that the distribution of intensity and position of individual lines in a band is almost random. Although line intensity and position are connected by definite quantum-mechanical formulas, nevertheless observations show that their variation is so irregular that it pays to build a statistical model for the absorption band. In such a hypothetical band the line intensity and position are determined by the probability theory. Consider, following Goody [14], the problem of radiation absorption in the case of a statistical model for the band. Let us calculate the transmission of radiation at the center of a frequency band of width nd, where n is the number of absorption lines and d is the average line spacing. Let nd be so large as to practically eliminate the influence of lines outside this frequency range on the absorption of radiation at the center of the band. If we assume, then, the absence of correlation between the intensity and position of lines, it is possible to determine the value of transmission, provided absorption coefficients are given and two vn) dv, dv, and Q ( S ) dS, the former functions are introduced: N(vl determining the probability of the line encounter at frequency intervals from v1 to v1 dv, , and from v 2 to v 2 dv, , etc., where for the latter, the probdS. Denote ability is that the line has intensity in the limits from S to S

- -.

+

+

. .-

+

e

TABLE 3.3

0 Q

Regions of Validity of Various Approximations for Band Absorption." Afrer Plass [22]

Approximation

Strong line approximation, Eqs. (3.38a) and (3.43a)

Weak line approximation, Eq. (3.38b)

Nonoverlapping line approximation, Eqs. (3.38~) and (3.43b)

Elsasser Model

x > 1.63 0.01 ; x > 1.63 0.1 ; x > 1.63 1; x > 1.35 10; x > 0.24 100; x > 0.024

= 0.001;

0.001; x < 0.20 0.01 ; x t 0 . 2 0 0.1 ; x < 0 . 2 0 l;x
Statistical Model. All Lines Equally Intense x x

> 1.63 > 1.63 > 1.63

x x > 1.1

x x

> 0.24 > 0.024

Statistical Model. Exponential Line Intensity Distribution

> 2.4 > 2.4 X O > 2.3 x , > 1.4 xo > 0.27 x , > 0.24 x, x,

x x

x, x, x,

x < m

x, x,

< 0.10 < 0.10 < 0.10 < 0.11 < 00

X < m

x

<m

< 63,000 < 630 x < 6.3 x < 0.22 x < 0.020 x < 0.0020

x < 80,000 x <800

< 0.20 < 0.20 < 0.20 x < 0.23 x

x

x

x t 8 x, < 0.23 x, < 0.020 x, < 0.0020

When x = S w/&a satisfiesthe given inequalities, the indicated approximation for the absorption i s valid, with an error less than 10 percent. = Sow/&cr.

For the exponential line intensity distribution, x,

3.1. General Principles of Selective Radiation Absorption

101

the absorption coefficient at a distance v, from the center of the line by Srf(v,, a ) where the function f(vr , a ) characterizes the line contour. Now we have, for the transmission of radiation at the center of the considered band at frequency v, and the absorbing mass w, pv = e-Srf(vpdw I

The probability of the presence of the line series r is n

N ( Y ~* **vn)dv1. * *dvn

n

Q(S,) dS,

1

For the integral band transmission value at all the possible permutations of lines in the band, we obtain

(3.41)

If all the changes are equally probable, then N is a constant value and will thus be reduced in (3.41). Assuming the equal probability of all line permutations and omitting the index r, we transform this formula in the following manner, assuming the equal validity of all integrations over v and, S,

Passing to the limit as n + m , we obtain

P

=

1

exp -

{

d

lr Q ( S )d S

ws: sw

Q ( S )[ 1 - e-sf(v,a)w]dS dv}

(3.42)

As shown by Kaplan [15], this formula may also be presented in the following simple form

P

=

(

Y?)

exp - -

(3.43)

where Fszis the average value Wsz for a single line. This formula can be derived if we take into consideration that at N = const when the probability

102

Radiation Absorption in the Atmosphere

of line position is n

P=nPpi i=l

where

Here Pi is the transmission function for the nth line in the interval Av = nd. We can also write

where Wsl,i,dV is the equivalent width of the ith line in the interval Av. Realizing a finite transition analogous with that of (3.42), we have

that is, we have a formula identical to (3.43). Integration over S in (3.42) can be performed only at a concrete form of the function Q(S). Goody calculated the function (3.44) where So is a certain parameter characterizing the mean line intensity. In this case, instead of (3.42) we have (3.45) With application of the Lorentz formula

for the line shape, we have finally r

1

(3.46)

3.1. General Principles of Selective Radiation Absorption

103

Calculations of the transmission function can also be made for other forms of Q ( S ) different from (3.44). Suppose that all line intensities are equal. This is equivalent to substituting (3.44) by the following formula

where 6 is the delta function (6 = 0 at S f So and 6 = 1 at S = So). In the case here, the transmission function for a single line of the absorption band will be determined by (3.43), in which the value Wsl = WSl corresponds to (3.33). It can be shown (see [22]) that the strong-line approximation for a statistical model is as follows (3.43b) where x is defined as

in the case of an exponential intensity distribution and x = SwJ2na in the case of all spectral lines being of equal intensity. The equation for the weak line approximation is found to be the same as in (3.38b). For the nonoverlapping approximation in the case of equally intensive lines, the formula (3.38~)is valid. In the case of a statistical model with an exponential intensity distribution, A

= j3xo(l

+ 2x0)-"*

(3.43c)

In Table 3.3 are given the values of parameters for the statistical absorption-band model which characterizes the possibility of fulfillment of various approximations.+ It is important for approximation of absorption functions that the influence of numerous weak lines on the value of absorption be as treated by Plass [140]. Also to be mentioned here is that Burch and Williams [143] have tested the applicability of Plass's approximations on measurement results of radiation absorption by N,O, CO, , CH, , and CO. The assumption of the equality of all line intensities is very approximate. This is illustrated by Fig. 3.1, which depicts the dependence of log S upon the total number of lines Z N with a given S as obtained by Godson [16] for different regions of the infrared water-vapor absorption spectrum. Later Goody suggested a three-parameter model of absorption (see [161]).

104

Radiation Absorption in the Atmosphere

FIG. 3.1 Cumulative probability distributions for water vapor line intensibiIities. er Gohon [16]. (1) 200-225 cm-l, 300'K; (2) 100-125cm-l, 300'K; (3) 100-125 cm-l, 220'K; (4) 15001525 cm-l, 287'K; (5) 1775-1800 cm-l, 287'K; (6) 1000-1050 cm-l, 287'K; (7) 900950 cm-', 287'K.

We see that in the first approximation there exists a linear dependence between log S and E N . Using this law of line-intensity distribution, Godson derived for the transmission function :

P

=

{ f -

exp - - [e UJ,(y) - 1

+ 2ye-v[JO(y)- iJl(y)]]}

(3.48)

Here

where Av is the spectral interval under consideration (summation extends over the entire interval Av), and J,, J1 are the Bessel functions of the imaginary argument. The radiation absorption values computed for the Elsasser and the statistical bands differ at high absorption values (Fig. 3.2). The Elsasser model gives exaggerated absorption values. Similar results of two considered absorption functions were obtained by Kozhevnikov [ 2 8 ] . Davis and Viezee [29] have worked out a semiempirical method for determination of the transmission function on the basis of theoretical formulas of the statistical model. The statistical transmission function is changed by the introduction of a new parameter, the water vapor content W,

3.1. General Principles of Selective Radiation Absorption

105

(varying as pressure and temperature vary) at which the radiation absorption makes 50 percent. A second new parameter is a value r = 6/nd, (6 is the mean line spacing, as is the line half-width at standard pressure and temperature). An empirical choice of the parameters w ,and r allows a more accurate definition of the theoretical transmission function. Analysis of experimental data has shown that r can be considered invariable for the whole band. The dependence of the parameter wh on pressure and temperature may be presented in the form 1

-- -

L.P”($)b”

wh

where L, is the value l / W h at the standard values of pressure p and temperature T. The exponent n is constant over the entire band and exponent 6, is the wave-number function. The parameters L,,, n, and by should be determined empirically. The methods given in [29] are applicable for spectral intervals of 50-cm-’ width. If the absorption is calculated for several (say, two) overlapping Elsasser bands, the results nearly coincide with the corresponding data for the statistical band (see Fig. 3.2). This has made Kaplan [15] and Plass [I71 suggest an absorption band model consisting of superpositions of randomly distributed Elsasser bands. A further development of this idea has found form in the “cluster” line model worked out by King (in this case lines of different intensity and half-width are clustered along the wavenumber scale). A quasistatistical model has been suggested by Wyatt et al. [26]. Zuyev and Tvorogov [149] have worked out an approximate

FIG. 3.2 Absorption as a function of /?x = (Su)/d for

/? = 0.1.

Afrer Plass [17].

(1) Elsasser band; (2) N = 2; (3) N = 2, S, = lOS,;(4) N = 5 , S, = S, = ( 5 ) statistical model, /? = 0.1.

= S,;

106

Radiation Absorption in the Atmosphere

method for computing the absorption function of a group of overlapping lines of equal half-width. In this case the absorption function appears to be expressed through the intensities of lines both within and without the considered interval. There are no limitations for the intensities or location of lines. All the foregoing expressions for the functions of transmission or absorption are derived on the assumption of homogeneity of the medium. In reality, pressure and temperature always vary along the ray’s path. Calculations show that the influence of temperature on radiation absorption may, as it appears, be neglected. This problem has not yet been studied enough. The influence of pressure, however, is quite notable, since the line half-width is directly proportional to pressure. Theory shows two ways of introducing a pressure correction. One consists in making use of the approximate relation (3.49)

where p o is standard pressure. Calculations give 0 5 n 5 1 if 0 5 P 5 1. In many cases it can be taken that n = & for the real atmospheric conditions. Only at very large P (P > 0.8) do the values of n differ significantly from Q. The second way, suggested by Curtis and Godson, consists in calculating the line shape in the variation of pressure along the ray’s path for the mean pressure value, determined as

p = , / 1p d w

(3.50)

where integration extends over the total length of the ray’s path in the layer of the absorbing mass w. If it is necesary to consider the temperature dependence of absorption, Plass [30] proposed to determine the effective temperature from

where Si is the intensity of the ith spectral line in the fixed spectral range containing N lines, and Sih is the equivalent intensity found in selecting the appropriate value of the equivalent temperature Th. Plass [33] has also discussed other possibilities of evaluating the dependence of absorption upon pressure and temperature.

3.2. Absorption Spectrum of Water and Water Vapor

107

Howard and Garing [31] show that the consideration of the inhomogeneity of the atmosphere in computing the infrared radiation transmission by the atmosphere is of extreme importance in certain cases. For example, calculations (performed by various authors with the use of different methods) of transmission of radiation of wavelength 2.7 p along the path in the atmosphere equal to 380 km between the observer at 20-km elevation and the target at the level of 87 km give values from 20 to 55 percent. Kondratyev and Timofeyev [1561 have realized certain numerical results that enable determination of radiation absorption peculiarities in different spectral regions by taking into account the real fine structure of the absorp tion spectrum. It should be noted that the reported results concern the directed radiation absorption exclusively. To compute the absorption of diffuse radiation, the general relation (1.31) must be used. It is possible, however, to compute approximately the transmission function for diffuse radiation from (1.32) and similar equations. Walshaw [18] has made theoretical calculations for the Elsasser band, which show that the diffusivity coefficient in (1.32) possesses a certain variability. In a wide range of P values, however, the degree of variability is not high. For example, in the variation of the water-vapor absorbing mass from 3x to 1.0 g cm-2, the diffusivity coefficient varies within the limits 1.64 to 1.73, thus making the average value 1.66 to 1.69 fairly satisfactory. 3.2. The Absorption Spectrum of Water and Water Vapor

As was already mentioned, the atmospheric absorption spectrum is extremely complicated. While the main gases of the atmosphere (nitrogen, oxygen) contribute only slightly to the radiation absorption, the variable atmospheric constituents such as water vapor, carbon dioxide, ozone, numerous nitrogen oxides, and hydrocarbon combinations have an enormous number of absorption lines and bands in various spectral regions. To show the general characteristic of atmospheric absorption, Fig. 3.3 schematically presents the most intensitive absorption bands of atmospheric gases in a wide range of wavelengths, from the vacuum ultraviolet to the far-infrared spectral regions. The upper curves in this figure characterize the energy distribution in the blackbody emission spectrum at 6000'K (the solar temperature) and 250'K (the average tropospheric temperature). The middle and the lower parts of Fig. 3.3 show the dependence of radiation absorption (in percent) upon wavelength at the surface level and at 1I-km elevation. As seen from Fig. 3.3, the most intensive and widest are the wa-

108

Radiation Absorption in the Atmosphere

600OO K

a

260. K

100

0s

d

E

o 100

8 m a

0 0.10.150.2

0.5

1.0

2

5

10

20

50

P

FIG. 3.3 Atmospheric absorption spectrum at the level of the earth's surface (the middle part of the figure) and at the altitude of 10 km (the lower curve). After Goody [34].

ter-vapor absorption bands. The domination of water vapor in atmospheric radiation absorption was stated as early as at the end of the past century. From that time onward there have been performed a great number of investigations with the purpose of experimentally and theoretically studying the quantitative characteristics of the absorption of radiation by water vapor and the structure of water vapor bands. We shall now give a survey of certain meteorologically essential applications of the results of the investigations mentioned. The interested reader can find more detail in the voluminous literature on the question (see [34-83, 1621). Specially undertaken physical investigations allow determination of the water vapor molecular structure and clarification of the origin of individual absorption bands. It has been found that in the unexcited state, the water vapor molecular configuration has the form of an isosceles triangle (see Fig. 3.4), with the side length So, = 0.958 %, (nucleus-to-nucleus hydrogenoxygen distance) and the apex angle 0 = 104'30'. The absorption of radiation by water vapor in the region from 0.54 to 9 p is caused by vibrational-rotational transitions and in the far-infrared (from 9 ,u to 1.5 cm) by purely rotational transitions. In the ultraviolet (A < 0.2 p ) the absorption is determined by electronic transitions. Three types of normal vibrations of the water vapor molecule (Fig. 3.5) are observed to have the following main frequencies : v1

= 3670

cm-l;

v2 = 1675 cm-l;

v8

= 3790

cm-I

3.2. Absorption Spectrum of Water and Water Vapor

109

All these types are active in absorption. The rotational water-vapor absorption spectrum is associated with rotational energy transitions caused by the molecule rotation around its axes a, b, c, asymmetrically and

lo

I FIG. 3.4 Structure of the water vapor molecule.

spinlike (see Fig. 3.4; the axis b in this figure is perpendicular to the plane of the drawing). Let us now consider the quantitative characteristics of the radiation absorption by water vapor for shortwave and longwave radiations.

u, * 3670cm''

v2=1675an-'

us= 3790cm-'

FIG. 3.5 Fundamental vibrations of the water vapor molecule.

1. The Absorption of Shortwave (SoZar) Radiation. Water vapor possesses a number of intensive absorption bands in the far-ultraviolet. The most intensive are the bands in the following spectral intervals: 160 to 1100 A, 1050 to 1450 A, and 1450 to 1900 A. The absorption of ultraviolet solar radiation by water vapor appears to be of importance to the energy of the upper atmosphere (see [163-1651). For the troposphere, however, the presence of an intensive water-vapor absorption spectrum in the farultraviolet is practically unimportant, since this radiation is fully absorbed in the upper atmosphere. Figure 3.3 does not show any water absorption bands in the visible range, but they are present, though very weak, and are known as "rain" bands within the wavelength interval of 572 to 703 mp.

110

Radiation Absorption in the Atmosphere

Of greatest importance to shortwave radiation absorption in the atmosphere is a number of intensive and wide water-vapor bands in the nearinfrared. The location and width of these bands are listed in Table 3.4. TABLE 3.4 Absorption Bands of Water Vapor in the Near-Infrared Spectral Region

Denomination of Band

Spectral Region, ,u

Band Center Position (Absorption Maximum), ,u

a 0.8 p e a t

0.70 -0.74 0.79 -0.84 0.9260.978 1.095-1.165 1.319-1.498 1.762-1.977 2.52O-2,845

0.718 0.810 0.935 1.130 1.395 1.870 2.68

v Y

52 X

In Fig. 3.6 is presented the dependence of generalized absorption coefficients I,, (see Eq. 3.39), computed by Yamamoto and Onishi [84, 851 for the above absorption bands and for a longer wavelength band near 6.3 p. The considered spectral region has another intensive water-vapor absorption band near 3.2 p.

l0fs

FIG. 3.6

12

10

8

6

-V

4

2

Generalized absorption coefficient of water vapor. After Yamamoto and Onishi [85].

As the preceding paragraph showed, the absorption of radiation depends on the pressure in the medium. Howard et al. [43, 631 have derived formulas from measurement data that enable computation of radiation in the con-

111

3.2. Absorption Spectrum of Water and Water Vapor

sidered bands, taking into account not only the water vapor content but also the pressure. The formulas are the following :

Jl: A , dv

s”a

A,, dv

”I

=

C

=c

Jll A,dv < A ,

+

~ ’ / ~ ( pe)k

s”a

+ D log w + K log@ + e )

A , dv

(3.51)

> A , (3.52)

”1

Here w is the water vapor content on the ray’s path with g cm-2 or centimeters of thickness of the “precipitated water” layer, “cm” (both these units are evidently identical); v1 and v2 are absorption band limits, p is the total pressure (mm), e is the partial pressure of water vapor (mm), and c, k, C, D, K are empirical constants. Integration here is extended over the entire frequency interval of the given absorption band. Knowing the band width Av, we obtain for the considered band the mean absorption function A from the formula A- = - 1 Av

1A,,dv

(3.53)

The formulas (3.51) and (3.52) are finite relations, valid for both weak and strong absorption. A , is an intermediate value J A , ,dv. Table 3.5 gives the values of all parameters in the foregoing formulas. From (3.51) we see that at small A, the absorption follows the square root law, and at large is a logarithmic function of the absorbing mass w. TABLE 3.5 The Parameters of the Formulas for Determining the Shortwave Radiation Absorption by Water Vapor. After Howard et al. [43, 631

Band

Spectral Region, cm-l

ea*

10,10&11,500 8,30&9,300

k

c

D

K

38 31

0.27 0.26

-

-

-

-

-

-

200 200

3.2,~

6,500-8, OOO 4,800-5,900 3,3404,400 2,800-3,340

163 152 316 40.2

0.30 0.30 0.32 0.30

202 127 337 -

460 232 246 -

198 144 150 -

350 275 200 500

6 . 3 ~

1,15&2,050

356

0.30

302

218

157

160

v w

9 X

C

A,

cm-l

TABLE 3.6 Absorption of Solar Radiation by Water Vapor, cal cm-= min-I. After MacDonald [88]

Band

Band Limits, p

Solar Radiation Flux Outside Atmosphere, cal cm-2 min-l

iz

Total Water Vapor Content on the Ray’s Path, g cm-a 0.5

1.0

2.0

3.0

4.0

5.0

6.0

8.0

c 1 . 0

D

b

6

a

0.70-0.74

0.0784

0.0016 0.0024 0.0047 0.0055 0.0071 0.0086 0.0102 0.0133

0.8 p

0.79-0.94

0.0777

0.0019 0.0031 0.0054 0.0062 0.0077 0.0093 0.0108 0.0140

z 3 B8’

e QT

0 . 8 M . 99

0.1578

0.0142 0.0205 0.0300 0.0363 0.0026 0.0490 0.0520 0.0584

p

97

1.03-1.23

0.1648

0.0165 0.0230 0.0329 0.0379 0.0445 0.0478 0.0510 0.0560

$

Y

1.24-1.53

0.1432

0.0538 0.0601 0.0686 0.0758 0.0801 0.0830 0.0844 0.0858

9F

s;,

1.53-2.10

0.1209

0.0350 0.0386 0.0423 0.0447 0.0458 0.0471 0.0496 0.0508

0.7428

0.1230 0.1477 0.1839 0.2064 0.2278 0.2448 0.2580 0.2783

Total

113

3.2. Absorption Spectnun of Water and Water Vapor

These conclusions have been confirmed by new investigations of Vasilevsky and Neporent [25, 32, 86, 871. After critically analyzing Fowle’s experimental data on solar radiation absorption by the whole thickness of the atmosphere, McDonald [88] obtained the following values of energy absorbed in different water vapor bands as given in Table 3.6. With available data on the absorption functions for individual water vapor bands, it is easy to calculate the integral shortwave radiation absorption function and also the amount of absorbed radiation in absolute units. Moller and Miigge [89] derived a simple empirical formula for the quantity of solar radiation flux A S (in cal cm-2 min-l) absorbed in the cloudless atmosphere, as follows :

AS

=0.172(m~,)~.~

(3.54)

where m is the atmospheric mass directed on the sun, w, is the total watervapor content in the atmosphere in the vertical direction (g cm-$). At a later time, Moller et al. [90] proposed a more precise formula to replace (3.54), based on radiation absorption data by Howard and others,

AS

=

exp{2.3026[- 0.740

+ 0.347 log (mw,)

- 0.056(10g(~~~,)~ - 0.006(10g(~~~,))~]}(3.55)

Computational results AS, obtained by Moller with the help of (3.55), are given in Table 3.7. According to McDonald’s data (Table 3.6), the coefficient 0.172 in (3.54) must be changed to 0.149. The data of Table 3.6 are also well described by a simple formula, suggested by Angstrom [91],

AS

= 0.10

+ 0.21(1 - e-‘.23mur-)

(3.54a)

Angstrom, however, also showed that all these formulas, which take into account only pure absorption, cannot be applied for calculation of solar radiation absorption in the real atmosphere, where scattering of solar radiation is also important. That is why another formula, which takes account of the scattering of solar radiation in the region of absorption bands, is more reliable:

AS

= 0.10

+ 0 . 02 .32~3-~+- /3 0.21(1

-

e7mb

e-0.23m,

)

(3.54b)

Here /Iis the optical atmospheric mass caused by scattering. Calculations

114 I-

0

d: 0

2

5 0

% 4

0

4 4

0

a

8 g

8

zz P-

2

w

m

4

0

m

s

m 0

m

m

2 s

2 0

0

3

p &

0

2

Radiation Absorption in the Atmosphere

0 e

* $1 m

X

2 W D)

X

2 m

I

N

$1 D)

$1 X m n

2

2

E

I

N

s”

brJ

E

3

115

3.2. Absorption Spectrum of Water and Water Vapor

show that even a t = 0.1 or 0.2, the results of determination of A S from (3.54b) notably differ from (3.54a). Usually the total water-vapor content in the vertical atmospheric column of unit section (1 cm2) w, varies within the range from several tenths to 1 to 2 g cm-2. Thus the amount of solar radiation absorbed by the cloudless atmosphere, is of the order of several tenths of cal cm-2 min-l. We shall see later that the total extinction of solar radiation in the atmosphere is far greater. This fact is related to the prime importance of scattering in solar radiation attenuation. We have already mentioned that radiation absorption depends on the pressure in the medium. This dependence is clearly exhibited in the case of solar radiation absorption. Table 3.8 gives the dependence of absorbed solar radiation values, AS on pressure p at different values of the watervapor absorbing mass w as illustrated by Moller et al. [90]. TABLE 3.8 The Dependence of Absorbed Solar Radiation on Pressure, Miiller et al. [90]

cal cm-2 min-I. Ajier

P, mb w, gkm2

10-4

10-3 10-2 10-1

100 10'

777

611

467

311

234

3.36 10.6 34.8 90.2 187.4 335.3

3.31 9.90 32.8 85.8 172.1 316.0

2.89 9.09 29.4 80.3 164.3 301.4

2.56 8.05 26.1 73.5 150.1 285.9

2.35 7.38 24.0 68.7 143.6 277.6

The dependence of AS upon p is quite marked. To take account of this dependence, Moller proposed to compute solar radiation absorption in the atmosphere for the mean pressure value p = {p,, , where po is the pressure at the earth's surface level. It must be noted here that Table 3.7 implies p = 777 mb ( p o = 1000 mb). The discrepancy between Tables 3.7 and 3.8 at small values of w results from the inaccuracy of the extrapolation of experimental data with the help of (3.55). And in general, the results of determination of AS by different authors do not coincide (especially a t small w). Shaw [139] contends that in the case of a gas with a constant mixing ratio, we may take p = ip,,.

116

Radiation Absorption in the Atmosphere

2. Longwave (Thermal) Radiation Absorption. Figure 3.3 presented the general features of the infrared water-vapor absorption spectrum in the atmosphere. We see that the thermal radiation, the main portion of which is usually localized at temperatures in the spectral region from 4 to 40 p, is almost completely absorbed by the atmosphere over the entire given range except the interval from about 8 to 12 p, where the absorption by water vapor is rather weak. To state it more accurately, the radiation absorption by water vapor is not great in the region of wavelength somewhat exceeding 12 p, but increases rapidly at wavelengths of about 15 to 20 p and longer. This increase is due to the presence in this region of an intensive carbon dioxide absorption band. The interval 8 to 12 p is often called the atmospheric transparency window, or simply the atmospheric window. Since attenuation of radiation in this transparency window is to a great extent caused by the effect of continuous absorption, the exponential absorption function appears to be

0.3

-

U

s

'I¶ Y

I

9

10

x

II

12

I3

FIG. 3.7 Absorption coefficients in the atmospheric transparency window 8-12 p. After Kondratyev et a[. [I. 221. (1) Anthony (1952); (2) Adel (1939); (3) Roach and Goody (1958); (4) Saiedy (1958); ( 5 ) Kondratyev et al. (1963).

3.2. Absorption Spectrum of Water and Water Vapor

117

quite satisfactory in this case, thus enabling determination of logarithmic absorption coefficients. One of the possibilities of determining absorption coefficients consists in the use of measurement data on the transparency , of the whole atmospheric thickness for solar radiation in the 8 to 12 u range. In Fig 3.7 are shown the results by Kondratyev et al. [169] in comparison with data of other authors. In the interval 8.7 to 10.5 p, the results of Kondratyev et al. [169] coincide with those of Roach and Goody [39] in the best possible manner. Remember now (Fig. 3.6) that, besides the main atmospheric window in the range of 8 to 12 p, a number of transparency windows are observed in the near-infrared region. Adel [92] also found an atmospheric window corresponding to small radiation transmission values in the range 16 to 24 ,u. As to the far-infrared region, it is characterized by the presence of a very intensive rotation band located near 50 p. This is seen in Fig. 3.8,

FIG. 3.8 Generalized coefficients of water vapor in the far infrared. After Yamamoto and Onishi [84].

which shows the dependence of a generalized absorption coefficient upon frequency in the far-infrared as calculated by Yamamoto and Onishi [84]. Later these data were made more accurate by Elsasser [20], Plass et al. [81], who theoretically calculated on a large scale the water vapor radiation

118

Radiation Absorption in the Atmosphere

absorption at a different values of pressure and temperature. The ultrashort radio waves show an intensive absorption line near 1.35 cm. The generalized absorption coefficients I,., mentioned as quantitative characteristics of absorption, were determined in the assumption of the real water-vapor bands being schematically presented as Elsasser bands. In reality, the asymmetrical property of the water vapor molecule as a vibrator and rotator leads to the irregularity of the absorption spectrum from the viewpoint of the dependence of line absorption and intensity on frequency. Analyzing experimental data, then, one comes to the conclusion that a statistical model is by far the more perfect model of absorption bands. However, quantitative characteristics of absorption for the statistical band have not yet been determined for the entire infrared spectrum. We should mention that, according t o Howard [76] and his collaborators, the quasimonochromatic transmission function for all water-vapor absorption bands in the near-infrared and also for the band near 6.3 ,u may be presented by the following empirical formula as it derives from (3.46): 1.97(W/Wo) P , = exp [l 6.57(w/~,)]'/~

{

+

= 3. The value w o is the function Here w,,is a value of w at which P,, of frequency and total pressure in the medium. Zuyev et al. [149], using the statistical model formulas and the Howard formula given above, conducted a number of computations of the absorption function in each spectral interval 0.1 ,u in the range 1 to 10 ,u for different values of pressure and precipitated water layer. The derived results are presented in the form of practical graphs. It is possible to use these graphs for direct determination of the transmission function for any distance up to 100 km at any altitude in the lower 30-km atmosphere layer. The description of absorption and transmission functions with the help of the analytical presentations, demonstrated in the preceding paragraph for the Elsasser and statistical bands, is fairly accurate though also cumbersome. It is convenient to use for this purpose an exponentially decreasing transmission function of the kind expressed in (1.23). Although, strictly speaking, a similar function is applicable only for pure monochromatic radiation (or for nonmonochromatic in the absence of the dependence of the absorption coefficient upon wavelength), nevertheless it may be used for an approximate description of the dependence of selective transmission upon the absorbing mass in application to spectral intervals of finite width.

119

3.2. Absorption Spectrum of Water and Water Vapor

In Table 3.9 are given average values of the so-called logarithmic absorption coefficients ki corresponding to the following analytical presentation of the integral transmission function : (3.56)

where p i is a portion of incident radiation in the spectral interval d l . Summation is performed over all the intervals of dl within the region under TABLE 3.9 Logarithniic Absorption Coefficients of Water Vapor. After Kondratyev [93]

k,, c m - 2g

Ea, cal cm-2 min-l

ki,cm-z g

EAA,

cal cm-2 min-l

5.0-5.5

40

123

19-20

43

568

5.5-6.0

118

188

20-21

23

502

6.66.5

198

287

21-22

58

455

6.5-7.0

156

349

22-23

64

408

7.0-7.5

46

439

23-24

75

364

7.5-8.0

12.8

502

24-25

80

319

8.W3.5

3.4

570

25-26

53

290

8.5-9 . O

0.10

585

26-27

93

256

9. 6 1 2 . 0

0.10

3880

27-28

116

232

12-13

0.25

1150

28-29

136

208

13-14

0.84

1088

29-30

152

190

14-15

1.30

989

30-3 1

179

170

15-16

1.65

850

31-32

179

156

4.40

716

32-33

179

142

33-34

198

128

34-35

110

116

16-17 17-18

17.2

18-1 9

14.0

634

consideration. Also given, corresponding to the absorption coefficient values, are the black emission intensity values E,, at T = 290'K for the corresponding spectral intervals (cal cm-2 per unit solid angle). All these absorption coefficients values are to be treated as approximate. It appears that (3.56) may be substituted, with a sufficient degree of

120

Radiation Absorption in the Atmosphere

accuracy, by an analogous but simple relation in which n = p , = p 4 = $, as follows:

= 4,

p1 = pz

4

P,(w) = $ C.

(3.57)

erkjw

j=l

where ki are mean absorption coefficients obtained by averaging kj over chosen spectral intervals Ail corresponding to absorption coefficients of the same order of value. The values kj in (3.57) are as follows: k, = 0.10, k, = 1.14, k , = 19.6, k , = 114 (cmz g-l). The formula (3.57) can be used for evaluation of the integral thermal radiation absorption by water vapor. Kuznetzov [94] has worked out an approximate mathematical method for presentation of the transmission function with the help of exponentially decreasing functions. This method permits description of radiation transmission within any absorption band in the following manner:

Pl(w)= y e-aw + ( 1- 7 )

(3.58)

e-@

where a, b, and y are the parameters determined from the measured values P,(w) with the help of the formulas derived by Kuznetzov. Table 3.10 displays the values of these parameters as calculated by Feugelson [95] for a number of the infrared water-vapor absorption spectrum intervals. TABLE 3.10 Parameters

Spectral Region, p

01,

/3, and y. After Feugelson [95]

B

O1

(cma g-l)

Y

5-8

9

0.9

8-1 1

1.6

0.12

0.24

4.8

0.14

0.18

-

1.o

11-13

>

17

2 10

0.87

3. Radiation Absorption by Water. Laboratory investigations have shown that water has far more intensive absorption bands than water vapor, with the absorption bands of water being somewhat displaced towards longer wavelengths. This fact is well illustrated by Table 3.11. As seen, the shifts of maxima are observed in all bands situated in the

3.2. Absorption Spectrum of Water and Water Vapor

121

TABLE 3.11 The Posifion of the Centers of the Absorpfion Bands of Water and Wafer Vapor

Band

X

Water

Water Vapor

0.745 0.85 0.98 1.18 1.46 1.79 1.98 2.52 2.97 4.69 6.1

0.720 0.83 0.935 1.13 1.396 1.87 2.68 6.04 6.50

wavelength region 1 > 3 p. The water bands 1.79,2.52, and 4.69 p, however, have no analogues in the water vapor spectrum. As to the band near 6 p, in the case of water vapor it has a double maximum. Figures 3.9 and 3.10 present the dependence of logarithmic water absorption coefficients upon wavelength, as shown by Kondratyev et al. 1961. The absorption coefficients values are expressed in “cm”-l (cm2 g-l). The wave-number values are plotted on the abscissa. The data of these figures clearly illustrate the great intensity of the absorption by water in the in-

Y

cm-‘

FIG. 3.9 Absorption coeficierits of water in ihe infrared spectral region (13,200-5600 cm-l). After Koridraiyev et al. [96].

122

Radiation Absorption in the Atmosphere k,cm-‘

250,

k,cm-’ I2000

.2400

200.

10000

:2000

150T

E

0

x

100 -

8000

I

50-

1000 ucm-l

FIG. 3.10 Absorption coeficients of water in the infrared spectral region (7200-400 cm-’). Afrer Kondratyev et al. [96].

frared spectral region. The latter leads to the fact that liquid water possesses practically no transparency windows of the kind observed in the case of water vapor. Liquid water, as does water vapor, absorbs radiation not only in the infrared but also in the visible spectral region. The absorption here, however, is far less intensive, which is demonstrated by Fig. 3.9. It should be noted further that the data given here for the liquid-water absorption coefficients were obtained from measurements of the “pure” absorption by water films of different thickness. In reality we are usually interested in the attenuation of radiation by water droplets. It is evident that in this case it is necessary to compute the attenuation of radiation on the basis of the theory of light scattering on absorbing particles. This problem will be treated in the next chapter. Of much interest is the absorption of solar radiation in water basins, the data for which are given in Table 3.12. In Table 3.12 the spectral energy distributions at different depths are calculated from the known energy distribution in the spectrum of solar radiation at the earth’s surface, taking into account the absorption of solar radiation. As seen from these data, only shortwave solar radiation penetrates the depths below 1 m.

123

3.3. The Absorption Spectrum of CO, TABLE 3.12

Distribution of Energy in the Solar Radiation Spectrum after Passing Water Layers of Different Thickness Water Layer Thickness, cm

AAP 0.001

0.3-0.6 0.64.9 0.9-1.2 1.2-1.5 1.5-1.8 1.8-2.1 2.1-2.4 2.4-2.7 2.7-3.0

Total

0.01

0.1

1

lo00

10,Ooo

237.0 237.0 237.0 237.0 236.2 236.2 229.4 172.9 9.5 359.7 359.7 359.7 359.0 353.4 304.9 128.6 8.2 178.8 178.7 178.1 172.2 122.8 86.6 86.1 81.8 63.3 17.1 80.0 78.2 63.7 27.0 10.9 25.0 23.0 18.9 1.1 25.3 24.5 2.0 7.2 6.3 0.4 0.2

13.9

10oo.o 993.7 952.1 859.6 730.2 549.3 358.0 182.4

13.9

0

10

100

3.3. The Absorption Spectrum of CO, Carbon dioxide is presented by several absorption bands in the infrared spectral region of the wavelength interval 1 > 1.3 p. Also known is the presence of radiation absorption by carbon dioxide in the far-ultraviolet region 1000 to 2000 A. The intensity of solar radiation in both these regions is very low, and the absorption of solar radiation by carbon dioxide is rather weak. This is not with respect to longwave radiation because it is essential to consider longwave radiation absorption by carbon dioxide. o

c

o

- . - c

uI = 1361~6'

o

t

c

+

o

i

u2 = 673cm-'

o c o +-+us = 2378cm-I

FIG. 3.11 Fundamental vibrations of the carbon dioxide molecule.

The carbon dioxide molecule is a linear molecule (see Fig. 3.1 1) with the nucleus-to-nucleus carbon-oxygen distance 1.15 A. The intensive bands of carbon dioxide absorption in the infrared are due to vibrational transitions. Three types of normal vibrations of the carbon dioxide molecule

124

Radiation Absorption in the Atmosphere

are observed, corresponding to the main frequencies v1 = 1.361 cm-I, v2 = 673 cm-l, and v3 = 2.378 cm-l (see Fig. 3.10). However, only two types of vibration with frequencies v2 and v3 are active in absorption. To the main frequencies correspond two intensive absorption bands near 4.3 and 14.7 p. Besides these main bands, CO, absorption bands, centered at 1.4, 1.6, 2.0, 2.7, 4.3, 4.8, 5.2, 5.4, 10.4 p, are also found. The band near 9.4 and 10.4 p is very weak and cannot practically be considered. As to the shortwave bands, they are positioned in the wing of the curve of the spectral blackbody radiation intensity distribution at temperatures observed in the atmosphere; thus, although the 4.3-p band is fairly intensive, its effect on the absorption of longwave terrestrial or atmospheric radiation may be ignored. The essential consideration, then, is only the absorption in the region of a wide (12.9 to 17.1 p ) band at 15 p. Table 3.13 is intended to clarify this conclusion by presenting logarithmic absorption coefficients at the centers of the bands together with relative values of the black radiation flux at different T (in percent to the total radiation B = 0T4) corresponding to the wavelength intervals of the first line. TABLE 3.13 The COzAbsorption Coefficients and the Distribution of Energy in the BIackbody Radiation Spectrum

T

2.6-2.8 (2.7)

4.104.45 (4.3)

9.1-10.9 (10.)

12.9-17.1 (14.7)

k (l/m of air at normal pressure and temperature)

0.025 193 213 233 253 273 293 313 313

-

0.03

0.33

2.10-6

0.083

0.1 0.1 0.2 0.4 1.1

5.1 6.9 8.5 10.0 11.1 12.0 12.6 13.2

18.5 19.9 20.5 20.6 20.2 19.6 18.9 15.5

125

3.3. The Absorption Spectrum of CO,

It is clearly seen from this table that in no case may the absorption by carbon dioxide be ignored. The band near 15 ,u is the only practically important absorption band. Recent years have produced a great number of investigations devoted to the influence of this band on atmospheric radiative heat exchange. The result of these investigations is a detailed presentation of the 15-,u band structure, which was found to be comparatively simple. The fine rotational structure of the 15-,u CO, band is quite regular, owing to the linear dislocation of atoms in the CO, molecule and the symmetrical vibrational and rotational properties of the molecule. This has enabled performance of theoretical calculations of radiation absorption by CO,, assuming that the Elsasser model is satisfactory enough in the given case. The line intensities in the 15-,u band vary considerably, but the regular position of lines makes the Elsasser model more preferable than the statistical model, especially if (instead of one isolated Elsasser band) two or more overlapping bands with different line intensities are considered. Howard and collaborators have presented the results of the measurement of the absorption in CO, bands with the help of empirical formulas (3.51) and (3.52). Table 3.14 gives the constant values of these formulas. TABLE 3.14 The Parameters of the Formulas for the Absorption of Radiation by Carbon Dioxide. After Howard et al. [43,631

Band, p

14.7 5.2 4.8 4.3 2.7 2.0 1.6 1.4

Spectral region, cm-l

550-800 1870-1 980 198CL2160 216CL25OO 348CL3800 475CL5200 600&6550 6650-7250

C

k

c

0.44 - 68 0.40 .._ 0.37 ... ... ... 275 3.15 0.43 -137 0.492 0.39 -536 0.063 0.38 0.058 0.41

3.16 0.024 0.12

D

K

55

47

...

...

...

...

34 77 114

31.5 68 114

A, cm-l

50 30 60 50 50 80 80 80

If we take the width of the 15-,u absorption band equal to 250 cm-' (which corresponds to the limits of the band 550 to 800 cm-I), with the

126

Radiation Absorption in the Atmosphere

help of (3.52) we obtain for the absorption function of the band, 2:

,&

=

-!250

A,,dv

=

-0.27

+ 0.22 log[& + p’)0.855]

(3.59)

where p‘ is the partial pressure of carbon dioxide and u is the content of CO, (cm). For the absorption function of the entire 15-p band, several other simple empirical relations have been proposed. For example, according to Shekhter [97], it is possible to use the following analytical presentation A-I -- 1 - eaUb

(3.60)

where a = 0.32, b = 0.4. The determination of the constants of this formula, made by Kondratyev and Nedovesova [98] from experimental data of Howard et al., relating to p = 1000 mb, has given somewhat different constant values: a = 0.256, b = 0.41. Kondratyev and Nijlisk [99] suggested the following empirical formula :

A-I --

(3.60a)

Io(a+bucP

where a = 1.44, b = - 0.309, c = - 0.287, and d = 2. Feugelson [95], using the Kuznetzov formula (3.58), has obtained constant values in this formula, corresponding to the different spectral regions as given in Table 3.15. TABLE 3.15 The Constants of Kuznetzov’s Formula for the 1 5 p Carbon Dioxide Absorption Band. After Feugelson [95]

Spectral Region, ,u

12.5-13; 17-17.5 13-14; 16-17 14-16

B

a (cmz g-l)

Y

22.1

0.19

0.1

30

1.25

0.5

143

1.1

0.95

In Fig. 3.12 are given the results of experimental and theoretical investigations of the absorption function of the 15-p band according to the summary made by Yamamoto and Sasamori [35, 361.

3.3. The Absorption Spectrum of CO,

127

FIG. 3.12 Comparison of absorption functions for 1 5 p carbon dioxide band. (1) after Elsasser and King; (2) after Howard (strong line approximation); (3) after Howard (weak line approximation); (4) the most probable curve; (5) after Yamamoto and Sasarnori (results of calculations); (6) after Howard ef al. (measurements for pressure 1.25 atm); (7) old data collected by Callendar; (8) after Kaplan and Eggers (pressure 1.25 atrn).

On the axis of the abscissa (Fig. 3.12) are plotted values log u, where u is the carbon dioxide content expressed in centimeters of a gas layer at normal pressure (1000 mb) and temperature (300'K). The value u can be determined from the formula

where integration extends over the entire ray path in the absorbing layers and eEoo,is the volume concentration of carbon dioxide in normal conditions. All the data of Fig. 3.12 relate to a homogeneous medium at normal pressure and temperature. Yamamoto and Sasamoti consider that the data of the curve 4 are the most reliable. At u < 10 cm, the curve 4 coincides with theoretical computations of Yamamoto and Sasamori and also with experimental data of Callender, Eggers, and Kaplan. At large contents of carbon dioxide, the curve 4 is built on the points corresponding to experimental results of Howard, Burch, and Wililams. The exaggeration of computational results in (3.51) and (3.52) is explained by the insufficient accuracy of these formulas and

128

Radiation Absorption in the Atmosphere

also by the fact that experimental data of Howard et al. are probably, too high at small values of u. The influence of pressure on the absorption of radiation by carbon dioxide can be taken into account with the help of (3.59). The empirical constants of this formula, however, are not yet reliable enough. Yamamoto and Sasamori, for example, give the exponent in (3.59) to be 0.83 rather than 0.855. For computations of radiation absorption by carbon dioxide, the data relating to individual intervals of the 15-p CO, bnad are interesting. Such data, derived by Yamamoto and Sasamori for normal conditions from theoretical calculations (the line half-width a is taken to be 0.064 cm-l), are presented in Table 3.16, which also gives values J A , dv for the entire absorption band. The investigation of the dependence of absorption on pressure for individual intervals of the 15-p band shows that in this case the dependence is more pronounced than in the case of the entire considered band. The same refers to the dependence of absorption on temperature. Calculations of Sasamori [58] have shown that even for the integral absorption in the 15-p band of CO,, the temperature dependence of absorption is significant, especially in great thicknesses of the absorbing gas layer. With the increase of temperature in the medium, a notable increase of absorption takes place. That is why, strictly speaking, in computations of the absorption of radiation by carbon dioxide, the dependence of absorption on pressure and on temperature must be taken into consideration. As is shown by Yamamoto and Sasamori, the introduction of these corrections is facilitated because of their additive nature. They can be taken into account when there are available data that characterize the separate dependences of absorption on the carbon dioxide content at different values of p and T. The same authors recommend that the effective values of the absorbing mass, the pressure, and temperature be determined from the formulas :

Elsasser [20] proposed the following dependence for approximation of the generalized absorption coefficient at the wings of the 15-p CO, absorption band : I(Y

- Yo) =

~

const C exp[ - -(v - 4 T T

2 1

TABLE 3.16 The Absorption Functionfor Different Regions of the 15 p Carbon Dioxide Absorption Band (in Portions of Unity). Afrer Yamamoto and Sasamori [35, 361 u, crn

ww

Spectral Region, ,u 0.01

0.03

0.1

0.3

1

12-13

3

10

30

100

300

1000

0.001

0.004

0.012

0.029

0.075

0.192

0.662

0.822

0.939

13-14

0.001

0.003

0.008

0.021

0.048

0.105

0.262

0.450

14-15

0.027

0.059

0.136

0.266

0.480

0.718

0.921

0.000

15-16

0.017

0.039

0.107

0.233

0.440

0.682

0.893

0.984

1.000

16-17

0.002

0.005

0.013

0.025

0.061

0.137

0.304

0.510

0.783

0.948

1.000

0.001

0.003

0.010

0.025

0.072

0.181

0.330

0.575

17-18 12-18 J A , av, cm-l

4

t

rn

51

0.008

0.017

0.043

0.088

0.166

0.266

0.387

0.486

0.584

0.665

0.751

2.2

4.7

12.0

24.4

46.1

73.9

108

135

162

185

209

s

130

Radiation Absorption in the Atmosphere

where vo is the frequency corresponding to the center of the band and c is constant. If I, is the generalized absorption coefficient value corresponding to To (for example 293’K), log I

= log

lo - a’

~

TO - T (v T

- Yo),

T + log TO

where a’ = c/To. Hence log I

= -

u(v - Y o ) ,

+b

(3.61)

At T = 293’K and the constant b = 0.10, a = 3.4 x lop4 (for the shortwave wing) and a = 4.6 x (for the long-wave wing). Using these formulas it is possible to calculate the dependence of the generalized absorption coefficient upon wave number at any temperature. Numerous theoretical computations of radiation absorption by carbon dioxide have been made by Stull et al. [80]. All the preceding data characterize the absorption of directed radiation. To determine the absorption function of diffuse radiation, one can use either a precise formula (1.31) or an approximate relation (1.32). As shown by Kondratyev and Nijlisk [99], in a very wide range of carbon dioxide content, can be taken to be equal to 1.66. Taking into consideration the data on the absorption of long-wave radiation by water vapor, it becomes evident that in the region of the 15-p CO, band, the infrared absorption spectra of water vapor and carbon dioxide overlap. This means that in the considered spectral region the atmosphere should be treated as a bicomponent absorption medium. The determination of the monochromatic transmission function for a bicomponent (and, in general, multicomponent) medium presents no difficulty, since in the given case the following obvious formula is valid: i=l

where k s i , wi are the mass coefficients of absorption and of the mass of absorbents, respectively. Summation extends over all n of the absorbing components. The problem of determination of the transmission function of a multicomponent medium for wide spectral intervals is more complex. It is solved in the case in which the transmission functions for wide spectral intervals are assumed to be exponential. In this case, according to calculations of Kondratyev [170], the mean absorption coefficients values k, and k, in

3.3. The Absorption Spectrum of CO,

131

(3.57) must be increased up to k, = 4.96, k, = 19.6, to make possible the absorption of radiation by carbon dioxide. Thereby w in (3.57) should be continued to be treated as the water vapor mass. A similar approach to the consideration of the overlap of water vapor and carbon dioxide absorption bands is suitable for approximate evaluation only. The transmission function of a gaseous mixture, too, is equal to the product of the transmission function components if the transmission functions are representative of the statistical absorption-band model. Figure 3.13 gives more accurate results of computation of the atmospheric transmission function in the 15-,u CO, band region as obtained by Kondratyev and Niilisk, who took into account the diffusivity of radiation. For comparison, the figure also gives values for the transmission function, obtained (with consideration of radiation absorption by water vapor) from data of Table 3.9 and of Yamamoto and Onishi [35, 361, and Sasamori [58]. It is seen that data on the absorption of radiation by water vapor are of essential importance, especially at low contents of carbon dioxide, u (cm), and high contents of water vapor, w (cm). U

OD004

I

.................. 2

FIG. 3.13 Trasmissionfunctions of carbon dioxide and water vapor for diffuse radiation. After Kondratyev and Nglisk [lOO]. (1) after Kondratyev Ch. 1 [ 21). and Yarnarnoto and Sasamori [35];(2) afteryamamoto and Sasarnori [35] and Yamarnoto and Onishi [84].

132

Radiation Absorption in the Atmosphere

3.4. The Absorption Spectra of Ozone and Oxygen Ozone is the third important radiation-absorbing component in the atmosphere. Its spectrum is abundant in intensive absorption bands in the regions of both short and long waves. Oxygen shows an intensive ultraviolet radiation absorption. 1. Absorption of Shortwave Radiation. It was hypothesized at the end of the past century that the observed “discontinuity” of the solar spectrum in the region of wavelengths shorter than 0.3 ,LA was due to the absorption of the ultraviolet solar radiation by ozone. Later observations, conducted under both natural and laboratory conditions, confirmed this supposition and demonstrated that ozone does have several absorption bands in the solar radiation spectral region, with the most intensive band occurring in the ultraviolet interval. Figure 3.14, borrowed from Prokofyeva’s monograph [I01 1, gives a schematic presentation of the ozone absorption bands in the ultraviolet and visible solar radiation spectral region. Also presented are two absorption bands of oxygen. On the axis of abscissas of Fig. 3.14 are plotted wavelengths in Angstroms; on the axis of ordinates, decimal absorption coefficients of different scales for various absorption bands (the decimal coefficients of absorption are values of kAin the relation S, = So,A 10-ki.z, Schurnonn-Runge region

Oxygen obsorplion

k2423

Ozone absorption boundory of oxygen dissoclatlon

FIG. 3.14 Absorption coefficients of ozone and oxygen in the ultraviolet and visible spectral regions (on different scales). After Prokofyeva [loll.

3.4. The Absorption Spectra of Ozone and Oxygen

133

where x is the ozone or oxygen content expressed as thickness of a radiation absorbing layer in centimeters at O°C and 1000 mb of pressure). As seen, the solar ultraviolet possesses a very intensive absorption band, called the Hartley band, extending from about 2200 to 3200 A. The presence of this band explains the discontinuity of the solar spectrum in the range il < 3000 A. The neighboring bands, the so-called Huggins bands, occupy the region 3000 to 3450 A. In the visible region is present an array of less intensive bands, the Chappuis bands (4400 to 7500 A). The influence of various ozone bands on solar radiation attenuation may be illustrated with the help of Fig. 3.15, by Kalitin [102]. Its lower curve describes the spectral distribution of solar radiation at the surface level;

FIG. 3.15

Absorption of solar radiation by ozone. Afrer Kalitin [102].

its upper curve determines the energy distribution in the black emission spectrum at T = 6000°K, which in the first approximation may be considered identical with the spectral distribution of solar radiation outside the atmosphere. Number 1 stands for the extremity of the Hartley band, numbers 2 and 3 mark the Huggins and Chappuis bands, respectively. The absorption of solar radiation caused by the latter bands is seen to be quite insignificant, but the Hartley band determines the complete solar radiation absorption in the shortwave region. Ozone also has fairly intensive absorption in the far-ultraviolet (I050 to 2000 A). The value of the decimal absorption coefficient here varies from several unities to about 100 cm-'. An essential contribution to the absorption of ultraviolet radiation by ozone is made by the absorption in the

134

Radiation Absorption in the Atmosphere

continua of Herzberg (2026 to 2420 A) and Schumann-Runge (1050 to 1750 A). The computation of the total solar radiation absorption by ozone shows that the integral solar radiation flux variation due to this absorption is insignificant. The portion of solar radiation absorbed by ozone varies from about 1.5 to 3 percent, depending on the solar elevation and the total ozone content in the atmosphere. The mean value of the integral flux of solar radiation absorbed by atmospheric ozone is taken as 2.1 percent. This value is averaged for the entire Northern hemisphere, the difference in average atmospheric mass between individual points of the hemisphere being considered in arriving at the average value. In the chapter on the spectral composition of direct solar radiation, we shall dwell again on the absorption of solar radiation by ozone and give special attention to the influence of ozone on the ultraviolet spectrum of solar radiation. lmportant results on ozone absorption are contained in Vigroux et al. [166-1681. There are two oxygen bands in the visible solar radiation spectrum: one centers at 0.69 p (band B), the other at about 0.76 p (band A). Their influence on solar radiation absorption, however, is not great. Apart from these two bands, oxygen also possesses several systems of absorption bands in the far-ultraviolet. One of them, the Schumann-Runge system, is localized in the region 1750 to 2026 A; another, the Herzberg system, fills the 2420 to 2600 8, range (see Fig. 3.14). Also important are intensive oxygen bands in the 1050 to 1750 A interval and in the region of still shorter wavelengths, but these bands have not been fully investigated. In the interval 850 to 1050 A, for example, were found the Hopefield bands and a number of unidentified bands. Wavelengths shorter than 850 A are characterized by the presence of continuous absorption, with maximum near 400 to 600 A. Toward wavelengths of the order of 100 A, the oxygen absorption coefficient falls to very low values. The above systems, as do bands A and B, cause the absorption of a very small portion of the total direct solar radiation flux. In spite of this, the absorption due to the considered band systems is of great importance for a number of problems. It is known, for example, that the radiation of 2420 A affects the degree of oxygen dissociation and that this process is connected with the formation of ozone in the atmosphere. Since the former of the mentioned oxygen band systems is located at 1 < 2420 A and the lower limit of the latter is 1 = 2420 A, they are the ones responsible for the dissociation. Thus, the investigation of solar radiation absorption in the far-ultraviolet is essential in dealing with the problem of the causes and regularities of the atmospheric ozone origin.

3.4. The Absorption Spectra of Ozone and Oxygen

135

It has also been stated that the absorption of solar radiation in the 1100 to 2200 A interval considerably affects the thermal regime of the upper atmospheric layers. For example, computations of solar radiation absorption by oxygen at different altitudes in the atmosphere within the 90 to 125-km range have shown that the variations in the air temperature due to this absorption can reach several tens of degrees per hour. Nitrogen, atomic oxygen, and some other gases intensively absorb radiation in the far-ultraviolet. The portion of the integral solar radiation flux absorbed by these gases is quite small. However, the absorption of solar radiation in the far-ultraviolet region is of really great importance to various processes taking place in the upper atmosphere. To characterize the absorption of ultraviolet solar radiation in the atmosphere, Fig. 3.16 gives the dependence upon wavelength of the height at which the solar radiation intensity decreases by e times, as compared with the corresponding value outside the amosphere. We see ozone, oxygen, and nitrogen as having the dominating influence on the absorption of ultraviolet solar radiation. It should be noted here that the dispersion of points in Fig. 3.16 is caused by the discrepancy between experimental data.

FIG. 3.16 Absorption of ultraviolet solar radiation in the atmosphere. After Watanabe [103].

2. Absorption of Longwave Radiation. Oxygen and nitrogen have no absorption bands in the infrared region of the spectrum. The absorption and emission of long-wave radiation by ozone are usually ignored when computing radiative heat transfer or radiative heat exchange in the at-

136

Radiation Absorption in the Atmosphere

mosphere. Although there are grounds for ignoring this as we shall see later, it may be necessary to account for absorption and emission of longwave radiation by ozone in the solution of certain problems such as formulating the theory of the thermal regime of the stratosphere. The ozone molecule structure is not yet clearly understood. Neither is identification of the infrared ozone band quite simple at present. As follows from data of Kondratyev and Yakovleva, the ozone molecule, like the molecule of carbon dioxide, must be linear. The connection angle of the ozone molecule has been lately reported to equal 118' and the length of the lateral sides 1.278 f 0.003 A. A general idea about the infrared ozone absorption spectrum can be formed from Fig. 3.17, which presents laboratory measurement data to show the dependence of transmission of radiation by ozone on wavelength

FIG. 3.17 Absorption of radiation by ozone in the infrared spectral region (see Kondratyev , Ch. I [2]

in the 0.6- to 16-p region. In the given region are clearly seen absorption bands centered at the following wavelengths: 2.7, 3.28, 3.57, 4.75, 5.75, 9.65, and 14.1 p (the last band has a double maximum at 13.8 and 14.4 p). Later another ozone band near 9.1 p was also found. In atmospheric conditions only a narrow but intensive 9.6-p band, overlapping with 9.1-p band, is sufficiently pronounced for observation. The rest are overlapped

3.4. The Absorption Spectra of Ozone and Oxygen

137

by more intensive water vapor and carbon dioxide bands. In this connection we shall confine our band characteristic to the consideration of the 9.6-,u ozone band. The fine structure of the 9.6-p band has been given a detailed investigation in both natural and laboratory conditions. As regards the quantitative characteristics of the absorption of radiation due to this band, they can be determined from Strong’s [I041 laboratory data. Strong showed that the total long-wave radiation absorption caused by the 9.6-p ozone band can be described by the square root law with a sufficient degree of accuracy. However, Kondratyev and Matroshina [lo51 found that the use of the exponential absorption function is quite satisfactory in this respect. The determination of the mean absorption coefficient for the whole band gives k = 5 cm-1 at pressure p = 1000 mb. The dependence of the absorption coefficient upon pressure may be described by the following simple formula: (3.62)

In the given case the band width is assumed to be 0.5 ,u (9.4 to 9.9 p). A more complex dependence of the integral radiation absorption for the 9.6-p band was found by Epstein et al. [106], who derived the following transmission function : PI

= [1

+ 100.4019+0.7736 log z(p/p0)0.253I-1

(3.63)

where x is the total ozone content in centimeters. A still more complex analytical presentation to characterize the dependence of the integral absorption due to the 9.6-p 0,band on the content of ozone has been derived by Walshaw [78] from laboratory measurements. According to Walshaw, JA,dv

=

dv[l

-

10-pf(p)]

(3.64)

where integration in the left side is performed over all frequencies of the 9.6-,u band. Here, dv is the band width assumed to equal 138 cm-l; p = xE; e, = xEa/p; x is the ozone content in centimeters; a is an empirical constant; E(x) is an arbitrary function x; and the pressure p is in millimeters Hg. It can be shown that in a particular case where 5 = a, f(p) = A(l ( A , B are constants), the formula (3.64) becomes identical with the analogous formula for the statistical model.

+

138

Radiation Absorption in the Atmosphere

From Walshaw’s data, a = 2.11 and the function t ( x ) for various x values may be approximated in the following manner:

I

+ +

1 0 . 1 0 2 5 ~. 1 1 . 6 1 ~’

x

0.984 x 10-0.532;

.f(tp)

is expressed by f(tp)

The values of

(3.65)

5 x 5 0.4

x 2 0.4

0.3 1~ x - O . ’ ~ ;

The function

0.1

5 0.1

= 1.185(1

~ ( t p ) are

+ 734tp)41’2’7(pl)

(3.66)

presented in Table 3.17. TABLE 3.17

Values of the r](p) Function. After Walshaw [78]

log rl

-4.0

1

-3.9

-3.8

-3.7

-3.6

-3.5

-3.4

-3.3

-3.2

-3.1

0.998 0.995 0.993 0.989 0.984 0.982 0.980 0.977 0.977

log

-3.0

7 (Y)

0.977 0.979 0.982 0.989 1.002 1.016 1.030 1.045 1.057 1.069 1.079

-2.9

-2.8

-2.7

-2.6

-2.5

-2.4

-2.3

-2.2

-2.1

-2.0

The mean calculation error in (3.64) is 2.4 percent and the maximum error is 5 percent. It appears that this formula should be considered the most precise of all available for computation of the integral absorption in the 9.6-p ozone band. To describe radiation absorption in individual regions of the 9.6-p band, the statistical band transmission function (3.46) fits satisfactorily. Walshaw and Goody [107], however, contend that there are better results if the transmission function is multiplied by a factor

where

v-----F

a a2+-

3.5. Minor Radiation-Absorbing Components of the Atmosphere

139

and K is an empirical constant depending on frequency and varying from -1 to + l . In Fig. 3.18 is given the dependence of the generalized absorption coefficient (the Elsasser band) for the 9.6- and 9.1-p bands of ozone at different temperature values as derived by Elsasser [20] from experimental data of M. Summerfield. The contours of the 9.6- and 9.1-,.u bands may be approximately described by (3.61). For the 9.6-p band, b = 0.40, a = 14 x lo-, and 5 x lo-, for the shortwave and long-wave branches, respectively. In the case of the 9.1-p band, b = 1.0, a = 40 x lo-, and 14 x lo4.

v,P

FIG. 3.18 Generalized absorption coefficient for the 9.6 ,u and 9.1 ,u bands of ozone. After Elsasser [20].

3.5. General Characteristic of Minor Radiation-Absorbing Components of the Atmosphere The characteristic of the main absorbing atmospheric constituents has been given above. There are also numerous gases in the atmosphere whose absorption bands are weak or very narrow. They are especially numerous in the infrared spectrum, which has thousands of spectral lines of various gases. Since the energetical aspect of their absorption and emission or long-wave radiation is of no practical concern, we shall merely enumerate them. Nitrogen oxides (NO, NzO, N,O,, N,O,) and a number of hydrocarbon combinations (C,H, , CzHs, C,H, , CH,) possess the largest number of lines in the infrared spectrum. Besides these lines were also found the bands of sulfurous gas, heavy water, and other substances. Table 3.18 illustrates the main features of the infrared spectrum of these secondary constituents.

140

Radiation Absorption in the Atmosphere TABLE 3.18

The Absorption Coefficient of Various Atmospheric Cases (cm-'). After Sutherland and Callendar [lo91

Gas

Band, ,u

K, l/cm

Notes

3.0 6.1 10.55

0.09 0.33 0.2

Superposed by H,O

3.3

7.7

0.2 0.3

3.3 5.3 6.9 10.5

0.1 0.2

3.4 6.8 12.2

0.15

3.4 1.0

6.0 1.5

4.5 13.5

0.01

3.0 1.0 14.0

0.2 0.4 3.0

Superposed by H,O

Superposed by H,O

Superposed by H,O

?

5.3 5.7 2.5 13.3

Superposed by H,O

0.33 0.4 0.25

Superposed by H,O Superposed by CO, Superposed by HzO Superposed by N,O Superposed by CO,

3.5 Superposed by H,O

5.7 6.7 7.5 8.6 9.6

0.04 0.02 0.02 0.02

2.4 1.15

0.03

?

Superposed by N,O Superposed by 0, Superposed by H,O Superposed by HzO

3.6. Integral Transmission Function for Thermal Radiation

141

The coefficient k values in the formula A1 =

k

1

+ kw

are given for several gases, the contents of which are assumed to be determined by the gas layer thickness in centimeters at normal pressure and temperature. We see that in some cases the absorption coefficients are quite large. However, in view of the negligible amount of these gases in the atmosphere their absorption of radiation is of no practical importance.+ Table 3.18 is but a fragmentary summary. Besides the bands mentioned above, the following were also found: CH, (3.53, 3.84 p), NO, (2.87, 2.97, 3.57, 3.9, 4.06 p), CO (4.6 p), H D O (3.7 p ) etc.t

3.6. The Integral Transmission Function of the Atmosphere for Thermal Radiation Section 3.5 treated the absorption spectra of the main atmospheric constituents together with the corresponding quantitative characteristics of absorption. These data are a starting point for many investigations related with study of atmospheric radiative transfer. Of special interest here is the determination of the integral shortwave and long-wave radiation absorption. In the case of shortwave radiation the absorption by water predominates. So, as was mentioned above, the integral solar radiation absorption can be determined with the help of (3.54) or (3.55). An essential role in the absorption of longwave radiation belongs also to carbon dioxide. Secs. 3.2 and 3.3 gave a detailed account of the data enabling computation of the absorption of thermal radiation by water vapor and carbon dioxide for different spectral regions. Using these data, it is possible to determine also the integral absorption of thermal radiation by compiling the corresponding integral absorption functions. There is, however, another way of finding this integral function, based on the use of measurement data on the integral absorption of thermal radiation in the atmosphere. In t In the lower atmospheric layers and for wide spectral regions.

t Burch et a/.[59] have derived numerical experimental data on the infrared radiation absorption in the N,O, CO, CH,, bands, which resulted from the investigation of the N,O band at 3.9,4.06,4.5,7.78,8.55,14.5, and 1 7 . 0 ~the ; CH, band at 3 . 3 3 , 6 . 4 5 , 7 . 6 5 ~ ; the CO band at 2.32 and 4.67 p. Both pure gases and mixtures with nitrogen and other gases have been measured. The concentration of N20,CH,, and CO varied within very wide limits.

142

Radiation Absorption in the Atmosphere

summary, the data given above permit obtaining the integral absorption function values in various independent ways, thus securing a more reliable determination. During the past two decades there have been many attempts to find the integral transmission function for diffuse radiation, which would be important in practical applications. Figure 3.19 displays a summary of the results of determination of A F ( w ) by different authors (the scale in Fig. 3.19 is logarithmic). All curves AE(w) in the region of small values

lot

0.9

0.7

0.4

o.3[

0.2

0. 0. r! 0

c!

9-0

0

0 0 0

0

o h -

-

--w, g/crn‘

FIG. 3.19 The integral absorption function. Afrer Kondrafyev and Yelovskikh [110]. (1) after Brooks; (2) after Deacon; (3) after Yelovskih; (4-5) after Kondratyev; (6) after Robinson; (7) after Shehter; (8) after Elsasser.

of w < 0.1 g cm-2 are obtained from the use of spectral absorption characteristics, which in some cases were supplied by the integral absorption measurement data, as in the Elsasser, Deacon, Robinson curves. At large w, the curves Ap(w) are obtained by the “spectral” method (the Elsasser, Kondratyev, Shehter curves) as well as by the “integral” method (the Brooks, Yelovskih, Robinson curves). In the latter case, to determine the absorption function, data on the angular distribution of the atmospheric emission intensity in a cloudless sky were used. It is assumed therefore that the relative emissivity of nonisothermal atmosphere in the direction of the zenith angle 0 equals ge/(o/7c)T4and is equivalent to the relative

3.6. Integral Transmission Function for Thermal Radiation

143

emissivity (absorptivity) of an isothermal atmosphere at the temperature T (T is the air temperature near the earth’s surface) containing a quantity of water vapor equal to

Although this method of “reducing” emissivity is not strictly founded, one may reckon that nevertheless it yields satisfactory results (which can be seen in particular from the comparison of different curves in Fig. 3.19). From Fig. 3.19 it follows that there are considerable discrepancies between the integral transmission function values derived by different authors, which are especially marked at small values of AF(w).However, it should be mentioned that a high exaggeration of the Elsasser and Robinson data at small w, in comparison with the other data, is due to an approximate consideration of the absorption of radiation by carbon dioxide (these authors assumed carbon dioxide to absorb 18.5 percent of the incident radiation at any small thickness of absorbing layers). This makes the values AF(w), obtained by Elsasser and Robinson for small w, certainly unreliable. Quite notable is the difference between the A F ( w ) curves in the region of intermediate absorbing masses of the order - lo-’ g cm-2. This is due to the fact that for the intermediate 10 values, the absorption function is the least reliable. The difference between the four Deacon curves is caused by the different assumptions as regards the carbon dioxide content. In the region of values w 2 1 g cmP2 there is a better coincidence of the absorption functions. Since the absolute differences are comparatively high, the relative divergences stay within the limits of 10 to 15 percent. The problem of choosing the most reliable function from among those of Fig. 3.19 is a difficult one, as there are no simple criteria for such a choice. The Brooks function, however, appears to be the preferred one because, in the region w 5 0.001 g cm-2, it agrees well with data of Deacon. It is noteworthy that this author is very scrupulous in determining AF(w) at small w. to lo-’ g cm-2, the Brooks In the interval of the w values of the order curve is situated almost at the center of the whole variety of curves. At large w > 1 g cm-2, the data of Brooks show good agreement with the values obtained by Yelovskih, Kondratyev, and Elsasser. Later, Kondratyev and Nijlisk [l I I ] reviewed the data on the integral absorption function, taking into account the most reliable information about the quantitative characteristics of the absorption of radiation by water

144

Radiation Absorption in the Atmosphere

vapor, carbon dioxide, and ozone. If the integral absorption functions for diffuse radiation is to be expressed in portions of unity, then the following relation takes place: Pr.(w, u, m) = O.OOI[P(w,u )

+ d P ( w , m)]

where w, u, m are the contents of water vapor, carbon dioxide, and ozone, respectively. P(w, u ) is the function of transmission of the water vapor and carbon dioxide mixture expressed in thousands of unity. d P ( w , m) is a correction for the absorption by ozone (also in thousandths of unity). In Tables 2 and 3 of the Appendix are given the values of the functions P(w, u ) and d P ( w , m) which enable computation of the integral transmission function. Allowance for the dependence of absorption on pressure is made possible by introducing the effective masses of the radiation absorbent, calculated from

where n = 0.5 in the case of water vapor, n = 0.8 for carbon dioxide, and n = 0.2 for ozone. Nijlisk 1141] has investigated the temperature dependence of the integral transmission function and has shown that this dependence is most apparent at low temperatures of the order of 200 to 25O0K. Zuyev’s works [ I 6 1 4 9 1 are devoted to the influence of the source temperature on the value of the integral transmission function and also to various problems of determination of the integral transmission function. 3.7. Absorption Spectroscopy of the Atmosphere as a Method for Investigation of the Atmospheric Composition At present, laboratory absorption spectroscopy is an effective method for quantitative spectral analysis and also for investigation of the physical and chemical properties of liquids and gases. It is natural that this method also be applied for quantitative spectral analysis of the content of radiation absorbents in the atmosphere. It is obvious, however, that such analysis is more complex than in the case of laboratory conditions. This results from both the nonhomogeneity of the atmosphere (the pressure, temperature, and concentration of the absorbent vary along the ray path) and the difficulty of establishing reliable calibration measurements of the atmospheric properties. Certain other factors also affect the performance.

3.7. Absorption Spectroscopy as a Method for Investigation

145

Since the atmosphere contains a great number of gases with intensive absorption bands, it is possible to use the method of absorption spectroscopy for determination of the total content and vertical distribution of various gases. Practically important, however, and more detailed are methods for determining the total content and the vertical distribution of ozone and water vapor. Only these methods will be briefly surveyed below. 1. Ozone. As follows from Sec. 3.4, ozone has intensive absorption bands in the ultraviolet and infrared regions. The method for determination of the total ozone content in the atmosphere from the absorption of solar radiation in the ultraviolet region is well worked out and widely used. The attenuation of solar radiation in the ultraviolet may be presented as

Here S, and So,, are monochromatic solar radiation fluxes at the surface level and outside the atmosphere, respectively; a (cm-') is the coefficient of absorption of ozone in the ozone layer thickness; p is the coefficient of attenuation of solar radiation due to molecular scattering; 6 is the attenuation coefficient of solar radiation on account of aerosol scattering; p, m are the atmospheric masses for the ozone layer and for the whole atmospheric thickness, respectively. In the first approximation m = p = secO,, where 8, is the solar zenith distance. If we assume that m = ,u = sec 8, and 6 = 0 (the latter holds only in high mountains), (3.67) may be rewritten as

Measuring S, in its dependence on sec8, and graphically giving the results, we have a line, the inclination angle y of which is related to the total ozone content in the atmosphere in the following manner: X =

tany-p a

(3.69)

The value of the absorption coefficient a is known from laboratory data and the coefficient p is easily computed theoretically. The use of (3.69) forms the basis of the so-called method of linear analysis for determination of the total ozone content. In principle, this method may be used for any wavelength of the ozone absorption bands (in the interval where S , f 0). Practically, however, the Huggins band is the most convenient. Other bands are less convenient because they have either

146

Radiation Absorption in the Atmosphere

too weak (the Chappuis bands) or too intensive (the Hartley band) absorption. Since, as a rule, 6 # 0, the method of linear analysis has not found a wide practical field. The so-called method of two wavelengths is more effective which allows us to exclude the effect of aerosol attenuation of solar radiation, thus resulting in a more general and more satisfactory procedure. Choose in the solar radiation spectrum the two wavelengths A and A', where A' > A and the absorption for A is far greater than for .'2 Suppose that d1 = dl,. This supposition may be considered valid if the wavelengths A and A' are sufficiently close. On the basis of (3.67) we may have, for the two considered wavelengths,

+ 6)m log s'a= log SiJ - a'xp - (p + 6)m log SA

= log

S0,J- axp - (B

Hence

or, introducing the notation SO,1. = Lo, log 7 SO#a

sa = L log 7 sa

we have X =

Lo - L - @-B')m (a - a')p (a - a')p

(3.70)

In this formula only the value L has to be measured; the others have already been determined. If the often used A = 3110 A and A = 3300 A are chosen, instead of (3.70) we have X =

Lo - L - 0.085 1.17

(3.71)

The preceding method is the most common in ozonometric practice. In carrying out this method, the measurements of monochromatic solar radiation fluxes are made with photoelectric spectrophotometers. A more perfect model for this purpose was developed by Dobson. In network use are common photoelectric ozonometers with filters that serve for the

3.7. Absorption Spectroscopy as a Method for Investigation

147

general ozone content measurement. A simple construction of this kind is presented, for example, by Rodionov, Osherovich, and Bezverhny (see [1121). There are other methods but they are not in wide use. We refer the interested reader to the monographs by Prokofyeva [loll and Gushchin [112, 1131, which also give information about the methods applied in determination of the vertical ozone concentration distribution in the atmosphere from measurement data on the absorption of solar radiation in the ultraviolet range. Unfortunately these methods are not yet sufficiently developed. The most reliable of them are direct methods for measuring the vertical ozone distribution with the help of sounding ballons and rockets. Recent years show that much attention has been paid to working out the method for determination of the total content and vertical distribution of the ozone concentration from measurements of the solar radiation absorption in the 9.6-,u band and from the thermal emission of the ozone layer. Such data are also used for determination of the mean altitude and temperature of the ozone layer. Technically, however, the infrared ozone spectroscopy is more difficult and less accessible than measurements in the ultraviolet region [ 167, 1681. 2. Wuter Vapor. It has been shown above that selective absorption of

direct solar radiation in the atmosphere is an important factor in attenuation of solar radiation. There is considerable absorption of solar radiation by water vapor. Even measurements of the total direct solar radiation flux show that it decreases noticeably with the increase of absolute humidity as the result of the increase of the absorption of solar radiation by water vapor. The influence of water vapor selective absorption is even more pronounced if observations are conducted with filters, which isolate the infrared portion of the solar spectrum; or, the effect can be even more observed with the help of spectral instruments. It has already been mentioned that the connection between the value of integral solar radiation flux and the total water vapor content in the atmosphere may be used in working out methods for determination of the latter on the basis of actinometric measurements. Let us now consider this in more detail, for it is of great interest for actinometry and meteorology in general. Since filter measurements in this case are far from perfect, we shall give only a characteristic of spectroscopic method for determination of the total water vapor content in the atmosphere (w,). This method is based on the use of the dependence of the infrared solar radiation absorption upon the water vapor content in the atmosphere. The first to describe

148

Radiation Absorption in the Atmosphere

this method was Fowle (1912-1913). Later, many other studies were devoted to this problem. We sthall mention some of them. Herzing’s work [I141 for determination of the water vapor content in the atmosphere used observations performed with the help of a double quartz monochromator. The monochromator had a radiation transmission region 0.28 to 3.0 p. A thermopile connected to a sensitive mirror galvanometer, set on photoregistration, served as a radiation detector. As characteristic values of absorption of radiation of different wavelengths by water vapor were used values of the relative absorption band-depth s determined from the relation s=-

Sa

so.a

x 100

(3.72)

Here S, is the monochromatic flux of direct solar radiation corresponding to the band maximum. In view of the finite slit width, a flux for the spectral interval of finite width d l was measured rather than the monochromatic flux. The flux So,a (more precisely, So,,, = CS,,,) was measured in the region between the bands where radiation absorption could be ignored. The value So,Acorresponding to the band maximum wavelength can be found by interpolating values So,,measured in the region between the bands. The measurement of values determining the band depth occupies only 1 to 2 min. To determine the water vapor content from the depth of the absorption band, it is necessary to plot first a calibration curve that characterizes the dependence of s on mw,. Such curves were built by Herzing for all the main water-vapor absorption bands eat, 9,y , The w, curve, relating to the p t band, was used as the basic one. Simultaneous measurements of s for different absorption bands at different solar heights and aircraft soundings of the atmosphere, on the basis of which was determined the variation of specific humidity q with height, were carried out with the purpose of plotting the calibration curves. As the total water vapor content in the vertical atmospheric column is

a.

00

ww

=

Io

ew

dz

where Sw is the absolute humidity and mixing ratio q = (e,/e), for which e is the air density, to calculate w, from the specific humidity distribution the following formula may be used (3.73)

3.7. Absorption Spectroscopy as a Method for Investigation

149

where p and p o are the pressure at the height z and at the surface level, respectively. The application of this formula for computation of w, demands, however, knowledge of the distribution of specific humidity up to high elevations. Aircraft sounding, meanwhile, is confined to small altitudes. The value w for the atmospheric column, with the upper boundary at the sounding top limit, was thus determined immediately from measurement data. The amount of water vapor above the peak point of sounding was evaluated from extrapolation. In this way the calibration curves of Fig. 3.20 were obtained. The presence of such curves allows determining the value mw, directly from the absorption band depth measurement. Having computed the atmospheric mass corresponding to the moment of observation, we obtain the total water

m w fcm H20 0

FIG. 3.20

Variation of the depth of @, Y,f2 absorption bands in dependence on water vapor content. After Herzing [114].

(1) single values; (2) means from two single values; (3) means from three single values; (4) the same from four values; (5) the same from five to seven values, (6) from eight to

ten values; (7) from ten values.

150

Radiation Absorption in the Atmosphere

vapor content in the vertical column of atmosphere. According to Herzing, the determination of w, in this way is performed with an accuracy of about 4 percent. It must be stressed, though, that the curves of Fig. 3.20 are not universal, as they are dependent on the properties of the applied spectral instruments. It is beyond doubt that, to a certain degree, the character of the calibration curves must also depend on the state of atmospheric transparency. An important consideration is the dependence on elevation of the observational point above sea level, which is caused by the dependence of absorption on pressure. A somewhat different (as regards detail) method for spectroscopic measurements of the total water vapor content in the atmosphere has been suggested by Hand [115]. In this method the radiation receiver is a thermocouple, one junction of which is illuminated by radiation of the wavelength 1 = 0.935. (band maximum eat), and the other junction of which is exposed to radiation of the wavelength il = 0.956 ,u (the area between the bands). The readings of the galvanometer attached to the thermocouple are thus directly proportional to the band depth. In the given case a single measurement suffices for determination of the water vapor content at the given moment from the corresponding calibration curve. Liquid water, when present in the atmosphere, may lead to an error in these measurements, so the spectroscopic method is allowed only in the case where the absorption of solar radiation is caused by water vapor exclusively. Surface measurements of the water vapor content in the atmosphere were given further development in Toropova et al. [120-1241. In recent years Hand’s method has been used for aircraft and balloon measurements [I 16-1 19, 125-1321. Neporent [I 16, 1171 headed the work on construction of instruments (which have proved to be perfect) for measuring the water vapor content in the upper troposphere and stratosphere with the help of an automatic balloon. Neporent and collaborators have built a vacuum monochromator with a diffraction grating that isolates the wavelengths 1.40 nd 1.88 p in the absorption bands and three wavelengths (1.24, 1.50, and 2.23 p ) outside the bands. As the receiver of radiation a cooled photoresistance of sulfide lead is used. This installation has made possible the investigation of the water vapor content in different atmospheric layers up to 17.5 km. Still greater heights (about 30 km) have been reached in analogous investigations by Gates et al. [125-1271, Murcray et al. [128-1301, and Kondratyev et al. [132]. As the latter mention [132], the influence of the contamination of the balloon container by water vapor played an important role in some previous investigations.

References

151

In a number of works a spectroscopic method for the absolute humidity determination has been proposed. This method is based on the light absorption measurement in the most intensive bands of water vapor. Hammermesh et al. [133], for example, have used a spectrometer with a diffraction grating as a spectroscopic hygrometer. The light from a tungsten incandescent lamp was passed through a certain air volume and was then directed to the spectrometer employed for spectral resolution of light. A vacuum thermocouple in contact with a sensitive galvanometer was a radiation receiver. The radiation intensity measurement was carried out at wavelengths 1 = 1.380 p and 1 = 1.250 p after passing through the measurement volume. The first of these wavelengths corresponds to the center of the band y , and the second lies in the region of a very narrow interval where radiation absorption is insignificant. It is easily understandable that, from the value of the light intensity ratio, Zl.380/11.250, by characterizing the radiation transmission value of the wavelength 1 = 1.380 p, one can judge the water vapor content on the ray path. If the spectroscopic hygrometer has been calibrated beforehand (that is, the curve of the dependence of the ratio of galvanometer’s deviations on the water vapor content have been derived), then on the basis of the calibration curve it is possible to determine the water vapor content on the ray path immediately from the ratio N,,380/N,.,,o.With the length of the ray path being known, it is easy to determine the mean absolute humidity of the measured volume. The hygrometer enables determination of the mean absolute humidity of an arbitrary air volume with a sufficient degree of accuracy. Foskett et al. [138] have realized a similar installation, which is different only in that the radiation receiver is not a thermocouple but a photoelement connected to a multiplier. Thus the instrument’s sensitivity is considerably increased. With the help of such a photoelectric hygrometer, short-period variations in absolute humidity from 0.14 to 0.56 g cm-2 have been measured. Analogous installations have been put in operation by Wood [134, 1351, Bocharov [136], and Yelaghina [137]. The spectroscopic hygrometers described are indeed complex instruments, and therefore their application has been so far confined to laboratory investigations.

REFERENCES 1. Elyashevich, M. A. (1962). “Atomic and Molecular Spectroscopy.” Fizmatgiz, Moscow. 2. Korolev, F. A. (1953). “Spectroscopy of High Resolving Power.” Gostekhizdat, Moscow.

152

Radiation Absorption in the Atmosphere

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97. Shekhter, F. N. (1950). On the calculation of radiant heat fluxes in the atmosphere. Proc. Main Geophys. Obs. NO. 22 (84). 98. Kondratyev, K. Ya., and Nedovesova, L. I. (1958). On thermal emission of carbon dioxide in the atmosphere. Proc. Acad. Sci. USSR, Geophys. Sect. (English Transl.), No. 12. 99. Kondratyev, K. Ya., and Nijlisk, H. J. (1963). On the problem of carbon dioxide thermal radiation in the atmosphere. Probl. atmospheric Phys. No. 2. Leningrad Univ. 100. Kondratyev, K. Ya., and Nijlisk H. J. (1960). On the question of carbon dioxide heat radiation in the atmosphere. Geofs. Pura Appl. 46. 101. Prokofyeva, I. A. (1951). “Atmospheric Ozone.” Acad. Sci. U.S.S.R., MoscowLeningrad. 102. Kalitin, N. N. (1938). “Actinometry.” Gidrometizdat, Leningrad. 103. Watanabe, K. (1958). Ultraviolet absorption processes in the upper atmosphere. Advan. Geophys. 5. 104. Strong, J. (1941). On a new method of measuring the mean height of the ozone layer in the atmosphere. J . Franklin Inst. 231,No. 2. 105. Kondratyev, K. Ya., and Matroshina T. D. (1953). On the influence. of long-wave ozone emission on the radiation balance of the earth’s surface. and atmosphere. Proc. Main Geophys. 0 6 s . NO. 41 (103). 106. Epstein, E. S., Osterberg, C. and Adel, A. (1956). A new method for the determination of the vertical distribution of ozone from a ground station. J . Meteorol. 13, No. 4. 107. Walshaw, C. D., and Goody, R. M. (1954). Absorption by the 9.6 p band of ozone. Proc. Toronto Meteorol. Con$ Toronto. 108. Walshaw, C. D., and Goody, R. M. (1956). An experimental investigation of the 9.6 p band of ozone in the solar spectrum. Quart. J . Roy. Meteorol. SOC.82,No. 352. 109. Sutherland, G., and Callendar, G. (1943). The infrared spectra of atmospheric gases other than water vapor. Rept. Progr. Phys. 90. 110. Kondratyev, K. Ya., and Yelovskikh, M. P. (1956). On the atmospheric absorption function for thermal emission. Proc. Leningrad Univ., Ser. Phys. No. 9. 111. Kondratyev, K. Ya., and Nijlisk, H. J. (1961). A new radiation chart. Geofis. Puru Appl. 49. 112. Gushchin, G. P. (1963). “Investigation of Atmospheric Ozone.” Gidrometeoizdat, Leningrad. 113. Gushchin, G. P. (1964). “Ozone and the Aerosynoptic Conditions in the Atmosphere.” Gidrometeozidat, Leningrad. 114. Herzing, F. (1937). Die Bestimmung des Wasserdampfgehaltes der Armosphare aus Registrierungen des Sonnenspektrums. Beitr. Geophysik 49, No. 1-2. 115. Hand, J. F. (1940). An instrument for the spectroscopic determination of precipitable atmospheric water vapor and its calibration. Monthly Weather Rev. 68, NO. 4. 116. Neporent, B. S., Dmitrievsky, 0. D., Zaitzev, G. A., and Kiselova, M. S . (1956). instrument for the direct high-altitude determination of the concentration of ozone and water vapor in the atmosphere. Opt. Mech. Ind. No. 2. 117. Neporent, B. S., Belov, V. F., Dmitrievsky, 0. D., Zaitzev, G. A., Kastrov, V. G., Kiselova, M. S., Kudriavtzeva, L. A., and Patalahin, I. V. (1957). An experiment of direct measurement of the high-altitude distribution of atmospheric humidity according to the spectral method. Proc. Acad. Sci. USSR, Geophys. Sect. (English Transl.) No. 4.

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118. Kiselova, M. S., Neporent, B. S., and Fursenkov, V. A. (1959). Spectroscopic determination of air humidity in the upper atmospheric layers. Opt. Spectr. 6,No. 6. 119. Houghton, J. T., and Seeley, J. S. (1960). Spectroscopic observations of the water vapor content of the stratosphere. Quart. J. Roy. Meteorol. SOC.86,No. 369. 120. Toropova, T. L. (1955). Determination of the water vapor content in the atmosphere. Proc. Acad. Sci. Kazakh Rept., Astrophys. Insr. 1, Nos. 1-2. 121. Makarova, E. L., and Sitnik, G. F. (1963). Some optical properties of the earth’s atmosphere and its water vapor content as shown by observations at Kuchino and during the high-altitude expedition of the Steinberg Astronomical Inst. Bull. State Astron. Inst. No. 126. 122. Shifrin, K. S., and Avaste, 0. A. (1963). On the spectral determination of the total water vapor amount in a vertical atmospheric column. Invest. Atmospheric Phys. No. 4. Tartu. 123. Kamiyama, K., Eguchi, M., Yatabe, Y., and Moriguchi, M. (1951). Determination of precipitable water in the air by near infrared spectroscopy. Geophys. Mag. 23, No. 1. 124. Vassy, A., and Vassy, E. (1950). Recherches sur l’epaisseur d‘eau condensable de I’atmosphere. J. Sci. MeteoroI. 2, No. 7. 125. Gates, D. M. (1956). Infrared determination of precipitable water vapor in a vertical column of the earth’s atmosphere. Meteorol. 13,No. 4. 126. Gates, D. M., Murcray, D. G., Shaw, C. C. and Herbold, R. J. (1958). Near infrared solar radiation measurements by balloon to an altitude of 100,000 feet. J. Opt. Soc. Am. 48, No. 12. 127. Gates, D. M. (1960). Near infrared atmospheric transmission to solar radiation. J. Opt. SOC.Am. 50, No. 12. 128. Murcray, D., Brooks, J., Murcray, F., and Shaw, C. (1958). High altitude infrared studies of the atmosphere. J. Geophys. Res. 63,No. 2. 129. Murcray, D. G., Murcray, F. H., Williams, W. J., and Leslie, F. E. (1960). Water vapor distribution above 90,000 feet. J. Geophys. Res. 65, No. 1 1 . 130. Murcray, D. G., Murcray, F. H., and Williams, W. J. (1962). Distribution of water vapor in the atmosphere as determined from infrared absorption measurements. 3. Geophys. Res. 67, No. 2. 131. Dobson, G. M. B., Brewer, A. W., and Houghton, J. T. (1962). The humidity of the stratosphere. J. Geophys. Res. 67, No. 2. 132. Kondratyev, K. Ya., Badinov, I. Y., Ashcheulov, C. V., and Andreyev, S. D. (1965). Instruments for investigation of infrared absorption spectra and atmospheric emission. Proc. Acad. Sci. USSR, Ser. Atmospheric. Phys. Oceanol. No. 2. 133. Hammermesh, B., Reines, F., and Korff, S. A. (1947). A photoelectric hygrometer. Bull. Am. Meteorol. Soc. 28, No. 7. 134. Wood, R. C. (1958). Improved infarared absorption spectra hygrometer. Bull. Am. Meteorol. SOC.29, No. 1. 135. Wood, R. C. (1959). The infrared hygrometer as a potential meteorological aid. Bull. Am. Meteorol. SOC.40, No. 6. 136. Bocharov, E. I. (1959). Radiation absorption by humid air. The optical hygrometer. Proc. Acad. Sci. USSR,Geophys. Sect. (English Transl.) No. 1. 137. Yelaghina, L. G. (1962). An optical instrument for measuring turbulent humidity pulsations. Proc. Acad. Sci. USSR,Geophys. Sect. (English Transl.) No. 8.

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138. Foskett, L. W., Foster, N. B., Thishstun, W. R., and Wood, R. C. (1953). Infrared absorption hygrometer. Monthly Weather Rev. 81, No. 9. 139. Shaw, J. H. (1963). Empirical methods for computing the integrated absorptances of infrared band of atmospheric gases at nonuniform pressures. Appl. Opt. 2, No. 6. 140. Plass, G. N. (1964). The influence of numerous low-intensity spectral lines on band absorptance. Appl. Opt. 3, No. 7. 141. Nijlisk, H. J. (1963). On the dependence of the atmospheric transmission function on temperature. Invest. Atmospheric Phys. No. 4. Tartu. 142. Burch, D. E., Singleton, E. B., and Williams, D. (1962). Absorption line broadening in the infrared. Appl. Opt. 1, No. 3. 143. Burch, D. E., and Williams, D. (1964). Test of theoretical absorption band model approximations. Appl. Opt. 3, No. 1. 144. Zuyev, V. E., Eliatberg, M. E., and Safonova, G. A. (1960). Transparency of small atmospheric thicknesses in the region 1-13 p. Bull. High Schools, Phys. No. 5. 145. Zuyev, V. E. (1961). The integral absorption of two overlapping bands. Bull. High Schools, Phys. No. 3. 146. Zuyev, V. E. (1961). The integral absorption function of long-wave radiation in the atmosphere. Bull. High Schools, Phys. No. 3. 147. Zuyev, V. E. (1962). On the role of the source temperature in the investigation of the integral absorption function of long-wave radiation in the boundary layer of the atmosphere. Bull. High Schools, Phys. No. 6. 148. Zuyev, V. E. (1962). The integral absorption function of long-wave radiation in the atmosphere. IV. Bull. High. Schools, Phys. No. 3. 149. Zuyev, V. E., and Tvorogov, S. D. (1966). “Atmospheric transparency for visible and infrared rays.” Sovetskoye Radio, Moscow. 150. Brounstein, A. M. and Kazakova, K. P. (1964). Experimental investigation of the infrared transmission function 111. Results of measurements at mean air humidities. Actinom. Atmospheric Opt. Gidrometizdat. 151. Zuyev, V. E., Nesmelova, L. I., Sapozhnikova, S. A., and Tvorogov, S. D., (1964). Calculations of atmospheric transparency for the infrared emission. Actinom. Atmospheric Opt. Gidrometizdat. 152. Antipov, B. A., Genin, V. N., Zuyev, V. E., Kohanenko, P. N., Sedacheva, T. P., and Sonchik, V. K. (1964). Transmission functions of long-wave radiation in the boundary layer of the atmosphere. Actinom. Atmospheric Opt. Gidrometizdat. 153. Antipov, B. A., Zuyev, V. E., Kabanov, M. V., Kohanenko, P. N., and Nekrasov, Y. I. (1964). New instruments for measuring atmospheric transparency in the infrared region. Actinom. Atmospheric Opt. Gidrometizdat. 154. Zuyev, V. E. und Tvorogov, S. D. (1965). Absorption function of a single line in a finite interval. Bull. High Schools, Phys. No. 1. 155. Zuyev, V. E., and Tvorogov, S. D. (1965) On the calculation of absorption function for overlapping lines. Bull. High Schools, Phys. No. 2. 156. Kondratyev, K. Ya., and Timofeyev, Y. M. (1965). Absorption function for the rotational band of water vapor. Proc. Acad. Sci. USSR,Ser. Phys. Atmosphere Ocean No. 12. 157. Mayot, M., and Vigroux, E. (1965). Application de I’approximation Curtis-Godson a I’ozone atmospherique. Ann. Geophys. 21,No. 1. 158. Plass, G. N. (1965). Equivalent width due to two overlapping lines. Opt. SOC.Am. 55, No. 1.

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SCATTERING OF RADIATION IN THE ATMOSPHERE

Absorption and scattering of radiation are the main processes managing radiative transfer in the atmosphere. In the case of long-wave radiation the medium's emission should also be taken into account. However, as already shown in Chapter 1, application of Kirchhoff's law simplifies the solution of the problem of determination of the medium's thermal emission, provided the quantitative characteristics of radiation absorption are known. After having studied the absorption of radiation in the atmosphere, let us now turn to the basic regularities of scattering of radiation.

4.1. The Solar Ray Path in the Atmosphere The attenuation of a flux of direct solar radiation S = J," S, dil during its passage through the atmosphere can be described by the following formula :

Where S , = flux of solar radiation at a level z in the atmosphere, So = J," So,Adil = flux of solar radiation outside the atmosphere, a, = kA (F, = mass coefficient of attenuation of solar radiation, e = density of air, and ds = element of the ray path. The quantity JZw a,@ds = @,(f3,) is usually called the optical thickness of the atmosphere for rays of wavelength I at the zenith solar distance €Jo(the quantity a,@is known as optical density). For determination of the total attenuation of solar radiation from (4.1), it is necessary to calculate the optical thickness O , ( Q along the ray path. To perform similar calculations for each individual case would seem unnecessary. It is more con-

+

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162

Scattering of Radiation in the Atmosphere

venient to calculate the optical thickness in the direction of the vertical @,(O) and compile a table of values of the function

We shall then be able to express inclined optical thickness through optical thickness in the direction of the vertical. This method is used in actinometry for calculation of attenuation of direct solar radiation in the atmosphere. It is quite obvious that this method can be applied to calculation of attenuation in the atmosphere of light of various kinds (direct, diffuse radiation, moonlight, starlight, artificial sources). The value of the ratio

within the spectrum of shortwave radiation is almost independent of wavelength; it is therefore possible to introduce in place of the above ratio, the following function for shortwave radiation as a whole:

The function determined by the relation (4.2) is called the atmospheric mass rn(0,). The strict mathematical determination of this function was given by Staude [l]. The term atmospheric mass is the most common in actinometry. It should be emphasized in this connection that the concept of atmospheric mass has nothing in common with the general idea of mass. Atmospheric mass in the given case is a dimensionless quantity indicating by how many times the optical thickness in the direction of inclination exceeds the optical thickness in the direction of the vertical. Consider now how the atmospheric mass can be calculated. For actinometry it is of interest to calculate m for different 8 a t the surface level. Therefore we begin with this particular case. It can be shown that the atmospheric mass, expressed by the ratio of optical thicknesses

is at the same time the ratio of the light-ray path in the direction of inclination to its path in the vertical direction. This conclusion will become clear if, as noted by Staude [2], one remembers that the attenuation of light in any medium in different directions can be formally substituted by

4.1. The Solar Ray Path in the Atmosphere

163

the attenuation in an infinitely thin homogeneous layer of equal thickness situated domelike over the point of observation. The calculation thus can be reduced to calculation of the length of the light ray in the atmosphere in the direction of inclination. Consider (Fig. 4.1) an element ds of the path of a light ray passing through the atmosphere from a certain light source (for example sun,

0

FIG. 4.1 The derivation of the expression for atmospheric mass.

star, moon). Let z be the height of the considered path element above the earth’s surface and q~ the angle of incidence of the ray on the layer dz. We have dz ds = (4.3) cos qI On the other hand, we can write the following so-called light-ray equation 131: rn, sin 8 = (r z)n sin 9, (4.4)

+

where r is the earth’s radius; n, no are the indices of air refraction at a height and at the surface level, respectively; 8 is the visible zenith distance of the light source. On the basis of (4.3) and (4.4) we write dz

(4.5)

164

Scattering of Radiation in the Atmosphere

Now denote the air density at the height z and at the surface level by e and eo, respectively. Multiplying both parts of (4.5) by the value @ / P o and integrating over all atmospheric layers, we obtain

I“ eo ds 0

=

(eleo) ds

m 0

(4.6)

r

The integral

where the function dz

=

P =H 0 eog

is the scale height H (the height the atmosphere would have, given its density to be invariable with height and equal to eo). Taking account of this fact, we have, from (4.9,

It is known that the refraction index n is the function of air density and that this dependence is expressed as follows:

n2 = 1

+ 2a- eeo

where the constant a = no - 1 depends, generally speaking, upon wavelength and averages 0.0002932 for white light. Note here that just this dependence of a, and consequently of h, upon wavelength makes necessary the introduction of “monochromatic” masses. Since, however, this dependence is not essential, it will be ignored in the computation for white light, that is, for the entire shortwave radiation region. According to (4.8), n dn -= d($) a

From the main equation of atmospheric statics we also have

4.1. The Solar Ray Path in the Atmosphere

165

provided the atmosphere is isothermal. Taking into consideration these relations and also that H =p,/e,,g, we have in place of (4.7), n dn

m = - J :1 a

(.+zr

1/1-

(4.9) $)‘sin28

The formula (4.9) was first derived by Laplace. After finding this formula it is possible to express atmospheric mass through the value of astronomical refraction (abbreviated here as “refr.”). Computations of astronomical refraction give the following relation for this value :

Refr.

=

s:”

(+)sin 8 dn

(*)

(4.10)

From comparison of (4.9) with (4. lo), the following differential equation is obtained: dm Since [(r mass :

21

1

an, sin 8

+

r

‘n3 d(refr.)

+ z ) / r ] n 3= 1, we obtain the Laplace formula for atmospheric m=-- 1 an,

refr. sin8

(4.1 1)

When deducing (4.1 1) the atmosphere is assumed to be isothermal. In reality the vertical temperature distribution in the atmosphere is far from being isothermal, and accounting for this circumstance affects the results of computation of m. The dependence of m on pressure should also be taken into account. More precise formulas for m, which took into account the atmospheric structure, were derived by Bemporad and Staude [4-71. The influence of atmospheric structure appears to be notable only in computing m for large zenith distances, the calculation of rn being possibly confined to m for a certain standard atmosphere only. Table 4 of the Appendix gives computed atmospheric masses for a standard atmosphere (p = 1000 mb; to = OOC).

8

The numerical results of computation of m show sufficient accuracy at we take (4.12) m = m, = sec8

< 60’ if

166

Scattering of Radiation in the Atmosphere

The formula (4.12) is an exact expression for (4.12) in the case where the atmosphere is a flat layer with the absence of refraction. At the same time it is clear that the curvature of the amtosphere and the refraction will least affect the value of m at small zenith distances. This explains why (4.12) follows from (4.11) as a finite relation valid for small 8. At 6C0 < 8 < 80' the atmospheric curvature must be taken into account, whereas the effect of deflection of the light ray due to refraction may be neglected. Elementary geometric consideration (see Fig. 4.1) leads in this case to the following relation:

m

= m, = AC -

AB

2/r2 cos28 + 2rH + H 2 H

-

r cos 8

(4.13)

For 8 > 60' but not too close to 90°, the following empirical formula is quite satisfactory: 2.8 m=m2sec8(4.14) (90 - e y

where 8 is expressed in degrees of the arc. The above calculations of atmospheric mass refer to the case where the light from a source outside the atmosphere is observed at the earth's surface, at which normal values of pressure p a = 1000 mb and temperature To = 273OK. If observations are conducted in mountain conditions or in the free atmosphere, it is natural that the pressure a t the point of observation may differ considerably from 1000 mb. Noticeable pressure variations are observed at sea level. It is perfectly clear that pressure variation will not affect the value of m determined according to (4.2) because O(8) and O(0) are approximately directly proportional to the value of pressure at the observation point. This fact follows from the optical atmospheric thickness 0 = J," a@ds being approximately proportional to the physical mass J," e ds of an atmospheric column of unit section, which in turn is proportional to the pressure at the level z. As regards the influence of temperature variation on the value m, it has been stated above that it develops only at significant zenith distances. In actinometric practice, observations at such large distances are performed solely in exceptional cases. The dependence of m upon temperature may therefore be neglected. In actinometry, along with the determination of atmospheric mass from (4.2), the following definition is also used: (4.15)

4.1. The Solar Ray Path in the Atmosphere

167

Jr

where Oo(0)= a@dz is the optical thickness in the vertical direction determined for p o = 1000 mb. When defining the concept of atmospheric mass in the above way, pressure corrections must be introduced into the values of m‘ if the pressure is not 1000 mb. One can see that m‘ is related with p in the following manner:

m’=-m P

(4.16)

Po

This relation follows from the formula @(O)

=

( p / p o ) @,(O)

where

O0(@ = J,“ a@ds is the optical thickness in the direction of inclination for p a = 1000 mb. As was noted by Averkiev [8] in the consideration of the problem of atmospheric transparency, it is more natural to use the concept of “relative” mass than “absolute” mass m’. Actually, when introducing the absolute mass, we always compare the atmospheric transparency in a definite direction (optical thickness in this direction) with the transparency of a certain standard atmosphere in the vertical direction (with the standard atmosphere optical thickness in the vertical direction). In this case it turns out that if the pressure is different at two observation points, then for these points equal values of atmospheric mass correspond to different solar zenith distances. That is why at the same factual transparency of the atmosphere at the considered points in a certain definite direction, a different transparency at equal mass values is found. A similar result is, of course, illogical, since it is most natural to expect equally transparent atmosphere in two different cases when the factual transparency is the same at a definite value of atmospheric mass to which corresponds a quite definite solar zenith distance. This, however, will take place only in the case when the concept of “relative”, not “absolute,” mass is used. It should also be noted that (4.16)and the relation @ ( O ) = (p/p0)@,(O) are strictly valid only in the case where attenuation of radiation is caused by molecular scattering alone. Since in conditions of real atmosphere this does not strictly hold, judicious use of the concept of “relative” mass is indicated. However, in practice, most cases d o not require distinction of these concepts. The matter is to be considered primarily in the processing of high mountain observational data on direct solar radiation. The calculations mentioned above are made on the assumption that light is being attenuated along the entire ray path through the atmosphere. It is known, however, that the atmosphere has certain layers (ozone, aerosols) of a higher radiation-attenuating activity. In this connection it is important to find relations, analogous to those mentioned above, for

168

Scattering of Radiation in the Atmosphere

the optical thicknesses of a certain spherical layer located at some definite height above the earth's surface. Multiplying both parts of equation (4.5) by e', the concentration of radiation-attenuating substance, and by integrating over all atmospheric layers and taking into account

we obtain for the mass of this substance p(8) in the direction constituting the angle 8 relative to the vertical:

where x = J," e' dz is the total content of radiation-attenuating matter in the atmosphere. Since e' # 0 only within a certain layer, the integral in (4.17) extends over a limited height interval only. As before, for zenith distances t9 < 60°, the formula (4.17) is reduced to a very simple dependence: p(8) = p@)

=

sec 8

(4.18)

For 60' < 8 < SOo, the deflection (refraction) of the light ray in the atmosphere may be neglected, but it is necessary to take into account the atmospheric curvature. The obvious geometric consideration leads to the following result: A.

I

where H I is the height of the lower boundary of the considered layer. Rodionov [9] has computed the values of p(8) at different values of the argument for the case of the absorption of solar radiation by an ozone layer, and has presented values of p(8) and m(8). Figure 4.2 presents Rodionov's curves p(8) and m(O), computed from (4.17) for curve 4, (4.19) for curve 3, (4.18) for curve 1, and curve m(8) for curve 2, plotted after

4.2. Scattering of Radiation (General Considerations)

169

Bemporad’s tables. Figure 4.2 gives a spectacular analysis of the relation between functions and p(0) and m(0).

l9

FIG. 4.2 Dependence of atmospheric masses m and p upon the zenith distance.

4.2. Scattering of Radiation (General Considerations) Scattering of radiation appears when the medium is not homogeneous. More precisely, scattering takes place in the presence of the spatial inhomogeneity of the dielectric constant. In atmospheric conditions, the main factors that determine scattering are air density fluctuations and aerosol particles (water droplets, dust particles, etc.). The scattering due to air density fluctuations is commonly named molecular scattering, for in this case the dimensions of the refraction index inhomogeneities that provoke scattering are of the same order of value as the dimensions of molecules. The fact that the dimensions of scattering inhomogeneities (particles) are small in comparison with the wavelength is a characteristic peculiarity of this type of scattering. The scattering in which the dimensions of the scat-

170

Scattering of Radiation in the Atmosphere

tering medium are notably smaller than the wavelength is called Rayleigh. Not only molecular scattering but also scattering of ultrashort radio waves by clouds (provided the wavelengths are not too small) may be considered to be Rayleigh scattering. Let us now clarify what quantitative characteristics may be employed to describe scattering of radiation. Imagine that a parallel beam of rays falls on a volume element dv of a scattering medium possessing physical properties different from the properties of the environment. Then, besides the beam of rays that has already passed through the given volume, scattered radiation that leaves the considered volume element in all directions will also be observed. The expression for the quantity of radiant energy, dF,(p,), scattered by the volume element dv in the direction constituting an angle n - p,, with the incident ray within the solid angle d o , may be presented in the following manner: dF2 = ( Z A S, ~ dv d o (4.20) where S, is the quantity of radiant energy incident on the volume element dv and a l p is the volume coefficient of radiation scattering in the direction p,. The scattering coefficient u , p shows what portion of radiation S, is scattered by unit volume in the direction p, within unit solid angle. The value a,? may be presented as

Here y(p,) is the scattering function that characterizes the scattered lightintensity distribution in the direction p,; 1c is a parameter depending upon the medium’s optical properties. It is evident that this value must be proportional to the number of scattering elements per unit volume. Taking account of (4.21), the formula (4.20) can be rewritten as follows: = XY(p,)SA

dv dw

(4.22)

On the basis of (4.22) we find that the radiant flux FAscattered over all by the volume element dv, will be determined as FA= 2nxASA dv Given dv

=

y(p,) sin p, dp,

(4.23)

y(p,) sin p, dp,

(4.24)

1, the value

Fa = 2nx, aA = -

s,

is a volume-scattering coefficient.

J:

4.3. Rayleigb Scattering

171

The determination of a A is of the greatest interest to us. As seen from (4.24), for determination of the scattering coefficient it is necessary to know To have an the value of the parameter x1 and the scattering function ~(9). idea about these values, the theory of light scattering in the atmosphere should be learned and observational data on atmospheric radiation scattering should be studied. Before proceeding to this, it should be noted that in the process of scattering of radiation, a change of the state of polarization always takes place. Since the incident radiation is unpolarized, the scattered shows polarization, and in the scattering of polarized radiation a change of the state of polarization is observed. In this connection, scattering theory takes an important place in the investigation of the characteristics of scattered light polarization (the degree of polarization, the position of the plane, and the ellipticity of polarization). At present the theory of radiative transfer in the atmosphere that takes account of polarization is only in the stage of primary development. We shall therefore give only a brief treatment of the problem of radiation polarization in scattering. The scattering on an individual particle or “cell” of the dielectric constant inhomogeneity is an elementary process of scattering of radiation. So we begin with the study of the main regularities of radiation scattering on isolated particles. A detailed treatment of the theory of light scattering by particles can be found in monographs by Shifrin [lo], van de Hulst [ll], Bullrich [12], and collected papers [13]. It may be mentioned that one of the most recent results has been the discovery of difference in scattering functions for coherent and incoherent light (see [13a]).

4.3. Rayleigh Scattering Theory of light scattering in the atmosphere first appeared in connection with the attempt to explain the blue color of the sky. The most important contribution to this field was made by Lord Rayleigh, who contended that air molecules were the cause of light scattering. This premise of Rayleigh’s theory, however, was erroneous, as Mandelstam [141 showed. Actually, the so-called molecular scattering of Rayleigh is the scattering of light caused by density fluctuations. The investigations by M. Smolokhovsky and A. Einstein developed this theory of light scattering by density fluctuations, which can be observed in small volumes of liquid or gas, owing to local condensation or rarefaction of molecular systems. The quantitative results of the theory of scattering by density fluctuations coincide, in the case of gas, with the quantitative

172

Scattering of Radiation in the Atmosphere

conclusions of Rayleigh’s theory. Further investigations showed that not only density fluctuations, but also fluctuations of molecular unisotropy can determine light scattering. Let us turn to the elementary theory of Rayleigh scattering as presented by Hvostikov [15]. This theory describes the general case of Rayleigh scattering and molecular scattering. For more detail see Shifrin [lo] and Volkenstein [16]. We start with the following fundamental assumptions : (a) The dimensions of scattering particles are small in comparison with the wavelength. The particles are spherical (not necessarily, however). (b) The scattering particles and the medium are not conductors and do not contain free electric charges. (c) The dielectric constants of the scattering particle and of the medium differ by a small value. The refraction index of the particle is not too high, so always ne < 1 (e is the particle density). (d) The particles scatter light independently of each other. Imagine now that a nonconducting sphere of a radius with the dielectric constant E is inserted into a periodically variable electric field of the light wave EO. Since dimensions of the sphere are small in comparison with the incident light wavelength, the field over the sphere extension may be considered constant and periodically varying in time. In the four assumptions given, the phenomenon of scattering of light on a small sphere may be accounted for as follows. Under the influence of the electric field Eo in the considered sphere, a dipole is induced that is a source of a secondary field and consequently of scattered light. Thus, the electric field E in the outer (relative to the sphere) space will be expressed as

E = Eo + E’

(4.25)

where E’ is the field created by the sphere (the dipole field). It should be emphasized that E’ can be taken as the dipole field only in the case where the dimensions of the sphere are small compared with the wavelength. In the opposite case (when the particle dimensions are comparable with the wavelength) it becomes necessary also to take into account the fields of higher orders (quadrapole, octapole, etc.). Let the scattering particle be at a point 0 (Fig. 4.3) and let the direction of the dipole moment of the particle constitute angle 8, with the direction of the radius vector R connecting the point 0 with the point of observation M . Then the field created by the scattering particle at the point 0 can be

4.3. Rayleigh Scattering

173

determined (see [17]) from

IE'J= IH'I

=

- -P- s i n 8 c2R

(4.26)

where p is the second derivative in time from the dipole moment p , which is a function [t - ( R / c ) ] ;t is the time; H' is the magnetic field tension; c is the velocity of light. Note that (4.26) determines the electromagnetic field of the particle only at a great distance from it (in the so-called wave zone).

FIG. 4.3 Dipole emission.

For the radiant flux determined by the Poynting vector S we have S

=

IS1

C = - IE,

4n

HI

In our case the radiant flux passing through the point M will be expressed as S

C = - IE'I2

4n

Taking account of (4.26), we obtain (4.27)

174

Scattering of Radiation in the Atmosphere

It was assumed above that the small sphere under consideration is in a periodically variable field. Assuming that incident monochromatic light of frequency co = (2nc/A) is on the sphere, we have for the dipole moment p = po cos cot; that is, in the given case, the dipole moment must vary according to the law of simple harmonious fluctuation. For p we have p

-p0o2 cos cot = -pw2

=

Thus the formula (4.27) in our case has the form S=

cos2 cot sin20 4nc3R2

p0204

Averaging this expression over time per period, we obtain (taking into account that the mean over the period from cos2cot is equal to h), (4.28) Let us now resolve the vector of the incident-wave electric field into two components (Fig. 4.4), one of which is perpendicular to the sighting plane II \ \ \

\

u FIG. 4.4

To the problem of molecular light scattering.

AOM (direction I) and the other lying in this plane (direction 11). Denote q by the angle of scattering. For direction I, then, we obtain

4.3. Rayleigh Scattering

8

=4

2 , sin 8

=

175

1; for direction 11, 8

n 2

=-

+ p,

sin 0

=

-cos p

We see that the radiant flux passing through the point M can be presented as a half of the total of fluxes corresponding to electric fluctuations in the directions I and 11. Thus it can be written

On the basis of this expression for the radiant flux scattered by unit medium volume in the direction cp, we have (4.29) where summation covers the entire unit medium volume. Let us now take into consideration that 2 p o , the electric moment of unit volume for the sufficiently rarefied gas in the field Boycan be expressed as follows:

CPO

n2 - 1

=

7BO

If N is a number of particles per unit volume, then from the latter formula we find n2 - 1 Po = 4nN 1 0 ~

Thus

So instead of (4.29) we obtain

-

s1 =

+

nc(1 cos2p) (n2 - 1) (17~)~ 16R2NA4

(4.30)

This expression determines the quantity of scattered energy passing per unit time per unit surface at a distance R from the scattering particle perpendicular to the direction of propagation of the scattered light, which constitutes the angle 9 with the direction of the incident flux of radiation. Taking account of (4.30), it is clear that the quantity of radiant energy scattered by

176

Scattering of Radiation in the Atmosphere

unit volume in the direction p within unit solid angle will be determined by (4.31)

The mean (over period) value of the Poynting vector for incident light equals C so= (p)z

8n

using this and equation (3.41), we find the following expression for the volume-scattering coefficient : (4.32)

Thus, in the given case, the parameter XL =

nynz - 1)2 2N A ~

+

and the scattering function y(p) = 1 cos2 p. The Rayleigh scattering function is presented in Fig. 4.5 (outer curve). The innermost curve depicts the function for the scattered light component in the sighting plane (plane AOM in Fig. 4.4). In drawing scattering functions, the scattering volume is assumed to be at the crossing point of lines of different directions, and the radius length of the vector starting from this point and describing the function is proportional to the intensity of radiation scattered in the given direction. The incident light propagation direction is indicated by an arrow. As seen from (4.32) and Fig. 4.5, in Rayleigh scattering the most and equally intensive scattering of light is in the directions forward and

FIG. 4.5

RayIeigh scattering function.

177

4.3. Rayleigh Scattering

backward. Scattering minima take place at p, = 90' and p, = 270°, with the scattering intensity at minima being half of the maximum intensity. The formula (4.32) also demonstrates that in Rayleigh scattering, the volume-scattering coefficient appears to be inversely proportional to the fourth power of the wavelength. The foregoing results enable computation of the polarization of light in Rayleigh scattering. As known, the degree of polarization, P is determined by the relation (4.33) where II and 111determine the scattered light intensity calculated for two reciprocally perpendicular polarized (in the directions I and 11) components. According to the above results, 1 - COS2Q) cos2 p,

P =

1

(4.34)

+

From (4.34) it follows that in Rayleigh scattering, the polarization degree equals zero at p, = 0' and 180", and makes 100 percent in the directions perpendicular to the incident beam of rays (p, = 90°, 270'). Having in mind (4.24) and (4.32), let us find the expression for the volume coefficient of radiant-energy attenuation due to Rayleigh scattering. We have a),= 2n Since

I:

a ) ,sin ~ p, dp, =

2n3(n2

s," (1 + cos2 p,) sin p, dp, a),=

-

112

I

=

2 ~ ~ 40 =

(1

+ cos2 p,) sin p, dp,

813, we obtain

8ns(n2 3NA4

+

Or taking into consideration that n 1 2: 2 and nz - 1 it is possible to write 32ns(n a),= 3NA4

+

(4.35) 2:

2(n - l), (4.36)

Remember now that n = 1 a(e/eo),where a = 0.0002932. If we insert eo = 1.27617 x g ~ m (To - ~= 273'K and Po = 1000 mb), we have n - 1 = cle, where c1 = 0.22904. We have, then, a). =

32ns~2~12 3Nit4

V. G . Fesenkov had in this formula c1 = 0.22607

178

Scattering of Radiation in the Atmosphere

Also to be noted is that air possesses notable dispersion, and generally speaking it is necessary to take account of the dependence of the diffraction index n upon wavelength. According to V. P. Koronkevich, for dry air in the visible spectrum the following dispersion formula is valid :

(n - 1) x lo6 = 272.543

+

1.5450 ~

12

0.01431 +7

where 1 is in microns. The following table characterizes the dependence of n upon wavelength, as shown experimentally by B. Adlen. 4P

0.2

(n-1) lo-'

341.9 307.6 298.3 294.3 292.2 290.2 289.2 288.7 288.3 288.0 287.7

0.3

0.4

0.5

0.6

0.8

1.0

1.2

1.5

2.0

4.0

If the dependence of the diffraction index upon wavelength is taken into account, then, as F. Linke showed, the formula for the Rayleigh scattering coefficient, computed for the entire vertical atmosphere column of unit section, will assume the form aR,, = a,H

= 0.008791-4.09

(4.37)

where H is the height of a homogeneous atmosphere computed for T = 273'K and 1 is in microns. When taking account of molecular anisotropy, a correction f must be introduced into (4.36) and (4.37), which is known as the anisotropy coefficient. The numerical value is f = 1.061; that is, the correction for anisotropy is 6.1 percent (see [18-221). Calculations show that if we take the scattering coefficient equal to unity for red light (1= 0.72 p), then for shorter wavelengths,

Orange Yellow Green Blue Violet Ultraviolet

0.62 0.57 0.52 0.47 0.44 0.30

1.6 2.2 3.3 4.9 6.4 30.0

As known, this significant increase in light scattering with the decrease of wavelength accounts for the blue of the sky. In Table 5 of the Appendix are given the Rayleigh scattering-coefficient values as obtained by R. Penndorf.

4.3 Rayleigh Scattering

179

Observations of the attenuation of direct solar radiation in the atmosphere show that in the conditions of maximum clarity of air, the attenuation of solar radiation in the visible spectrum is to a great degree caused by Rayleigh scattering. The scattering of centimeter radiowaves by clouds may be considered Rayleigh, too. Having compared the results of computation of the Rayleigh and aerosol scattering coefficients, Bullrich showed that the influence of Rayleigh scattering becomes important only at a meteorological visibility range exceeding 5 km, being most important in the region of scattering angles near 130’ and increasing as the wavelength decreases. In the surface layer of the atmosphere the contribution of Rayleigh scattering to the attenuacan reach 50 percent. tion of radiation of the wavelength 0.4 to 1 . 0 ~ It is interesting, however, that even at great altitudes the aerosol scattering can be quite important. Balloon measurements by Newkirk and Eddy [22a] show that the observed sky brightness (wavelength 0.44~)at the 25-km level at the scattering angle 2.4’ (relative to the sun) is twice as large as Rayleigh’s obtained from computations. It is otherwise at greater scattering angles : for 10’ the observed brightness exceeds Rayleigh’s only by 10 percent; at 20’ they are practically equal. We shall see later that this pronounced dependence is caused by the strong elongation of the aerosol scattering function in the direction of propagation of light. Rocket measurements of the zenith sky brightness for the range 0.32 to 0.38 p, performed by Mikirov [22b] have shown that aerosol scattering is of much importance even at 80 to 500 km. Moreover, the contribution by Rayleigh scattering to the observed sky brightness above 100 km becomes negligible. All the above results refer to “pure” scattering on nonconductors whose dimensions are small in comparison with the wavelength. It appears that in the case of small conducting absorbents the regularities of scattering are essentially different. As Shifrin [lo] showed, in this case the attenuation coefficient (due to both absorption and scattering) calculated per single particle is (4.38)

Here x is the absorption index ( x = a1/4n, where a is the volume absorption coefficient), m = n - k is a complex refraction index, and v is the particle volume. From (4.38) it follows that in the case of small absorbing particles A; l/1, if n and x are independent of wavelength (that is in the absence

-

180

Scattering of Radiation in the Atmosphere

of dispersion). We see that the attenuation coefficient aA'is directly proportional to the particle volume. It can be proved meanwhile that the Rayleigh scattering coefficient, calculated per single particle, is proportional to v2. Since our concern is scattering by small particles vz

v

F
the conclusion is that the attenuation coefficient in the case of conducting absorbents must by far exceed the scattering coefficient. Thus in the presence of small absorbers, absorption dominates over scattering. Shifrin has analyzed the finite case of a small absolutely reflecting particle as 1 m I 1. In this case, as in Rayleigh scattering, the scattering coefficient is inversely proportional to A4. The scattering function and the distribution of polarization over different directions, however, are quite distinct. The scattering by ideal reflectors is usually backward-traveling. The ratio of the light intensity scattered forward to the light intensity scattered in the reverse direction is in the Rayleigh case 119 instead of unity. The preceding discussion is based on certain conclusions with respect to the theory of scattering of light on small homogeneous particles whose electrical properties differ from the corresponding properties of the medium. Of those cases considered, the case of Rayleigh scattering on a small nonconducting sphere corresponds most closely to the phenomenon of molecular scattering in the atmosphere. When mentioning the comparison between computations and observations, it was stressed that the latter refer to the conditions of a clean and dry atmosphere. The cause for this is in the fact that the presence of a comparatively small number of any coarse particle admixtures (dust, water, droplets, etc.) in the atmosphere is sufficient to nullify the part of molecular light scattering. It is necessary, therefore, to turn now to the study of the regularities of light scattering on large particles whose dimensions are comparable with wavelength or exceed it.

>

4.4. Scattering of Radiation on Large Particles

The most important investigations in this area were made by G. Mie, V. V. Shuleikin, K. S . Shifrin, H. C . van de Hulst, R. B. Penndorf [23], and others. The theoretical aspects of these investigations are intricate mathematically and therefore we consider here only certain fundamental results.

181

4.4. Scattering of Radiation on Large Particles

The main difference between light scattering on large and small particles consists in the following physical aspects. If the particle dimensions are comparable with the wavelength of the incident light, we cannot, as in the Rayleigh case, consider the field to be constant over the particle path of travel. The proper radiation field of the particle, therefore, cannot be taken to be dipole. Depending on the relation between the dimensions of particles and the incident light wavelength, it is necessary to take account of fields of higher orders, such as quadrapole or octapole. The angular distribution of the scattered radiation relative intensity, f(e, v, m),is determined in the given case by the following Mie formulas: (4.39) where

e = 2na/l

(a is the particle radius) = angle of scattering m = complex refraction index il , i2 = intensities of scattered radiation polarized in two perpendicular planes.

The intensities il, iz are determined by the relations (4.40a) (4.40b) where

for which

-G[

= 1c762-

(1 - xZ)n,;

x = cos v;

i

=2/--T

182

Scattering of Radiation in the Atmosphere

Here the Zl+l,2 are cylindrical functions with a semi-integral index and are Legendre polynomials. The function f(e, 97, m) is identical to the earlier scattering coefficient a A p . That is why, taking account of (4.24) we obtain all = 2?c

I" f(e, 0

q4 m) sin 97 d v

Many investigations [23] have been dedicated to the search for approximate (simplified) formulas for computing scattering functions and coefficients. Numerous variants of such formulas are suggested [for example, see [23a, 23bl). The regularities of light scattering on large particles appear to be distinct from the Rayleigh case. Large particles scatter mostly forward (the so-called Mie effect); the degree of polarization of scattered light decreases in comparison with its value in Rayleigh scattering. Large particles suspended in the atmosphere may be divided into two categories: opaque particles (dust, n = w) and transparent particles (water droplets, n = 1.33). Consider certain main results obtained with the theory of light scattering on the large particles included in these two categories of atmospheric admixtures. In Fig. 4.6 are given theoretical data on the scattering function for opaque, absolutely reflecting particles (n = w) at various parameter values e = 2na/il, where a is the particle radius. P.O.1

FIG. 4.6

90"

Scattering function of opaque absolutely reflecting particles. After Shifin [lo].

The function for e = 0.1 of Fig. 4.6 fully corresponds to the function of a finitely small opaque particle. In this case, light is scattered mainly backward. An approximately similar picture is observed also at e = 1. At e = 5 and e = 10, however, the form of the scattering function changes markedly; such large particles direct scattering exclusively forward. As regards the distribution of the degree of polarization in direction, it appears quite complex in the case of e = 5 and e = 10. It must be mentioned that the angular distribution of radiation intensity possesses a complicated fine astructure [11, 12, 231.

4.4. Scattering of Radiation on Large Particles

183

The attenuation coefficient aA' for an isolated particle is usually presented as a;

= na2K(e)

(4.41)

where K ( e ) is a certain function of e = 2na/il. In the given case the attenuation coefficient a; is the ratio of the quantity of radiant energy scattered by the particle in all directions, to the quantity of radiant energy incident on the particle. Figure 4.7 presents a graph K ( e ) for absolutely reflecting (n = co) spherical particles as given by Shifrin [lo]. The interrupted line depicts the development of a similar curve from the earlier computations by Gotz. The new computations show that the curve K ( e ) is of fluctuating nature for the finitely large particle (at e 00, K ( e ) -+ 2). ---f

Y

222.0

1

1

2

3

4

5

6

P

FIG. 4.7

Function K(e) for absolutely reflecting spherical particles. After Shifin [lo].

It is understood that the inclination of K ( e ) shows (at an invariable particle size) how strong is the dependence of the attenuation coefficient upon wavelength. Let us present the attenuation coefficient in the form (4.42)

where c and b are certain constants. It is evident that in the upward region of the curve K ( e ) , the value b > 0 and equals zero at maximum. In the downward section, K ( e ) , b < 0, and b = 0 at minimum. With the increase of particle size (at constant wavelength) b + 0. Table 4.1 displays certain observed data on radiation scattering in the spectral region 0.5 < A < 2.5 p in various smokes. Let us now pass to consideration of the regularities of scattering on a large transparent sphere, which in atmospheric conditions will correspond to scattering on water droplets.

184

Scattering of Radiation in the Atmosphere

TABLE 4.1 Values b According to Laboratory Measurement Data Substance Cigarette smoke NH,CI Pipe smoke Lampblack Spores

a, P

b

0.10 0.25 0.30 0.60 2.00

2.6

1.9 1.6

1.3 0.2

At the present time there are a great number of computations of scattering functions and coefficients for various refraction indices and a wide range of the parameter e variations (see [23]). These computations show that with the increase of the parameter ,o, the scattering functions strongly elongate “forward” (in the direction of the incident light). This is seen in Fig. 4.8.which presents functions of light scattering by water droplets (n = 1.33) at e = 4, 8, 15, 30. 900

I \ FIG. 4.8 Scattering function of water droplets. After Shifin [lo].

Figure 4.9 gives the results of computation of the function K ( e ) performed by Penndorf and B. Goldberg [23] for n = 1.33 and n = 1.50. As in the

a-

FIG. 4.9 Function K(e) for the values of refraction index n = 1.33 and n = 1.50. After Penndorf and Goldberg [23].

4.4. Scattering of Radiation on Large Particles

185

case of opaque particles, the new computations show that the curve K(p) is of fluctuating character. K. S. Shifrin affirms that the cause for this is the interferential nature of light scattering in both cases. The extreme positions of the curves K ( e ) for spherical particles can be determined with the help of simple empirical formulas. Maltzev [23c] showed that if k is the extreme’s ordinal number, then the corresponding value will be determined from

k=-

n2 - 1

4

( a - 1)

The curves of Fig. 4.9 can be used for computation of scattering functions in the visible range where n = 1.33, and in the ultraviolet where the refraction index increases with the decrease of the wavelength. It is clear from Fig. 4.9 that the form of the function K ( e ) is seriously affected by dispersion. As t o the infrared, it shows a wide variation of n from 1.154 (A = 11 p ) to 2.059 (A = 152 p), depending on the wavelength. Of still greater importance is the necessity to take account of absorption in the infrared region, which complicates the calculations of scattering functions and the function K ( e ) . We have considered certain results of the theory of radiation scattering on particles whose dimensions are comparable with wavelength or larger. It would be of interest to dwell now on the case of finitely large particles for which e -+ 00. It follows from the previous consideration that in this case the finite value of the attenuation coefficient per single particle is 2 n d (since limQ+w[K(p)] = 2). This means that the finitely large particle scatters a double quantity of radiant energy in relation to the amount of energy per its unit section. This conclusion may seem rash, especially if we assume the possibility of application of geometric optics to the calculation of the light scattering by a finitely large particle. However, Shifrin [24-271 has proved that even in investigations of light scattering on such particles, the approximation of geometric optics does not suffice, but an evaluation of the diffraction effect is needed. Calculations show that the total amount of light diffracted by a finitely large particle is identical with the amount of light scattered by the particle in the result of refraction and reflection. The whole of this “diffracted-scattered” light propagates forward within a very small angle, /I,- l / a . The scattering functions therefore have a forward-directed narrow “tongue.” The finite value of the attenuation coefficient in this case must equal 2na2, owing to the diffraction effect. All that has been said above refers to spherical particles. The actual

186

Scattering of Radiation in the Atmosphere

particles in the atmosphere are not ideal spheres, which necessitates study of the peculiarities of light scattering by nonspherical particles. This has been done by K. S. Shifrin on the basis of his approximate theory of light scattering by particles [26, 281 and also by Atlas et al. [29]. Shifrin has studied the scattering of light by ellipsoid particles, flattened or sticklike. The scattering functions in this case are quite peculiar. Figure 4.10 shows the scattering function for an ellipsoid whose longer axis exceeds the shorter by ten times and 2na*/il= 1 (a* is the radius of a sphere whose volume is equal to that of the ellipsoid). The curve a is the Rayleigh scattering function. The curve b presents the function in the case where the light falls along the major axis. The curve relates to the case of the light falling across the major axis. This function is calculated for a plane passing through the major axis and the incident ray. 900

FIG. 4.10 “Stability” of the Rayleigh scattering function. After Shifrin [lo]. (1) Rayleigh function; (2, 3) functions for the case when the ratio of the larger axis of the ellipse to the smaller is 2,3 and 10.

Shifrin’s results were also interesting in the investigation of scattering on small nonspherical particles (2na*/il< 1). It appears that the Rayleigh scattering function is comparatively stable in relation to formal variations ; for example, the contraction of a small spherical particle by two to three times has almost no effect on the form of the function in comparison with the Rayleigh. K. S. Shifrin has computed also the “mean” scattering function for the case of different orientations of nonspherical particles. In the case of the sticklike particle, the function averaged over directions coincides with the function of a sphere of the volume u* = u a ) , where o is the particle volume and Y(E) is a certain function of the ellipsoid eccentricity. Fedorova [29a] has performed an interesting experimental investigation

4.4. Scattering of Radiation on Large Particles

187

to determine the influence of the form of transparent particles (glass fragments) on the character of scattering functions. It has been found that the intensity of the light scattered by glass fragments has a practically constant value within the limits of scattering angles 50' < q~ < 180°, and rapidly increases forward. Fragments scatter three to five times more light in the backward direction than do spherical particles. Especially great are differences in the degree of polarization of the light scattered by fragments and spherical particles. For spherical particles the degree of polarization in certain directions may reach 100 percent; in the case of fragment it does not exceed to 10 to 15 percent. Practically of equal importance is the solution of the problem of light scattering on nonhomogeneous particles (for example, on double-layer spheres), since the actual scatterers in the atmosphere are often nonhomogeneous (sleet, hail particles wetted by rain). A similar problem was solved by Shifrin [30], A. Guettler, M. Kerker et al. [23], and Fenn and Oser [30a]. It turned out that even in the presence of a thin water film around the ice nucleus, such a particle behaved in scattering as a water droplet. Besides molecular scattering and scattering on large particles of different admixtures, the atmosphere also shows scattering on large-scale (in comparison with the light wavelength) turbulent density pulsations. The latter accounts for the scintillation of stars. It may be assumed that turbulent pulsations also affect the scattering of direct solar radiation (see [31-361). We have considered the most essential regularities of light scattering as confirmed by theoretical calculations and laboratory measurements. Let us now get acquainted with the results of experimentation in solar radiation scattering in the atmosphere. Barteneva [36a] has carried out an extensvie research of the scattering function in the surface layer of the atmosphere. Her investigations show a marked dependence of the form of scattering function upon atmospheric transparency. It was found, however, that there were different forms of function at the same transparency. This means the absence of a simple function-transparency dependence. Having studied the peculiarities of scattering function in different atmospheric transparency conditions (including fogs) in several geographical regions, Barteneva suggested a ten-division classification of functions. Each division has definite values of the scattering assymnietry coefficient K (the ratio of the forwardscattered radiant flux to the flux scattered backward), varying from 1 up to 25 to 35, and meteorological visibility range from 220 up to 0.4 to 0.5 km. The elongation of the scattering function rapidly increases with the decrease of the visibility range. In some cases (at K > 3.5) the functions of

188

Scattering of Radiation in the Atmosphere

the given type must be in turn subdivided into distinct groups of “slanting” and “acute” types. Voluminous material is now available on the atmospheric scattering functions referring to the whole thickness of the atmosphere. In this connection numerous attempts have been made at finding the most expedient analytical presentation of the real function. Fairly common is the empirical formula proposed by Schoenberg : Y(T> = 1

+ p cos T + q cos2

Q,

(4.43)

where p and q are certain parameters depending on atmospheric state (turbidity above all) and determined from observations. Krat [37, 391 found that the Schoenberg function was not satisfactory and, based on his own observations, obtained the formula (4.44) Here, as in (4.43), the values P, and QA are empirical coefficients determined from observations, and the function f ( ~ = ) e-39 - e-(3/2)n = e-3q - 0.009, where cp is expressed in radians. Observations with light filters show that these coefficients depend not only on the composition of the scattering medium but also on the wavelength. Livshitz [38a] proposes the following presentation of (4.44) : Y(T)

+

= o ~ ( 1 COS

T)

+ a,[l + k(e-3p - 0.009)]

(4.44a)

which gives a spectacular expression of the physical sense of the individual terms of the function corresponding to the Rayleigh and aerosol scatterings (oR,o, , are the Rayleigh and aerosol coefficients, respectively, and k is an empirical coefficient). From (4.44) the value y ( n / 2 ) = 1. The numerical values of the coefficients P, and Q,, therefore, can be derived from observation of the relation

for different q~ by the method of the least squares. Here, Z,(cp) designates the intensity of the light of the wavelength ;Z scattered in the direction p. The variability of P, and Q , in dependence on the wavelength is quite great. For example, Piaskowska-Fesenkova [39] obtained the results in the following table:

4.4. Scattering of Radiation on Large Particles

476 546 625

PA

QA

3.64 4.99 7.02

0.65 0.55

189

0.44

V. A. Krat notices that (4.44) cannot be applied to the narrow interval of scattering angles from 0 to 2.3' (the area of the circumsolar aureole), where the function y ( y ) has a very high maximum (the Mie effect). Piaskowska-Fesenkova [39] performed a series of experimental and theoretical investigations in the scattering of light in the atmosphere. Numerous observations were conducted by her in order to measure the intensity of diffuse radiation (the sky brightness in terms of the photoelectric unit system) at the solar almucantharat at different angular distances from the sun. The observations were carried out by means of a photometer with blue, green, and red filters constructed by V. G . Fesenkov. The effective wavelengths for these filters were 0.476, 0.546, and 0.625 p, respectively. Figure 4.1 1 gives atmospheric scattering functions as observed by Piaskowska-Fesenkova at different geographical points, but at the same atmospheric optical mass [O(O)= 0.291. For comparison are also given the spherical and Rayleigh functions. It is seen that large particles greatly influence light scattering (forward-elongated function). It is also important that at the same optical thickness (equal transparency) the scattering functions of the same wavelength may have different form, especially at small angles. Piaskowska-Fesenkova gave particular consideration to the atmospheric scattering function asymmetry. Her data show that the forward-directed elongation of the function increases with the increasing wavelength. The reverse elongation will decrease with the increase of the wavelength. The ratio of the quantity of light scattered forward and backward for symmetrical scattering angles, Ip/In-9 (I9 is the intensity of light scattered in the direction of the angle of scattering y ) , is introduced to characterize quantitatively the asymmetry of the atmospheric scattering function. Experimental data show that the dependence of IT/ZnW9 upon wavelength is linear, with all lines crossing in the vicinity of a point characterized by the coordinates IJIn-+, = 1, A = 0.300 p, as is shown by the extrapolation of lines toward the region of short wavelengths. This means that in the proximity of A = 0.3 p, the asymmetry of the atmospheric scattering function

190

Scattering of Radiation in the Atmosphere

vanishes, and in the region il < 0.3 p, the light is scattered mainly backward (the negative Mie effect). The latter effect has not yet been satisfactorily explained.

t t't t FIG. 4.11 Scattering functions of the whole atmospheric thickness. After PiaskowskaFesenkova [39]. (1) vicinity of Moscow, 1 = 546 m p ; (2) Vladivostok, I = 546 m p ; (3) vicinity of Alma-Ata, 1 = 476 m p ; (4) near Moscow (Ivanovo); (5) Rayleigh scattering function; (6) spherical scattering function.

In treating observational data of Piaskowska-Fesenkova, the first-order scattering theory is employed. Generally speaking, taking account of multiple scattering may be corrective in this process. Piaskowska-Fesenkova, however, has shown that the influence of higher-order scattering on her results is practically insignificant. Basing on the above observations, V. G. Fesenkov suggested the following formula for the scattering function :

y(v) = 1

+ u cos v + b

v + c C O S ~v

COS~

(4.45)

where a, b, and c are empirical coefficients determined from observations.

4.4. Scattering of Radiation on Large Particles

191

Piaskowska-Fesenkova gives the coefficient values a = 0.049, b = 1.093, and c = 0.704, corresponding to the mean atmospheric function at il = 0.476 p. The same observations show that the Krat formula (4.45) can describe the atmospheric scattering function with a sufficient degree of accuracy. Both formulas, however, need correction at very small scattering angles (in the vicinity of the circumsolar aureole). The variation of the form of the scattering function with height is important. So far there are few available preliminary data on the scattering function in the free atmosphere. In Fig. 4.12 are given the functions at 5,20, and 100 km of the horizontal visibility range from data by L. Foitzik

CP

FIG. 4.12 Scattering functions in the free atmosphere. (1) after Foitzik and Zschaeck; (2) after Reeger and Siedentopf; (3) after Bullrich and Moller.

and H. Zschaeck, E. Reeger and H. Siedentopf, K. Bullrich and F. Moller, generalized by the authors of [40]. The different data show good coincidence. The most interesting feature of the results is a marked elongation of the scattering function even at 100-km visibility range. Balloon measurements by Belov [40a] and Chayanov [42] also confirm the great elongation of the scattering function in their range of measurement up to 22 km. It is of interest that the form of the function in the upper troposphere and stratosphere shows little variation with height. Sandomirsky et al. [42a] have conducted aircraft measurements of the

192

Scattering of Radiation in the Atmosphere

sky brightness at the height of 8.0 to 17.5 km by means of an electrophotometer with a light filter (Aeff = 0.444 p, 0.554 p, and 0.635 p). According to Sandomirsky et al. [42a], the elongation of the scattering function increases with the increasing height and wavelength (anomalous increase of the function elongation near the tropopause). Scattering functions in the atmospheric layer above are characterized by a marked variability from day to day with the increase due to the increase of wavelength. This shows the presence of significant. aerosol at 10 to 20 km, whose variations cause the variability of the scattering function. We should also mention an interesting dependence of the scattering coefficient upon wavelength for different scattering angles q~: a x H = %A,HT

(v)

where p(v) is the scattering function averaged over the whole thickness of the atmosphere, and H is the height of the homogeneous atmosphere. Let axH be, analogously with (4.42), a I H = cl-b

(4.46)

As shown by observations of Piaskowska-Fesenkova [39] at Mount Kumabel (z = 3100 m) in the vicinity of Alma-Ata, for scattering angles 9I loo, the dependence of logaxH upon log A is always linear [in accordance with (4.46)]. Table 4.2 gives the dependence of the exponent b upon scattering angle v. TABLE 4.2 Dependence of the Exponent b upon the Scattering Angle. After Piaskowska-Fesenkova [39]

b

vo 10 15 20 40

60 80 90 120 140

8- 19-49

8-23-49

8-12-49

094 195 2.3 2.9 3.1 3.0 3.2 3.5 -

192 290 2.3 2.5 2.7 2.6 2.7 3.2

190 1,8 2.4 3.2 3.4 3.6 3.6 2.6 3.7

-

4.4 Scattering of Radiation on Large Particles

193

As seen from this table, the exponent b increases with the increase of p. We should look for the cause of this in the fact that at small scattering angles, the influence of large particle scattering is most essentially marked. The exponent b, as we have already noticed, always has smaller than Rayleigh b = 4 values for large particles. At larger angles the role of Rayleigh scattering increases and also increases the exponent b. A more complex picture can be seen in the determination of the scattering coefficient a;,= for small angles within the circumsolar aureole. In this case the dependence of log azB on log 1 is not linear, and therefore b depends not only on p but also on 1.As stated by Piaskowska-Fesenkova, at p = 2’ in a dusty atmosphere, the attenuation of red light is more intensive than that of green light. This anomaly is not observed in a clean and humid atmosphere. The expression for the coefficient of attenuation due to scattering calculated for a vertical atmospheric column of unit section, may be presented analogously with (4.44). In this case, as experiments show, a great variability of the coefficients c and b is observed, with c varying from 0.012 to 0.239 and b from 0.48 to 1.83, which corresponds to the variation of al,Hfor 1 = 0.55 p from 0.026 (c = 0.019, b = 0.48) to 0.571 (c = 0.239, b = 1.46). According to Schiiepp [43] and Volz [44], the values b = 1.37 and b = 1.48 can be taken as the mean values for mountainous and plain regions, respectively. In measurements of the attenuation of ultraviolet radiation, anomalous extinction ( b < 0) is often observed. The cause of this is that, for short wavelengths (1 < 0.3 p), the parameter e > 6, and so with the increase of wavelength the function K ( e ) and therefore the scattering coefficient increase. Also important is the dependence of the aerosol attenuation coefficient upon altitude in the conditions of a cloudless atmosphere; Penndorf [23] shows that, on the average, the attenuation coefficient exponentially decreases with height down to 4 to 5 km:

where ah, a. are the coefficient values at a height h and near the earth’s surface, respectively. For the H D the height of the “homogeneous dust atmosphere,” measurements give values of the order of 1 to 1.5 km. Analogous results were obtained by Rabinovich [45], who recommends the

194

Scattering of Radiation in the Atmosphere

following parameters of (4.46a) for the visible spectrum: Winter:

HD =

1 km,

a0 = 0.25 km-'

Summer:

HD =

1.4 km,

a. = 0.15 km-l

The values of a0 refer to A = 0.55 p. It is natural in individual cases (and sometimes over long time intervals) that the vertical distribution of the attenuation coefficient shows greater complexity than that given by (4.46a). Waldram [46] has obtained certain data from his four-series aircraft measurements (Fig. 4.13) which illustrate the above statement. The curves 1 to 4 correspond to different conditions of observation. In recent years the first investigatious of aerosol scattering in the upper atmosphere were taken from spacecraft [46a].

SCATTERING COEFFICIENT,

FIG. 4.13

km-'

Vertical projiles of the attenuation coefficient. After Waldram [46].

4.5. Computation of the Attenuation of Radiation in the Atmosphere due to

Scattering

In the previous sections we considered the main statements of the theory of light scattering. Let us now turn to some applications of this theory related with calculation of the attenuation of radiation due to scattering caused by the presence of water droplets or opaque aerosol particles in the atmopshere.

4.5. Computation of the Attenuation of Radiation

195

1. Theoretical Method for Computation of the Attenuation of Radiation due to Scattering. As already mentioned, the coefficient of attenuation due to scattering, calculated per single particle, can be presented as

al'

= na"(e)

If there are n particles of equal size per unit volume, assuming that the the particles scatter independently, the volume attenuation coefficient a1 is a1 = a{n = nna2K(e)

(4.47)

The attenuation of radiation in a monodispersing aerosol is determined by the simple relation (4.48)

where I is the ray path. In computing the attenuation of radiation due to scattering it is also essential to evaluate the relation between Rayleigh (molecular) and aerosol scattering. Table 4.3 gives some Rayleigh attenuation coefficient values calculated for the visible region, taking into account the depolarization and dispersion. TABLE 4.3 The Rayleigh Attenuation Coefficient for the Visible Spectral Region

4P

aa 1/km

4P

0.400 0.450 0.500 0.550 0.600

0.04358 0.02715 0.01750 0.01190 0.00833

0.650 0.700 0.750 0.800

ad,

l/km

0.00554 0.00446 0.00364 0.00260

The computation of aerosol attenuation coefficients show that even at a very small number of large scattering particles, the aerosol attenuation by far exceeds the Rayleigh; for example, in the case of water droplets at n = 200 ~ m - a~ = , 1 p, the value aA= 6 km-l for il = 0.55 p. We remember that in the case of scattering of ultrashort radio waves by clouds, the matter stands otherwise; the scattering on water droplets in this case may be considered practically Rayleigh, since the parameter e = 2na/il is small.

196

Scattering of Radiation in the Atmosphere

The actual aerosol is in most cases polydispersional, thus necessitating the account of the size distribution of particles in the computation of radiation attenuation. If the size distribution of particles is characterized by the function dnlda =f(a) (where n is the number of particles per unit volume and a is the radius of particles), we have for the volume attenuation coefficient, (4.49) where a,, a, are the minimum and maximum radii of the scattering particles. By giving a concrete form to the distribution function f(a), it is possible to calculate from (4.49) the radiation attenuation coefficients of different aerosols. C. Junge, for example, determined the distribution function for dust particles (dry aerosol) in the size interval 5 x lo6 < a < cm to be of the form dn = ~ a da - ~ (4.50) where c is constant. Substituting this expression in (4.49), we have (4.51) where c1 = 2n2c and the function K4 = sa;K(e)e-z dp depends only upon the complex refraction index nz and the integral limits a, , a,. Actually we may take a, = 0, and at large a the function K + 2. This means that K4 is practically dependent upon m only. As seen from (4.51), the coefficient of radiation attenuation due to scattering by dust is inversely proportional to the wavelength. Although, as was shown above, the spectral dependence of aerosol attenuation on wavelength may vary, nevertheless this conclusion can hold for the visible spectral region. F. Shmolinsky asserts that at weak and intermediate atmospheric turbidity the spectral dependence of the attenuation coefficient in the visible range is satisfactorily described by (4.42) a t b = 0.92 f 0.25. Similar computations of the coefficients an in the 0.5 to 14-p regions, making use of (4.51) and (4.50), were performed in the work of V. E. Zuyev, V. V. Sokolov, and S. D. Tvorogov (see Chapter 3 [149]), where atmospheric aerosol particles are treated as water spheres with radii within a, = 0.1 and a, = 1.0 in accordance with C. Junge's data. Computations of aA were performed for different values of the meteorological visibility range. The formula for the attenuation coefficient assumes a particularly simple

4.5. Computation of the Attenuation of Radiation

197

form in the case where the dimensions of scatterers are large enough to give K ( e ) = 2. This takes place, for example, in the determination of the transparency of rains. For raindrops, the parameter variation lies within the range 600 to 3000. Poliakova and Shifrin [47] used the following formula for the size distribution of raindrops: (4.52) where A, /3 are empirical parameters. Substituting this expression for f ( a ) in (4.49), we obtain a

=

I

m

0

2na2Aa2e d a da

48nA

= -

B2

(4.53)

The wavelength index has been omitted here, since a is independent of wavelength. The preceding method for computation of radiation attenuation due to scattering employs the quantitative results of the scattering theory. This method, however, is applicable only in the case where the nature, quantity, and dimensions of scatterers are known. In most cases any concrete data on the amount and size of atmospheric scatterers are either absent or approximate, which accounts for the fact that the above method has not to date been widely used in actinometry and is applied mainly for solution of various optical problems of clouds and fogs. A complete investigation of this kind was made by Zuyev et al. [48-521, who had computed coefficients aAin the interval 0.5 to 1.4 ,u for water clouds and fogs with different characteristics of microstructure, taking into consideration the complex water refraction index. We shall now pass to more common actinometric methods which are semiempirical in their nature. 2. Semiempirical Methods for Computation of Attenuation of Radiation in the Atmosphere due to Scattering. The method proposed by Angstrom [53], is the most popular of all methods for computing attenuation due to scattering. Let the attenuation coefficient be

where a a , R is the coefficient of the attenuation due to Rayleigh scattering. The attenuation of the solar radiation flux caused by scattering can then be determined from

198

Scattering of Radiation in the Atmosphere

where So,Ais the spectral energy distribution of solar radiation outside the atmosphere, S,’ is a flux of solar radiation near the earth’s surface (taking into account the attenuation due to scattering, m is atmospheric mass in P1dz is the so-called turbidity factor. the direction on the sun, and B = Given the values of the parameters fl and b and the function So,>,, it is possible, with the help of (4.55), to calculate the attenuation of radiation due to scattering. Angstrom’s original suggestion was to take the value b = 1.3. Then, provided So,Ais known, the problem of determination of radiation attenuation is reduced to the determination of the turbidity factor B. To determine the turbidity factor, the usual actinometric observations of direct solar radiation can be used. Having in mind that the total attenuation of solar radiation in the atmosphere is caused not only by scattering but also by absorption, we can write for the solar radiation flux S , , measured by actinometer, S,

=

S,’ - AS

where A S is the value of the portion of direct solar radiation absorbed in the atmosphere. The value A S may be determined by means of empirical formulas relating this value with the absolute humidity near the earth‘s surface or with the total water vapor content in the atmosphere (see Chapter 3). Thus the turbidity factor P is the only unknown value in (4.56) presenting an integral equation for B. Equation (4.56), however, does not provide for the determination of B with sufficient accuracy, as the value A S cannot be always satisfactorily computed from the empirical formulas mentioned. This accounts for the fact that determination of the turbidity factor based on the use of data of actinometric filter measurements has found a wider field of application. As we have seen, the attenuation of direct solar radiation caused by absorption takes place mainly in the range of wavelengths 1 > 0.7 p. If we measure a flux of direct solar radiation, shading the actinometer with a filter that “cuts off” all radiation of il < 0.7 p, we can write

4.5. Computation of the Attenuation of Radiation

S,

=t

/lS,,,{exp[

- m J m ~ A , Rdz]}{exp[-

1

mj3A-1.3]dA - t A S

199 (4.57)

Here S, is a solar radiation flux measured with a shaded actinometer (A, is the lower filter transmission limit), and t is the mean (integral) transmission coefficient of the filter. Dividing both parts of (4.57)by t and subtracting the obtained relation from (4.56), we have 1 SI = s,--s,

t

This expression for solar radiation flux S, excludes the influence of radiation absorption. It is possible, thus, to avoid computation of A S by combining measurement of the direct solar radiation flux with the S, flux measurement. In actinometric practice for determination of the value S1, Schott filters RG 2 (red) and O G 1 (yellow) are used. These “cut off” the radiation of the wavelengths shorter than 0.630 p and 0.525 p, respectively. Table 6 of the Appendix gives values calculated for 1, = 0.630 p at different B for the range of atmospheric masses from 1.0 to 6.0 (in application to the 1956 pyrheliometer scale). Knowing the measured value of S1, it is possible with the help of Table 6 to find the corresponding value of the turbidity factor /?. Measurements show that the turbidity factor varies within very wide limits from several hundredths (good transparency) to several tenths (low transparency). In connection with the use of (4.58)for computing the turbidity factor, it is of interest to trace the role of Rayleigh scattering in the attenuation of radiation by the whole atmospheric thickness. A spectacular presentation of this kind is provided by Table 4.4, compiled from data on atmospheric optical thicknesses in the vertical direction with account taken only of Rayleigh and aerosol (to,Jscattering for three different values of the total (integral) optical thickness at A = 0.55u , characterizing atmospheric transparency. We see that in a cloudless atmosphere the importance of Rayleigh scattering in the visible region is quite obvious even at a significant aerosol turbidity of the atmosphere. In the infrared region the Rayleigh scattering (see Zuyev et al., Chapter 3 [149])is of no importance as compared with the aerosol scattering.

200

Scattering of Radiation in the Atmosphere

TABLE 4.4

me Integral Aerosol and Rayleigh Atmospheric Optical Thicknesses. After Shifin and Minin [54]

I

0.2

0.3

0.5

0.400

0.349

0.144

0.281

0.556

0.450

0.217

0.128

0.250

0,494

0.225

0.444

0.500

0.140

0.115

0.550

0.0952

0.105

0.205

0.405

0.600

0.0666

0.0960

0.188

0.371

0.650

0.0443

0.0886

0.173

0.342

0.700

0.0356

0.0823

0.161

0.318

0.750

0.0291

0.0768

0.150

0.297

0.800

0.0208

0.0719

0.140

0.278

As became evident in practice, the most vulnerable aspect of the Angstrom method is its assumption of the stability of the parameter b value. Schiiepp [43] has based his generalization of this method just on this point, proposing to engage two parameters: and b for characterization of atmospheric transparency. In this case, however, the application of the semiempirical method for radiation attenuation determination is greatly complicated. Angstrom [ 5 5 ] showed that a high accuracy of measurements and the use of filters are of great importance in the determination of the turbidity factor. The accuracy being less than 1 percent, the results become unreliable. This is a difficult and not always feasible requirement, especially if we take into account that along with measurement errors caused by the inaccuracy of instruments, it is necessary to consider the influence of geometric parameters of instruments on the results of measurements. The latter fact has been given attention by Zuyev et al. [48,56-58]. 4.6. Elementary Theory of Radiative Transfer, Including Multiple Scattering

In Sec. 4.5 we considered some results characterizing radiation scattering by certain particles. The computation methods that followed were based on the assumption that radiation attenuation is determined by the primary scattering influence only. In real conditions, however, the scattering of the

4.6. Elementary Theory of Radiative Transfer

201

already scattered light takes place; that is, multiple scattering takes place and its influence increases in proportion to the optical thickness of the scattering medium. To solve the problem of radiative transfer while taking account of the multiple scattering, it is necessary to consider the precise integrodifferential transfer equation (1.80) with the corresponding boundary conditions. The general form of this problem is intricate and unwiedly. To solve it, various approximate methods have been offered. We confine our consideration, however, to two approximate methods in application to the case of shortwave radiation transfer, which are common in actinometry and atmospheric optics. 1. The Method of Successive Approximations. Let the transfer equation

(1.81) be rewritten as follows:

cose--azuaz

=@O1.

4n

J

ZJ,(Z,

r‘)yA(Zyr‘, r ) do‘ -

@UAZA- @&JA

(4.59)

where ,omis the density of the radiation absorbent. Introducing the notation and @on= aA = PAaA,we have instead of (4.59) aL = @oL

+

81” PA% cos e -4n

J

aZ

With optical thickness

t=

zA(zYr’)yA(z;r‘, r ) do‘ - aAzA

(4.60)

J,”a,dz, we obtain

where B(z, 8,

v) = A 4n

zA(t,r’)yA(t;r‘, r ) d d

(4.62)

In (4.62) let us isolate the term corresponding to the direct solar radiation. If the zenith distance and the solar azimuth are 8, and v,, respectively, and and P1 are scattering angles, then, assuming the scattering function to be independent of height, we obtain

where So,ais the “monochromatic” solar constant, and ZA is the intensity of the scattered radiation. Assuming, for simplification, that the albedo of the earth’s surface equals

202

Scattering of Radiation in the Atmosphere

zero, let us add the following boundary conditions to (4.61) z,(O, 8,

v)

z,,(to,8,

at 8

n <2

at 8

>2

(4.64a)

n

(4.64b)

where to = J," a, dz is the atmospheric optical thickness. It can be shown that Eqs. (4.61) and (4.63) are reducible to a common integral equation. Solving (4.61) relative to I, and substituting the result in (4.63) after simplifying the scattering function to be spherical (in this case y, = 1;B depends upon t only; I, upon t and O), we obtain the following:

where m

E,(x) =

Jl

t-'

dr.

This equation was first derived by 0. D. Hwolson in the investigation of light scattering in matt glass. There are a number of solutions of Eq. (4.65) according to the method of successive approximations, one of the most common consisting in the use of the free term (4.65) as the first approximation: (4.66) Substituting this first approximation in the first part of (4.65), we obtain the second approximation, likewise the third, etc. This solution of the transfer equation is equivalent to the consecutive account of the scattering of the first, second, and higher orders. As Sobolev [59] showed, the solution in (4.65) is defective in that at t 2 1 and a small (that is, in strong absorption), it is necessary to assume too many approximations. This can be seen, for example, from the following table containing the results of the brightness coefficient e computation for the radiation passing through the lower boundary of the medium as performed by Sobolev : TO

0.2 1 .o 2.0 03

el

e2

0.041 0.108 0.123 0.125

0.052 0.171 0.205 0.212

Precise 0.055

0.27 0.47 1.06

4.6. Elementary Theory of Radiative Transfer

203

It is seen from this table that already at t = 1 the second approximation e2 greatly differs from the exact solution. Kuznetzov [60]after carefully analyzing the problem of agreement of the method of successive approximations, reported that more satisfactory results are obtained in the case where an approximate solution of the transfer equation according to Schwarzschield (one of the rough approximations of (4.65)), is taken as the first approximation. Kuznetzov has developed a solution of the transfer equation in application of the method of successive approximations, which is feasible for the cases of considerably anisotropic scattering (elongated scattering functions). In this case the expression for the scattering function is presented in the form of a finite series into the Legendre polynomials: N

y ( r ; r', r) =

z ci(z)Pi[cos(r', i=l

r)]

(4.67)

where c i ( t ) are the resolution coefficients, Pi[cos(r', r ) ] are the Legendre polynomials. The solution in this case is sought in the form of a trigonometric series : N

I,dr, r ) = Ao(t, 0)

+Z k=l

Ak(~70)

cos kv

(4.68)

Here A , and A k are the coefficients of the resolution into a series. Substituting (4.68)in the transfer equation, we are led to a system of equations for the coefficients A k . This system is solved according to the method of successive approximations, with the first approximation being the solution that takes account of the first-order scattering only. Computations of the intensity of scattered radiation of the cloudless sky have shown that the influence of multiple scattering considerably increases with the increase of the atmospheric optical mass to.For example, at to= 0.8,the error due to neglecting multiple scattering may reach 70 percent. Even for a slightly turbid atmosphere ( t o= 0.2), this error may exceed 25 percent. It is still more essential to take account of multiple scattering in the study of radiative transfer in clouds.

2. Method of Reducing to Diferential Equations. The variants of the method of successive approximations permit attaining a very high accuracy. For this, however, some cumbersome computations are required. To simplify and shorten these computations, many different variants of reduction of the integrodifferential transfer equation to differential equations have

204

Scattering of Radiation in the Atmosphere

been proposed. The integration of the latter is a simpler problem than the solution of complex integrodifferential or integral equations. The main idea of the deduction of the most often used approximate transfer equations consists in substituting the exact integrodifferential equation for radiant intensity by common differential equations for the upgoing and incident radiation fluxes. The general solution of this problem has been given by Kuznetzov [61]. Having performed the transformation of the exact transfer equation describing the transfer of unpolarized radiation in the absorbing, scattering, and emitting medium, Kuznetzov derived the following two transfer equations for the upgoing (F,) and downcoming (F,) radiant fluxes: dF1 - 2kB - m,(k

e dz

1 dF2

+ orl)F, + m,oI’,F2

2kB - m,or,F,

--=-

e dz

where

m,

=

m2 =

+ m,(k + ar2)F2

J Zl(z, r‘) do’

l Zl(z, r’) cos 8’ do’ J Z 2 ( Z , r’) do’ J Z 2 ( 2 , r‘) cos 8‘ do’

(4.69a) (4.69b)

(4.70) (4.71)

r’)Bi(J) do’ r, = J ZI(Z, l Zl(z, r’) do’

(4.72)

, r’)Bz(r‘) do’ r2= J ZJ2 ( zZ2(z, r’) dw‘

(4.73) (4.74)

Pa@’)

1 =

y 2 , , ( z ;r‘, r ) d o

(4.75) (4.76) (4.77) (4.78) (4.79)

Note here that the wavelength index should be added to all the values of (4.69) to (4.79), since the transfer equations (4.69) are valid only for monochromatic radiation. We omit this index for brevity and also because

205

4.6. Elementary Theory of Radiative Transfer

Eq. (4.69) is actually used for description of nonmonochromatic radiation transfer by introducing the concept of the effective wavelength. The coefficients of absorption, k, and scattering, 0,of the transfer equation are, generally speaking, functions of the height z. The values m , , m 2, r,, r2are also functions of the height z. Eqilations (4.69) can thus be treated as common differential equations. It appears, however, that this is expedient only if the dependences m, , m 2, , T2upon z are equal at any Z,(z, r ) and Z2(z, Y), that is, at any angular distributions of radiant intensity. Moreover, it should be emphasized that the values I', , r2may vary, depending on the form of the scattering function. This means that I-', and r2must be universal functions of z independently of the character of the radiant intensity angular distribution and the form of the scattering function. Thus the problem of the use of Eqs. (4.69) as common differentials is reduced to the problem of the dependence of the coefficients m, and m2 upon the angular radiant intensity distribution and of and also upon the form of the scattering function. As Kagan and Yudin [62] show, the coefficients m, and m2 are quite simple (and universal) functions of the optical thickness of the atmosphere in the case of isotropic scattering (spherical scattering function). Since, however, the scattering of light in the real atmosphere is always anisotropic, it is vital to know the importance of this fact. Certain useful conclusions can be derived on the basis of qualitative analysis of Eqs. (4.69). Let us consider such derivation, following the author's works [63, 641. Introducing an optical thickness t = J,"oe dz, and taking the case of shortwave radiation transfer in a nonabsorbing atmosphere, instead of (4.69) we have

r,

r,

r,

(4.80) In computating total radiation in a cloudless sky, the following transfer equations obtained by nonexact deduction were often used :

" ' dt

- dF2 -

dt

E,

sec &(F, - F,)

(4.81) Y

where el is the portion of light scattered backward. As seen from Eq. (4.80) the above equations are erratic. In reality, in the case of the computation of total radiation, the upgoing radiation may in the first approximation be considered isotropic (if particular situations with specular reflection are excluded). This means that

206

Scattering of Radiation in the Atmosphere

-

m, = 2. The angular distribution of the downcoming radiation has a very marked maximum at 8 = 8, and consequently m, see 8,. As regards the values T,and T,,it is easily understandable [see (4.72) and (4.73)] that they characterize the portion of radiation scattered backward. If we take F, = T,= E, then Eq. (4.80) in application to the case of the total radiation computation will be of the form dF1 dF2 - -= E , sec 8,F,

dz

dz

- 2&,F,

(4.82)

As can be seen, these equations do not coincide with (4.81). The author showed that similar conclusions are derived in the analysis of certain other variants of approximate equations deduced from nonexact consideration. It will be shown later that equations of the type of (4.82) can be defined more accurately on the basis of quantitative consideration. Let us now turn to the quantitative analysis of the variability of the coefficients m,, m,, F,, F,, as required by the problem of shortwave (diffuse, total) radiation transfer in the atmosphere. Unfortunately, at present there are no data available on the angular shortwave radiation intensity distribution at different altitudes. We can therefore use data of measurements near the earth’s surface. At a fixed z the values m, , m,,T,, T,must be constant in the case of the possibility of using (4.69) as common differential equations. The variability of the said values will characterize the possibility of approximation of the transfer equations (4.69). We begin our consideration with analysis of the variability of the coefficients m, and m,. Note first of all that, as follows from (4.70) and (4.71), the coefficients m, and m, are certain mean secants of zenith distances that determine the direction of the “center of gravity” of radiation. The author and Senderikhina [64] have computed the coefficients m, and m 2 for the total radiation (that is, in the case when F, and F, are the upgoing and downcoming total radiation fluxes) and also the coefficient m, for the diffuse radiation, which will further be denoted by mB. Once we have computed m D ,we shall have no difficulty in determining m 2 from an obvious formula that follows from (4.71): m 2=

+ +

S’ sec 8, Dm, S‘ D

(4.83)

where S‘ and D are the fluxes of direct solar and diffuse radiation on a horizontal plane, respectively.

4.6. Elementary Theory of Radiative Transfer

207

Let us first give some results relating to the case of a cloudless sky. The coefficient m, in this case is apt to be greatly variable. It has been stated, however, that in the first approximation, m, is a linear function of see 8,; that is, m, = a sec 8, b (4.84)

+

According to the author and Senderikhina, a = 0.21, b = 1.78, and the mean error of computation from (4.84) is 3 percent. The dependence of m, upon sec8 shows the influence of anisotropic reflection of radiation increasing with the increase of the solar zenith distance (at small 8, the coefficient m, approaches the “isotropic” value equal to 2). It is important, however, that b a ; that is the above approximation is to be preferred to the approximation m, 21 sec 8,. It is natural that the dependence of mDupon 8, must be still more markedly expressed. Observations confirm this conclusion. In the given case we have

>

m,

= a’ sec 8,

+ b’

(4.85)

From observations, a‘ = 0.54, b‘ = 1.09. The average error is 3 percent. As seen, b‘ < b, which means that the anisotropy of scattered radiation is greater than that of reflected radiation. Still more anisotropic is total radiation in a cloudless sky. In this case the coefficient m 2 may be considered to be practically equal to sec 8,:

m 2= sec 8,

(4.86)

The mean departure of m 2 from sec8, is about 4 percent. It is obvious that such a result is a consequence of the scattered radiation of a cloudless sky being small in comparison with the direct solar radiation. Only at significant solar zenith distances can m 2 differ notably from sec8,. Quite distinct are the results of computation of m,, mD, and m 2for the conditions of overcast sky. In this case, m, differs but slightly from m 2 , since reflected radiation is practically isotropic. However, mD still shows a great variability, which ranges from 1.47 to 2.12, the dependence between m, and 8, being absent. In the given case the coefficient m, varies greatly at a slightly varying solar height, owing to the peculiarities of the cloud cover. Consider now the results of the computation of I‘,and I‘, (in a cloudless sky). As was already said, to compute the values I‘, and I‘, it is necessary to have data not only on the angular distribution of scattered and reflected radiation but also on the scattering function. As regards the mean atmospher-

208

Scattering of Radiation in the Atmosphere

ic scattering function, it can be approximately determined from measurement data on the diffuse radiation intensity at the solar almucantharat. Thus, knowing the angular distribution of the diffuse radiation intensity, it is possible to find the scattering function; this method was applied in the given case. Computations show that the values rl and r2(particularly the latter) vary, depending on the scattering-function character, in a fairly marked way. Also marked is the dependence of rl and r2upon the solar zenith distance at a constant scattering function. The vaiarion for T,and T2 is 0.25 to 0.34 and 0.05 to 0.39, respectively. The coefficients m, and m2 increase with the increase of the solar zenith distance. These values come closer only at very long zenith distances. The remaining cases show that the portion of radiation scattered backward is much different for the upgoing and downcoming radiant fluxes. The above results have demonstrated that even in the approximation of isotropic scattering, when rl = T2 = 112, a considerable variability of the coefficients m, and m 2 under the influence of the variation of the angular shortwave radiation intensity distribution, takes place. This variability, however, can be taken into account by substituting the values m, , m,, and m2 in the transfer equation in the form (4.84), (4.85), (4.86). If the scattering anisotropy, always present in the real atmosphere, is taken into consideration, the situation becomes more complex: the variability of rl and do not correlate with 8, and is quite great. The latter does not permit introducing the averaged values and T,because considerable variations of individual values, relative of to the mean, will undoubtedly lead to great errors in the approximation of (4.69). To evaluate these possible errors, the variations of the total radiation flux at the surface level due to the variability of m, , m, ,I',, r2have been calculated. The results are given in Table 4.5 (on the assumption of a purely scattering atmosphere, that is, k = 0). From the consideration of Table 4.5 it follows that the errors of the total radiation computation exceed the errors of measurement many times over. We see thus that the approximation of the transfer equations (4.69) as common differentials may lead to great errors in the results. Therefore a sufficiently reliable method for quantitative computations of shortwave atmospheric radiant fluxes cannot be developed on the basis of the above approximate treatment. Approximate equations of the considered type are valid only for rough estimates. We shall consider later some examples of application of similar equations. More detail can be found in Malkevich's work [65].

r,

r,

209

4.7. Structure of Atmospheric Aerosol

TABLE 4.5 Average, 6 F2 (0),andMaximum, amaxF, (0),Relative(Percent) Departureof TotalRadiation Fluxes from the Mean due to the Variability of the Coefficientsm,, m,, TI,and T',at Different Values of the Albedo of the Underlying Surface A . After Kondratyev and Senderikhina [64]

I

25 50

i

A=0.0

A=0.4

24 45

26 44

1

i

A=0.0

A = 0.4

40 71

40 73

4.7. Radiation Scattering and the Structure of Atmospheric Aerosol We have seen that the regularities of radiation scattering are essentially dependent on the microstructure of the scattering medium, which enables solution of the inverse problem of determination of microphysical characteristics of aerosol (fog, clouds, dust aerosol) from observations of the peculiarities of the radiation scattering by the given aerosol. Effective methods for solution of this problem can be found in the works of a number of Soviet scientists (K. S. Shifrin, E. M. Feugelson, A. E. Mikirov, and G. D. Petrov) and scientists of other countries (as of [66, 671 etc.). Let us present the angular intensity distribution Z,(p) of the radiation scattered on polydisperse aerosol and characterized by the curve of size distribution of particles f ( a ) , using the following formula, similar to (4.49) :

where Zo,a is the incident radiation intensity, and F(p, a) is the optical cross section of scattering particles. Having computed the dependence of F(p, a) upon wavelength for a fixed scattering angle p and measured the energy distribution in the spectrum of diffuse radiation for the same angle p, it is possible to find f ( a ) by converting the integral in (4.87). The general form of this problem, however, is quite complex. As Shifrin [68] showed, the solution is simplified if the scattering on sufficiently large particles (e 2 1 ) over small angles (p i 3-5') is investigated. In this case

210

Scattering of Radiation in the Atmosphere

Thus, in place of (4.87), we obtain (4.88)

where Jl(e, p) is the Bessel function of the first order. It is expedient to transform the latter formula as

Treating the right-hand side integral as a generalized Fourier transformation for the function f ( e ) e , by converting this integral it is possible to obtain (4.89)

Here F(x) = xJl(x)Yl(x),where Y,(x) is the Bessel function of the second kind. It should be mentioned that this formula has been derived without taking account of the influence of multiple scattering of light, which limits its application as regards the permitted optical density of the given aerosol. In practice (4.89) can be used in a dual way: either by immediate application to the computation of f(e)e-at a fixed wavelength 1 equivalent to f(a)a-from the measured angular distribution of the scattered light intensity, or by giving a definite form tof(a) and the following determination of the distribution function parameters. The latter way is simpler because it reduces the solution of the problem to the determination of several parameters instead of finding numerous f(u) values. It is known, for example, that the size distribution of droplets in clouds is satisfactorily described by the following empirical formula : f ( a ) = Aap e-ca where A , p, c are parameters. Having in mind (4.90), in place of (4.88) we derive

(4.90)

4.7. Structure of Atmospheric Aerosol

Here, A r a p e-ca da ume) and therefore

=

21 1

N ( N is the total number of droplets per unit vol-

Thus

where

Introducing a new variable z

where C* = (A/2np)c; ,u* = ,u to the following results: 1 Q)jl*=l

=

2 4 1 - .2)1'2

=

(2np/A)a, we have

+ 2. The computation of the integral leads ((2 - t Z ) E ( t ) - 2(1

-

Here E ( t ) and K ( t ) are full elliptical integrals. The subsequent values pp*(c*) can be found from the following recurrent relation :

Using the above relations, Mikirov [69] computed the values of 12(p) for three scattering angle values at the ,u parameter from 1 to 8 and the c from 1 to 20 (for the visible spectrum). After that, two families of the curves Z(pl)/Z(pz) and I(p1)/Z(p3)were plotted for different ,u and c (Fig. 4.14). Measuring the above relations for the scattered radiation intensities, it is possible, with the help of the graphs of Fig. 4.14 [69] to find the ,u and c values and consequently the droplet size distribution function. Shifrin and Golikov [68] have shown, however, that the method for determination of the droplet size distribution function, based on measurements of the scattered light intensity at two fixed angles, may sometimes lead to significant errors. Far more reliable results are obtained from data

212

Scattering of Radiation in the Atmosphere

on the scattering function at small angles of scattering with the immediate use of (4.89). In this case the accuracy of determination of the aerosol spectrum according to the method of scattering measurement at small angles is about 20 to 30 percent.

I

0

1

2

3

4

c

1

2

3

4

5

6 c

FIG. 4.14 Determination of aerosol microstructure according to optical method. After Mikirov [69].

Feugelson [l 11 has developed a method for determination of the aerosol microstructure which takes into account the influence of multiple scattering. This, however, leads to complex and cumbersome results, although the problem is stated more precisely. A simple semiempirical variant of determination of the fog microstructure from data on transparency and water content has been proposed by Barteneva and Pliakova [71].

REFERENCES 1 . Staude, N. M. (1946). The Bemporad function and its significance in atmospheric optics. Proc. Acad. Sci. Kazakh Rep., Ser. Astron. and Phys. No., 2. 2. Staude, N. M. (1949). The twilight method for investigation in the stratosphere. Proc. Acad. Sci. USSR,Ser. Geograph. Geophys. No. 4. 3. Tverskoy, P. N. (1939). A Course in Geophysics. Gostechizdat, Moscow. 4. “Handbuch der Geophysik,” Vol. VIII. 1943. 5. Staude, N. M. (1929). On the determination of the transparency coefficient of the earth’s atmosphere. Proc. Lesgaft Znst. 15, Nos. 1 and 2. 6. Krat, V. A. (1942). Some problems of the theory of light scattering in the earth’s atmosphere. Astron. J, 19, iss. 6.

ReferenceS

213

7. Staude, N. M. (1949). Illumination of the atmosphere (aureole) from terrestrial sources of light. Proc. Acad. Sci. USSR,Ser. Geograph. Geophys. No. 1. 8. Averkiev, M. S. (1951). On the determination of the maincharacteristics of atmospheric transparency. Hydrometeorol. Directorate’s Bull. No. 1. 9. Rodionov, S. F. (1950). Atmospheric transparency in the ultraviolet spectral region. Proc. Acad. Sci. USSR,Ser. Geograph. Geophys. No. 4. 10. Shifrin, K. S. (1951). “Scattering of Light in a Turbid Medium.” Gostekhizdat, Moscow. 11. Van de Hulst, H. C. (1957). “Light Scattering by Small Particles.” Wiley, New York. 12. Bullrich, K. (1964). Scattered radiation in the atmosphere and the natural aerosol. Advan. Geophys. 10. 13. Proc. Interdisciplinary ConJ Electromagnetic Scattering, 1963. Pergamon Press, Oxford. 13a. Carrier, L. W., and Nugent, L. J. (1965). Comparison of some recent experimental results of coherent and incoherent light scattering theory. Appl. Opt. 4, No. 11. 14. Mandelstamm, L. I. (1948). “Complete Works,” vol. I. Acad. Sci. U.S.S.R., Moscow. 15. Hvostikov, I. A. (1940). Theory of light scattering. Advan. Phys. Sci. 24, NO. 2. 16. Volkenstein, M. V. (1951). “Molecular Optics.” Gostekhizdat, Moscow. 17. Tamm, I. E. (1946). “The Basic Theory of Electricity.” Gostekhizdat, Moscow. 18. Tihanovsky, I. I. (1927). Investigation of the sky polarization. J. Geophys. Meteorol. 4, No. 2. 19. Tihanovsky, I. I. (1928). Theory of the sky polarization and brightness for the absolutely clean earth’s atmosphere. Proc. Crimean Pedagog. Znst. 2. 20. Tichanowsky, I. I. (1927). Theorie der Lichtzerstreuung in der Erdatmosphare. Physik. 28, 680; 29, 442 (1928). 21. Chandrasekhar, S., and Elbert, D. (1951). Polarization of the sunlit sky. Nature 167. 22. Rosenberg, G. V. (1949). Polarization of the secondary scattered light in ,the case of molecular scattering. Proc. Acad. Sci. USSR, Ser. Geogrph. Geophys. NO. 2. 22a. Newkirk, G., Jr., and Eddy, J. A. (1964). Light scattering by particles in the upper atmosphere. J. Atmospheric Sci. 21, No. 1. 22b. Mikirov, A. E. (1965). Investigation of atmospheric brightness at 120-150 km. Space Invest. 3, No. 2. 23. Bibliography on numerical computations of scattering and absorption of electromagnetic radiation for spherical particles based on the Mie theory. Prepared by R. B. Penndorf, Avco Corp., Wilmington, Mass. 1962. 23a. Smart, C., and Vand, V. (1964). Approximate formula for total scattering of electroAm. 54, No. 10. magnetic radiation by spheres. J . Opt. SOC. 23b. Smirnov, A. S., and Maiov, I. P. (1964). An approximate expression for the coefficient of light scattering by dielectric nonabsorbing spheres. Opt. Spectr. (USSR) (English Transl.), 1, No. 1. 23c. Maltzev, Y. V. (1960). On the comparison of functions of light scattering by spherical particles. Opt. Spectr. (USSR) (English Transl.) 8, No. 5. 24. Shifrin, K. S. (1950). The coefficient of light scattering by large particles. Proc. Acad. Sci. USSR, Ser. Geograph. Geophys. No. 1. 25. Shifrin, K. S. (1951). Light scattering by a large transparent particle of the finite size. Proc. Main Geophys. Oh.No. 26 (88). 26. Shifrin, K. S. (1950). On the theory of light scattering in the atmosphere. Proc. Main Geophys. No. 19 (81).

214

Scattering of Radiation in the Atmosphere

27. Shifrin, K. S. (1950). Light scattering by large water droplets and polarization of light in rainbows. Proc. Acad. Sci. USSR, Ser. Geogrph. Geophys. No. 2. 28. Shifrin, K. S. (1947). The function of light scattering by atmospheric admixture particles in the form of sticks and disks. Proc. Main Geophys. Obs. No. 6 (68). 29. Atlas, D., Kerker, M., and Hitschfeld, W. (1953). Scattering and attenuation by nonspherical atmospheric particles. J. Armospheric Terrest. Phys. 3, No. 2. 29a. Fedorova, E. 0. (1955). A method for determination of the functions of light scattering by large particles of arbitrary form. Light Techn. No. 4. 30. Shifrin, K. S. (1952). Light scattering by double-layer particles. Proc. Acud. Sci. USSR, Ser. Geophys. No. 2. 30a. Fenn, R. W., and Oser, H. (1965). Scattering properties of concentric soot-water spheres for visible and infrared light. Appl. Opt. 4, No. 11. 31. Krasilnikov, V. A. (1949). On fluctuations of the angle of income in the phenomenon of stellar scintillation. Rept. Acud. Sci. USSR 65, No. 3. 32. Obukhov, A. M. (1949). The temperature field structure in a turbulent flux. Proc. Acad. Sci. USSR, Ser. Geograph. Geophys. No. 1. 33. Megaw, E. C. S. (1950). Scattering of electromagnetic waves by atmospheric turbulence. Nature 166. 34. Staras, H. (1952). Scattering of electromagnetic energy in randomly inhomogeneous atmosphere. J . Appl. Phys. 23, No. 10. 35. Obukhov, A. M. (1953). The influence of weak atmospheric inhomogeneities on the propagation of sound and light. Proc. Acad. Sci. USSR, Ser. Geophys. NO. 2. 36. Gillford, F., and Mikesell, A. H. (1953). Atmospheric turbulence and the scintillation of starlight. Weather 8, No. 7. 36a. Barteneva, 0. D. (1960). Light scattering functions in the boundary layer of the atmosphere. Proc. Acad. Sci. USSR, Ser. Geophys. No. 12. 37. Krat, V. A. (1943). The function of light scattering in the earth’s atmosphere. Asfron. J., 20, NOS. 5-6. 38. Krat, V. A. (1946). Some problems of theory of visibility of terrestrial targets from aircraft. Proc. Main Astron. Obs. No. 135. 38a. Livshitz, G. Sh. (1965). “Light Scattering in the Atmosphere,” Part I. Nauka, Alma-Ata. 39. Piaskowska-Fesenkova, E. V. (1957). “ Investigation in Light Scattering in the Earth’s Atmosphere.” Akad. Nauk. USSR, Moscow. 40. Feugelson, E. M., Malkevich, M.S., Kogan, S. Y., Koronatova, T. D., Glazova, K. S., Kuznetzova, M. A. (1958). “Computation of the Atmospheric Light Brightness in Unisotropic Scattering,” Part I, Proc. Inst. Atmospheric Phys., Acad. Sci., U.S.S.R., Moscow. 40a. Belov, V. F. (1957). Investigation of scattering functions in the troposphere and lower stratosphere. Proc. Main Aerol. Obs. No. 23. 41. Belov, V. F. (1956). “Measurement of the Main Optical Characteristics of the Boundary Air Layer.” Gidrometeoizdat, Moscow. 42. Chayanov, B. A. (1962). Measurements of local scattering functions in a free atmosphere. Proc. Main. Aerol. Obs. No. 45. 42a. Sandormirsky, A. B., Altovskaya, N. P., and Trifonova, G. I. (1964). Investigation of the Daylight Sky Brightness a t 8.0-17.5 km Altitude.” Actinom. Atmospheric Opt., Nauka, Moscow. 42b.. Shifrin, K. S., and Perelman, A. Y. (1964). “Calculation of the Spectrum of Particles

References

215

from Spectral Transparency Data.” Actinom. Atmospheric Opt., Nauka, Moscow. 43. Schiiepp, W. (1949). Die Bestimmung der Komponenten der atmospharischen Trubung aus Aktinometermessung. Arch. Meteorol., Geophys. Bioklimatol. B1, Nos. 3 and 4. 44. Volz, F. (1954). Die Optik und Meteorologie der atmospharischen Trubung. Ber. Deut. Wetterdienstes 2, No. 13. 45. Rabinovich, Y. I. (1961). Vertical distribution of aerosol attenuation in the troposphere. Proc. Main Geophys. Obs. No. 118. 46. Waldram, J. M. (1945). Measurements of the photometric properties of the upper atmosphere. Quart. J. Roy. Meteorol. SOC.71, No., 309 and 310. 46a. Rosenberg, G . V., and Nikolayeva-Tereshkova, V. V. (1965). Stratospheric aerosol as measured from spacecraft. Proc. Acad, Sci. USSR, Ser. Phys. Atmosphere and Ocean 1, No. 4. 47. Poliakova, E. A., and Shifrin, K. S. (1953). The microstructure and transparency of rains. Proc. Main Geophys. Obs. No. 42 (104). 48. Zuyev, V. E. and Kabanov, M. V. (1964). “Attenuation of a Light Signal with Taking Account of Diffuse Scattering in the medium.” Actinom. Atmospheric Opt., Nauka, Moscow. 49. Zuyev, V. E. (1964). “Spectral Transparency and Microstructure of Natural Fogs.” Actinom. Atmospheric Opt., NaJka, Moscow. 50. Hmelevtzev, S. S. (1964). “Installations for Continuous Measuring Aerosol Microstructure According to the Method of Small Angles.” Actinom. Atmospheric Opt., Nauka, Moscow. 51. Zuyev, V. E., Koshelev, B. P., Tvorogov, S. D., and Hmelevtzev, S. S. (1965). Attenuation of visible and infrared radiations by artificial water fogs. Proc. Acad. Sci. USSR, Ser. Phys. Atmosphere and Ocean, 1, No. 5. 51a. Dave, J. V. (1964). Importance of higher order scattering in a molecular atmosphere. J. Opt. SOC.Am. 54, No. 4. 51b. Dave, J. V. (1964). Meaning of successive iteration of the auxiliary equation in the theory of radiative transfer. Astrophys. J. 140, No. 3. 52. Zuyev, V. E., Kabanov, M. V., Koshelev, B. P., Tvorogov, S. D., and Hmelevtzev, S. S. (1964). Spectral transparency and microstructure of artificial fogs. I. Instruments and results of experimental investigations. 11. Calculation of attenuation coefficients and comparison with experimental data. Bull. High Schools, Phys. Nos. 2 and 3. 53. Angstrom, A. (1963). The parameters of atmospheric turbidity. Tellus 16, No. 1 . 54. Shifrin, K. S., and Minin, I. N. (1957). On the theory of nonhorizontal visibility. Proc. Main Geophys. Obs. No. 68. 55. h g s t r o m , A. (1961). Techniques of determining the turbidity of the atmosphere. Tellus 13, No. 2. 56. Zuyev, V. E., Kabanov, M. V., and Borovoy, A. G. (1963). Extinction of a light signal in a scattering medium. I. Results of computation of emission once scattered from aside. Bull. High Schools, Phys. No. 6. 57. Zuyev, V. E., and Kabanov, M. V. (1964). Extinction of a light signal in a scattering medium. 11. Experimental investigations in the artificial fogs chamber. Bull. High Schools, P h y . No. 1. 58. Zuyev, V. E., Kabanov, M. V., and Savelyev, B. A. (1964). Extinction of a light signal in a scattering medium. 111. On the applicability of the exponential law of extinction. Bull. High Schools, Phys. No. 5.

216

Scattering of Radiation in the Atmosphere

59. Sobolev, V. V. (1963). Radiative Transfer. Van Nostrand, Princeton, New Jersey. 60. Kuznetzov, E. S. (1951). General method for composition of approximate transfer equations. Proc. Acad. Sci. USSR,Ser. Geophys. No. 4. 61. Kuznetzov, E. S. (1942). On the problem of approximate transfer equations in a scattering and absorbing medium. Rept. Acad. Sci. USSR 37, Nos. 7 and 8. 62. Yudin, M. I., and Kagan R. L. (1956). An approximate solution of the equation of light scattering. Proc. Acad. Sci. USSR,Ser. Geophys. No. 8. 63. Kondratyev, K. Ya. (1956). On the Schwarzschild approximation in the theory of radiative transfer. Proc. Leningrad State Univ., Ser. Geophys. No. 9. 64. Kondratyev, K. Ya., and Senderikhina, I. L. (1959). On the approximate radiative transfer equations. Proc. Acad. Sci. USSR,Ser. Geophys. No. 5. 65. Atroshenko, V. S., Glazova, K. S., Kogan, S.Y., Koronatov, T. D., Kuznetzov, M. A., Malkevich, M. S.,and Feugelson, E. M., (1962). “ Computation of the Atmospheric Light Brightness in Unisotropic Scattering,” Proc. Inst. Atmospheric Phys., Acad. Sci., U.S.S.R., No. 3. Moscow. 65a. Laktionov, A. G. (1964). On the relation of light scattering in a free atmosphere to the vertical distribution of the aerosol particles concentration. Proc. Acad. Sci. USSR, Ser. Geophys. No. 6. 66. Hodkinson, J. R., and Greenfield, J. R. (1965). Response calculations for light-scattering aerosol counters and photometers. Appl. Opt. 4, No. 11. 67. Foitzik, L. (1965). The spectral extinction of the atmospheric aerosol by Mie particles with different Gaussian distributions. Beitr. Geophys. 73, No. 3. 68. Shifrin, K. S., and Golikov, V. I. (1964). Measurements of microstructure with the method of small angles. Proc. Main Geophys. Obs. No. 152. 69. Mikirov, A. E. (1959). On small angles of the scattering function. Proc. Acad. Sci. USSR,Ser. Geophys. No. 2. 70. Feugelson, E. M. (1954). Deterpination of the water content in clouds from their scattering capacity. Proc. Geophys. Znst. No. 23 (150). 71. Barteneva, 0. D., and Poliakova, E. A. (1965). Investigation of attenuation and scattering of light in natural fogs in dependence on their microphysical properties. Proc. Acad. Sci. USSR,Ser. Phys. Atmosphere Ocean 1, No. 2.

5 DIRECT SOLAR RADIATION

5.1. Distribution of Energy in the Solar Spectrum at the Earth’s Surface Level 1. General Characteristics. Except for a few recently performed rocket investigations, all measurements of the energy distribution in the spectrum of solar radiation were conducted at the surface level. Therefore we shall start consideration of the solar radiation spectral composition by studying measurement data at the earth’s surface. The previous sections state that the attenuation of solar radiation in the atmosphere is essentially selective. Hence it follows that the spectral composition of solar radiation will apparently be different, depending on the state of atmospheric transparency and the solar zenith distance (that is, at different atmospheric masses). The variation of the solar radiation spectral composition is mainly due to three factors: (1) aerosol scattering on large particles (dust, water droplets, etc.) ; (2) molecular scattering, which is most intensive in the region of short wavelengths; (3) selective absorption by water vapor in the near-infrared. In the ultraviolet range, ozone decisively affects the spectral composition of solar radiation. The peculiarities of the variation of the solar radiation spectral composition find spectacular illustration in Figs. 5.1 and 5.2, which have been plotted from observational data provided by the Smithsonian Institution [l]. On the axis of abscissas is plotted the wavelength in Angstroms; on the axis of ordinates, the values of monochromatic solar radiation fluxes are given in conventional units. The curves of Figs. 5.1 and 5.2 have been obtained from spectrobolometric measurement data. Up to the past decades this method for measuring the energy distribution in the solar radiation spectrum was most 217

218

Direct Solar Radiation

FIG. 5.1 Energy distribution in the solar spectrum with clear skies and at different atmospheric masses; w, = 0.05 g/cma.

common. Recent years have introduced electrospectrophotometry as a method for investigating the solar spectrum. Reference to instruments for such measurements was given in Sec. 2.6 (see also [2, 31). Figure 5.3 presents the scheme of a spectrobolometer. The solar beam, reflecting from mirrors A and B, passes through the slit of the spectrometer C and, being reflected from the collimator D, hits the prism E. The refracted monochromatic beam is directed by its reflection from the mirrors F and G to the bolometer H positioned in the focus of a spherical concave mirror G. The bolometer’s main part consists of two blackened metallic laminae, one of which is a radiation detector and the other of which is protected against the effects of radiation. Both laminae are components of a Wheatstone bridge. In the heating of the lamina irradiated by solar radiation, the resistance of the radiation detector changes, which leads to the breakdown of electric equilibrium in the bridge. The reading of the galvanometer, registering the appearance of electric current, serves as a measure of the observed radiant

WAVELENGTH,

A8

FIG. 5.2 Energy distribution in the solar spectrum in the conditions o f a turbid atmosphere; w, = 1.37g/cm2.

GI

E -&? F H

-4D

FIG. 5.3 The scheme of the spectrobolometer.

flux value. The spectrum obtained after the passing of radiation through the prism E is gradually transported through the bolometer's receiving lamina by means of a clock mechanism. By setting the galvanometer on photorecording, it is possible immediately to obtain in relative units the curve of the energy distribution in the solar radiation spectrum. In Fig. 5.1 is shown the energy distribution in the solar spectrum, plotted from spectrobolometric measurement data taken in high mountains. A high atmospheric transparency in this case was observed together with a low water vapor content in the thickness of the atmosphere, amounting to w, = 0.05 g/cm2. The five lower curves were plotted on measurement data at atmospheric masses m equal to 1, 2, 3, 4, 5 (the case m = 5 corresponds to the lowermost curve). The uppermost curve is derived by extra-

220

Direct Solar Radiation

polation from the measured values of solar radiation monochromatic fluxes outside the atmosphere. This curve, therefore, approximately characterizes the energy distribution outside the atmosphere. The results of analogous measurements of the solar radiation spectral composition at sea level are presented in Fig. 5.2. In this case the atmosphere was far more turbid, the total water vapor content being w, = 1.37 g/cm2. From the comparison of Figs. 5.1 and 5.2 it is seen that the variation of atmospheric transparency is an essential factor in the spectral energy distribution near the earth's surface. Especially spectacular is the notable deepening of water vapor bands in the infrared associated with the increase of the total water vapor content in the atmosphere. Both figures also demonstrate a considerable variation of the solar radiation spectral composition dependence upon the height of the sun (that is, at different atmospheric masses m). The peculiarity of the spectral energy distribution curves in Figs. 5.1 and 5.2 is their very sharp decline in the region of short wavelengths. As already mentioned in Chapter 3, this sharp decline results from the absorption of ultraviolet solar radiation by ozone. It may be of interest to compare the above curves derived from measurement data with analogous curves for an ideal (dry and clean) atmosphere as obtained from computations on the basis of the theory of molecular scattering. Figure 5.4 shows the curves of energy distribution in the solar spectrum near the earth's surface under conditions of ideal atmosphere with atmospheric masses equal to 1.1 (h, = 65O), 2(h, = 30°), 3(h, = 19O), 4(h, = 11.3O) as given in [4]. The upper curve in this figure characterizes the spectral distribution of solar radiation outside the atmosphere. Table 5.1 gives numeri-

FIG. 5.4

Energy distribution in the solar spectrum in the conditions of an ideal (clean and dry) atmosphere.

5.1. Distribution of Energy in the Solar Spectrum

22 1

cal results of somewhat more accurate computations of the energy distribution in the solar spectrum in the ideal atmosphere, performed for rn varying from 0 to 10 (Chapter 4, Ref. 4). The energy distribution in the solar spectrum outside the atmosphere corresponds here to the solar constant value 1.94 cal/cm2 min. Table 5.1 also characterizes the variation of the portion of ultraviolet ( A < 0.40 p ) and infrared (A > 0.74 p ) radiation related to the variation of atmospheric mass. The last figures demonstrate the increase of the portions of infrared radiation with the increase of atmospheric mass (that is, with the decrease of the solar altitude) caused by the effect of molecular scattering. Table 5.1 shows in detail the dependence of the variation of solar radiation spectral composition on solar height under the conditions of an ideal atmosphere. The figures in the horizontal lines of the table characterize the variation of monochromatic radiant fluxes in this or other spectral regions, depending on the height of the sun. The totals of the column values determine the spectral composition of solar radiation at different atmospheric masses (solar height). Kalitin [4],using computational data of V. A. Berozkin, which are similar to those of Table 5.1, plotted a graph (see Fig. 5.5) following the variation of the absolute and relative distribution of energy in the solar spectrum, depending upon the height of the sun. The curves in Fig. 5.5 determine

col/cm2 min

FIG. 5.5

Energy distribution in the solar spectrum in dependence upon solar height.

222

Direct Solar Radiation TABLE 5.1

Distribution of Energy in the Solar Spectrum at Different Masses for a Dry and CIean Atmosphere (MolecuIar Scattering) in cal ern+ min-I.

28-30 30-32 32-34 34-36 36-38 38-40 40-42 4 2 4 44-46 46-48 48-50 50-52 52-54 54-56 56-58 58-60 60-62 6244 W66 66-70 70-74 74-80 80-90 90-100 1w200 > 200 Integral radiation

2.6 11.5 22.8 23.7 30.5 40.0 55.0 57.0 61.0 62.9 62.5 59.7 57.3 55.5 54.6 54.3 52.8 50.3 48.4 92.3 83.1 106.4 143.4 113.4 426.0 113.0

1.3 7.0 15.5 17.6 24.0 33.2 47.2 50.2 54.9 57.7 58.0 56.0 54.4 53.0 52.6 52.5 51.2 49.0 47.2 90.7 81.8 105.0 142.4 112.8 425.2 113.0

0.7 4.2 10.6 13.1 19.0 27.4 40.4 44.3 49.5 52.8 53.8 52.7 51.5 50.7 50.4 50.7 49.7 47.7 46.1 88.8 80.6 103.8 141.0 112.2 424.6 113.0

0.2 1.6 4.9 7.2 11.8 18.8 29.7 34.4 40.2 44.3 46.6 46.5 46.2 46.2 46.6 47.3 46.8 45.2 44.1 85.4 78.1 101.5 138.8 111.1 423.0 113.0

0.6 2.3 4.0 7.4 12.9 21.7 26.8 32.6 37.1 40.3 41.1 41.5 42.2 43.1 44.2 44.0 42.8 42.0 82.2 75.7 99.1 136.5 110.0 421.6 113.0

uv

-

-

0.2 0.7 1.8 4.2 8.7 12.5 17.6 22.0 26.0 28.2 30.0 32.2 34.0 36.3 36.6 36.6 36.6 73.1 69.0 92.6 130.4 106.5 417.1 112.7

-

0.2 0.7 2.0 4.7 7.6 11.6 15.5 20.1 22.0 24.3 26.8 29.1 31.6 32.6 32.9 33.3 67.2 65.0 88.7 126. 105.9 413.9 112.7

0.0 0.3 0.9 2.6 4.7 7.6 10.9 14.2 17.1 19.7 22.4 24.8 27.4 28.8 29.5 30.5 62.6 61.4 84.0 122.5 102.5 411.0 112.6

1940.0 1853.4 1779.3 1659.5 1564.7 1487.7 1365.6 1274.5 1198.0

The portions of ultraviolet ( A < 0.40 p) and infrared spectrum are in percent:

IR

0.2 1.1 2.2 4.6 8.8 16.0 20.8 26.6 21.2 35.4 36.2 37.3 38.6 39.9 41.4 41.5 40.7 40.1 79.0 73.4 87.1 134.3 108.8 419.7 112.8

6.7 46.5

5.3 48.4

4.2 50.0

2.7 53.5

1.8 56.2

(A > 0.74 / I ) 1.1 58.1

radiation in this

0.5 63.0

0.2 66.6

0.1 69.6

the variation of monochromatic radiant fluxes in cal/cm2 min of certain wavelengths, whose values are indicated at the upper scale, in dependence on solar altitude. The figures in the intervals between these curves characterize the percentage of the integral radiant flux as distributed over spectral sections at the given altitude of the sun. The absolute values of radiant

5.1. Distribution of Energy in the Solar Spectrum

223

fluxes for individual spectral intervals can be determined from the difference between the abscissas corresponding to the ends of the considered interval. We find, for example, that at h, = 90' the portion of infrared radiation constitutes 49.1 percent of the total solar radiation flux; at h, = 10' it increases to 60.5 percent. Similar results are obtained with the help of Table 5.1. Birukova [5] has performed theoretical calculations of the energy distribution in the solar spectrum at different heights and at different atmospheric masses for the conditions of a dry and clean atmosphere, taking into account the solar radiation attenuation due to molecular scattering and absorption by ozone. These calculations have been based on the solar radiation spectral distribution outside the atmosphere, presented by F. Johnson (see Sec. 5.3). The results are summarized in Tables 6 and 7 of the Appendix. The comparison of the above-mentioned curves (Figs. 5.2, 5.3, 5.4) makes it quite evident that the distribution of energy in the solar radiation spectrum at the earth's surface, computed for an ideal atmosphere, differs much from the observed energy distributions. This difference becomes especially marked in the presence of a sufficient quantity of atmospheric water vapor and aerosols, which makes necessary the careful analysis of the available measurement data on the solar radiation energy distribution at the earth's surface. The curves in the figures give a sufficiently reliable general characteristic of the solar radiation spectral composition. Let us now discuss in more detail the energy distribution in the ultraviolet, visible, and infrared spectral regions.

2. The Ultraviolet Spectral Region. The amount of solar radiant energy corresponding to this spectral region is quite small. Figure 5.6 shows the dependence of ultraviolet flux of solar radiation in the wavelength interval 1 5 3132 A on the value of the integral direct solar radiation flux as given by Kalitin [4]. It is seen that the ultraviolet radiation in the considered range is from 0.01 percent of the full flux S at S = 1 cal/cm2 min to 0.08 at S = 1.35 cal/cm2 min. As small values of S correspond to small solar heights, it follows from Fig. 5.6 that the portion of ultraviolet radiation increases with the increase of the solar height. Although the portion of ultraviolet radiation in the total solar radiant energy flux is insignificant, nevertheless its investigation is quite vital. It is known, for example that ultraviolet solar radiation greatly affects the upper layers of the atmosphere. The absorption of ultraviolet radiation, in particular, is one of the main causes for high air temperatures in the upper atmospheric layers.

224

Direct Solar Radiation

The ultraviolet radiation of chromospheric flares has been stated to be responsible for ionospheric perturbations. It is also known that the dissociation of oxygen molecules under the influence of ultraviolet radiation of wavelengths iE < 2420 A accounts for the formation of the stratospheric ozone layer. The ultraviolet radiation of longer waves provokes dissociation of ozone and consequently contributes to the destruction of the ozone layer.

FULL FLUX OF-SOLAR R A D I A T I O I U , C O ~ / & ~ ~ ~

FIG. 5.6 The relationship between the ultraviolet and full fluxes of solar radiation.

Quite important are investigations in various biological effects of ultraviolet radiation. Short wavelength rays, for example, are effective bactericides, cause hemolysis and erythema, act as albumen coagulants, and affect living organisms in some other ways. Table 5.2 summarizes intensities of various biological effects of ultraviolet radiation according to its wavelength. The maximum effect is taken to be 100. Relations between ultraviolet radiation and the living activity of various organisms partly accounts for this area of research becoming the concern of biophysicists and bioclimatologists. Let us now consider the energy distribution in the ultraviolet solar spectrum at the earth’s surface. Note here, that, to date, this problem has been given quite unadequate attention and only in recent years have there been attempts to conduct appropriate investigations, employing modern rocketborn spectroscopic instruments. The first to measure the energy distribution in the ultraviolet range 2850 to 3300 A in absolute units were Rodionov et al. [ 6 ] .They performed these measurements in the Elbrus at 2000 m above sea level. The results of the measurements, obtained in 1935, are given in Table 5.3.

TABLE 5.2 Biological Effects of Ultraviolet Rradiation

Wavelength, mp 320

315

310

305

300

295

290

285

280

275

270

265

260

255

250

245

240

Erythema

0

1

11

33

83

98

31

9

6

7

14

25

42

54

56

57

56

Hemolysis

0

4

Bactericidal effect Coagulation of albumen

6 0

2

10

3

23

8

7 7

20

40

28

43

60

12

10

53

65

80

80

92

100

15 100

96

90

25 82

65

60

235

5’

2

226

Direct Solar Radiation TABLE 5.3 Energy Distribution in the Ultraviolet Region of the Solar Spectrum

A

Ultraviolet Radiation, erg/cma sec. A

2850 2900

2.1 x 10-6 7.1 x 10-4 2.5 x lo-' 5.6 x lo-' 7 . 1 x 10-1

3100 3200 3300 ~

~~~

Stair [7, 81 has obtained energy distributions in the ultraviolet at m = 1.39 and m = 1.43, which are shown in Fig. 5.7. The values of spectral radiant fluxes, plotted on the axis of ordinates, are expressed in relative units. As seen from this figure the energy distribution in the ultraviolet spectrum is complex. A great number of dips in the distribution curve results from the 22 20

-

18 16 -

g. ,14-.h

-B

12:

pr nl o - 8 6-

42-

-

Wovelength,

FLG. 5.7

Amp

Energy distribution in the ultraviolet solar spectrum at sea level.

5.1. Distribution of Energy in the Solar Spectrum

227

influence of various Fraunhofer lines. The curve falls in a particularly marked way in the vicinity 1 = 390 mp. When given a detailed investigation (as in [9]), it can be shown that the real case presents a still more complex picture than that shown in Fig. 5.7. For instance, in the narrow spectral interval 2935 to 3060 A 665 individual lines were found. This indicates that the curves of Fig. 5.7 should be treated as very smoothed. In general, however, a rapid decrease toward zero of the monochromatic radiant flux value in the ultraviolet spectrum at sea level, at 1 = 290 mp, has been noted. Of considerable interest, especially as viewed biologically, is the location of the lower boundary of the ultraviolet spectrum. A number of Soviet scientists, Mamontova [lo], Poliakova [ll], Yampolsky [12, 131, and others have contributed to the investigation of this problem. As already mentioned, the interruption of the solar spectrum from the side of short waves is caused by the absorption of solar radiation by ozone. Experiments show that the shortening of the solar spectrum on this side is the greater, the lower the solar height. According to Poliakova’s observations [l 1] at Pavlovsk (near Leningrad), the location of the ultraviolet spectrum’s lower boundary is characterized by the wavelength values given in Table 5.4. TABLE 5.4 Location of the Lower Boundary of the Ultraviolet Solar Spectrum. After Poliakova [ l l ]

Solar Height, deg

1 2 3 5 7 10 15 20 25 30 35 40 45 50

Boundary of the Spectrum,

v 420 382 352 327 318 312 306 304 302 300 298 297 296 295

228

Direct Solar Radiation

The absolute values ;Iminare determined by the sensitivity of the instruments used. The observed shortening of the minimal spectral wavelength ;Iminwith the increase of the solar height is related to the shortening of the solar ray path through the absorbing ozone layer. Certain variations of ;Iminat the same time are caused by tropospheric transparency fluctuations. According to the averaged observational data, it appears possible to present the dependence of Amin upon the height of the sun in a simple analytical form : ;Imin= u - b log (sin h,) At Pavlovsk the constant values u and b are equal to 290.0 and 25.8, respectively. At these values, ;Iminis expressed in millimicrons. When investigated experimentally, the dependence of the lower ultraviolet spectral boundary upon elevation of the observation point above sea level has been found to be very insignificant. This appears to be due to the presence of ozone in the upper atmospheric layers, where it absorbs ultraviolet radiation and causes the discontinuity of the solar spectrum. Figure 5.8 presents the energy distribution in the ultraviolet region at different altitudes exceeding 10 km. It can be seen that above this limit, a rapid increase of solar radiant flux in the ultraviolet spectrum is observed. We shall consider later some results of rocket solar spectrum investigations, which enable more careful study of the ultraviolet spectrum at high elevations.

A, A

FIG. 5.8 Energy distribution in the ultraviolet solar spectrum at heights over 10 km.

229

5.1. Distribution of Energy in the Solar Spectrum

The invariability of I min with height in the troposphere, however, does not mean that the ultraviolet flux is independent of height. Since attenuation of solar radiation due to the scattering decreases with height, it becomes clear that at a constant I the ultraviolet solar radiant flux will increase with height. Since the position of the ultraviolet spectrum lower boundary determined by the value I min depends considerably upon the solar height, it is natural that ultraviolet solar radiation will experience diurnal and annual variation. The minimal I min values are observed during afternoon hours and during the warmer part of the year; however, in autumn, not in summer, because of the minimal ozone content in the atmosphere in autumn months. At that time, solar radiation abonds in ultraviolet rays. Table 5.5 gives some numerical values characterizing the variation of il in dependence on solar height and season obtained in Switzerland. TABLE 5.5 Variation of lmi,(mp) according to Solar Height and Season Solar Height, deg Season

Spring Summer Autumn Winter Year

10

20

30

40

50

60

316 319 312 314 316

308 310 305 308 308

304 306 302 307 304

302 301 300 302

298 299 299

298 298 -

298

The decrease of Imin value with the increase of solar height means a simultaneous increase of the total flux of ultraviolet solar radiation. The daily range curve of ultraviolet radiation is experimentally shown to possess a marked maximum about noon, the maximum flux being observed somewhat earlier than at true noon. It may be associated with a higher atmospheric transparency before noon, compared with that in the afternoon.

3. The Visible and Infrared Spectral Regions. Energy distribution at the earth’s surface in these regions can be generally characterized by Figs. 5.1 and 5.2 and Table 5.1, which will be supplemented now with more data. Figure 5.9 gives energy distribution curves in the visible and (partly) in the ultraviolet and infrared regions from direct measurements and cal-

230

Direct Solar Radiation

culations made by different authors (see [14]). The curves 1, 2, and 3 are plotted from near-noon spectrophotometric measurements, with a radiant flux value at il = 560 mp assumed to be unity. Curves 4, 5, and 6 were calculated from a known energy distribution in the solar spectrum outside the atmosphere and from spectral atmospheric transparency. Curve 4 was calculated for m = 2 (h, = 30') during a dry clear day; curve 5 was calculated for the same altitude of the sun, but at a greater water vapor content in the atmosphere. Curve 6 is for m = 1.59 and the mean atmospheric transparency conditions. Curve 7 presents energy distribution in the blackbody emission spectrum at T = 5200°K. In plotting the curves 4, 5, 6, and 7, a radiant flux value at 1 = 560 mp was assumed to be unity. From consideration of Fig. 5.9 it follows that the spectral composition of solar radiation in the visible and ultraviolet regions undergoes significant variations caused by variations in atmospheric transparency. However, the differences between the calculated curves may be also due to the inequality of data on which curves 4, 5, and 6 were based. In spite of the variations of the solar radiation spectral composition in the visible region, all energy distribution curves here are fairly close to the curve of energy distribution in the blackbody emission spectrum at T = 5200°K. The same is observed in the near-infrared spectrum (700

FIG. 5.9 Energy distribution in the solar spectrum in dij%erent conditions.

5.1. Distribution of Energy in the Solar Spectrum

23 1

mp < A < 1000 mp). In the latter case, however, it is necessary to take account of the fact that the curves 4 and 5 of Fig. 5.9 were computed on the assumption that the attenuation of solar radiation is due to scattering only. Selective absorption will complicate the form of these curves, depending on wavelength (see Figs. 5.1 and 5.2). The visible and infrared spectral regions being of the chief importance in solar radiant energy distribution, it appears interesting to study the relation between their respective distributions of radiant energy. Kalitin [4,151 made numerous measurements of infrared radiation flux with wavelengths 1 > 0.65 p, conducted by means of a Schott RG 5 filter. Table 5.6 gives annual ranges of the integral S and infrared SIRfluxes of solar radiation and their ratio SIR/S. It is seen from this table that over the entire year the infrared radiation constitutes 60 percent of the integral solar radiation flux, being especially great during winter when solar height reaches minimum. In general, the dependence of the ratio SIRISupon solar height is quite pronounced. Kalitin [15] investigated this dependence and obtained observational data at Pavlovsk (near Leningrad), deriving the results given in Table 5.7. The annual range of infrared solar radiation incident on a perpendicular surface is determined by the joint influence of variations in solar height and transparency of the atmosphere. According to the observations at Karadag (Table 5.6), the maximum SIRis observed in winter, which may be explained by the higher atmospheric transparency and the minimal solar elevations at this time of the year. The observations at Pavlovsk give a contrary result: During winter the radiant flux S,, amounts to 0.68 cal/cm2 min; in summer it equals 0.70 cal/cm2 min. This decrease from summer to winter is due in the given case to the predominating influence of increasing solar radiation attenuation, owing to the decreasing solar height (increase of atmospheric mass) in winter. Interesting information on energy distribution over individual spectral intervals was obtained by Yaroslavtzev [16, 171 from actinometric measurement data (Mihelson actinometer) at Tashkent with application of glass Schott OG 1 and RG 2 filters. Mean annual relative values (in percent of the integral flux) of solar radiant fluxes on a perpendicular surface in difIt is seen from Table 5.8 that at Tashkent, as elsewhere, the portion of infrared radiation in the total flux is quite large. Another important feature of data by Yaroslavtzev is a comparatively low year-to-year variability of mean relative values of solar radiant fluxes in different spectral intervals. This certifies the stability of the mean relative, spectral, solar-energy distribution and the absence of selectivity of solar

TABLE 5.6 Relation between the Mean Monthly Noon Values of the Integral Flux and Infrared Solar Radiation Flux on a Perpendicular Surface from Observational Data at Karadag (Crimea) in 1936-40. Ajler Kalitin [15]

8

8

Flux

Jan.

Feb.

March

April

May

June

July

Aug.

Sept.

Oct.

Nov.

Dec.

S

1.20

1.35

1.32

1.32

1.31

1.25

1.23

1.22

1.24

1.25

1.24

1.21

!

SIR

0.84

0.90

0.85

0.84

0.82

0.78

0.76

0.76

0.79

0.80

0.82

0.82

c1

SIRIS

0.689

0.667

0.644

0.636

0.626

0.624

0.618

0.623

0.637

0.640

0.661

0.677

I'

g

233

5.1. Distribution of Energy in the Solar Spectrum

TABLE 5.7 Dependence of the Ratio SIRISupon Solar Height. After Kalitin [15] h, SIRIS

2 5 10 15 20 25 30 35 40 45 50 0.80 0.79 0.71 0.67 0.64 0.62 0.61 0 . 6 0 0.59 0.58 0.57

ferent spectral intervals at h, = 10'3' are given in Table 5.8. For comparison is given the energy distribution over the corresponding intervals outside the atmosphere. TABLE 5.8 Mean Annual Relative Values of Solar Radiant Fluxes in Different Spectral Intervals from Observations at Tashkent (193640).Afrer Yaroslavtzev [16]

Spectral Intervals, rnp Year

1936 1937 1938 1939 1940 Mean, % Outside the atmosphere, %

290-525

525-617

617-3000

18 17 15 18 20 17

13 16 17 17 17 16

69 67 68 65 63 67

27

12

61

radiation attenuation at a given solar height and a varying atmospheric transparency. Table 5.8 demonstrates clearly that, outside the atmosphere, solar radiation is far richer in short wavelengths than at the earth's surface. Further on we shall give a more detailed treatment of this matter. Yaroslavtzev [161 also investigated the solar radiation distribution over different spectral intervals. The corresponding observational data at h, = 19' are given in Table 5.9. We see that in northern latitudes the portion of infrared radiation (at the same solar height) notably increases. The main cause for this is the decrease of atmospheric humidity northward, which results in decrease of solar radiation attenuation in the infrared region. Because of the decreasing

234

Direct Solar Radiation TABLE 5.9

Percent Relative Distribution of Solar Radiation over Different Spectral Intervals, Depending upon the Latitude of an Observational Point

Observation Point

Ashkhabad Tashkent Yevpatoria Cape Schmidt Matochkin Shar

Latitude, deg

38.0 41.3 45.2 68.9 73.3

Pressure of Water Solar Vapor at the Height, deg Earth’s Surface, mm

8.9 7.9 6.9 2.5 1.8

19.3 19.3 19.3 19.1 19.1

Spectral Interval, mp

290-617

617-3,OOO

67 67 70 77 80

33 33 30 23 20

atmospheric humidity with height the infrared radiant flux increases as the altitude of the observational point above sea level increases. Table 5.10, compiled from observational data analyzed by Popov [18], illustrates this. TABLE 5.10 Dependence of the Ratio

Z ,

sIR/supon Altitude above Sea LeveLa After Popov

Observation Point

Altitude above Sea Level, m

sIR/s%

Yevpatoria Pavlovsk Ashkhabad Tashkent Alma-Ata Heirabad

-

61 64 67 67 72 83

30 219 478 848 2028

[181

The solar height is hm = 19.3’ for all cases.

5.2. Spectral Atmospheric Transparency Section 5.1 has given the basic information on energy distribution in the solar radiation spectrum at the earth’s surface. When known, similar energy distributions at different atmospheric masses permit obtaining the dependence of spectral radiant fluxes upon atmospheric mass. Making use of such

5.2. Spectral Atmospheric Transparency

235

dependence, it is possible to calculate atmospheric transparency coefficients for various narrow spectral intervals. Such calculation is based on the use of the following formula: sm,AA

=S~.~AP%

where the subscript A l means that the corresponding values relate to the spectral interval A l within which these values may be considered constant; S,,,, and So,,, are solar radiant fluxes at the earth’s surface and outside the atmosphere, respectively; pAnis the transparency coefficient. Applying logarithms in both parts of the preceding equation, we find log sm,AA

= log

sO,AA

$-

log P A 2

(5.1)

As seen from (5.1), the transparency coefficient can be determined graphically. The study of spectral atmospheric transparency is particularly important in connection with the determination of energy distribution in the solar spectrum outside the atmosphere and the subsequent finding of the integral solar radiation flux outside the earth’s atmosphere (the so-called solar constant). Spectral transparency was fully investigated at the Smithsonian Institution where the spectrobolometric method has been employed. The use of other methods has given voluminous material on the subject, as in the works of Lugin [19], Popov [18], Koocherov [20], Sokolova [21], Poliakova [22], Nikitinskaya [23], Toropova [24], Rabinovich and Guseva [24a], etc. Table 5.1 1, borrowed from Savostyanova’s work [14] presents a summary of spectral atmospheric transparency coefficients from observations at Kuchino (near Moscow, N. P. Lugin’s data), also at Tucson (Arizona, Pettit’s data) and Washington, D.C. (Abbot’s measurements). N. P. Lugin used photographic photometry with application of a quartz spectrograph. Pettit’s investigations were conducted at 760 m above sea level, with application of a double quartz monochromator. Abbot employed the spectrometric method. Abbot’s data (the “Washington” column of Table 5.1 1) were obtained by means of averaging results of observation over 20 days during the fall, a time at which the total water vapor content in the atmosphere was varying from 0.145 g/cm2 (“dry” days) to 1.46 g/cm2 (“humid” days). It is essential to mention that transparency coefficients in the infrared spectrum had been calculated by Abbot without taking account of selective absorption

236

Direct Solar Radiation

TABLE 5.11 Dependence of the Atmospheric Spectral Transparency Coefficient upon Wavelength as Observed at Different Points. After Savostyanova [14]

Standard Transparencv Coefficient

Kuchino Wavelength, Tucson m/l ~

292 295 300 305 310 315 320 325 330 340 350 360 370 380 390 400 420 450 450 500 550 600 650 700 750 800 850 900 950 lo00

0.02 0.03 0.07 0.17 0.25 0.30 0.35 0.39 0.43 0.49 0.54 0.58 0.61 0.64 0.66 0.68 0.71 0.74 0.74 0.76 0.78 0.79 0.79 0.79 -

-

-

0.825 -

-

-

-

-

-

Washington After 1141

After [24a 1

~

~

0.153 0.230 0.331 0.400 0.435 0,474 0.546 0.595 0.632 0.672 0.710 0.710 0.760

July 1933

July 1933

May 1933

0.040 0.142 0.260 0.336 0.380 0.422 0.464 0.547 0.578 0.641 0.672 0.710 0.710 0.760 0.790 0.800 0.825 0.830

0.159 0.240 0.298 0.352 0.379 0.429 0.455 0.472 0.500 0.524 0.550 0.570 0.607 0.645 0.645 0.695 0.720 0.728 0.760 0.760

-

-

-

-

-

-

-

-

-

-

-

0.540 0.587 0.640 0.640 0.700 0.735 0.760 0.805 0.840 0.855 0.867 0.877 0.866 8.893 0.899

0-080 0.150 0.240 0.300 0.350 0.380 0.430 0.455 0.475 0.495 0.510 0.530 0.545 0.582 0.640 0.640 0.700 0.735 0.760 0.805 0.840 0.855 0.867 0.877 0.866 8.893 0.899

0.602

0.694 0.800 0.860 0.855

by water vapor (from measurements of solar radiation fluxes in the spectral intervals between water-vapor absorption bands). As seen from Table 5.1 1, the common characteristic of all observations is the rapid decrease of atmospheric transparency in the region of short

237

5.2. Spectral Atmospheric Transparency

wavelengths down to very small values at 2 = 300 m p where the ozone absorption band begins. In the visible region the dependence of the transparency coefficient upon wavelength is comparatively weaker. This dependence is graphically presented in Fig. 5.10, according to Toropova [24], who conducted observations in the vicinity of Alma-Ata. In this figure, curve I represents the average dependence of the transparency coefficient upon wavelength, based on data of 1952. Curves I1 and I11 are drawn from analogous data for 1952 and 1954, isolating time intervals before and after noon. For comparison, Abbot's data are also given (the Mount Wilson Observatory, curve IV) together with V. S. Sokolova's data (AlmaAta, circles). Although the range of the transparency coefficient p 1 in dependence upon wavelength is approximately the same in all cases, the absolute values of p A, as Table 5.11 and Fig. 5.10 show, differ considerably. These differences are caused, naturally, by the atmospheric transparency variations. One may conclude that even at near-noon hours, the values of spectral transparency coefficients differ enough to be noticed; the increasing turbidity of the atmosphere in the afternoon leads to the decrease of transparency coefficients. An interesting peculiarity of the p n curves, referring to Alma-Ata, is the presence of a weak minimum in the region 500 to 600 m p due to solar radiation attenuation in the ozone absorption bands (the Chappnis bands). Analyzing observational data, Savostyanova [141 attempted to compile a table of characteristic values of spectral transparency coefficients (col-

......._...... A

+m

+- I a- 2

l

3 A- 4 0-

"'

""I 4

I'

am1

400

0-

I 500

I

600

I

700

I

000

I

900

5

I

loo0

FIG. 5.10 Spectral transparency of the atmosphere. (1) average morning data for 1954; (2) average afternoon data for 1954; (3) average data for 1952; (4) Abbot's data; (5) Sokolova's data.

238

Direct S o h Radiation

umn for Ref. 14 of Table 5.11). Analogous values were obtained by Rabinovich and Guseva (column for Ref. 24a of Table 5.11). For solution of certain practical problems it is convenient to characterize data on atmospheric transparency by a dependence of optical atmospheric thickness upon wavelength. This dependence in its most general form for spectral regions outside the absorption bands can be expressed as follows : = aA4

+ bA-‘@’ + c

(5.2)

where a, b, c are constants, and B(1) is a certain function of wavelength. According to Mirzoyan’s [25] data obtained from observations on clear days, the following formula gives satisfactory results :

K. S. Shifrin and I. N. Minin [Chapter 4, Ref. 541, when working out a standard atmosphere radiation model, came to a conclusion that the most characteristic peculiarities of spectral atmospheric transparency are well described by the first two terms of (5.2), where it is possible to take B = 1 (see Table 4.4). Toropova [24] has shown not only that aerosol attenuation may be a decreasing wavelength function, or neutral, as expressed by the last two terms of (5.2), but also that in certain cases a maximum of aerosol scattering in the wavelength interval 460 to 520 mp is observed. The introduction of standard spectral coefficients of atmospheric transparency appears to be quite useful in a number of cases. However, one should remember that the real coefficient values may at times greatly differ from the standard. Consider now the relationship between the standard and the ideal and real atmospheric transparency coefficients. Table 5.12 (according to data of [26]) gives optical thickness values of an ideal (dry and clean atmosphere 0, in the direction of the vertical and the transparency cofficient p a = e-@&The value @, has been computed from the following formula : 32 H 0, = --(n - 1y3 NA4 where H = 7.991 x lo5 cm is the atmospheric scale height, and N = 2.70 x 1019 is the number of molecules per unit volume. ~ , by taking account In Table 5.13 are gathered values P ~ , obtained solely of scattering of solar radiation by water vapor in a vertical atmospheric column containing 1 g/cm2 of water vapor.

239

5.2. Spectral Atmospheric Transparency

TABLE 5.12 Spectral Transparency of the Ideal (Dry and Clean) Atmosphere PA

0.28 0.29 0.30 0.31 0.32 0.33 0.34 0.35 0.36 0.37 0.38 0.39 0.40 0.42 0.44 0.45

1.555 1.335 1.15 1 .oo 0.872 0.768 0.676 0.598 0.530 0.472 0.422 0.378 0.340 0.277 0.229 0.208

0.211 0.263 0.316 0.368 0.418 0.464 0.508 0.551 0.589 0.623 0.656 0.685 0.712 0.758 0.795 0.812

0.46 0.48 0.50 0.52 0.54 0.55 0.56 0.58 0.60 0.62 0.64 0.65 0.66 0.68 0.70 0.75

0.190 0.160 0.155 0.115 0.0988 0.0915 0.0852 0.0738 0.0645 0.0564 0.0497 0.0466 0.0438 0.0388 0.0345 0.0262

0.827 0.852 0.874 0.891 0.906 0.913 0.918 0.929 0.938 0.945 0.952 0.955 0.957 0.962 0.966 0.974

0.80 0.85 0.90 0.95 1 .OO 1.10 1.20 1.40 1.50 1.60 1.80 2.00 2.50 3.00 3.50 4.0

0.0201 0.0158 0.0125 0.0101 0.00821 0.00560 0.00395 0.00213 0.00162 0.00125 0.00078 0.00051 0.000208 0.000100 O.OOOO54 0.oooO32

0.980 0.984 0.988 0.990 0.992 0.995 0.996 0.998 0.998 0.999 1 .Ooo 1 .000 1 .000 1 .Ooo 1 .Ooo 1 .Ooo

TABLE 5.13 Attenuation of Solar Radiation due to the Scattering by Water Vapor

0.342 0.350 0.360 0.371 0.384 0.397 0.413

0.920 0.926 0.944 0.940 0.945 0.949 0.953

0.431 0.452 0.475 0.503 0.935 0.574 0.624

0.957 0.961 0.964 0.968 0.972 0.970 0.975

0.686 0.764 0.864 0.987 1.146 1.302 1.452

0.981 0.984 0.986 0.987 0.987 0.987 0.987

1,P

pa,w

1.603 1.738 1.870 2.000 2.123 2.242 2.348

0.987 0.987 0.987 0.986 0.985 0.984 0.983

Comparing Tables 5.12 and 5.13, we see that the ideal atmosphere is far more transparent than the standard real atmosphere. Let us consider in this connection the variations of the real atmospehric transparency coefficients in relation to the standard values. Nikitinskaya [23] has provided some relevant data of much interest. Nikitinskaya has measured solar radiation fluxes in individual narrow

240

Direct Solar Radiation

spectral intervals by means of an actinometer shaded with combined interferential and Schott glass filters. This combination allows isolation of certain narrow intervals of the solar spectrum. In the given case, filter combinations have been obtained that give transmission bands with maxima near 504, 608, 708, 772, 928, and 1094 mp. The effective width of transmission bands varied from 11 to 39 mp. Measurements of solar radiation fluxes in the region of these wavelengths at different atmospheric masses have enabled determination of spectral atmospheric transparency coefficients for the corresponding wavelengths. It has been found in this connection, however, that the use of Eq. (5.1) introduces serious difficulties in the mentioned determination and appears to be well grounded solely for the case of the atmospheric transparency (that is, transparency coefficient P A , ) that remains constant over the entire period of observations and gives the dependence of log S,,,, upon my a condition that often cannot be fulfilled. At the same time it is essential, as was stated by Nikonov [27], that the presence of a linear dependence between log Sm,Aa and m is not at all a criterion of the stability of atmospheric transparency. Indeed, suppose logp,, = a (blm), that is, the transparency coefficient is not constant but depends upon atmospheric mass. Then, instead of (5.1) we have

+

log Sm,Ad

= log

So,A,

+ am + b

(5.la)

that is, the dependence between log S m , A a and m remains linear. It follows from the above that a reliable determination of the atmospheric transparency coefficient from Eq. (5.1) can be realized only in the presence of a stable state of transparency. In actual fact, even on days that exibit stable transparency, the transparency coefficient (and consequently, the spectral transparency) shows a pronounced daily variation. Figure 5.1 1, for example, presents the daily range of transparency coefficients in the wavelength region 506, 708, 772 mp, according to observations of N. I. Nikitinskaya on Aug. 21, 1947. It can be seen that the coefficients in question, especially in the shortwave region, vary considerably during the day, showing noticeable departures from the standard values. The standard value being 0.700 at il= 500 mp, the corresponding real coefficient varies even within a single day period, with stable transparency from 0.705 to 0.775. Figure 5.12 illustrates similar variations in a still more spectacular way. The solid curve of this figure represents the dependence of S,,,,, on atmospheric mass, calculated from the standard value of the transparency coefficient at il = 504 mp (s504,, is the monochromatic solar radiant flux value at il = 504 mp). The dots

5.2. Spectral Atmospheric Transparency

24 1

700I

1

1

FIG. 5.11 Daily variation of spectral atmospheric transparency on Aug. 21, 1947.

in this figure determine the measured S504,m values. It is seen that the observed values are in many cases far from the corresponding values computed for the conditions of the standard atmosphere, with the variations of transparency being very irregular. Popov [18] has investigated the annual range of atmospheric transparency coefficients in different spectral regions from measurement data obtained by means of the Michelson actinometer with Schott filters OG 1 and RG 2. The results are given in Fig. 5.13. The curves 1 , 2, 3, 4, 5 refer to the respective spectral intervals 290-525, 525-628, 290-5000, 525-5000, and 6285000 mp. The transparency coefficients of the corresponding spectral intervals all show a wide annual range. It has already been mentioned that in the shortwave spectral region, the

E

. ; .' '.. .: . . .. . . . . . .....

FIG. 5.12 The measured values of solar radiation intensity near 504 mp in comparison with Savostyanova's data.

242

Direct Solar Radiation

0.750 9

0.700

FIG. 5.13 Annual variation of atmospheric spectral transparency.

atmospheric transparency variations are the greatest. In this connection it is of much importance to investigate atmospheric transparency in this region, which has been done in the works by Rodionov [28] and Poliakova [221. Rodionov has studied the transparency of the lower atmospheric layers at the 3 to 3.5-km height by spectrophotometrically measuring the brightness of the sunlit snow summit of the Elbrus at 10-km distance. A sharply marked attenuation band due to selective attenuation of light by aerosols has been found. Figure 5.14 gives the results of a series of Rodionov’s measurements. On the axis of ordinates are plotted the measured aerosol attenuation coefficient S calculated for the light ray path equal to 1 km; on the axis of abscissas, the wavelength is in microns. The interrupted curve characterizes a theoretically calculated dependence of S upon A. Figure 5.14 shows that the lower atmospheric layer transparency does not monotonously vary in dependence upon wavelength but has a minimum near 0.4 p. It is essential that at il < 0.4 p the transparency should decrease as the wavelength increases. Rodionov states that as the result of this dependence, the so-called anomalous atmospheric transparency appears in the ultraviolet region 0.295 to 0.320 p. The same author has also found that at large zenith solar distances the atmosphere in the above spectral region becomes relatively more transparent for shortwave than for longwave radiation.

243

5.2. Spectral Atmospheric Transparency 0 05 004

$ 0 03 00

0 02 0 01 I

03

05

04

06

ASP

FIG. 5.14

The band of aerosol aftenuation.

Rosenberg [29], however, suggested another interpretation of Rodionov’s results, explaining the observed effect of anomaly by the influence of multiscattered light hitting the instrument along with direct solar radiation. Rosenberg also contends that this factor is quite important for the applicability of Buger’s law to determine spectral atmospheric transparency. At large optical thicknesses equal to 8 to 10 (large zenith solar distances), the contribution of diffuse radiation becomes so very significant that deviations from Bouger’s law, even in the case where the solid angle of the instrument approximately corresponds to the solar angular dimensions, are too great to be neglected. The above data on atmospheric transparency in the ultraviolet relate to wavelengths iZ > 2900 A. Solar radiation of shorter waves cannot be found at the earth’s surface. Although certain authors reported the presence of solar radiation in the range near 2100 A during observations in high mountains, such supposition should be judged erroneous. Only rocket investigations of recent years (see [30-371) have made it possible to derive experimental data on atmospheric transparency for ultraviolet radiation of wavelengths shorter than 2900 A. Table 5.14 gives the observed and computed (on the assumption that oxygen is fully dissociated above 100 km) values of the heights at which penetration of solar radiation of different wavelengths into the atmosphere takes place, which characterize spectral atmospheric transparency. The data in Table 5.14 illustrate the presence of a higher atmospheric transparency (transparency “windows”) near 1 i5: 1200 A and 1 - 2100 A, which appears to be essential from the standpoint of the influence of ultraviolet radiation on the processes in the ionosphere. Also interesting are data on the variation of spectral atmospheric trans-

244

Direct Solar Radiation TABLE 5.14

Penetration of Solar Radiation of Various Wavelengths into the Atmosphere. After Mandelstamm [32] Height of Penetration, km Spectral Region, 8,

795-1050 1050-1 240 1100-1350 1240-1 340 1400-1550 1750-2100

Observed

Computed

88-127 80-90 7 0 3 ~5 90-125 95 (50) 7

90-100 80-90 80-100 100 25

parency with height. Having analyzed the available experimental data, E. M. Feugelson [Chapter 4, Ref. 651 proposed the following formula for the average vertical profile of the optical atmospheric thickness z(z) above 6.5 km: z(z) = t(zo)e-ar-

+ b(z - z,,)

(5.2a)

where z(zo) is the optical thickness at the initial low level zo = 6.5 km, and a and b are constants. Assuming that scattering over 30 km is Rayleigh, determining b from this condition, and also taking a = 0.13, we obtain a variation of optical thickness with height as given in Table 5.15. TABLE 5.15 Variation of Optical Atmospheric Thickness with Height. After Feugelson [Chapter 4, Ref. 651 z, km

6.5

= (4

0.143

8 0.117

10 0.0895

15 0.0455

20 0.0221

25 0.0094

30 0.0024

Elterman [38a] has built a standard optical model of the atmosphere based on the Rayeligh aerosol and total attenuation coefficients for the spectral range 0.4 to 4.0 m p at elevations up to 30 km with 1-km interval. Although we have mentioned a supposition about a purely Rayleigh scattering at altitudes over 30 km, the aerosol scattering remains important at still greater altitudes. This has been demonstrated in the discussion of

5.3. Energy Distribution in the Solar Spectrum

245

data on the formal variation of the scattering function with height (see Fig. 4.12). For example, according to Morozov [39], the ratio r = t / z R (t,tR are the total and Rayleigh optical thicknesses, respectively) increases with height up to 25 km (these data relate to Aeff = 0.5 p). Mirtov [40], analyzing data on the vertical distribution of the meteor dust concentration, came to the conclusion that in the region of heights near 92 to 96 km is situated a transition boundary between the zone of aerosol scattering domination and the Rayleigh scattering zone. The latter is comparatively narrow, with a thickness of the order of several tens of kilometers. Still higher up, the aerosol scattering is again dominant, owing to a very low air density and a significant concentration of aerosols of meteor origin. Considerable aerosol scattering at high altitudes has been found in rocket measurements of the sky’s luminosity, a study conducted by Mikirov [41]. 5.3. Energy Distribution in the Solar Spectrum outside the Atmosphere The preceding sections of the present chapter have dealt with the results of investigation of energy distribution in the solar spectrum near the earth’s surface and of spectral atmospheric transparency. Making use of these results, it is possible to study the solar radiation spectral composition outside the earth’s atmosphere. After the dependence of log s,,,, upon ni is obtained from observations and extrapolated until m = 0, the value is found. Similar computations for a sufficient number of various narrow spectral intervals d l are performed, and then we obtain the curve of energy distribution in the off-atmospheric solar radiation spectrum. If the coefficients p d Aand values S,,,, are known, the values of spectral solar radiant fluxes outside the atmosphere can be directly calculated from the formula Sm,na = So,,, p y A . The described method was used for obtaining data on energy distribution in the solar spectrum outside the atmosphere in the ultraviolet (partly) and also in the visible and infrared spectra. It cannot be applied to ultraviolet radiation of wavelengths shorter than 290 mp because the radiation of these wavelengths fails to reach the earth’s surface. In the given case the only reliable source of information on solar radiation are rocket measurements. The most convincing data on energy distribution in the solar spectrum outside the atmosphere, obtained by means of extrapolating from the surface measurements results, are those by Dunkelman and Scolnik [42], who had realized spectral solar radiation measurements of long duration at Mount Lemmon at about 2400 m elevation in September and October of 1951.

246

Direct Solar Radiation

However, only one day (October 4), owing to otherwise unstable atmospheric transparency, was chosen as satisfying the main requirement of applicability of Bouger’s method. The authors had employed a double quartz monochromator with photoelectric spectrum registration. Calibration and sensitivity control were realized by means of a standard tungsten lamp. Figure 5.15 shows the results of measurements by Dunkelman and Scolnik (3) in comparison with earlier 1955 data by R. Stair and R. Johnston (1) and 1940 data by E. Pettit (2).

3000

4000

5000

6000

7000

Wavelength,

FIG. 5.15

Energy distribution in the solar spectrum outside the atmosphere (the visible region).

Figure 5.16 shows the energy distribution in the solar spectrum outside the atmosphere for the interval 0.2 to 20 p as derived by Johnson [43] from both surface measurement data (extrapolated) and rocket results (in the ultraviolet spectrum). During recent years this distribution has been explored in a number of works (see [44-581). Two series of high-mountain observations by Sitnik [48] produced the data of Table 5.16 for solar radiant flux values outside the atmosphere. As regards the effects of solar radiation on the upper atmospheric layers, it is of interest to investigate the Ultraviolet and X-ray solar spectra outside the atmosphere. At present, exploration of outer space makes it possible to characterize the ultraviolet and X-ray solar radiation spectral composition outside the atmosphere. It should be stressed that even rocket measurements do not fully eliminate the necessity of taking into account the radiation absorption by the atmosphere. The results of rocket exploration of solar radiation spectral distribution in the area 850 to 2600 8, are given in Table 5.17 by Detwiler et al. [50]. These measurements employed a diffracted double quartz monochromator

5.3. Energy Distribution in the Solar Spectrum

247

URVE FOR BLACKBODY ct 59000K

Oo 02 a4 ~ 6 oa . ID 1.2 1 4 16 la 20 22 24 26 20 30 3.2 WAVELENGTH (p)

FIG. 5.16 Energy distribution in the solar spectrum.

with photographic spectrum registration. The installation was elevated on the “Aerobi-Hi” rocket to the height of 235 km. It is interesting that even so high up there was observed strong radiation absorption by molecular nitrogen in certain spectral intervals. Table 5.17 gives flux values for wavelength intervals of 50 8, width relating to the mean sun-earth distance. Compared with Johnson’s results, these data show that for radiation of wavelengths longer than 240 mp, a satisfactory agreement exists. In the spectral region 1 -240 mp, Johnson’s data should be taken as lower (at 1 = 220 m p spectral radiation flux values differ by twice). In Fig. 5.17 is given a graphical presentation of data for intervals of 10-8, width and the mean sun-earth distance, with corrections introduced to take into account the radiation absorption above the level of observations. The dashed curves of Fig. 5.17 follow the energy distribution in the blackbody spectrum at different temperatures. From consideration of Fig. 5.17 and Table 5.17 it follows that in the region of wavelengths shorter than 200 8, the radiation temperature of the sun decreases from 5000’K at 1 = 2085 8, to 4900’K at A = 1800 8, and 4750’K at 1 = 1500 A. In the interval 1450 to 1280 8, the minimum, T = 4700’K, is observed. An important feature of the spectral region considered here is the presence of a great number of emission lines overlying continuous emission. In the given case not all lines are resolved, owing to a rather rough averaging over spectral intervals of 10-8, width. The most

TABLE 5.16 Values of Monochromatic Luminosity and Illumination in the ContinuousSolar Spectrum. Luminosity Ba and Fa Are in erglcm' see ern ster; Illumination En in erg/emz see p . Ba Is the Solar Disk Luminosity at Center; Fa Is the Mean Luminosity. After Sitnik [48] 1

3280 3300 3350 3380 3400 3450 3500 3550 3600 3650 3700 3750 3800 3900 4000 4200 4400 4600 4800 5000 5200 5400 5600 5800 6000 6200

Ea

Ba 24.9 x lOla 25.0 25.1 25.2 25.3 25.7 26.1 26.4 26.9 27.6 28.1 31.4 36.0 42.0 46.5 50.05 52.3 52.8 50.05 47.5 44.45 41.9 39.65 37.4 35.4 33.6 X 10"

17.3 x 10" 17.4 17.6 17.8 17.9 18.3 18.7 19.0 19.5 19.9 20.2 2.5 25.8 30.1 33.5 36.8 52.3 40.4 38.8 37.2 35.2 33.5 32.0 30.4 28.9 27.6 x 10ls

11.8 x 106 11.8 12.0 12.1 12.2 12.4 12.7 12.9 13.3 13.5 13.7 15.3 17.5 20.5 22.5 25.0 39.3 27.5 26.4 25.3 23.9 22.8 21.8 20.7 19.7 18.8 x lo6

I 6,400 6,600 6,800 7,000 7,500 8,000 8,500 9,000 9,500 10,000 11,000 12,000 13,000 14,000 15,000 16,000 26.7 18,000 19,000 20,000 21 ,000 22,000 23,000 24,000 25,000 26,000

Ba

Fa

Ea

32.2 x 1Ola 30.9 29.5 28.15 24.8 21.85 18.95 16.65 14.65 13.1 10.4 8.1 6.62 5.57 4.71 3.94 17,000 2.69 2.20 1.77 1.50 1.28 1.08 0.920 0.790 0.680 x 10l3

26.7 x 1Ola 25.8 24.7 23.7 21.2 18.8 16.5 14.5 12.9 11.7 9.24 7.25 5.96 5.04 4.30 3.63 2.98 2.50 2.05 1.65 1.40 1.20 1.01 0.863 0.742 0.640 x 10'3

18.2 x lo6 17.5 16.8 16.1 14.1 12.8 11.2 9.86 8.77 7.96 6.28 4.22 4.05 3.43 2.92 2.47 2.03 1.70 1.39 1.12 0.952 0.816 0.687 0.587 0.505 0.435 X 10

%

Pa 8'

P

8

249

5.3. Energy Distribution in the Solar Spectrum

TABLE 5.17 Energy Distribution in the Solar Spectrum outside the Atmosphere in the Region 850 to 2600 A. Afrer Detwiler et al. [50]

A

*

SQ,A h

25 A

erg/cm2 sec (50

2600 2550 2500 2450 2400 2350 2300 2250 2200 2150 2100 2050 2000 1950 1900 1850 1800 1750

A

A)

700 560 380 390 340 320 3 60 350 310 240 145 90 70 55 41 28 19 12

* 25 A

SQ,A h erg/cm2 sec (50

A)

8.2 5.0 3.2 1.7 0.95 0.50 0.26 0.26 0.18 0.15 5.7 0.08 0.06 0.10 0.18 0.15 0.25 0.11

1700 1650 1600 1550 1500 1450 1400 1350 1300 1250 1200 1150 1100 1050 1000 950 900 850

10 0-

t-

z

.

I

800

I

I

1000

1

I

1200

I

I

14.00

I

I

1600

1

,

I800

I

1

*

2000

A. A

FIG. 5.17 Energy distribution in the solar spectrum outside the atmosphere (the ultraviolet region).

250

Direct Solar Radiation

intensive is the Lyman line centered at A = 1216 A. Beginning with A = 1280 8, aml-tmwd the region of short wavelengths, the solar radiation temperature first increases (to about 6500’K at the center of the Lyman line) and then rapidly decreases down to near 5200°K, ultimately increasing at 1 > 1040 A. A comparatively complete set of data for the emission line intensities at the interval 835 to 1892 A is given in Table 5.18. As in the previous cases, the line intensity is reduced to the mean sun-earth distance and the effect of radiation absorption by the atmosphere is excluded in the final processing. The intensity of the La line is given for a spectral interval of 1-A width. The L, line is clearly seen to dominate, as regards intensity. Just this fact accounts for the attentinn paid to the investigation of this line during recent years. The energy connected with the La line is approximately equal to the entire amount of energy in the whole spectrum of wavelengths shorter than 1500 A. The remainder of the Lyman series makes only 0.12 erg cm-2 sec-l, and the energy flux in the area of Lyman’s continuum (A < 912 A) is approximately 0.24 erg cm-2 sec-’. The accuracy of such measurements in the region 1400 to 2000 8, appears to be f.20 percent. In any case, one can be sure that the errors do not exceed 50 percent. At A < 1300 A, the measured values may differ from the true values by two times, and possibly may be even more in some cases. Such low accuracy and a considerable variation with time of the hard ultraviolet radiation, due to absorption in the atmosphere mean that the given results should be treated as preliminary. The continuous ultraviolet emission of the sun appears to be invariable with solar activity. Fluctuations in the corpuscular emission take place and are especially great in the X-ray spectral region [49]. For example, in the interval of wavelengths less than 8 A, the fluxes of corpuscular radiation may differ by 600 times. Also observed are short-period fluctuations in the X-ray emission during solar flares; for example, on Aug. 24, 1959, the radiant flux of wavelengths of 2 to 8 8, increased by 26 times, in 11 min after the flare, and on Aug. 31, 1959, by 100 times in 4 min after the flare. The measurements taken during the flares recorded solar radiation of 0.15 A. According to data of satellite measurements made from July to November 1960, the usual value of the X-ray radiation in the band 2 to 8 A is less than the limit value (0.006 x lo-* W/cm2) of the receiver’s sensitivity. An excess above this limit accompanied by rapid fluctuations in emission (during about a minute’s time) was observed only when various perturbations occurred on the sun. In the cases when the radiant flux exceeded 0.02 x lo-* W/cm2, ionospheric perturbations were taking place. Obser-

25 1

5.3. Energy Distribution in the Solar Spectrum

TABLE 5.18 Intensities of the Strongest Emission Lines in the Solar Spectrum (835 to I892

Identification

1892.03 1817.42 1808.01 1670.81 1657.00 1640.47 1561.40 1550.77 1548.19 1533.44 1526.70 1402.73 1393.73 1335.68 1334.51 1306.02 1304.86 1302.17 1265.04 1260.66 1242.78 1238.80 1215.67 1206.52 1175.70 1132.89 1085.70 1037.61 1031.91 1025.72 991.58 989.79 977.03 949.74 937.80 835

I11 (I) I1 (I) I1 (I) I1 (2) c 1 (2) H L I1 (12) c 1 (3) c IV (I) c IV (I) Si I1 (2) Si I1 (2) Si IV (1) Si IV (I) c I1 (I) c I1 (I) 0 1 (2) 0 1 (2) 0 1 (2) Si I1 (4) Si I1 (4) N V N V H La Si I11 (2) c I11 (4) C I (20-23) N I1 (I) 0 VI (I) 0 VI (I) HLB N I11 (I) N I11 (I) c I11 (I) H La H Le 0 11, I11

Si Si Si A1

0.10 0.45 0.15 0.08 0.16 0.07 0.09 0.09 0.11 0.041 0.038 0.013 0.030 0.050 0.050 0.025 0.020 0.013 0.020 0.010 0.003

0.004 5.1 0.030 0.010 0.003 0.006 0.025 0.020 0.060 0.010 0.006 0.050 0.010 0.005 0.010

A)

252

Direct Solar Radiation

vations of the fluctuating emission in the Lyman a-line with a varying Xray emission showed that even if these fluctuations really existed, they did not exceed 18 percent. 5.4. The Solar Constant

1. General Notes. The solar constant is the amount of solar radiant energy from outside the atmosphere passing per unit time (1 min) through unit surface (1 cm2) perpendicular to the sun’s rays and at a distance of the mean radius of the terrestrial orbit. In other words, the solar constant is a flux of solar radiant energy on a surface perpendicular to the rays outside the atmosphere at a mean sun-earth distance. The determination of the solar constant is of fundamental importance to the determination of the regularities of solar radiation attenuation in the atmosphere. Given a known solar constant value, it is possible to find radiant flux values at the earth’s surface level. The first attempts at such determination were made in the 1850s. Only at the beginning of the twentieth century were works produced that provided means of computing sufficiently reliable solar constant values, mainly at the Smithsonian Institution [60]. In the U.S.S.R. too, this problem was initially investigated by Kalitin [61, 621, Fesenkov [63-651, and Krylov [66]. At the Smithsonian Institution, two methods (known as the “long” and the “short”) were worked out. 2. The “Long” Method. The essence of this method may be presented as follows. Using spectrobolometric measurements of energy distribution in the solar spectrum at the earth’s surface level, taken at various atmospheric masses and at different daytime hours, it is possible to determine the spectral solar radiation distribution outside the atmosphere by means of the extrapolation method, as given in the preceding section. Knowing the energy distribution curves outside the atmosphere and at the earth’s surface and computing the area underlying these curves, one thus obtains the integral solar radiation fluxes outside the atmosphere and near the earth’s dA and J Sm,AdA in relative units. However, such surface, equal to J values are not exact enough because the measurements do not take into consideration the extreme limits of the ultraviolet and infrared spectral regions. For more precision the so-called “ultraviolet” and “infrared” corrections must be introduced (see later sections). If simultaneously with spectrobolometric measurements the solar radiant flux at the earth’s surface level, S , , is measured in absolute units (using the pyrheliometer), then for

5.4. The Solar Constant

253

the solar flux outside the atmosphere, S,,’, expressed in absolute units, we have

Reducing So’ to the mean sun-earth distance for the solar constant S o , we get R2 so= So’

R02

where R, is a mean sun-earth distance and R is this distance at the moment of measurement. These are the principal features of the “long” method. Let us now dwell on a more detailed consideration of certain important aspects of this method, especially as regards computation of the “infrared” and “ultraviolet” corrections and the area under the curves of spectral solar radiation distribution. The underlying area was computed in the following way (according to [60]). First the area under the smoothed distribution curve, plotted to exclude dips corresponding to absorption bands, was computed. Then the area of absorption bands was substracted, from this area, which gave the desired result. Such procedure appears to be the most convenient because of its simplicity and sufficient accuracy in determining the area under the smoothed spectral distribuition curve and also because it permits easy determination of the absorption-band area (it has been found that the latter is a simple function of the y-band area). Therefore it is necessary only to measure the y-band area and then make a corresponding empirical correlation in order to find the total band area. After the area under the measured curve of energy distribution in the solar spectrum has been determined in the described manner, it is necessary to make corrections to compensate for the fact that the glass prism spectrometer used at the Smithsonian Institution does not transmit a portion of solar radiation in the ultraviolet and infrared regions (the measurements include the interval from 0.546 to 2.4 p). Also important is the fact that a considerable part of ultraviolet and infrared radiation fails to reach the level of the earth’s surface. The infrared correction is determined with the help of a spectrometer having a rock-salt prism, which measures the energy distribution in the infrared up to il = 10.9 p and leads to evaluation of the portion of infrared radiation (A > 2.4 p ) in relation to the solar radiation in the region

254

Direct Solar Radiation

0.704 to 2.4 p ; that is, this measurement is the basis of the infrared correction value. Table 5.19 gives infrared corrections in percent, obtained in the above way, according to the total atmospheric water vapor content (g/cmz). The right-column values are ratios (in percent) of the area under the energy distribution curve in the spectral region 2.4 to 10.9 p to the area under the smoothed curve in the region 0.704 to 2.4 p. As seen from Table 5.19, at rn = 0 (outside the atmosphere), the infrared correction amounts TABLE 5.19 Infrared Correction Values. From Ref [60]

Water Vapor Content, g/cm2

Infrared Correction,

% 3.95 3.28 2.77 2.35 2.03 1.80

to about 4 percent. Note here that the area under the entire computed distribution curve in the solar spectrum outside the atmosphere, obtained from field observations, exceeds by almost exactly twice the corresponding area computed for the 0.70 to 2.4-p region. This means that for the integral solar radiation outside the atmosphere, the infrared correction is about 2 percent of the total measured flux of solar radiation. When computing the ultraviolet correction it was first assumed that the energy distribution in the solar spectrum outside the atmosphere is identical with the blackbody spectral energy distribution at 6000OK. Approximate corrections were introduced to take into account the radiation absorption caused by the Fraunhofer lines in the solar atmosphere. Thus obtaining the solar radiation energy distribution outside the atmosphere enables determination of the ultraviolet correction. The above method for determining the ultraviolet correction should be considered very approximate, as it is based on a rough estimate of the solar radiation spectral composition outside the atmosphere. Another attempt to evaluate the ultraviolet correction consisted in direct measurements of the spectral composition of solar radiation in the ultraviolet re-

255

5.4. The Solar Constant

gion il < 0.345 p. For this purpose, data on the spectral distribution of solar radiation outside the atmosphere, obtained from extrapolation of the surface measurement results relating to the range il > 0.295 p, were used. The two methods yield considerably differing results, although neither is sufficiently reliable. The final values of ultraviolet correction were averaged from data derived in the application of both methods. Table 5.20 illustrates similar calculations performed for three different states of atmospheric transparency. The transparency coefficient value at Iz = 0.413 p is taken as the parameter, that characterizes the state of atmospheric transparency. Table 5.20 shows that the ultraviolet correction values outside the atmosphere are 3.44 percent of the total measured flux of radiation (note here that the region 0.346 to 0.704 p outside the atmosphere includes almost half of the total radiant flux). TABLE 5.20 Ultraviolet Correction Values (Percent Relative to the Radiant Flux in the Region 0.344 to 0.704 p). After [60]

Atmospheric Transparency Coefficient, A = 0.413 p Atmospheric Mass 0.790

0.760

0.730

6.88 3.90 2.22 1.28 0.73 0.45

6.88 3.39 1.79 0.96 0.56 0.33

6.88 2.90 1.36 0.64 0.38 0.21

It should be mentioned that the ultraviolet correction obtained in the above way has been calculated on the assumption of the solar spectrum’s being “cut off” in the region of wavelengths il < 0.295 p. From the data presented earlier, it follows that there is no observed “discontinuity” of the solar spectrum above the ozone layer. The apparent discontinuity obviously results from the absorption of radiation by ozone, and the calculated ultraviolet correction, therefore, should be considered c c ~ ~ b o ~ ~ n a l . ” To determine the solar constant with a higher degree of accuracy, it is necessary along with the ultraviolet correction to take into account the influence of the solar radiation absorption by ozone. That is why recent investigations consider the ultraviolet correction taken from data of rocket

256

Direct Solar Radiation

measurements of the solar radiation spectral composition above the ozone layer. Another factor in addition to the infrared and ultraviolet corrections is the absorption of radiation by ozone not only in the ultraviolet ( A < 0.290 p), but also in the visible spectrum where the Chappuis bands (0.480 to 0.630 p ) are located. The correction for the absorption in the Chappuis bands is, however, insignificant and is not more than 0.20 to 0.45 percent of the solar constant. From the preceding discussion we see that the “long” method for determination of the solar constant requires long-time spectrobolometric measurements (for several hours) in an extremely stable state of atmospheric transparency. However, even high mountainous conditions cannot provide the required stability. That is why, and also because of its complexity, the “long” method has been found to be inconvenient and not reliable enough. The “short” method, which demands little time for single measurements of the solar constant, is more widely used. 3. The “Short” Method. The basic idea of this method consists in finding an approximate way of determining the solar radiation spectral composition outside the atmosphere by extrapolating from the results of a single masurement of the energy distribution in the solar spectrum near the earth’s surface. Such extrapolation requires knowledge of spectral atmospheric transparency coefficients. To determine the latter, the “long” method uses protracted measurements of energy propagation in the solar spectrum at different atmospheric masses, a most inconvenient process. But these coefficients can be determined in a simpler way: It is known that the attenuation of solar radiation in the atmosphere is caused by the processes of scattering and absorption. As has been mentioned in Chapter 4, the attenuation of radiation due to scattering can be presented as the intensity function of the circumsolar aureole. The absorption of solar radiation is primarily related to the total water vapor content in the atmosphere. Investigations of the Smithsonian Institution have shown that this dependence on vapor content allows spectral coefficients of atmospheric transparency to be treated as simple functions, having the following value :

F=

W,

+ QE

where w, is the total water vapor content in the atmosphere, E is the difference between the observed and “normal” intensities of the circumsolar aureole, and Q is a certain factor characterizing the variation of w, in dependence upon E .

257

5.4. The Solar Constant

If the transparency coefficients pa are expressed in tenths of unity, the w, value should be expressed as g/cm2, with E as lo-* cal/cm2 min, when interpreting the angular reading on the instrument. Observations at the Montezuma (altitude 2745 mi, Chile), for example, give the following expressions for the F value at three different gradations of the total atmospheric water vapor content and atmospheric mass of 1.5 9 m 9 2.5:

woo(

g/cm2) 0-45 45-75 75

F

w, w, w,

+ 1.34 + 1.06 + 1.84

E E E

For determination of the total water vapor content in the atmosphere, the spectroscopic method was employed by investigators at the Smithsonian Institution. (See Chapter 3.) Measurements of the intensity of the circumsolar aureole were conducted with the help of a specially designed instrument. All the foregoing remarks show that by finding the empirical relations of all pa on F for a given point of observation, it is possible to compute, with the help of these factors, all values of the transparency coefficient in different spectral intervals, using only the F value obtained from observations. Such a method for determining spectral atmospheric transparency eliminates the need for long-time measurements of energy distribution in the solar radiation spectrum at different atmospheric masses. Therefore the duration of observations is considerably shortened (from several hours to 10 min). With the results of a single measurement of the solar radiation spectral composition at the earth‘s surface and the spectral coefficients of atmospheric transparency found as described above, it becomes possible to determine the energy distribution in the solar spectrum outside the atmosphere and to compute the solar constant value in the same way as the “long” method. The application of the “short” method also shortens the time necessary for data processing. To compute one value of the solar constant according to the long method, two observers must spend 8 h each, whereas it takes them only 1 h in the case of the “short” method. Numerous observations performed at the Smithsonian Institution have shown that the “short” method is also more accurate, and that the discrepancy between two respective values of the solar constant (from the “short” and the “long” method data) is not great, not more than 4 percent. During more than 50 years of observational activity, a really great num-

258

Direct Solar Radiation

ber of solar constant values have been obtained at the Smithnonian Institution. Time has shown that these values vary within 0.5 to 2 percent of the real solar constant. In this connection many investigators tried to find the direct relation among these variations of the solar constant, the solar activity, and meteorological processes. However, the low percentage of variations do not exceed the limits of possible measurement errors, and therefore the determination of such a relation does not seem to be vital at present. The average (over many years) value of the solar constant, as obtained at the Smithsonian Institution, taking into account the infrared and ultraviolet corrections and also the improvement of the pyrheliometric scale (see Chapter 2), is 1.94 cal/cm2 min. The same mean value of the solar constant was derived in earlier investigations made during the 1920s. The identical value found by earlier and recent studies is explained by the fact that the later work was more precise because it applied two reciprocal compensating corrections : first, the pyrheliometric scale was improved to reduce its error deviation from 2.4 & 0.1 percent; second, later investigations produced different infrared and ultraviolet corrections. Thus both corrections totaled 5.44 percent of the full flux of solar radiation, whereas earlier corrections had been estimated at 2.13 percent. Using the 1913 American pyrheliometric scale, measurements would give the integral solar flux outside the atmosphere (in the accessible region 0.346 to 2.5 p ) a value 1.90 cal/cm2 min. The earlier corrections (2.13 percent) would then give the solar constant So = 1.94 cal/cm2 min, but since the old scale was faulty by a 2.4 percent error, the corrected solar radiation flux value in the range 0.346 to 2.5 p would then be 1.85 cal/cm2 min. Now taking account of the more accurate infrared and ultraviolet corrections (5.44) of later studies, we have So = 1.95 cal/cm2 min; that is, the solar constant remains practically the same, the difference of 0.01 cal/cm2 min being insignificant. For this reason, some authors conclude that the old value of the solar constant may be kept unchanged, although it is evident that the identical values found at different times is coincidental. The most recent measurements of the solar radiation flux in the infrared region [67] indicate, however, that the Smithsonian Institute value of So = 1.94 cal/cm2 min is somewhat lower. This has been shown by rocket measurements of the spectral distribution of solar radiation in the ultraviolet. Thus the problem of more accurate determination of the solar constant remains important (see [6-9, 67-69]). A substantial and well-grounded revision of the available solar constant values has been carried out by Nicolet [68] and Johnson et al. [43, 491.

+

5.4. The Solar Constant

259

Making use of the new correction data on energy distribution in the solar spectrum outside the atmosphere, Nicolet [68] has come to the conclusion that the value of the above-mentioned ultraviolet correction (3.4 percent) may be accepted practically unchanged, whereas the infrared correction should be somewhat increased. Nicolet’s final value of the solar constant is 1.98 cal/cm2 min. Johnson’s solution of the same problem took into account the surface measurement data in the region 1 0.3 ,u and rocket measurements for the ultraviolet (0.22 to 0.34 p). Figure 5.16 presents the energy distribution in the solar spectrum outside the atmosphere; these data form the basis of determination of the solar constant. According to Johnson, So = 2.00 f 0.04 cal/cm2 min. Note here that according to Johnson’s data, the ultraviolet correction is not 3.44 percent (in the Smithsonian Institution estimate) but 4.75 percent. The integral solar radiation flux for the 0.346 to 2.4-,u region equals 1.841 cal/cm2 min. For the infrared correction, Johnson has found a value 8.1 percent instead of 3.95 percent. Thus in absolute values, the main solar constant components are 1.841 0.085 0.076 = 2.002 cal/cm2 min. It can be seen that for the determination of a more reliable solar constant, it was important not only to obtain a new ultraviolet correction (from rocket measurement data) but also a revision of the infrared correction. The International Radiation Commission recommended the value 1.98 cal/cm2 min according to the International Pyrheliometric Scale (1956) as the standard solar constant. The preceding results show that at present no absolutely reliable value has been found for the solar constant. The only dependable fact is that the mean value is about 2 cal/cm2 min. Further investigations are needed to improve methods of absolute measurements of solar radiation flux and to realize more exact measurements of the solar radiation spectral distribution at the earth’s surface above the ozone layer. The solar constant discussed in this section may be called energetical, since it characterizes the quantity of radiant energy passing per unit time per unit area perpendicular to the direction of the sun’s rays and situated at a mean sun-earth distance. Also interesting is the determination of the luminous solar constant characterizing the visual illumination by the sun’s rays outside the atmosphere of a surface located perpendicular to the beam of rays at a mean sun-earth distance. Different methods are used to determine the luminous solar constant. Sharonov [70], after applying three methods for determination of the luminous solar constant and comparing the obtained results with those

+

+

260

Direct Solar Radiation

by other authors, proposed the value 135,500 lux as the most probable. To the energetical solar constant, equal to 2 cal/cm2 min from F. Johnson's data, corresponds the luminous solar constant equal to 13.67 lumen/ cm2, or 137 kilolux. According to Nicolet [68], it equals 142 kilolux. Stair and Johnston [71] have found a higher value of 150 kilolux. 5.5. Total Attenuation of Solar Radiation in an Ideal Atmosphere Remember the following relation, which determines the attenuation of a monochromatic radiant flux Sa along the path ds: dS2

= -

aAS,e ds

where aa is the mass factor of attenuation of radiation of wavelength I , and e is air density. Designating by ,So,, the monochromatic solar radiant flux outside the atmosphere, and integrating the above relation over the entire beam's path in the atmosphere, we have

where S, is the monochromatic flux at the earth's surface. The value 0(&)= J" a,@ds is the optical thickness of the atmosphere along the ray's path, depending on the solar zenith distance BO. With introduction of the optical atmospheric thickness in the vertical direction 0, = S," a,@dh, wa can write

o(e,)

=

@@(en)

(5.4)

where rn(0,) is atmospheric mass in the direction on the sun. We can now rewrite (5.3) as follows:

s1 - s0,Ae-8,m(B@) -

(5.5)

Introducing the value p,

= e-80

which is known as the atmospheric transparency coefficient for radiation of wavelength I (or, in short, the monochromatic transparency coefficient), and taking account of (5.6), instead of (5.5) we have

26 1

5.5. Total Attenuation of Solar Radiation

Integrating both terms of (5.7) over all wavelengths from 0 to

s

m

S,

=

00,

we find

m

S, dil

=

SO,,pdmdil

(5.8)

Here S , is the full (integral) flux of direct solar radiation at the earth’s surface level in the case when the atmospheric mass in the direction on the sun is m. Formula (5.8) is basic for calculations of the total attenuation of direct solar radiation in the atmosphere. Knowing from observations the energy distribution of solar radiation outside the atmosphere, So,,, and the dependence of the monochromatic transparency coefficient p , upon wavelenght, it is possible to use (5.8) to compute the full flux S , a t the earth’s surface for any mass of m value. Such computations are usually performed by means of numerical integration. Of interest is a method of analytical presentation of the values So,Aand p , as wavelength functions, with subsequent derivation of an analytical dependence of the flux s, upon atmospheric mass. Let us now consider computations of this kind beginning with the study of an idealized case of direct solar radiation attenuation in a dry and clean atmosphere. The problem of the transparency of a dry and clean atmosphere for solar radiation has two interesting aspects. First, theoretical calculations of radiation attenuation in this case can be rather easily brought to final numerical results. Second, useful results are obtained from comparison of data on the ideal and the real atmospheres. The most important contributions to the investigation of this problem have been made by Kastrov (72-741, Averkiev and Riazanova [75, 761, and some others. Taking into account the radiation attenuation due to Rayleigh scattering and also due to the absorption by ozone and constant gases, we obtain the following expression for the integral flux of solar radiation at a level where atmospheric pressure is P and at atmospheric mass m :

Here k, is the ozone absorption coefficient, IT,is the Rayleigh scattering coefficient [Kastrov took oA= 1.14 x 104(n - l)2il-4], and f3(rn, (P/Po)) is a part of the flux absorbed by constant gases (oxygen above all). The , is an absolute mass for the ozone layer and is determined by (4.19). value u Kastrov based his computation of S(m,P) from (5.9) on the energy distri-

262

Direct Solar Radiation

bution in the solar spectrum outside the atmosphere as obtained by Johnson (see Sec. 5.3). Table 5.21 gives the results of computation of the ozone radiation absorption in the visible (fi) and ultraviolet (fJspectral regions. Also given are F. Fowle’s data on the absorption by constant gases. TABLE 5.21 Absorption of Solar Radiation by Ozone and Constant Gases (cal/cmzmin). After Kastrov [741

1

2

P

1

1.98

2.95

3.86

5.60

fi fz

0.010

0.020

0.030

0.039

0.056

0.036

0.042

0.046

0.048

0.052

rn

3

4

6

fi +fz

0.046

0.062

0.076

0.087

0.108

f3

0.010

0.012

0.013

0.014

0.016

Table 5.22 shows the results of computation of the integral fluxes at different heights and different atmospheric mass values. TABLE 5.22 Solar Radiant Fluxes in the Conditions of a Dry and Clean Atmosphere (cal/cmz min). After Kastrov [74] m PIP,

1

2

3

4

6

0

1.946

1.930

1.917

1.906

1.895

0.25

1.894

1.837

1.786

1.743

1.663

0.50

1.845

1.758

1.681

1.616

1.509

0.75

1.802

1.688

1.595

1.520

1.398

1 .oo

1.766

1.628

1.525

1.440

1.309

It should be stressed that the above data correct the earlier analogous results. Kastrov has shown that these data can be described with a high

5.6. Characteristics of the Real Atmospheric Transparency

263

degree of accuracy (error not exceeding 0.003) by the following empirical formula : -'(my p, - 1.041 - 0.160 SLl

In Table 7 of the Appendix are given Sivkov's [38] results of detailed calculations of the spectral composition of solar radiation in a dry and clean atmosphere. An important characteristic of the ideal atmosphere is the integral transparency coefficient qm. Table 5.23 presents the values qnl from data of Feussner and Dubois, Kastrov, and the International Radiation Commission (see [38, 74, 781). Their results show very little difference. TABLE 5.23 The Integral Transparency Coefficient of a Dry and Clean Atmosphere

m

After Feussner and Dubois, 1930

After Kastrov [731

International Radiation Commission, 1956

1 2 3 4 6

0.907 0.915 0.921 0.926 0.925

0.906 0.914 0.921 0.927 0.935

0.906 0.916 0.922 0.982 0.936

5.6. Quantitative Characteristics of the Real Atmospheric Transparency Of the many suggested quantitative characteristics of atmospheric transparency, we shall discuss here only those that have found wide acceptance. 1. The Transparency Coeficient. According to (5.8) we have, for the flux of direct solar radiation at the level of the underlying surface,

Introduce a value of the integral transparency coefficient averaged over the whole of the spectrum. Then it is possible to write m

s m = Pm"

joSQ,,dl.

=

Sopm"'

(5.11)

264

Direct Solar Radiation

The integral transparency coefficientpmcan be determined empirically and calculated for the ideal atmosphere on the basis of the theory of molecular light scattering in the atmosphere. Some results of such calculation have been given above for the conditions of a dry and clean atmosphere (see Table 5.23). As seen from these data, the coefficient q m cannot be treated as a simple quantitative characteristic of atmospheric transparency, for at a given and stable state of transparency, the value q m increases with the increase of atmospheric mass, which is caused by the selective character of solar radiation attenuation in the atmosphere. For better understanding of the dependence of the transparency coefficient on atmospheric mass, let us consider the following schematic case. Imagine that the attenuation of solar radiation in the atmosphere results from the presence in the atmosphere of a certain attenuation band (see Fig 5.18) as an attenuating medium. In this case it is clear that at small m (when the solar ray path through the atmosphere is comparatively short),

-x FIG. 5.18

Wavelength

To the problem of the selective attenuation of solar radiation in the atmosphere.

attenuation of solar radiation is observed mainly in the region of wavelengths of the central, most intensively absorbing part of the attenuation band. Since the radiation attenuation in the region near A, is quite great even at comparatively small m, almost complete attenuation of radiation will take place here. At a further increasing m, the increase of attenuation will be slower because it is now caused by the influence of the far less intensively attenuating wings of the band.

5.6. Characteristics of the Real Atmospheric Transparency

265

Thus we see that in the considered schematic case, the increase of the total attenuation value will somewhat slow down as the atmospheric mass increases. Hence it apparently follows that the mean transparency coefficient calculated for the entire attenuation band must increase with the increase of m. An analogous picture is observed in the case of the real atmosphere the difference being that the attenuation of radiation here is caused by the presence of a great number of attenuation bands. As already been mentioned, the selective absorption of solar radiation in the atmosphere presents an especially complex function of wavelength. Also significant is the dependence of the molecular scattering coefficient upon wavelength. The performed analysis of the dependence of p m on atmospheric mass assumes that the fluxes Sm and So are measured actinometrically. Since in this case the receiving surface is perfectly black, the integral transparency coefficient can by found from the formula

The results are different for visual observations. For this case p m should be found from

where &A is the spectral sensitivity of the eye. Calculations of Fessenkov and Piaskowska-Fessenkova [79], employing the latter formula and the known S o s Ap, A , and E ~ have , shown that in visual observations the dependence of p m on atmospheric mass is insignificant. This conclusion is confirmed by observations. Data of actinometric observations find the existence of a clearly expressed dependence of the transparency coefficient p m on atmospheric mass, but this dependence is not specifically important. It becomes important only when variations of the transparency coefficient due to the varying atmospheric transparency have the same order of value as the variations associated with the dependence of p m on atmospheric mass. It is necessary, therefore, to consider transparency coefficient values obtained from observational data of different conditions with the purpose of

266

Direct Solar Radiation

stating the degree of sensitivity of the transparency coefficient as a characteristic of the state of atmospheric transparency. Table 5.24 presents an annual variation of the transparency coefficient p m , calculated by Kuznetzov [80] from data of measurements by means of the Mihelson actinometer for three values of atmospheric mass: 1.5, 2, and 3. This table shows that during the year, the mean monthly values vary only within the limits 0.73 2 p1.52 0.69; 0.81 2 p 2 2 0.69; 0.82 2 p 3 2 0.71. At the same time the variations of the transparency coefficient due to the dependence upon mass reach 0.05 to 0.06. Thus it is absolutely impossible to compare transparency coefficient values obtained from observations at different atmospheric masses; for example, in May, p z = 0.723 and p3 = 0.752. The latter value corresponds to the value of p 2 in March or October, although both p z and p 3 have been computed for May. This appears to confirm the impossibility of obtaining a correct presentation of the actual variation of atmospheric transparency during the day as based on the variation of the transparency coefficient for different solar heights. To make possible the comparison of transparency coefficients, obtained at different atmospheric masses, it is necessary first to exclude the dependence of these coefficients upon mass. Toward this purpose many methods for reducing the transparency coefficients to some given mass have been worked out. We shall treat the most reliable of them. The method used by Tverskoy [81] has shown that the transparency coefficient values at different atmospheric masses are connected by the following relation : Pm

=Pi

(3 - ai

(5.12)

where i is the value of atmospheric mass at which the coefficient p i value was observed, rn is any other mass, and ai is a certain constant at a given state of atmospheric transparency. Further analysis of observational data showed that the dependence between ai and p i may be considered linear, with a sufficient degree of accuracy: a . = A 2. - B i P z. (5.13) where A i and Bi are constant for the given atmospheric mass. At a high atmospheric transparency (pi> 0.8) the formula (5.13) turns out to be not precise enough and must be substituted with the following relation : (5.14) ai = A i - B g i Ci(pi - Pi)3

+

TABLE 5.24 ; i

Mean Annual Variation of the Transparency Coeficient for Different Atmospheric Masses, Baku (1932-39). After Kuznetzov [80]

Mar.

Apr.

May

June

July

Aug.

Sept.

Oct.

Nov.

Dec.

Year

-

0.737

0.661

0.702

0.708

0.691

0.656

0.703

0.728

-

-

0.698

0.764

0.755

0.720

0.723

0.732

0.700

0.689

0.724

0.751

0.733

-

0.736

0.815

0.769

Jan.

Feb.

0.809

3.

p % I

F w

Pl.6

Pa

!3 U

P PS

0.823

0.797

0.791

0.748

0.742

0.755

0.734

0.712

0.746

0.775

0.781

. i .

268

Direct Solar Radiation

Here Pi is the value of the transparency coefficient beginning with which it is necessary t o introduce a correction term to (5.13), and Ciis constant for the given atmospheric mass. Values of the constants A i, Bi, Ci, and Piin dependence on atmospheric mass are given in Table 5.25. With the help of (5.12) and (5.14) and of Table 5.25, it is possible t o reduce the transparency coefficient to any given atmospheric mass. In practice, however, there is no uniformity in actinometric calculations as regards the reduction of the transparency coefficient. Different authors reduce it to different values of atmospheric mass from m = 1 to m = 3. The value m = 2 should be considered the most expedient in this case because, for the majority of observation points, it is possible t o determine directly the transparency coefficient p z from measurement data on direct solar radiation flux. Somewhat more precise than Table 5.25 is the method for reducing the transparency coefficient, proposed by Sivkov [82]. Using observational data of eight points on the territory of the U.S.S.R., Sivkov compiled a table of mean values of direct solar radiation flux S , at different mass numbers and different states of atmospheric transparency. Thus the S , values were obtained for m varying from m = 1.5 to m = 8.0. Interpolating between the observed S , value at m = 1.5 and the solar constant So = 1.88 cal/cm2 min, Sivkov computed values of direct solar radiation flux for m = 1 and m = 0.5. This permitted computation of the transparency coefficients for the mentioned eight observation points in the range 0.5 < m < 8. The results were averaged over the entire considered territory, which gave average transparency coefficients for different mass values and different states of atmospheric transparency, that is, averaged over both time and territory. To eliminate the necessity of computations in the reduction of the transparency coefficient to a given mass, Sivkov compiled a table of all possible values of transparency coefficients at the variation of atmospheric mass from m = 0.5 to m = 8. This table (Table 9 of the Appendix) helps to determine the transparency coefficient value at any mass from the given coefficient at some definite mass. Comparing results obtained by this method with the results derived from application of Tverskoy’s formulas gives an insignificant discrepancy; for example, the reduction to p , according to Tverskoy yields somewhat higher values of p , than those given by Sivkov. For the same purpose the following empirical formula, describing the relation between fluxes of solar radiation at the masses m, and m 2 ,can also be used:

TABLE 5.25 Dependence 0/ Ihe Conslanl Values A;, B i , C,., and Pi upon Atmospheric Mass

1.5

20

2.5

3.0

3 5

4.0

4.5

5.0

5.5

6.0

6.5

70

7.5

8 0

A,

0.336

0.364

0.390

0.411

0432

0.450 0.468

0.485

0.502

0.519

0.535

0.552

0.569

0.586

0.602

B,

0.358

0.385

0.411

0.432

0453

0.471

0.506

0.524 0.541

0.558

0.575

0.592

0.609

0.626

c,

10

P,

O.H20

10

11

0.830 0.840

12

o 850

0.489 15

0.860

20

25

30

35

0.870

0.880

0.885

0.890

270

Direct Solar Radiation smz - (l/ml) In -1.91 (1/m2)

+ 0.15 + 0.15

Sm, In 1.91

(5.15)

With the help of this formula it is possible to find from a measured

Sm,value the value of solar flux SmZat any other atmospheric mass (for example m = 2). Then the coefficient p z is easily obtained from (5.11). Since in (5.1 1) we find the solar constant, its meaning should be deciphered for this case. As it is seen from the definition of the solar constant, whose values have been given in Sec. 5.4, it cannot, strictly speaking, be used for computation of atmospheric transparency and radiant fluxes from (5.1 1) or of analogous relations that will be treated later. Indeed, in the determination of such an “astronomical” solar constant, an account of the spectral components of the integral flux, which are never observed at the surface level, is made. It is obvious that there is no need of such accounting in the determination of the solar constant of (5.11). The latter constant must quantitatively differ from the astronomical, and Georgi [69] has proposed that it be called “meteorological.” It should be determined either according to the method presented in Sec. 5.4 (but without the infrared and ultraviolet corrections) or on the basis of formulas of the type of (5.11). In Georgi’s [69, 831 estimation the meteorological solar constant equals 1.80 cal/cm2 min. Sivkov [84] evaluated it at 1.82 cal/cm2 min. This latter value is expressed in the International Pyrheliometric Scale (1956). In the new scale the values of the solar constant, without taking account of the infrared and ultraviolet corrections, are 1.854 cal/cm2 min, according to the Smithsonian Institution; 1.855 cal/cm2 min according to Nicolet; and 1.841 cal/cm2 min according to Johnson. Certain discrepancy between these values requires their clarification, and therefore a new simple method for calculation of the transparency coefficients depending on the solar constant value is needed. Murk [85-881 has noted that after differentiating (5.1 I), (5.1 6a) It is easily seen that Ap, = 0 if ASm/Sm= ASn/Sn. This means, for instance, that old tables may well be used for determining the transparency coefficient from the measured S , values if the latter are changed in corresponding proportion.

5.6. Characteristics of the Real Atmospheric Transparency

27 1

It is also possible, of course, to calculate the transparency coefficients directly from (5.16b) Table 5.26 indicates the values A S , / S , corresponding to the transition from three different solar constant values to So = 1.98 cal/cm2 min of the standard adopted by the International Radiation Commission. It should be noted that the introduction of similar corrections does not greatly affect the final result. This can be seen from Table 5.26, which presents the ideal atmosphere transparency coefficients computed from different basic solar constant values. This is also shown in Table 5.27, which gives the results of computation of A p , from (5.16a), performed by Murk for the case of the transition from So = 1.88 to So = 1.98 cal/cm2 min. TABLE 5.26 AS,,lSm, %

So, cal/cm2 min

1.88 1.94 1.91

-5.1 -2.0 -3.0

TABLE 5.27 Corrections to the Transparency Coefficients. After Murk [84]

rn Pm

0.85 0.80 0.75 0.70 0.65 0.60

1

1.5

2

3

4

5

6

-0.031 -0.009 -0.007 -0.024 -0.022 -0.021

-0.021 -0.020 -0.018 -0.017 -0.016 -0.015

4.016 -0.015 -0.014 -0.013 -0.012 -0.011

-0.011 -0.010 -0.010 -0.009 -0.009 -0.008

-0.008 -0.008 -0.007 -0.007 -0.007 -0.006

-0.006 -0.006 -0.006 -0.006 -0.006 -0.005

-0.004 -0.004 -0.004 -0.004 -0.004 -0.004

Note here that the described reduction method can be applied also for other characteristics of atmospheric transparency, depending on the relation S,/So.

272

Direct Solar Radiation

The work on methods for reduction of the transparency coefficients to a given mass allowed, to a certain degree, the elimination of the unrepresentative quantitative characteristic of atmospheric transparency associated with its dependence on atmospheric mass. However, the methods for reduction can be applied only for processing the averaged observational data. As to the coefficients calculated from unaveraged data, their validity becomes rather doubtful. That is why many investigators try to obtain other characteristics of atmospheric transparency independent of mass. Below are some results of such attempts. 2. The Turbidity Factor. Widely accepted is a characteristic of atmospheric transparency proposed by F. Linke [Chapter 4, Ref. 41 and known as the turbidity factor. We have already seen that the attenuation of solar radiation in the atmosphere is caused mainly by three factors: molecular scattering, scattering and absorption of radiation by water vapor and liquid water droplets, and scattering and absorption of radiation by dust. Taking this into account, present the optical atmospheric thickness in the vertical direction 0, as

where k , is the mass coefficient of molecular scattering, Q is air density, aw,Ais the mass coefficient of the radiation attenuation by water vapor, pW is water vapor density (absolute humidity), ad,, is the mass coefficient of the radiation attenuation by dust, and @ d is dust concentration. Determine now the monochromatic turbidity factor T, from the relation

As seen, T,is the ratio of the optical atmospheric thickness in the direction of the vertical 0,to the corresponding optical thickness calculated kA@dh with account of molecular scattering only. Since the values 0, and may be also considered to be radiation attenuation coefficients, calculated for the entire vertical atmospheric column, Ta can be determined as the ratio of the total attenuation coefficient to the coefficient of attenuation due to molecular scattering. The turbidity factor T, characterizes the correlation between the transparency of the real atmosphere and that of the ideal atmosphere in which the attenuation of solar radiation is caused solely by molecular scattering. With (5.18) taken into account, the formula for determination of the

sow

5.6. Characteristics of the Real Atmospheric Transparency

273

total attenuation of the monochromatic direct solar radiation flux at a given solar zenith distance (given atmospheric mass m) can be presented in the form

Or, having in view that the value qA= exp(J; kAedh) is the monochromatic transparency coefficient of the ideal atmosphere, it can be written as Sm.2

= So,aqamTA

(5.19)

The relation (5.19) permits giving another and very spectacular explanation of the turbidity factor TA. As seen, the turbidity factor indicates by how many times the ideal atmosphere mass should be increased to ensure that the total attenuation of solar radiation in such an atmosphere will be identical with the one in the real atmosphere. On the basis of (5.19) the following expression for the full flux of direct solar radiation, Sm, is obtained:

If the values qm and T,, averaged over the whole solar radiation spectrum, are introduced, we find that

s,

= SoqmmT

(5.20)

Here T is the integral turbidity factor. The comparison of (5.11) and (5.20) shows that the integral turbidity factor T can be expressed through the transparency coefficients p m and qm in the following manner: (5.21)

where 8, and are the optical thicknesses of the real and ideal atmospheres, corresponding to the direction determined by the mass m value; OO(O)and @O,ideal (0) are the optical thicknesses of the real and ideal atmospheres in the vertical direction. This formula can be used for computation of the turbidity factor T from the theoretically calculated q(m) value and the empirical value of p m . In computations of this kind the following factors should be remembered. As we have already seen, the ideal atmosphere transparency coefficients

274

Direct Solar Radiation

are computed only for the case of a certain standard atmosphere. In actual fact, the values q m , corresponding to a given real atmosphere, vary in dependence upon the pressure Po value at the earth’s surface level. So, with m being understood as “relative” mass, it is necessary to calculate qm for the given Po value in each case. With the use of only one “normal” qm value, the meaning and value of the turbidity factor change. Averkiev [Chapter 4, Ref. 81 has computed the ratio K = T/T’ for different Po values, where T is the turbidity factor computed with account taken of the dependence of qm upon Po, T’ is the ccabsolute”turbidity factor, obtained with the use of the same standard qm value corresponding to Po = 760 mm. The results of these computations for m = 2 are given in Table 5.28. TABLE 5.28 Values of the Ratio of the Turbidity Factors K = TIT’ at rn [Chapter 4, Ref. 81

Po,

780 770 760 750 740 730 720 710 700 690 680 670 660 650 640 630 620

K 0.980 0.991 1.Ooo 1.012 1.024 1.036 1.049 1.062 1.073 1.085 1.099 1.113 1.128 1.143 1.157 1.172 1.189

Po,

610 600 590 580 570 560 550 540 530 520 510 500 450 400 350 300

=

2. After Averkiev

K 1.205 1.220 1.238 1.256 1.275 1.294 1.311 1.329 1.350 1.369 1.391 1.417 1.547 1.711 1.923 2.221

Table 5.28 shows that at small pressure variations (near 20 to 30 mm), the difference between T and T’ is insignificant, considerably increasing, however, at greater variations of pressure.

5.6. Characteristics of the Real Atmospheric Transparency

275

The turbidity factor values can be calculated with the help of two relations following from (5.20): T=

1 so In sm - Q, In m In qm SO Sm

(5.22)

and

T=

In S, - In So In Sm,ideal - In SO

(5.23)

Here Q, = - (1 /mIn q,) is a value that can be computed theoretically. The same refers to S,,ideal, the flux of direct solar radiation at the earth’s surface in the case of the ideal atmosphere. In Table 10 of the Appendix are given Q, values corresponding to the International Pyrheliometric Scale (1956) (S = 1.98 cal/cm2 min). It should be stressed that when using these formulas for computation of T, the tabulated values of q , or S,,ideal obtained for the standard ideal atmosphere cannot be applied the same as with (5.21). Strictly speaking, it is necessary each time to calculate q , and Sm,ideal specially for a given P value. It is easily understandable, however, that such detailed computations have no meaning when q , and S,,ideal are determined as corresponding to the value of m understood to be the “absolute” mass. For example, if we determine q, from the table for the standard atmosphere at m, = m(P/Po),then the obtained q , value corresponds to the factual atmospheric transparency in the direction that is determined by the “relative” mass m value at a pressure at the surface level equal to P. It may seem that the value of the turbidity factor is independent of the pressure at the earth’s surface, for according to (5.21), this value is determined as a ratio of optical thicknesses. However, it should be remembered that if the optical thickness of the ideal atmosphere is directly proportional to pressure, this relation is not valid for the real atmosphere. As to the values m,p , , and S, of the formulas (5.21) to (5.23), they must all correspond to the “relative” m values at a given height of the sun. This latter fact is particularly important because some authors have computed the turbidity factor T with all the values of the above formulas corresponding to the values of the “absolute” mass. Such a method is wrong. The introduction of the turbidity factor was made in order to derive a quantitative characteristic of atmospheric transparency whose value would be independent of atmospheric mass (solar height). We see that the dependence of the transparency coefficient on mass is mainly caused by the influence of the selectivity of molecular scattering and the absorption of solar

276

Direct Solar Radiation

radiation. The turbidity factor is determined as a quantitative characteristic of atmospheric transparency, indicating the degree to which the real transparency, differs from the transparency of the ideal atmosphere. It is natural, therefore, that by introducing this “relative” transparency characteristic, we exclude the effect of the dependence upon mass, which is caused by the selectivity of molecular scattering. It should also be expected that the turbidity factor would show less dependence on atmospheric mass than would the transparency coefficient. This is indeed so, although the turbidity factor is still dependent on atmospheric mass; this can be seen from Table 5.29, which was made up from actinometric measurements at Tashkent by Yaroslavtzev [89]. Yaroslavtzev took m for an “absolute” mass, introducing a pressure correction in all cases. Although such method for calculation of m is not quite perfect, as we have already noted, the data of Table 5.29 can be trusted as far as the dependence of the turbidity factor upon m is concerned. The turbidity factor is seen to decrease with the increase of atmospheric mass. This is in accordance with the above qualitative conclusions about the dependence of transparency characteristics upon mass and with the empirical determination that transparency coefficient increases with increasing mass. TABLE 5.29 Mean Annual Values of the Turbidity Factor at Tashkent. After Yaroslavtzev [89]

rn Year

1926 1927 1928 1929 1930 1931

Mean

1.5

2

-

2.64 2.83 2.44 2.68 2.72 2.70 2.67

3.09 2.67 2.87 2.89 2.95 2.89

For elimination of the dependence of the turbidity factor upon mass, the values T referring to various m should be reduced to a definite atmospheric mass. Since, according to (5.21), the turbidity factor is determined as a ratio of the logarithms of the real and the ideal transparency coeffi-

5.6. Characteristics of the Real Atmospheric Transparency

277

cients, then to reduce the turbidity factor to a given mass, the described method for the reduction of the transparency coefficient can be applied. Using the formula of Tverskoy [81] as a possible method, we have

Here bi = ai/ln qi and ci = ai,ideal/lnqi are theoretically computable values that are constant for the given mass. The necessity of reducing the turbidity factor to a given mass in order to obtain comparable data greatly complicates its practical application as a characteristic of atmospheric transparency. In this connection, Linke [Chapter 4, Ref. 41 has proposed that the definition of the turbidity factor be changed with the purpose of decreasing its dependence upon mass. It has already been mentioned that the dependence of T upon m is mainly caused by the selectivity of absorption. If, therefore, the turbidity factor be referred not to a clean and dry atmosphere but to a clean atmosphere containing some quantity of water vapor, then it becomes clear that the effect of the dependence of the turbidity factor upon mass due to the influence of selective absorption can be eliminated to a considerable degree. In intermediate latitudes the mean water vapor content w, in an atmospheric column of unit section (1 cmz) is about w, = 1 g/cm2 (1 cm of “precipitated water”). Following these considerations, Linke offered to introduce the so-called new turbidity factor 0 in relation to a clean atmosphere containing 1 g/cm2 of water vapor. The turbidity factor 0 can be calculated from formulas similar to (5.21) to (5.23). The values q m , Om, Sm,ideal, however, must be calculated in this case for the above-mentioned ideal humid atmosphere. Although the new turbidity factor 13 possesses certain advantages in comparison with the T factor, the International Radiation Commission did not recommend its use, arguing that the inequality T > 1 always takes place, whereas I3 may be less than unity in a dry atmosphere. In actinometric practice the Angstrom coefficient for the turbidity is widely used (see Sec. 4.5).

3. Kastrov’s Formula. After finding approximate analytical presentations for the energy distribution in the solar spectrum outside the atmosphere and for the spectral atmospheric transparency, Kastrov [90] was able to

278

Direct Solar Radiation

obtain the following simple formula for the integral solar radiation flux at the surface level:

so s, = ___ 1 + crn

(5.25)

where c is a quantitative characteristic of atmospheric transparency. This formula has been widely accepted in actinometric calculations, for since the coefficient c is determined empirically, the formula allows satisfactorily calculation of the attenuation of direct solar radiation in the atmosphere. The Kastrov formula has been verified by Savinov [91], Korsak [92], and Yaroslavtzev [93], who showed it to be convenient and quite satisfactory as regards the accuracy in the calculation of solar radiation totals. Let us dwell now on the characteristic of numerical values of the coefficient c in Kastrov’s formula. Sivkov [82] has calculated c for different values of atmospheric mass and transparency, estimated from measurement data on the solar flux S , average for eight different observational points (U.S.S.R.). The state of atmospheric transparency has been characterized by the integral transparency coefficient p 2 , calculated for m = 2. The results are given in Table 5.30. The solar constant has been taken to be 1.88 cal/cm2 min. The coefficient c is shown to be a very “sensitive” characteristic of atmospheric transparency, exhibiting more variation in the considered cases than the transparency coefficient p z ; for example, the decrease of p z by 29 percent corresponds (at rn = 1) to the increase of c by approximately 3.5 times. At the same time, however, it is seen that the coefficient c is TABLE 5.30 The Coefficient c Values in Kastrov’s Formula. After Sivkov [82] Atmospheric Mass, m

PZ

0.60 0.65 0.70 0.75 0.80 0.85

1

1.5

2

3

4

5

8

0.775 0.577 0.474 0.367 0.289 0.218

0.827 0.605 0.490 0.376 0.292 0.222

0.862 0.631 0.494 0.365 0.281 0.207

0.879 0.641 0,487 0.356 0.271 0.203

0.974 0.664 0.488 0.353 0.267 0.198

1.079 0.710 0.516 0.360 0.269 0.199

1.381 0.854 0.576 0.383 0.272 0.194

5.6. Characteristics of the Real Atmospheric Transparency

279

not a unique value, for at a given state of atmospheric transparency (characterized by a definite p z value), it varies in dependence on atmospheric mass and consequently on the height of the sun. At a high transparency (p2= 0.85) the variation of c in dependence upon m is not great, c decreasing as m increases. At a lower transparency ( p 2 < 0.70; c > 0.4) the dependence of c upon m is more pronounced, and c increases with the increase of m. In the average conditions of transparency ( p z = 0.75) the value c has a minimum at m = 4, increasing toward both increasing and decreasing m values. It appears from Table 5.30 that the dependence of the coefficient c upon mass is comparatively low at m 5 3. In this range of m values the variations of c due to the variations in atmospheric transparency exceed by far the values of this coefficient related with its dependence on atmospheric mass. Thus, in the interval from m = 3 before noon to m = 3 in the afternoon, the coefficient c may be considered practically constant, independent of atmospheric mass. The same conclusion was reached by Yaroslavtzev [93], who had computed c for different m from observations at Ashkhabad. Since the most essential calculations of c are those from (5.25) for m 2 3 , it follows that the coefficient c as a quantitative characteristic of atmospheric transparency is rather feasible. The same holds for (5.25). 4. Murk’s Formula. Murk [85, 861 has shown that a simple description of the total attenuation of solar radiation in the atmosphere is possible only in the case where there are two, not one, parameters to characterize the atmospheric transparency. His formula has the following form :

S,

= SoplmmBm

(5.26)

Here p l , the transparency coefficient at m = 1, and the parameter B are quantitative characteristics of atmospheric transparency. It can be shown that with flux value measured at two different masses m and n on hand, we are able to compute the p 1 and B values from the formulas n log n(log S , - log So) - m log m(log S, - log So) (5.27) h P , = mn (log n - log m ) B=

m(log S,

-

log So) - n(log S,, - log So) mn(1og n - log m )

(5.28)

For more convenience in these computations, Murk has constructed a nomogram (Appendix 11) that enables easy determination ofp, and B.

280

Direct Solar Radiation

It can also be used to compute the transparency coefficient p , , the turbidity factor T, and a value t = BIB*, where B* is the parameter B related to the conditions of an ideal atmosphere. The isolines of this nomogram correspond to equal radiant flux values. The axis of abscissas presents atmospheric masses; the axis of ordinates, transparency characteristic values. If we find a point with the coordinates m, S , corresponding to the measured S , value, we can find the value p 1 sought at the point of intersection of the given isoline S and the axis of ordinates (scale p ) . The corresponding value p m is the ordinate of the considered point (m,s,) on the scale p . The parameter B is obtained by counting the ordinate of the intersection point of the line passing through the zero point (the scale in the right top corner of the form), the point m, S , , and the axis of ordinates on the scale p . Similarly the turbidity factor T, can be found. In this case the mentioned line must pass through the points 0 (right top corner of the form) and (m,S,). Let us illustrate this with the following example. Let m = 2.5 and S = 1.12 cal/cm2 min. Then, from the nomogram, p m = 0.813, T, = 2.53, and B = 0.0563. It is evident that the considered nomogram can also be used for the reduction of p , , T,, and S , to a given mass. To this effect the points (m,S,) and 0 must be connected with a line. After this, the values pm and I, at any other mass are obtained as ordinates of the line points considered, corresponding to the given atmospheric masses on the scales p and T. The intersection of the line with the isoline S at a given m, points out the S,,, value corresponding to the given m. For the above case the use of this nomogram provides the following values of p m , T, , and S , at different atmospheric masses: 1

rn

0.772 2.62 1.45

Pm Tm

Sm

2 0.803 2.55 1.22

3

0.821 2.49 1.04

4 0.835 2.46 0.91

5. The Turbidity Index of Makhotkin. Makhotkin [94-96, 96a] has proposed a characteristic of transparency somewhat resembling Linke’s turbidity factor, but basically different. Analyzing data of actinometric observations at Karadag (Crimea), Makhotkin proved that in the interval 0.1 S , 5 0.6, the following relation is valid: S,

+ 0.49(Sm - 0.81)3 = 1.425 - 1.121 log m

(5.29)

5.6. Characteristics of the Real Atmospheric Transparency

28 1

Let us assume that the atmosphere for which the relation (5.29) is valid is the standard atmosphere. If the measured flux value at a mass m* is S,, , determine from (5.29) the m value corresponding to S, = S,. Now introduce the following turbidity index as a characteristic of transparency : N = - m (5.30) m*

It is natural to meet the cases of N 3 1. The turbidity index has a clear physical meaning, for it points out how many standard atmospheres must be taken to obtain the observed radiant flux value at a given solar height. To simplify the computation of the turbidity index, Murk [86] has constructed a special nomogram that presents the isolines S in a coordinate system log N , log m (see Appendix 12). The value N corresponding to the measured S, is obtained here as the ordinate of the point (m,S,). In 1940 P. Moon proposed a model of a “standard summer atmosphere” for description of the mean atmospheric optical properties. Table 5.31 gives some turbidity index values calculated by Makhotkin for the standard summer atmospheric model. TABLE 5.31 The Turbidity Index for a Standard Summer Atmosphere. Affer Makhofkin [94] m

N

1 1.05

2 1.04

3 1.04

4 1.04

5 1.06

Mean 1.05 f 0.01

We see that the standard atmosphere by Moon closely corresponds to the standard atmosphere suggested by Makhotkin for the turbidity index calculation. On the other hand, Table 5.3 1 demonstrates practical absence of the dependence of N on atmospheric mass. Table 5.32 characterizes the variation of the turbidity index over the year for different places. Abundance of quantitative transparency characteristics (only the most important have been considered here) makes the problem of their comparison and the choice of the most rational very important. As has already been mentioned, the main criterion for such choice must be the relation between the “sensitivity” of characteristics to atmospheric transparency and their dependence on atmospheric mass. To express this relation quanti-

TABLE 5.32 AnnuaI Range of the Turbidity Index. After Makhotkin [94]

Latitude, deg

Elevation above Sea Jan. Level, rn

Feb. Mar. Apr.

May June July

Aug. Sept. Oct.

Nov. Dec. Mean

--.z U

Exdalemuir

55.3 N

Modena

44.6

51

Helwan

29.9

116

0.95 0.70 0.69 0.77 0.77 0.76 0.76 0.76 0.78 0.84 0.82 0.78 0.78

6.2 S

8

1.25 1.40 1.10 1.00 1.06 1.00 1.05 1.26 1.62 1.85 1.38 1.24 1.30

Aroza

46.8 N

1860

0.32 0.32 0.34 0.39 0.45 0.45 0.46 0.42 0.42 0.37 0.31 0.30 0.38

Aroza (reduced to sea level)

46.8

1860

46.8 0.40 0.43 0.49 0.56 0.56 0.58 0.53 0.53 0.47 0.40 0.39 0.48

244

0.69 0.79 0.88 1.09 0.94 0.83

1.02 0.87 0.98 0.87 0.73 0.77 0.69 0.86

F

1.27 0.94 1.21 1.08 1.34 1.24 1.06 1.26 0.85 0.80 0.84 1.07

PE 0

Djakarta

5.7. Variation of the Atmospheric Transparency State

283

tatively, Murk [87] has suggested introduction of the “rationality coefficient” rj , determined by the formula (5.31)

where f is a characteristic of transparency and the derivatives af/aS and af/am determine its dependence on solar radiation flux (atmospheric transparency) and atmospheric mass. The computation of the rationality coefficient for the above considered transparency characteristics has shown the Makhotkin and Murk characteristics to be most adequate.

5.7. Some Data of Observations on the Variation of the Atmospheric Transparency State The preceding section has given certain data on various transparency characteristics which make it possible to estimate the observed variations of transparency. Now we shall consider this problem in more detail. It should be mentioned, however, that the values of transparency characteristics given below were computed in different ways. The comparison of these values, therefore, can be only qualitative in a number of cases.

1. Diurnal Variation of Atmospheric Transparency. It is quite natural that the transparency of the atmosphere is not constant over the whole day, and therefore we can speak of a daily range of transparency. The investigation of this range is a complex task and has not so far been given enough attention. We know that all transparency characteristics have an apparent daily range caused by their dependence upon mass. On the other hand, the methods for reduction of transparency characteristics to a given mass, allowing exclusion of the dependence upon mass, are sufficiently reliable only for averaged transparency values. All this points out the necessity of such a transparency characteristic for determination of the daily range of atmospheric transparency, whose dependence upon mass would be minimal. The new turbidity factor 0 is one such characteristic. The daily variation of 0 [Chapter 4, Ref. 41 has been found [78,97] to depend in a very marked way on observational conditions. In the majority of cases, however, the maximum 0 takes place at the noon hours in summer, which results from a higher dust content of the lower atmospheric layers due to a greatly developed convection at these hours. In winter there is no such marked variation of atmospheric transparency, and in a number of cases the minimal

284

Direct Solar Radiation

turbidity factor (transparency maximum) has been found at noon. Both in summer and in winter the transparency of the atmosphere in the afternoon is usually lower than before noon. It has already been noted that the intensity of the circumsolar aureole is a very sensitive characteristic of atmospheric transparency. Its measurement shows that microfluctuations of atmospheric transparency take place during very short time intervals. The results of such measurements, conducted by Kalitin [Chapter 3, Ref. 1021, are given in Fig. 5.19 which shows the variation of the circumsolar aureole’s intensity during a period of M A Y 10,1931

C 00240 .-

E

“E

”

\

0

0.0220

t

MAY 27,1931

l

0.0180

0.0160 A

0.0140

0.0140

I 2 3 4 5 6 7 8 9 1011 12131415161718192021 rnin

FIG. 5.19

Variation of the intensity of the circumsolar aureole during the day.

21 min for three different days. As seen, the microfluctuation of atmospheric transparency are different on different days. On July 19, 1913, for example, they were very weak, but on the other two days they were quite strongly felt. In some cases (May 27, 1931), during very windy days, sharp leaps of atmospheric transparency are observed. Kalitin explains this by the transfer of air masses of different transparency before the sun. We see, therefore, that the daily range of atmospheric transparency is complex and much dependent on the conditions of observation (season, wind, and similar weather factors).

5.7. Variation of the Atmospheric Transparency State

285

2. Annual Variation of Atmospheric Transparency. Averaged characteristics of atmospheric transparency show a comparatively simple annual range, with a maximum of transparency in winter months and a minimum during summer. Being computed over shorter time intervals, they are much more complex. For illustration, let us examine the results of detailed computations of the annual transparency coefficient variation by Kalitin [Chapter 3, Ref. 1021 at Pavlovsk (Fig. 5.20). The solid curves of this figure characterize the “detailed” variation, and the dashed curved show the average variation of the transparency coefficient in 1920 and 1921. The “detailed” range is complex and differs from year to year. This appears to be caused by the peculiarity of atmospheric transparency in different years; for instance, the sharp minimum of transparency at the end of August and the beginning of September, 1920, is explained by extensive forest fires in the Leningrad region at that time. It is also natural that since the annual range of atmospheric transparency varies from year to year, the mean annual values of transparency characteristics are different in different years. Kalitin [Chapter 3, Ref. 1021

-I

- ---2

FIG. 5.20 Annual variation of the transparency coefficient as observed at Pavlovsk. (1) detailed; (2) averaged p , for 1920 and 1921.

286

Direct Solar Radiation

calculated the mean annual transparency coefficient p 1 for Pavlovsk over 31 years (see Table 5.33). At the mean many-year coefficient p z = 0.745, the mean annual transparency coefficient varies from 0.570 to 0.770. Note here that the lower transparency in the period 1912-1914 was caused by pollution of the atmosphere with volcanic dust in the eruption of the Katmai volcano (Alaska) in 1912. Calculations of the annual range of the turbidity factor T lead to the same conclusion on the basic regularities of the annual atmospheric transparency variation as expressed by the above calculations of the transparency coefficient. Corresponding to the decrease in atmospheric transparency during the summer variation is found the maximum T at this time (see Table 5.35). 3. Geographical Variation of Atmospheric Transparency. Calculations of geographical distribution of the turbidity factor in the Northern Hemisphere show an increase of the turbidity factors T and @ southward; that is, a decrease of atmospheric transparency with the decrease in latitude. TABLE 5.33 Mean Annual Values of the Transparency Coeficient for Pavlovsk (Near Leningrad). After Kalitin [Chapter 3, Ref. 1021

Year

P1

1906 1907 1908 1909 1910 1911 1912 1913 1914 1915 1916 1917 1918 1919 1920 1921

0.754 0.710 0.741 0.770 0.741 0.738 0.570 0.690 0.710 0.734 0.738 0.714 0.741 0.751 0.721 0.748

I

PI

Year 1922 1923 1924 1925 1926 1927 1928 1929 1930 1931 1932 1933 1934 1935 1936 Mean

0.748 0.760 0.754 0.758 0.765 0.752 0.757 0.745 0.754 0.750 0.755 0.754 0.745 0.757 0.754 0.745

287

5.7. Variation of the Atmospheric Transparency State

The geographical distribution of the turbidity factor in the Northern Hemisphere from data of [Chapter 4, Ref. 41 is given in Table 5.34. The parentheses enclose numerical values of the turbidity factor 0. These data should be considered approximate; it seems that they somewhat exaggerate the dependence of T and 0 upon latitude. TABLE 5.34 Geographical Distribution of the Turbidity Factor. Afrer Kalitin [Chapter 4, Ref. 41 Latitude, deg 90-60

Dec. and Jan.

40-20

1.6 2.5 3.0 3.8

2 w

4 . 6 (2.9)

60-50 5040

(1.1) (1.6) (1.9) (2.4)

March 1.9 2.6 3.5 4.0 4.6

(1.3) (1.7) (2.2) (2.4) (2.8)

June and July 2.3 3.2 4.1 4.4 4.6

(1.4) (2.0) (2.5) (2.6) (2.8)

Sept. 2.2 2.9 3.6 4.2 4.6

(1.5) (1.9) (2.2) (2.5) (2.8)

It should be noted that the importance of the decrease in atmospheric transparency southward is exaggerated in some cases; for example, the statement that in the southern, coastal areas of the U.S.S.R. is observed a decreased transparency of the atmosphere, based on the comparison of actinometric measurement data from Pavlovsk (near Leningrad) and Feodosia (Crimea), was later negated by Sivkov [98], who had proved that the observations at Feodosia were not representative because they had been conducted under conditions of a town-polluted atmosphere. The actual variations of the mean transparency values over the vast territory from Pavlovsk to the south coast of the Crimea are very small. This can be seen in Table 5.35, which gives the annual range of the turbidity factor T a t m = 2 calculated by Sivkov from direct solar radiation observations at Pavlovsk, Kursk (Central Russia) and Karadag (Crimea). The turbidity factor values were computed from the formula T, = 12.86 log (So/S,),where So = 1.88 cal/cm2 min. The above data determine a much slower increase of the turbidity factor toward the south than is apparent from Table 5.34, this increase developing only as regards the mean annual T values. In isolated months, both increase and decrease of the turbidity factor southward are observed. There are reasons to beileve that the data of Table 5.34 exaggerate the dependence of the turbidity factor upon latitude, since they are compiled from approximate calculations, whereas Table 5.35 summarizes direct measurement values.

TABLE 5.35 Annual Variation of the Turbidity Factor at Pavlovsk (1906-1926), Kursk (1927-1933), Karadag (1933-1937). Afrer Sivkov [98]

Observation Point

Latitude,

Jan.

Feb.

March Apr.

May

June

July

Aug.

Sept.

Oct.

Nov.

Dec. Yearly

deg N

8 F

E Pavlovsk

60

1.90

2.19

2.42

2.65

2.94

3.04

3.15

2.85

2.75

2.51

2.02

1.81

2.52 I’

3 .

Kursk

52

2.02

2.19

2.10

2.61

2.94

3.04

3.31

3.21

2.94

2.56

1.90

1.81

2.55

Karadag

45

2.15

2.33

2.65

2.70

2.99

2.94

3.10

3.21

2.75

2.56

2.24

2.02

2.64

8

5.7. Variation of the Atmospheric Transparency State

289

Arctic and Antarctic are anomalous with respect to atmospheric transparency. As seen from Table 5.36, which presents Makhotkin’s [99] values of the turbidity index, these polar regions are characterized by an exceptionally high transparency. Zvereva [IOO] has shown that the small amplitude of the annual atmospheric transparency variation is an important peculiarity of arctic conditions. It is also interesting that the increase of transparency with latitude is observed only up to 80’N. Still farther to the north, the transparency is lower, owing to the increasing condensational turbidity. According to Subbotina [ l o l l the transparency coefficient in Antarctic varies within the range 0.80 to 0.90, with the highest value observed at the continental stations “Komsomolskaya” and “Vostok.” 4. Variation of Atmospheric Transparency witk Height above Sea Level.

It is quite evident that atmospheric transparency should increase with the increase of altitude above sea level. Some data on this fact have been given in Chapter 4. This conclusion is confirmed by measurement data on the integral atmospheric transparency. For instance, calculations of the transparency coefficient by Belinsky 11021 from balloon observations over direct solar radiation (Aug. 12, 1946) lead to the following conclusions. Below 900 m, the transparency coefficient varies in dependence on atmospheric mass from 0.578 to 0.686. The layer 900 to 2300 m shows the 0.728 to 0.756 variation; above 2300 m, the coefficient is 0.824 to 0.891. Thus, in the lower layers, a considerable increase of the transparency coefficient with height takes place. The above data were obtained by calculating the coefficient at an “absolute” mass m (corrected for pressure). Therefore they characterize the reduced, not factual, transparency. Since the “relative” mass values corresponding to the given “absolute” masses must be greater in the considered case, this means that the coefficients characterizing the factual atmospheric transparency must exceed those mentioned above. Very spectacular in its presentation of the variation of atmospheric trasparency with height is Fig. 5.21, which summarizes the results of aircraft measurements of the aerosol attenuation coefficient from data of Faraponova [I031 and Krug-Pielsticker [104], and also from certain data of observations in mountains (V. M. Kazachevsky and T. P. Toropova, S. F. Rodionov et al., V. S . Sokolova). The aerosol attenuation coefficients have been computed as a difference between the measured total attenuation factors and the computed Rayleigh scattering coefficients. The latter show insignificant variation with height, and therefore the curves of Fig. 5.21

N

TABLE 5.36 Atmospheric Transparency (Turbidity Index) in Arctic and Antarctic. Afrer Makhotkin [99] -

Station

Latitude, deg

Period Observation

No. of Observations

Mean

Maximum

Minimum

North Pole 2

76-79 N

Apr.-Sept., 1950

103

0.73

(1.7)

0.36

North Pole 4

81-83 N

Apr.-Sept., 1955

133

0.69

1.02

0.53

North Pole 5

83-87 N

May 1955-Apr. 1956

46

0.62

1.04

0.42

Mowson (Antartic)

67.6 S

March 1953-Jan. 1955

89

0.47

1.06

0.22

-3

E!

p 5FI

L

B 8

5.7. Variation of the Atmospheric Transparency State

29 1

are representative (with sufficient degree of accuracy) of the vertical profile of the integral attenuation coefficient as well. Figure 5.21 demonstrates that, on the average, the atmospheric transparency decreases with height, and especially rapidly in the lower 3 to 4 km. However, in individual cases (as was mentioned in Chapter 4) an uneven variation of the attenuation factor is observed (as, for instance, according to data by G. P. Faraponova) in the summer time and the 1 to 3-km layer shows the presence of a comparatively stable turbid portion. The data of Fig. 5.21 have a curious peculiarity in that they indicate a higher atmospheric transparency in mountains than in a free atmosphere at the same altitude.

a,, km-'

FIG. 5.21 Aircraft (1-3) and high-altitude (4-10) measurement data on the aerosol attenuation coefficient. Central Aerological Observatory: (1) I = 428 mp; (2) I = 495 mp Krug-Pielsticker; (3) I = 540 mp Rodionov et a/.; (4) I = 428 mp, morning and evening; (5) I = 495 mp, morning and evening; (6) I = 428 mp, noon; (7) 1 = 495 mp, noon, Sokolova; (8) 1, = 432 mp; (9) I = 500 mp, Kazachevsky and Toropova; (10) I = 540 mp.

Comparing the aerosol and the Rayleigh attenuation coefficients, we see that the latter becomes equal to, or exceed, the former at elevations above 3 to 4 km. However, even at about 5 km, the optical thickness of the atmosphere is higher than the value corresponding to the conditions of a dry and clean atmosphere. This means that the upper troposphere and the stratosphere are rather turbid. 5. Relation between Various Factor Determining Atmospheric Transparency. We have seen above that the attenuation of solar radiation in the atmosphere is caused by three main factors: molecular scattering, scattering

292

Direct Solar Radiation

on large particles (aerosol attenuation), and selective absorption (by water vapor, first of all). Let us now consider certain data characterizing the relation between these attenuating factors. Figures 5.22 and 5.23 give the results of computations by Kalitin [lo51 for Pavlovsk and Tashkent. The upper straight lines characterize the income of solar radiation on the outer atmospheric boundary (So = 1.88 cal/cm2 min). The distance between the topmost and the middle lines de-

FIG. 5.22 Attenuafion of solar radiation by water vapor and aerosols (Pavlovsk).

termines the attenuation of solar radiation in a dry and clean atmosphere at m = 2. The distance between the lines Sideal and S, is numerically equal to the value of atmospheric radiation absorption. The latter was computed ~ ~ . lower curve charfrom F. Moller’s formula A S = 0.172 ( r n ~ , ) ~ . The acterizes the annual range of direct solar radiation, observed at m = 2. It is obvious that the distance between the curves S, and S, determines the

FIG. 5.23 Attenuation of solar radiation by water vapor and aerosols (Tashkent).

5.7. Variation of the Atmospheric Transparency State

293

value of aerosol attenuation. Thus, we follow the variation of the relation between two different attenuating factors over a year. In both cases the influence of selective absorption and aerosol attenuation is at maximum in the summer time. The cause for this seems to be the higher moisture and dust content in the atmosphere during summer. The considered relation can also be conveniently characterized by means of the turbidity factor T. For this purpose the turbidtiy factor is usually presented as T=l+W+R

(5.32)

From the definition of the turbidity factor it follows that T = I if the attenuation of solar radiation is caused by molecular scattering only. The second term in the right-hand side of (5.43), the value W, is called the humid turbidity factor, and characterizes the influence of absorption and scattering of radiation by water vapor on the attenuation of solar radiation. The third term, R, is known as the residual turbidity factor, determining the effect of radiation attenuation due to absorption and scattering by dust and water droplets. It should be considered, strictly speaking, as consisting of two components, the dust and the condensational turbidity factors. The determination of the humid and residual turbidity factors is realized in the following manner. Observations over the direct solar radiation during days with stable and minimal residual turbidity are performed first. On the basis of such observations the turbidity factor T is computed, and then the dependence of T - 1 upon absolute air humidity at the surface level e, is investigated. Thus it has been found that the following dependence takes place : T-l=a+be,

(5.33)

where a and b are constants. It is natural that the absolute air humidity will affect the humid turbidity first of all. We can therefore assume that W = be, and then R = a. In all other cases (with a significant R value) the residual turbidity factor can be determined from the relation R = T - 1 - be,. However, as was shown by Sivkov [82], the above method for determination of the humid and the residual turbidity factors is not accurate enough because in the presence of some expressedly nonzero, residual turbidity, even though not great, the turbidity stability cannot be counted on. For example, according to Sivkov, at an absolute humidity value of

294

Direct Solar Radiation

about 2 to 5 mm and rn = 2, along with very high values of solar radiation flux of the order of 1.35 to 1.40 cal/cm2min, anomalously low values of 1.00 to 1.10 cal/cm2min are observed. These fluctuations of the flux of direct solar radiation at constant absolute humidity and atmospheric mass are caused exclusively by variations of dust and condensational turbidity. A similar phenomenon is observed at over 12 mm of absolute humidity. Taking into account the instability of R,Sivkov has offered to determine W from observations on such days when the residual turbidity may be considered zero. The degree of the zenith sky polarization will help to estimate the smallness of R in such cases. The degree of polarization must be almost independent of atmospheric water vapor content, but must at the same time be affected by the variation of atmospheric large particles and water droplets. Sivkov had chosen some cases of measured flux values when the degree of zenith polarization at rn = 3 exceeded 73 percent (Karadag observations). Reducing the corresponding S, values to the mean sun-earth distance and averaging them for the absolute humidity intervals 0-1, 1-2, 2-3, ... mm, Sivkov calculated the turbidity factor T, from T, = 12.86 s, SO

(5.34)

Since on the days considered the influence of residual turbidity was negligible, the turbidity factor can therefore be found from the relation W, = T2- 1. The dependence of the humid turbidity factor on absolute humidity can, in the given case, be presented as W, = o.50e00.43

(5.35)

If R # 0, by calculating W from (5.35) we find R from the relation = T - 1 - W. Making use of (5.34) and (5.35), Sivkov [98] calculated the annual range of T,, W,, and R, on the basis of observations at Pavlovsk, Kursk, and Karadag. The results of his calculations are given in Table 5.37. Although the application of (5.35), deduced for the conditions of Karadag, to calculations referring to the more northern Pavlovsk and Kursk introduces certain inaccuracy to the numerical values of Table 5.37, nevertheless this table gives a number of interesting and sufficiently reliable qualitative conclusions as to the relation between the turbidity factor components under different conditions. Table 5.37 also gives Zvereva’s results [loo] from observations at the “North Pole 6” station (1957). The given values of the integral turbidity factor T have already been given in Table 5.36. The variation of T over a vast territory from Pavlovsk

R

TABLE 5.37 Annual Variation of the Turbidity Factor and Its Components at Pavlovsk, Kursk, and Karadag. After Sivkov [82] and Zvereva [lo01

Station

Jan.

Feb.

Mar.

Apr.

May

June

July

Aug.

Sept.

Oct.

Nov.

Dec.

Year 01

Integral turbidity T

1.90

2.19

2.42

2.65

2.94

3.04

3.15

2.85

2.75

2.51

2.02

1.81

2.52

Kursk

2.02

2.19

2.10

2.61

2.94

3.04

3.31

3.21

2.94

2.56

1.90

1.81

2.55

Karadag

2.15

2.33

2.65

2.10

2.99

2.94

3.10

3.21

2.75

2.56

2.24

2.02

2.64

2.40

2.54

2.29

North Pole 6 (1957)

c 5.

s s

2.08

B

Humid turbidity W Pavlovsk

0.60

0.60

0.70

0.92

1.02

1.23

1.36

1.29

1.15

0.95

0.75

0.58

0.93

Kursk

0.53

0.53

0.53

0.90

1.20

1.36

1.40

1.45

1.24

1.04

0.84

0.58

0.97

Karadag

0.91

0.90

1.00

1.09

1.28

1.42

1.54

1.51

1.40

1.30

1.16

0.90

1.20

0.48

0.59

0.71

North Pole 6 (1957)

s

E3

"J

9

H

2

3

0.52

6

?,

Y

Residual turbidity R

?!

Pavlovsk

0.30

0.59

0.72

0.73

0.92

0.81

0.79

0.56

0.60

0.56

0.27

0.23

0.59

Kursk

0.49

0.66

0.53

0.71

0.74

0.68

0.91

0.76

0.70

0.52

0.06

0.23

0.58

Karadag

0.24

0.43

0.65

0.61

0.71

0.52

0.56

0.70

0.35

0.26

0.08

0.12

0.44

0.91

0.95

0.58

North Pole 6 (1957)

0.56

'

h, \o v1

296

Direct Solar Radiation

to Karadag has been found to be very small. If, however, we analyze the spatial variability of the turbidity factor components, we shall see some interesting peculiarities. As seen from Table 5.37 (see the last column), the low latitude variability of the integral turbidity factor results from the presence of the reciprocal latitudinal countervariability of the factors of humid and residual turbidity. The humid turbidity factor increases southward, whereas the residual turbidity factor decreases. The increase of W at this point somewhat exceeds the decrease of R, which results in the increase of the, mean annual integral turbidity factor toward the south. In isolated months the humid turbidity factor is shown to increase southward, too. The exception is found during cold months, when the W value at Kursk is smaller than that at Pavlovsk. The cause for this increase of the humid turbidity factor southward appears to be the increase of the absolute humidity in the same direction. Likewise, a somewhat decreasing humid turbidity at Kursk in winter (compared with Pavlovsk) is due to the domination of relatively dry masses of continental arctic air at this time. In general, according to (5.35), the geographical distribution of the humid turbidity factor follows that of the absolute humidity. Similarly, the annual range of the humid turbidity factor is associated with the absolute humidity variation over a year. From Table 5.37 we see that the W maximum at all points over the year is observed in the more humid summer months. For the Arctic area, a high transparency due in the main to the low humid turbidity factor values is characteristic. The residual turbidity factor shows a more complex picture of spatial and time variance, which is related with a more intricate nature of residual turbidity as determined by the combined effect of the dust and condensational turbidities. As was stated by Sivkov [98], the dust turbidity on the territory of the U.S.S.R. should be expected to increase toward the south, although there may be exceptions. On the other hand, the condensational turbidity shows a reverse dependence on latitude, decreasing southward because the state of water vapor in the north is closer to saturation than in the south. It follows from the above general considerations that the latitudinal fluctuations of the two components are mutually compensated to some degree. This compensation (see Table 5.37) is not complete and through the year the decrease of the residual turbidity factor southward dominates. The monthly latitudinal range is at times still more complex. The annual range of the residual turbidity factor at all three points is

297

5.7. Variation of the Atmospheric Transparency State

characterized by the presence of a double maximum in the spring-summer months, one being observed in May, the other in July-August (see Table 5.37, especially at Kursk and Karadag). Sivkov attributes the spring maximum to the sharp increase of condensational turbidity. The summer maximum appears to result from the increasing dust turbidity. Also to be noted is the asymmetry of the annual range curve: during the warmer halfyear (April-September) the residual turbidity values are higher than in the colder months. The minimal factor at all these points is observed during November and December. Very low R for Kursk and Karadag were obtained in November; this was caused by a high atmospheric transparency at that time (after a long period of fogs and drizzle cleaning the atmosphere of various coarse scatterers). 6. The Turbidity Factor and Air Masses. Since different air masses differ as regards their humidity and dust content, it is natural that the values of the turbidity factor and its components, obtained from observations in different air masses, are different, too, and can be therefore used to characterize the air masses actinometrically. This idea was developed in the 1930s by a group of Soviet investigators [106-1081. Table 5.38 summarizes the mean values of T, , W , , and R, for various air masses observed in Moscow, Pavlovsk, and Kursk [107]. The T,values were computed at all these points, but W , and R, were based only on Pavlovsk and Kursk data. As seen from this table, the integral turbidity factor has a maximum in arctic air masses and a minimum in the tropics. This is evidently due to the higher humidity and dust content of the tropical masses. The greatest variation of the integral turbidity factor over the TABLE 5.38 Values of the Turbidity Factor and Its Components for Various Air Masses as Observed in Moscow, Pavlovsk, Kursk. After Sivkov et al. [lo71

Season : Air Masses: Sea Artic Continental Arctic Sea polar Continental polar Continental tropical

Warm Period, May-Sept .

Transitional Period, Apr.-Nov.

Cold Period, Nov.-Mar.

T,

W,

R,

T,

W,

R,

T,

W,

Rz

2.44 2.65 2.84 3.08 3.59

0.59 0.88 0.92 1.13 1.32

0.86 0.79 0.92 0.96 1.23

2.37 2.51 2.57 2.64

0.38 0.39 0.53 0.60

0.99 1.06 1.04 1.04

2.16 2.26 2.38 2.31

0.23 0.14 2.29 0.33

0.92 1.08 1.09 0.98

298

Direct Solar Radiation

masses takes place in summer. In winter it is the smallest. The most variable is the humid turbidity factor, while the residal turbidity component changes but slightly. This leads to the conclusion that in the variation of the integral factor, the air humidity fluctuations predominate. In particular, the winter decrease of the integral factor value (Table 5.38) is caused by a considerable decrease of the humid turbidity factor. To evaluate the degree of representativeness of the turbidity factor as an air-mass characteristic, it is necessary to determine the possible limits of the turbidity factor fluctuations at a constant given mass. Table 5.39 gives average departures of the integral turbidity factor and its components from the corresponding means, computed on the basis of observational data at Pavlovsk and Kursk. TABLE 5.39 Average Departures of the Integral Turbidity Factor and Its Components from the Corresponding Means, as Observed at Pavlovsk and Kursk. After Sivkov et al. [lo71

Season : Air Mass: Sea arctic Continental arctic Sea polar Continental polar Continental tropical

Warm Period, May-Sep t

.

Transition Period, Apr.-Nov.

Cold Period, Nov.-Mar.

AT,

AW,

AR,

AT,

AW ,

AR,

AT,

AW, ARB

0.19 0.20 0.34 0.36 0.44

0.10 0.21 0.19 0.24 0.21

0.19 0.19 0.30 0.24 0.46

0.35 0.20 0.34 0.25

0.08 0.05 0.14 0.13

0.39 0.23 0.37 0.24

0.24 0.28 0.36 0.26

0.06 0.05 0.08 0.10

0.22 0.35 0.34 0.23

When we compare the two tables we see that in all cases, there is overlap of the values of the integral factor and its components relating to different air masses, which indicates that the turbidity factor is not a unique characteristic of the given mass. If we remember also that the classification of air masses is to a certain degree conventional, since in reality continuous gradation of their properties takes place, then it becomes clear that it is expedient to speak of connection between atmospheric turbidity and the passage of the front or air-mass transformations and not of any possible classification of air masses according to the turbidity factor. Poliakova and Sivkov [I081 have shown that the passage of the front or transformation of air mass always leads to a notable and quite definite variation of the turbidity factor and its components. In Table 5.40 are

299

5.7. Variation of the Atmosphere Transparency State TABLE 5.40

Variation of the Integral Turbidity Factor and Its Components in the Passage of a Cold Front. Kursk, July 30-31,1933. After Poliakova and Sivkov [lo81

July 30

Ta Wa R,

July 31

Hour: 07.50

18.10

18.30

05.40

06.40

07.50

3.42 1.73 0.69

3.36 1.38 0.98

3.42 1.38 1.04

2.34 0.86 0.48

2.16 0.88 0.28

2.38 0.97 0.41

presented data on such variations during the passage of a cold front (Kursk). During the night from July 30 to July 31 the turbidity of the atmosphere considerably decreased. Analysis of the synoptical chart for that period showed the passage of a cold front. Table 5.41 gives an example of variation of the same quantities in the process of air-mass transformation observed at Kursk from June 28 to July 3, 1933. We see here that the atmospheric turbidity increases in the process of transformation of a sea-polar air mass into polar-continental air mass. TABLE 5.41 Variation of the Turbidity Factor and Its Components in the Process of Air-Mass Transformation. After Poliakova and Sivkov [lo81

Date June June June July July July

28 29 30 1 2 3

Air Mass Sea polar

Sea polar Continental Continental Continental Continental

polar polar polar polar

Ta

Wa

Ra

2.63 2.92 3.34 3.39 3.28 3.39

0.97 1.09 1.07 1.09 1.10 1.35

0.66 0.83 1.27 1.30 1.18 1.04

The preceding examples demonstrate that calculations of the turbidity factor may be very useful in obtaining certain additional data to characterize the varying air-mass properties. It is beyond doubt, however, that such information will not suffice as the only basis for determination of these

300

Direct Solar Radiation

variations of various actinometric characteristics ; in particular, those determining atmospheric thermal emission should be used. This is a task for future investigation. 5.8. Attenuation of Solar Radiation by Clouds

We know (see Chapter 4) that large particles are intensive light scatterers. It is natural, therefore, that clouds consisting of a great number of water droplets or ice crystals must be very active in scattering the direct solar radiation passing through them. Table 5.42 gives values of solar radiation transmission by clouds of various forms (in percent) according to data from observations by Makarevsky [lo91 at Pavlovsk. The percentage of transmission was determined as a ratio of the actually observed flux value in the presence of cloudiness that hid the sun to that in a cloudless sky. The latter was found from interpolation over actinographic recordings. TABLE 5.42 Average Percentage of the Transmission of Direct Solar Radiation by Clouds of Different Form at Different Solar Heights. After Makarevsky [lo91

Cloud Form

Cirrus (Ci)

Cirrostratus (Cs)

Cirrus (Ci) with Cirrostratus (Cs)

Altocumulus (Ac)

Solar Height, deg

Average Transmission of Solar Radiation, %

5-15 15-25 25-35 35-45 45-55 5-15 15-25 25-35 35-45 45-55 15-25 25-35 3545 45-55 5-1 5 15-25 25-35 35-45

62 68 76 80 84 46 53 61 63 73 58 63 65 72 10 13 15 35

Number of Cases

Max. and Min. of transmission

2 7 4 6 3 3 6 22 13 10 5

86-37 88-46 89-66 96-57 93-67 69-26 62-28 84-85 8635 91-14 8340 77-3 1 8440 89-57 14-7 20-1 37-2

5

9 7 2 4 4 1

301

5.8. Attenuation of Solar Radiation by Clouds

The above data should be considered very approximate in view of a limited number of observations and the imperfection of the method for their determination. Nevertheless, the influence of the upper-layer cloudiness on the attenuation of radiation can be clearly followed. As is shown by Table 5.42, the radiation attenuation is already considerable in the higher layers, the intermediate layer (Ac) lets only a small portion of the incident flux pass, whereas the clouds of the lower layer are known to be completely impenetrable by solar radiation. Table 5.43 gives radiant flux values incident on a horizontal surface in a cloudless sky and in the presence of clouds according to Kalitin [110]. We can see that the stratus lower-layer cloudiness absolutely does not transmit solar radiation. TABLE 5.43 Flux of Solar Radiation on a Horizontal Surface in a Cloudless Sky and in the Presence of Clouds (cal/crnamin). After Kalitin [110] Solar Height, deg Cloud Form 0

Cloudless sky Cirrus clouds Altocumulus clouds Stratus clouds

0.00 0.00 0.00 0.00

5

10

15

20

30

40

50

0.06 0.00 0.00 0.00

0.13 0.00 0.00 0.00

0.22 0.04 0.00 0.00

0.33 0.11 0.00 0.00

0.59 0.32 0.00 0.00

0.84 0.60 0.12 0.00

1.10 0.90 0.31 0.00

Recently new data [111-1141 have been obtained by means of aircraft actinometric measurements, which ensures direct determination of the transmission, absorption, and reflection of solar radiation by clouds. For this purpose measurements of the upgoing and downcoming fluxes of shortwave radiation on a horizontal surface in the proximity of the upper and lower cloud boundary have been conducted. Let us introduce the following denomination for these radiant fluxes: FuJ is the downcoming flux of shortwave radiation (direct solar and diffuse) at the level of the upper cloud boundary, R, is the flux of radiation reflected from the upper surface of the cloud, FLJ is the downcoming flux of shortwave radiation (direct solar and diffuse or only diffuse) at the level of the lower cloud boundary, FL’ is the upgoing flux of shortwave radiation (diffuse atmospheric or reflected from the earth’s surface) at the lower cloud boundary level, AE is the albedo of the earth’s surface. Now, for the cloud albedo A , , the relative values of the transmitted P

302

Direct Solar Radiation

and absorbed (1 - P ) radiation, we have the following relations: A , = - RU Fu+

(5.36) (5.37) (5.38)

It should be noted that (5.37) cannot be used to determine the transmission of direct solar radiation by cloud and makes possible only approximate calculations of P, which is the ratio of a shortwave radiation flux leaving the lower cloud surface to a flux incident on the upper cloud boundary. The above measurements have shown that the cloud albedo varies within very wide limits from several percent to 90 percent and averages 50 to 55 percent. More detailed results will be given in Chapter 7. Transmission values P show considerable dependence upon amount and form of clouds; for example, according to measurements by Cheltzov [ I l l ] , at the cloud thickness 200 m, the P value is 43 percent for altocumulus (Ac) clouds and 59 percent for stratocumulus (Sc). At 500 m of stratocumulus clouds, the transmission decreases to 24 percent. In Fig. 5.24 the dependence of the transmission by stratocumulus and altocumulus clouds upon their thickness is presented graphically on the basis of Cheltzov’s measurements in the Arkhangelsk and the Moscow regions. It is seen to be very considerable. Cheltzov has shown that the presence of the above dependence raises

6QQl

E 5QQ

at

5

200

9

100

& &l 40

&3 60 ?0 00 TRANSMISSION COEFFICIENT P

b%

FIG. 5.24 Dependence of the transmission coefficient P upon cloud amount. (1) Sc 10, fights at Arkhangelsk; (2) Ac 10, the same flights.

5.8. Attenuation of Solar Radiation by Clouds

303

a problem in the use of actinometric measurements for determining cloud thicknesses. For this purpose even surface measurements can be utilized for the ratio of global radiation fluxes measured at the earth’s surface with a cloudy sky to those with a clear sky, for the same solar altitudes approximately characterize the value P and depend in the main upon cloud thickness. Plotting the “calibration” curve of the dependence of the above ratio upon cloud thickness for stratocumulus cloudiness, Cheltzov has shown that this curve (from data of measurements of radiation scattered by clouds) and a known possible global radiation flux value can determine the thickness of stratocumulus clouds with the accuracy & 45 m. The results of the determination of the radiation absorbed by clouds, differ considerably from author to author. For example, for the cases of uninterrupted cloudiness 360 m thick, Cheltzov has derived a mean value 1 - P = 3.5 percent (Arkhangelsk aircraft observations). The same author gives a mean 1 - P = 7.2 percent, at 530-m cloud thickness for the Moscow region. Neiburger [112] has obtained values of the radiation absorbed by stratus clouds, 1 - P = 7 percent, but he did not find any notable dependence of the absorbed radiation value upon cloud thickness. Nor was it found by Fritz and Donald [113], whose task was to determine the value of absorbed radiation for thick (5.5 to 7 km) cloud systems. The portion of the absorbed radiation in that case has been averaged at 20 percent greater than in any other investigations. In a number of cases an attempt has been made theoretically to calculate the reflection, transmission, and absorption of radiation by clouds. We shall describe some results of such research further on (see Chapter 7). At present we shall illustrate it only by Fig. 5.25, showing the dependence of re-

80

-

60

-

40

-

z 20-

. AMOUNT OF CLOUDS rn

FIG. 5.25 Dependence of the reflection ( I ) , transmission (2) and absorption (3) of solar radiation upon cloud amount.

304

Ditect Solar Radiation

flection, transmission, and absorption of radiation by clouds upon their thickness, as has been theoretically calculated by Hewson [114]. Although these calculations are roughly approximate, nevertheless the curves of Fig. 5.25 enable correct evaluation of the relation between the transmitted and absorbed solar radiation in dependence upon cloud thickness. We see, for example, that the transmission of radiation rapidly decreases with the increasing thickness of the cloud cover. Conversely the albedo shows a marked increase as the cloud thickness increases. As regards the absorbed radiation, it is only slightly variable and remains practically constant when the cloud thickness exceeds 500 to 600 m.

5.9. Theoretical Calculations of Irradiation of the Earth’s Surface by the SUn 1. General Notes. In Sec. 5.4 we considered the problem of the solar constant, which is a flux of solar radiation outside the atmosphere on a surface perpendicular to the sun’s rays at a mean sun-earth distance. This distance varies considerably during the year, and the value of the solar radiation flux outside the atmosphere at each given moment S,’ must be different from the solar constant So value. The relation between these two values can be determined as

4nR2S,’ = 4nRO2S, where R is the sun-earth distance a t a given moment of time, and R, is the mean sun-earth distance. On the basis of the above formula we have

(5.39) where e is a sun-earth distance expressed in units of the mean distance. Table 5.44 gives the results of applications of this formula to calculation of relative flux variations due to the variations of the sun-earth distance. Since solar radiant fluxes outside the atmosphere and at the earth’s surface are related as directly proportional, Table 5.44 characterizes the relative variation of S,’ and S , as well. It is seen that the amplitudes of relative flux variations due to variations of the sun-earth distance do not exceed f 3.5 percent. This table is sometimes used for reduction of the observed flux values to the mean sun-earth distance. This reduction, however, lacks any real meaning. Expedient

305

5.9. Theoretical Calculations of Irradiation

TABLE 5.44 Relative Variations of Solar Radiation Flux Due to the Variation of the Sun-Earth Distance (Percent of the Mean) Jan.

+ 3.4

Feb. Mar.

Apr.

May June

+2.8 t 1 . 8 +0.2 -1.5

-2.8

July

-3.5

Aug. Sept.

-3.1

-1.7

Oct. -0.3

Nov.

Dec.

t 1 . 6 t1.8

(and necessary) is only the reduction of direct solar radiation flux values theoretically calculated for different seasons on the basis of the same solar constant (referring to the mean sun-earth distance) to the given distance (season). Formula (5.39) determines the value of a solar radiation flux outside the atmosphere on a surface perpendicular to the sun's rays. Meanwhile, we are practically interested in the income of solar radiation on a horizontal surface. It is quite obvious that a horizontal surface of unit aera will receive less radiation than a corresponding surface perpendicular to the beam of rays. The decrease of the incoming radiation on a horizontal surface will be determined by the relation between the segments BC and AB (see Fig. 5.26). Since BC = A B cos 8, , where Oo is the solar zenith distance; then we have the following relation between the fluxes on a horizontal S,," and a perpendicular S,,' surfaces: =

SO cos 8, s,,'cos eo = -

ez

=

so e

* sin h,

(5.40)

- 0, is the height of the sun.

where h,

= 90

FIG. 5.26

To the derivation of the formula for direct solar radiation flux on a horizontal surface.

306

Direct Solar Radiation

Formula (5.40) characterizes the relation between solar radiation fluxes on a horizontal surface and one perpendicular to the sun’s rays outside the atmosphere. It is easily seen that a similar relation will hold for solar fluxes near the earth’s surface. Let us now consider calculations of irradiation of the earth’s surface by solar radiation. First will be considered an imaginary case of the absence of the atmosphere, to make possible further characterization of the influence of atmospheric solar radiation attenuation.

2. Irradiation of the Earth’s Surface in the Absence of Atmosphere. In the considered case (equivalent to the case of irradiation outside the atmosphere) we have, according to (5.40), for a solar radiation flux on a horizontal surface, So cos eo so”= -

e2

(5.41)

To use this formula for computation of the distribution of the incoming radiation over the earth’s surface and for exploring the time relation of incoming radiation it is necessary to express the solar zenith angle 8, in terms of the geographical or astronomical parameters affecting this angle. Toward this end consider the celestial sphere presented in Fig. 5.27. At the center M of the sphere is situated the earth; at point S on the spheric surface is the sun, moving in the plane of the ecliptic EE‘. The intersection points y and y’ of the celestial equator and the ecliptic EE‘ are the points

FIG. 5.27

The ceIestial sphere.

5.9. Theoretical Calculations of Irradiation

307

of the vernal and autumnal equinoxes. The earth's axis is piercing the sphere at the point N (north of the pole world); the plane HH' is the plane of the true (mathematical) horizon. The normal to this plane is piercing the celestial sphere at the zenith point Z. A large circle, HNZH'A in the plane of the drawing is the meridian of the given point M . The geographical latitude of the point is identical with the height of the pole above the horizon, that is, 9 = arc NH. The arc BC of the larger circle determines the inclination of the sun, arc BC = 6. The spherical angle Z N S is the sun's hour angle 52. Taking into account the above definitions, we have in the spherical triangle ZNS, arc NC

n 2

= --

6,

arc ZC = 8,,

n arc ZN = - - v 2

+ sin b sin c cos A

we obtain for the considered spherical triangle, cos 8,

= sin

sin 6

+ cos q~ cos 6 cos52

(5.42)

Taking account of this relation, instead of (5.41) we find So''

SO

= - (sin

e2

q~sin 6

+ cos v cos 6 cos52)

(5.43)

Formula (5.43) defines the solar radiation flux S," as the function of geographical coordinate (latitude v), season (incline 6), and time of day (hour angle 52). It must be stressed that this formula has meaning only at 8, 5 n / 2 when S," 2 0. If 8, > n/2, this means that the sun is under the horizon, and consequently Si' = 0. Making use of (5.43), it is easy to analyze the fundamental regularities of irradiation of the earth's surface by solar radiation in the absence of atmosphere. We see, for instance, that the maximal income of solar radiation takes place on the earth's surface when two such conditions are simultaneously fulfilled : cos52 = 1,

sin v sin 6

+ cos q~ cos 6 = cos(g7 - 6) = 1

Thus the maximum of irradiation takes place at 52 = 0 and

v = 6.

308

Direct Solar Radiation

The minimal irradiation will apparently be at such points of the earth's surface where the sun is at the horizon. In this case the following condition must be fulfilled: sin rp sin 6

+ cos rp cos 6 c o s 9 = 0

that is, cos52 = - tan rp tan 6. On the basis of (5.43) is easy to obtain the expression for the daily range of solar radiation flux. Since the solar inclination during the day varies insignificantly, the values

SO

-sin rp sin

e2

6

and

~

SO cos rp cos 6

e2

may be considered constant. Thus from (5.43) we obtain S,"

= A

+B C O S ~

(5.44)

If the time t is counted from noon and the angular speed of the earth's rotation is w , then the hour angle 9 = wt. Therefore, instead of (5.44) we can write S," = A B cos cot (5.45)

+

It should be borne in mind, of course, that the hour angle 9 or the time t can assume such values only when Sdr > 0; that is, values within the limits of the hour angles (moments of time) of sunrise and sunset. These values of the hour angle, and consequently of the moments of time, can be B c o s 9 = 0, from which calculated from the equation A

+

cos9,

A B

= - -= -

tan rp tan 6

The negative root of this equation, 9 = - cot0, corresponds to sunrise; the positive root, 52, = cot, to sunset. For all hour angle values (moments of time) outside the limits of the interval (- cot, , cot,) the solar flux S," value equals zero. Thus we see that the daily range of solar radiation flux on a horizontal surface in the absence of atmosphere is a simple periodical function of time. Let us now turn to the calculation of total solar radiation, using formula (5.43). It is easily understandable that the daily total of solar radiation on a horizontal surface 2 So'can be calculated from the following formula:

2 S,'

so

=QZ

-to

(sin rp sin 6

+ cos rp cos 6 cos cot) dt

(5.46)

5.9. Theoretical Calculations of Irradiation

309

Integrating in this formula, we have (sin v sin SQ,,

+ cos 31 cos 6 sin cot,)

(5.47)

On the basis of (5.47) we can calculate daily totals for any latitude q~ and different inclination 6. Figure 5.28 presents the space-time variability of daily radiation totals in the absence of atmosphere. The vertical coordinates of the surface are proportional to the daily totals for the corresponding latitude and season.

LATITUDE

FIG. 5.28 AnnuaI variation of the daily totals of solar radiation at different latitudes.

The latitudinal range of daily totals is seen to differ from season to season. In the winter half of the year the circumpolar area receives no solar radiation, while in summer the daily radiation totals in the polar regions exceed the equatorial totals. The same marked variation is observed in the latitudinal dependence of the annual range of daily totals. At the equator the annual range of these totals is rather weak and has double maxima and minima. Toward the poles the curve of the annual range sharpens upward with a maximum on the summer solstice day. Behind the polar circle there is no income of solar radiation during some period of time; for example at the North Pole the continuous duration of the polar day is 186 days, and of the polar night, 179 days. Knowing daily totals of radiation, it is possible to compute monthly or annual totals, or totals for any given time interval. The above remarks refer to the case of the absence of atmosphere. Let us now consider similar results obtained for the real conditions, that is, in the presence of atmospheric radiation attenuation.

310

Direct Solar Radiation

3. Irradiation of the Earth’s Surface in the Real Condition of a Clear Sky. Taking account of (5.1 l), obtain the following formula for calculation of daily radiation totals on a horizontal surface (atmospheric radiation attenuation considered) :

so 2 S’ = e2

s-to 10

pmm(sinp sin 6

+ cos y cos 6 cosQ) dt

(5.48)

Since the transparency coefficient Q, is a very complex function of time in the given case, integration in this formula can be only performed either graphically or numerically. The totals of solar radiation calculated on the basis of radiation attenuation in the atmosphere (assumed to be cloudless) are known as “possible” totals. In the more general case, using the quantitative characteristic of atmospheric transparency f as presented by Makhotkin [95], instead of (5.48) the following formula is obtained :

:1

ZS’=-1440 ne2

S[rn(Q),f ] sin[h(Q)] dQ

(5.49)

where the possible daily radiation total S’ is expressed in cal/cm2 day. Carrying out the mean flux value S[m(Q), f ] behind the integral sign, we have

S’

= 458

1

e

-

S[rn(Q),f]

JQ0

sin[h(Q)] dQ

or designating Qo* =

I,”” sin[h(Q)] dQ

We have 1

-

2 S‘ = 458 -S[m(Q),f ]Qo* e2

(5.50)

The main difficulty consists now in the determination of the mean atmospheric mass value m(Q). As was shown by Makhotkin when using the turbidity index N as a quantitative transparency characteristic, the formula (5.50) can be presented in the following form: Qo* 2 S’ = 458S(NAR0) e2

(5.51)

5.9. Theoretical Calculations of Irradiation

z

31 1

where R, = S/zS' is the ratio of daily totals of radiation on surfaces perpendicular to the rays (2 S ) and horizontal S') outside the atmosphere, and A is an empirical parameter. For N equal to 0.3, 0.5, 1.2, and 4, the lg A values are - 0.04, - 0.05,- 0.06, - 0.075, and - 0.09, respectively. This variation of this parameter is very limited. Since the e value is known and the computation of Q,* and R, is quite easy, we must now find the value S , corresponding to m = N A R , , from (5.29), and then calculate 2 S'. To simplify this process, Makhotkin has proposed a special nomogram. Consider now some results of computation of possible daily radiation totals. Figure 5.29 gives the latitudinal range of calculated totals on a horizontal surface on the summer solstice day at different transparency coefficient values. As seen from this figure, accounting for atmospheric radiation attenuation considerably changes daily radiation totals. For example, at p = 0.6, radiation totals in the vicinity of the equator decrease by two times as compared with the nonatmospheric case.

(x

1200

s" Kmo

3- \

600

c'

-g 4 0 0 z 200 0

Lotitudr,deg

FIG. 5.29 Latitudinal variation of solar radiation totars on the horizontal surface in diyerent conditions of atmospheric transparency (on the day of the summer solstice).

The decrease of incoming radiation with the decrease of the transparency coefficient p is not the same for all latitudes; the higher the latitude, the more pronounced is the decrease of the incoming radiation. With respect to the annual range of daily radiation totals at different latitudes, we can say that it corresponds to the case with absence of atmosphere. The absolute values of the income of radiation, however, when calculated on the basis of atmospheric radiation attenuation, are considerably different from those in the case of no atmosphere. At p = 1 (absence of atmosphere) the North Pole receives the maximum of solar ra-

312

Direct Solar Radiation

diation on the summer solstice day; at p = 0.7, the daily income of radiation at the North Pole i s less than at any other point of the Northern Hemisphere. 4. Semiempirical Methods for Computation of Actual Solar Radiation Totals. The discussion above has considered methods for computation of possible totals of solar radiation received by an underlying surface in the complete absence of clouds. It is natural that when clouds screen the sun, the incoming direct solar radiation on an underlying surface decreases. Since in the majority of cases there is always some cloudiness present, it becomes clear that the real radiation totals differ from the “possible” totals. The problem of considering the cloudiness effect is therefore important to the calculation of the real totals. The real and the possible totals of solar radiation on a horizontal surface have the following relation :

2 S‘ = 2 S’jiia)

(5.52)

P

T

where &!? and C,S’ are the real and the possible totals, respectively, and f(a) is a certain function of the parameter characterizing the sky’s cloudiness, so that in the case of a perfectly clear sky, f(a) = 1. In the simplest case the degree of the sky’s luminosity can be expressed by the mean degree of cloudiness ii or the relative duration of sunshine, sl. The latter is understood to be the ratio of the real sunshine duration s, to a possible value sp. From elementary physical considerations it is clear that in this case the parameter a need be expressed as a = s1 = 1 - A. As to the functionf(a), in the simplest case it must have the following form: f(a)

= s, = 1

-A

where A is expressed in portions of unity. Thus the formula (5.52) in the considered case can be presented as

x S‘

S‘sl =

=

x S‘(1 - 3)

(5.53)

P

7

When we compare (5.53) with observations, we find, however, that it is far too approximate. In particular, even in the calculation of mean annual s, and A, the equality s1 A = 1 is not sufficiently accurate.

+

Savinov’s Formula. As was shown by Savinov [115, 1161, the values

x S‘ x S‘

r P

and

s1

+ (1 - A) 2

5.9. Theoretical Calculations of Irradiation

313

show quite satisfactory coincidence. For different points located in different climatic conditions, we can write the following equation to give sufficient accuracy for approximate calculations :

c. S' = c. S' + (1 - A) s1

(5.54)

1

r

P

L

Thus neither s1 nor 1 - A are satisfactory in characterizing the sky luminosity and we must take, their half-totals. The formula (5.54) derived by Savinov is very simple and has the important advantage of ecxluding empirical coefficients. All values of (5.54) can be directly measured. It may be used, however, only for computing monthly totals of radiation because it has been derived from mean monthly values of s, ,A, and z r S / z p S ' . Kopylov [117] has shown that in this case the maximal error in the use of (5.54) for Pavlosk is 7.8 percent. As seen from (5.54), for the computation of the real radiation totals it is necessary first to find the possible totals. The real totals can be then determined from data on the mean degree of cloudiness and the relative duration of sunshine. However, Ukraintzev and Shepelevsky [118] and Galperin [I191 have shown that on the average, there is a linear dependence between 1 - (s1 f i ) n, = 2

+

and A. Making use of such dependence, it is possible to determine the parameter A, values from the known degrees of cloudiness, A. Sivkov's Formula. A somewhat different approximate method for calculating the real solar radiation totals, which does not require heliograph observations, has been suggested by Sivkov (see [117]). It is understandable that (5.54) and the relation s1 A = 1 would be valid in the case of equal distribution of perfectly nontransparent clouds over the celestial hemisphere. In actual fact, however, some clouds are semitransparent. That is why even an ideal noninertial heliograph must give s1 > 1 - A and, accordingly,

+

P

On the other hand it is clear that the inequality

P

314

Direct Solar Radiation

is valid, where n, is the degree of the lower-layer cloudiness. Thus the value &!?’/zpS‘ has the following limits :

P

Assuming the transparency of the upper and middle layers cloudiness to be 50 percent, and taking into account the latter inequality, zS‘

x S‘ -1--A+-

--

-A - nL

2

P

where -A - nL is the degree of the upper and intermediate layers cloudiness. The preceding formula can also be rewritten as -A r

+ nL

(5.55)

P

Comparison with observational data shows that the accuracy of Savinov’s and Sivkov’s formulas is almost the same. The Sivkov formula gives systematic exaggeration of z r S ’ / z p S r ,whereas the error in Savinov’s formula is of different sign. For example, the mean five-year annual total of the incoming solar radiation on a horizontal surface, calculated from Sivkov’s formula, has turned out to be in error by 8.1 percent. Taking account of the above systematic exaggeration, Kopylov [117] has proposed to reform the Sivkov formula as

+

xSr= r

x S’(0.97 P

~

2

(5.56)

The annual total of solar radiation at Pavlovsk, calculated according to (5.56), differs from the actually observed by less than 1 percent. Mean monthly totals, obtained in the same way, differ from the real by f 5 percent. For the general case, Kopylov has suggested the following relation : (5.57) where k is a certain empirical coefficient whose numerical value appears not to exceed 0.05 and depends mainly upon local cloudiness regime.

5.9. Theoretical Calculations of Irradiation

315

The Ukraintzev Formula. The above methods for calculating mean monthly totals are fairly perfect though not absolute. However, their application is confined to the case where there are known possible radiation totals. To determine the latter, it is necessary to have data of actinometric observations, which are in many cases either limited or nonexisting so that application of the Savinov and Sivkov formula is not always possible. Development of approximate methods for computation of solar radiation totals, which would demand the minimal number of empirical quantities, therefore appears to be extremely important. Interesting results in this field have been obtained by Ukraintzev [120]. Ukraintzev has shown that factual monthly totals of direct solar radiation on a horizontal surface are a linear function of monthly totals of sunshine duration. Therefore, to compute the monthly totals XJ', the following relation can be used:

XS'

+ b)

=~ ( s

(5.58)

I

where a and b are certain constants, and s is a monthly total of sunshine duration in hours. Table 5.45 gives some values of the constant a and b for a number of points. We see that (5.58) is quite simple and that, at known a and b, it allows determining the real monthly totals on the basis of heliographic sunshine duration records only. It should be stressed, however, that this formula is strictly empirical. Table 5.45 shows that the numerical values of a and b depend considerably on both time and location of a given point. This necessitates special determination of a and b for each point and indicates that care must be exercised in their extrapolation or interpolation. Meanwhile, the advantage of formula (5.58) can be appreciated only in the case where a and b can represent points for which there have been no actinometric observations (only heliograph records available), but for which interpolation or extrapolation has approximated a and b values from actinometric observations in the neighborhood of the points to be evaluated. Investigating the dependence of the real daily radiation totals on a horizontal surface upon daily duration of sunshine, Ukraintzev has shown that in this case the dependence of the type of (5.58) is not substantiated. Daily radiation totals are nonlinear functions of the duration of sunshine. When the latter increases, the daily totals increase, at first comparatively slowly (at small values of s) and then (at large values of s) much faster. This is evidently caused by the fact that at longer sunshine duration (little cloudiness), the atmosphere is more transparent than at shorter duration

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5.10. Variability of Fluxes and Totals of Solar Radiation

317

(greater cloudiness). At large s, therefore, a more rapid increase of daily radiation totals with the increase of s than at small s is observed. The problem of the relation between the duration of sunshine, cloudiness, and totals of direct solar and global radiation has been discussed in a number of works (see, for example, [121-1241).

5.10. Temporal and Spatial Variability of Fluxes and Totals of Solar Radiation The preceding section has given some methods and data of theoretical calculation of irradiation of the earth’s surface. Let us now consider the results of measurements of fluxes and totals of direct solar radiation. 1. Daily and Annual Variation of Direct Solar Radiant Flux. Numerous observations show that in the conditions of a cloudless sky (or in the presence of cloudiness not screening the solar disk during the whole day), the direct solar radiation has a very simple daily range with a maximum at about noon. In Table 5.46 are presented measurement data on a flux of direct solar radiation on a surface perpendicular to the sun’s rays and at different atmospheric masses (solar altitudes) in the colder and warmer half-years, obtained by Nebolsin [I261 with the help of the Mihelson actinometer at an agrometeorological station in the vicinity of Moscow. The table was compiled from data for 1926-1945. Systematic observations at all atmospheric masses were conducted only in 1935-1936. We see from the table that the solar flux increases before noon and then begins to decrease in the afternoon. It is quite obvious that the noon maximum is caused by the minimal ray path in the atmosphere at that time (and, consequently, minimal radiation attenuation). The variation of solar radiation over the day is not symmetrical in relation to noon. During the warm half-year, morning flux values at m > 2 somewhat exceed the corresponding afternoon values (see the first line of Table 5.46). In winter, on the contrary, afternoon values at m > 3 are higher than those before noon (the second line of Table 5.46). This asymmetry of the daily range of solar radiation results from different atmospheric transparency at the respective times. In summer the atmosphere is more turbid in the afternoon, and accordingly the afternoon radiation income decreases. The contrary variation of atmospheric transparency and incoming radiation is observed in winter. It must be noted, however, that similar regularities of the daily flux variation do not take place in all cases. For example, according to obser-

TABLE 5.46 Mean and Maximal Values of Direct Solar Radiation Flux (cal/cma min) on a Perpendicular Surface at Different Atmospheric Masses (Solar Heights) in the Cold and the Warm Half-Years. After Nebolsin [12q

Before Noon Atmospheric Mass

Values of Radiant Flux

Means in the warm half-year (Apr.-Oct.)

8

5

0.50

0.72

4

1.00

Afternoon Atmospheric Mass

Noon

3

2

1.5

1.01

1.05

1.10

1.14

1.5

2

3

4

5

8

1.13

1.07

0.90

0.77

0.68

0.49

%’ a

8

Pe

Means in the cold half-year (Nov.-Mar.)

0.50

0.67

0.95

1.06

1.13

-

1.02

-

-

0.92

0.96

0.85

0.62

Yearly means

0.50

0.69

0.97

1.03

1.06

1.12

1.09

1.12

1.07

0.93

0.87

0.73

0.57

1.08

1.18

1.15

1.29

1.34

1.38

1.33

1.29

1.22

1.24

1.01

0.78

1.08

1.12

1.11

1.16

-

1.28

-

1.17

1.12

1.14

1.01

0.70

Maximal in the warm half-year Maximal in the cold half-year

U

0.54

f

5.10. Variability of Fluxes and Totals of Solar Radiation

319

vations by Lileev [127] at Sverdlovsk, forenoon values of radiation flux on a perpendicular surface at all heights of the sun over the entire year, with the exception of June, were found to be smaller than in the afternoon. The two lower lines of Table 5.46 give maxima of radiant flux on a perpendicular surface during the warm and the cold halves of the year. The largest maxima take place in summer. The fullest picture of the regularities of the daily radiant flux variation is presented by graphs of isopleths of solar radiation. In Fig. 5.30 are given isopleths of solar radiation on a perpendicular surface, plotted by Kalitin [Chapter 3, Ref. 1021 from 20year observations at Pavlovsk; in Fig. 5.31 are given analogous isopleths according to observations by Yaroslavtzev at Tashkent [128]. The interrupted lines of Figs. 5.30 and 5.31 indicate the moments of sunrise and sunset.

Hour

FIG. 5.30 Zsopleths of direct solar radiation flux on a surface perpendicular to the sun’s rays (Pavlovsk).

The above figures show that the general character of the daily radiation range is the same at different geographical points : in all cases the maximum is observed near noon. However, there are also considerable latitudinal differences in the observed daily variation; for example, at a southern point (Tashkent) the noon maximum is more pronounced than in the north (Pavlovsk). The cause of this peculiarity becomes clear if we bear in mind that the more to the north the observation point is, the less is the daily variation of solar altitude. At the North Pole, for example, the daily variation of solar height is so insignificant that there is practically no daily range of solar radiation flux.

320

Direct Solar Radiation

4 5 6 7 8 9 10 II 12 13 14 15 16 17 18 I9 20 Hour

FIG. 5.31 Isopleths of direct solar radiation flux on a surface perpendicular to the sun’s rays (Tashkent).

The graphs of isopleths also allow characterization of the peculiarities of the annual radiation range on a perpendicular surface. The annual range is seen to depend considerably on the time at which the radiant flux value is being determined and also upon the latitude of the observation point. Experiments show that the most pronounced daily radiation range (equal for any time of the day) takes place at the pole. At the equator, on the contrary, the amplitude of the daily range is least observable, especially in the morning and evening hours. A peculiar double annual range of radiant flux is observed at the equator a t noon. The two maxima indicate the vernal and autumnal equinoxes. These features of the annual variation of the direct solar radiation flux on a perpendicular surface are mainly caused by the solar height variation, that is, when the length of the ray’s path in the atmosphere (and consequently the attenuation of radiation) varies. Let us follow it in the finite cases of the pole and the equator. At the pole the height of the sun during the polar day varies considerably, but this is equally true for any time moment within the day when it is constant throughout the day. Thus, there is observed a very marked, but independent of day time, annual range of solar radiation flux. The contrary is observed at the equator, where the morning and evening heights of the sun are almost invariable throughout the year, and the annual radiation range at these hours is thus very weak. In the afternoon the equatorial sun is at the zenith twice a year (vernal and autumnal equinoxes), which results in two maxima of the annual range of afternoon radiation values a t the equator. The solar altitude, however, is not the only factor determining the annual range of direct solar radiation. Also important is the variation in the

5.10. Variability of Fluxes and Totals of Solar Radiation

321

state of atmospheric transparency leading to the appearance of certain anomalies in the “normal” annual course of radiant flux. In Fig. 5.30, for example, we see that the isopleths of the solar radiation flux at Pavlovsk are not symmetrical around the moment of the summer solstice and are somewhat elevated in relation t o the dashed line indicating this moment. This means that the maximum flux occurs somewhat earlier than on the summer solstice day when the solar altitude is at its highest. This shift of the maximum is explained by the fact that at Pavlovsk the transparency of the atmosphere in the May-June period is higher than in the JulyAugust period. A similar shift of isopleths is still more pronounced in Fig. 5.31 for data a t Tashkent. In May-June the isopleth 1.25 is sharply concave toward the center, which indicates the rapid decrease of atmospheric transparency in these months. For a more detailed quantitative characteristic of the influence of atmospheric transparency variations on the annual radiation range, Table 5.47 presents the annual range of monthly means of direct solar radiation on a perpendicular surface at different solar altitudes, compiled from observational data by Kalitin at Pavlovsk. Since the annual radiant flux range at a constant height is caused by atmospheric transparency variations only, the considered table enables quantitative characterization of the influence of atmospheric transparency variations on the income of solar radiation. We see that in summer, radiant flux values at a given height of the sun turn out to be the smallest, which is the result of the decrease in atmospheric transparency at this time of the year. The amplitude of the annual flux range due to atmospheric transparency variations at low elevations of the sun is over 0.2 cal/cm2min, that is, amplitude is quite evident. The above data characterize the regularities of the daily and annual variations of radiant flux on a perpendicular surface. Of more practical interest is the investigation of the income of solar radiation on a horizontal surface. Let us now consider the annual and daily ranges of direct solar radiation on a horizontal surface. The radiant flux value on a perpendicular surface is usually directly measured. As to the flux on a horizontal surface, it can be computed from the familiar formula S,

=

S, sin h,

Figure 5.32 presents the mean curves of the daily range of solar radiation on the perpendicular and horizontal surfaces for July and January (Kalitin’s many-year observations at Pavlovsk). The decrease of incoming solar radiation on a horizontal surface, compared with that on a perpendicular

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5.10. Variability of Fluxes and Totals of Solar Radiation

323

surface, is very noticeable at the latitude of Pavlovsk (q = 59'41' N), especially in January. The noon values of radiant flux on the horizontal surface are lower than the corresponding values for the perpendicular surface by 20 percent in July and by 88 percent in January.

1

--

-2

Hour

FIG. 5.32 Daily variation of the flux of direct solar radiation on the perpendicular ( I ) and the horizontal (2) surfaces. Pavlovsk. July, January.

In Fig. 5.33 are given isopleths of radiant flux on a horizontal surface (Kalitin at Pavlovsk). Comparing these isopleths with those of Fig. 5.30 for the perpendicular surface, we find that the former show more symmetry and smoothness than do the latter. In particular, the isopleths of flux on a horizontal surface are quite symmetrical in relation to the summer sol-

FIG. 5.33 Isopleths of direct solar radiation flux on the horizontal surface as observed at Pavlovsk.

324

Direct Solar Radiation

stice. This symmetry and planeness of the course of isopleths of radiant flux on a horizontal surface is explained by the smoothing effect of the multiplication of radiant flux values on a perpendicular surface by sin ha. As in the case of the incoming radiation on a perpendicular surface, the variation in the incoming radiation on a horizontal surface results first of all from the varying solar height. However, in the latter case some influence on the incoming radiation is produced by atmospheric transparency variations. Table 5.48 gives quantitative characteristics of the effect of atmospheric transparency on the basis of monthly means of solar radiation on a horizontal surface for different solar heights (Kalitin’s observations at Pavlovsk). This table is analogous with Table 5.47 for a perpendicular surface. The values of Table 5.48 are obtained from multiplication of the corresponding values of Table 5.47 by sin ha. Therefore it is clear that the relative variations of direct solar radiation flux on the respective surfaces at a given solar height are equal throughout the year. The absolute values of the amplitude of the daily flux range on a horizontal surface appear to be significantly lower, and do not exceed the limits of measurement errors at small solar altitudes. The above data were obtained-underconditions of a cloudless sky, which is a rather rare case. The actually observed (with cloudy skies) daily range of solar radiation is more complex and irregular, and the total of incoming solar radiation notably decreases during the day. We shall give more attention later to the problem of the decrease in the incoming solar radiation from an overcast sky.

2. Influence of Towns on Incoming Direct Solar Radiation. It is perfectly evident that air pollution (dust and smoke) existing in a large town will lead to a considerable decrease of solar radiation as compared with that in rural areas. This problem has been quantitatively treated in a great number of works, which show that the value of solar radiation in town is, on the average, from 10 to 20 percent smaller than in the country. Averkiev [129, 1301 and Shubtzova [131] have explored the peculiarities of the Moscow radiation regime from data of actinometric observations in Moscow and in the outskirts. Table 5.49 summarizes Shubtzova’s results of measurement of direct solar radiation on a perpendicular surface in Moscow and outside (Kuchino of the Moscow region), made in 1937 and 1938. The above data demonstrate that the mean difference in S, value between Moscow and the vicinity is 0.1 1 cal/cm2 min. An anomalous relation

TABLE 5.48

Monthly Means of SoIar Radiant Flux (cal/crnz min) on the Horizontal Surface for Different Solar Heights as Observed by Kalitin at Pavlovsk

Solar Height, deg 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53

Jan.

Feb.

Mar.

Apr.

May

June

July

Aug.

Sept.

Oct.

Nov.

0.00 0.02 0.05 0.09 0.13 0.17 0.22 0.26 0.31 0.36

0.00 0.02 0.05 0.08 0.12 0.16 0.20 0.25 0.29 0.34 0.39 0.44 0.49 0.54 0.59

0.00

0.00

0.00

0.00 0.01 0.04 0.06 0.10 0.13 0.17 0.21 0.25 0.30 0.35 0.39 0.44 0.49 0.53 0.58 0.62 0.67 0.71 0.75 0.79 0.84 0.88

0.0 0.02

0.00 0.02 0.05 0.08 0.12 0.16 0.20 0.24 0.29 0.34 0.39 0.44 0.49

0.00

0.02 0.04 0.06 0.10 0.13 0.17 0.21 0.25 0.29 0.33 0.38 0.42 0.46 0.51 0.56 0.60 0.64 0.69 0.74 0.77 0.82 0.86 0.91 0.94 0.99 1.02

0.00 0.01 0.03 0.06 0.09 0.12 0.16 0.20 0.24 0.28 0.32 0.37 0.41 0.45 0.50 0.54 0.59 0.63 0.67 0.72 0.76 0.80 0.85 0.88 0.93 0.96

0.00

0.02 0.04 0.07 0.11 0.15 0.19 0.23 0.27 0.32 0.37 0.41 0.46 0.51 0.56 0.61 0.65 0.70 0.75 0.80 -

0.03 0.05 0.09 0.13 0.18 0.22 0.27

0.03 0.06 0.01

-

-

-

-

-

-

-

0.02 0.04 0.07 0.10 0.14 0.18 0.21 0.25 0.30 0.34 0.39 0.43 0.48 0.52 0.57 0.62 0.67 0.71 0.75 0.80 0.85 0.89 0.94 0.97 -

-

-

-

-

-

-

-

-

-

-

0.04 0.07 0.10 0.14 0.18 0.22 0.27 0.31 0.36 0.41 0.45 0.50 0.55 0.59

Dec.

VI

b

P

c

E -.

1 .r;r

TABLE 5.49 Measurements of Solar Radiation Flux on a Perpendicular Surface in Moscow and in the Moscow Region (Kuchino). After Shubtzova [131]

1937

1938

Observation Dataa June

July

Aug.

Sept.

Oct.

Nov.

Dec.

Feb.

Mar.

Apr.

Moscow

1.19

-

1.11

1.14

1.21

1.26

-

1.29

1.24

1.12

4!

Kuchino

1.10

-

1.17

1.21

1.29

1.44

-

1.36

1.36

1.34

Li

Difference

0.09

Number of cases

6

-

-0.06

-0.07

-0.08

1

2

4

-

-0.07

-

1 ~~

~~

~

rD

3

K

-0.12

-0.22

-0.22

3

6

2

~~~~~~

Mean monthly difference. over 1937 and 1938, 0.09; mean monthly difference, excluding June 1937, 0.11,

m .M

g

5.10. Variability of Fluxes and Totals of Solar Radiation

327

between the respective fluxes was observed in June, 1937, when the monthly incoming radiation in Moscow was higher than in the rural areas. This appears to have been caused by a very small quantity of precipitation in June, 1937. The dust turbidity of the atmosphere increases in the country, and everyday street washing in town prevents atmospheric dust pollution in such cases. Averkiev [129, 1301 has obtained a far smaller mean difference between the respective fluxes, 0.045 cal/cm2 min. This smaller value can be explained by the fact that the other observation point (besides Moscow) was in too close proximity to Moscow. The most significant town effect has been found in the result of actinometric observations in Berlin, where the radiant flux value on a perpendicular surface averaged - 20 percent as compared with the reference point (Potsdam). At small values of solar height this difference reached 50 percent. Observations show that the additional radiation loss in town conditions decreases from the center toward the suburbs. The town effect is the most pronounced in windless weather when the town atmosphere accumulates a large quantity of dust and smoke. 3. Daily Totals of Solar Radiation. As was shown above, daily radiation totals are determined, in the main, by the duration of sunshine, the altitude of the sun, and degree of cloudiness. Since all these factors vary over the year and depend upon the location of the observer, it is natural that the daily totals have annual variation and are dependent upon local climatic conditions. The longest and most complete continuous measurements of solar radiation flux to determine the observed radiation totals were performed by Kalitin at Pavlovsk. During 29 years, from 1912 till 1941 the observations were recorded by means of the Crova actinograph, with a detector in the form of a thermoelectric Savinov star. The curve of the annual variation of daily totals on a horizontal surface (curve 4), obtained in the result of 26 years of observation, is presented in Fig. 5.34. Observations of 1912 and 1941 were not used because they were not representative of the whole year; 1913 and 1914 were omitted, owing to the effect of the Katmai volcanic eruption during those years. For comparison, Fig. 5.34 also gives the curves of the annual range of daily radiation totals in the absence of atmosphere (curve l), in the ideal atmosphere (curve 2), and under condition of a clear sky (curve 3). As seen from this figure, the maximum in the annual variation falls in June and early July when the solar height is

328

Direct Solar Radiation

the greatest. The largest factual value of daily totals is, however, not observed on the day of the summer solstice (maximal solar height) but a t the beginning of June, which can be explained by the influence of cloudiness on the incoming solar radiation.

js \

750

m

O

i

i

C Month

FIG. 5.34 Annual variation of the daily totals of radiative heat on the horizontal surface.

Comparing the curves 3 and 4, we see that the influence of cloudiness on the incoming radiation is in general very great, and the real daily totals are far smaller than the possible totals. Values of the real daily totals on a horizontal surface, averaged over five days (1915-1940), vary within a year from 1.5 cal/cm2min in the first five days of Janury to 316 cal/cm2 min in the second such term of July, whereas the possible totals must be scores of times as large in the winter months and double the real value in summer. The above data characterize the daily totals of solar radiaticn on a horizontal surface. Table 5.50, compiled by Ukraintzev [120], displays values of the ratio xS‘/ZS from observations at various points on the territory of the U.S.S.R. From Table 5.50 it follows that at all points the values of the ratio ZS’/ZS are maximal in the spring-summer months, which points out that the main factor determining the relation between daily totals of radiation on horizontal and perpendicular surfaces is the altitude of the sun. However, variations in cloudiness and atmospheric transparency tell on the ZS’/ZS values as well. Ukraintzev notes that in all cases when the “radiation conditions” get worse, the ratio xS’/ZS is seen to increase. For example, in the decrease of atmospheric transparency, ZS’ decreases in a lesser degree than ZS, and consequently the ratio ZS‘/xS increases. This takes place owing to the fact that the increase of radiation attenuation with the decrease of atmospheric transparency is the most developed at

TABLE 5.50 The Ratio (Percent) of Daily Totals of Solar Radiation on the Horizontal Surface to the Corresponding Totals on the Perpendicular Surface. Ajier Ukraintzev [120] ~

Point Observation" Tashkent 1 2

~~

~~

~~

~~

~~

Jan.

Feb.

Mar.

Apr.

May

June

July

Aug.

Sept.

Oct.

Nov.

Dec.

Year

34 35

40

51 55

59 63

64 68

65 68

64

61 63

54 57

46

67

-

36 38

31 33

53 59

44

Tbilisi 1 2

35 36

Vladivostok 1 2

33 35

Kislovodsk N. Caucasus 1 32 2 33

34

52 55

59

64

64 70

61 71

65 71

42

51

-

-

-

-

44

55

65

71

71

73

50 54

57 63

63 65

65 69

64

44

27 40

46 52

54 59

62 66

64

-

42

44

62 66

56 59

48 49

37 39

31 34

53 60

62 68

53 56

42

44

32 34

29 30

52

67

59 65

52 57

42 44

32 37

29 31

51 54

63 65

60 63

51 54

41 45

33 -

28 32

50

YI>

w E

P

B

Feodosia (Crimea) 1 2

28 34

w

h,

W

w w

TABLE 5.50 (continued) Point Observation Yevpatoria (Crimea) 1

2 Odessa 1

2

0

Jan.

Feb.

Mar.

Apr.

May

June

July

Aug.

Sept.

Oct.

Nov.

Dec.

Year

29 32

39

49 53

59 62

63 66

66

-

64 65

60 61

52 53

43 44

33 34

25 29

51

-

(23) 30

37 39

47 49

56 59

62 66

65 66

65 67

58 61

50 53

39 43

31 31

18 27

50 57

-

30 33

41 44

52 54

58 60

59 63

60 64

55

24

58

46 49

35 37

25 26

20 20

28 30

39 41

48 52

52 57

55 60

55 59

48 55

41 47

30 33

20 21

11 14

20 24

32 36

43 47

50 54

54 56

52 56

46 51

38 42

25 29

9 11

20 21

32 33

42 44

50 53

55 55

51

45 48

41 39

24 26

20

50

13

16

42 53

13 16

8 9

39 48

11 13

(6) 8

41 44

Moscow region

1 2 Pavlovsk 1

2 Yakutsk (Siberia) 1

2

a.

E?

Irkutsk (Siberia)

1 2

U

(1) for cloudless sky; (2) for cloud conditions.

54

f

!iI’ 3.

P

33 1

5.10. Variability of Fluxes and Totals of Solar Radiation

small solar heights when the horizontal surface receives an insignificant amount (and the vertical a notable amount) in comparison with the daily total, quantity of solar radiation. The same is observed when cloudiness appears and increases or thickening of cloudiness takes place (not only at noon). That is why the ratios of the possible daily totals on horizontal and vertical surfaces are systematically lower than the corresponding ratios of the actual totals (see Table 5.50). 4. Monthly Totals of Radiative Heat. Knowing the daily totals of solar radiation, it is easy to compute monthly values. As has already been mentioned, the most reliable material for determination of radiative heat totals is provided by the 29 years of observation at Pavlovsk. In Table 5.51, made up by Kalitin [I341 from observational data at Pavlovsk (19141940), are given mean, maximal, and minimal monthly radiation totals on perpendicular and horizontal surfaces. It is seen from this table that maximum in the annual range of monthly totals is observed in June-July (maximal daylight and solar altitude). The difference between maxima

TABLE 5.51 Means, Maxima, and Minima of Monthly Radiative Heat Totals on the Perpendicular and the Horizontal Surfaces for Pavlovsk (caI/cmz).After Kalitin [134]

Monthly Radiative Heat Totals Month

Jan. Feb. Mar. Apr. May June July Aug. Sept. Oct. Nov. DK.

Perpendicular Surface

Horizontal Surface

Mean

Max.

Min.

Mean

Max.

Min.

1,290 2,670 7,080 9,400 14,300 14,480 14,570 10,750 6,790

2,920 5,620 13,170

380 400 3,690

13,820 20,950 19,670 22,440 16,770

4,600 9,600 7,390 9,250

180 620 2,560 4,400 7,620 8,140 8,130 5,440 2,790 950 170 70

310 1,340 4,650 6,990 1,200

50 80 1,520 1,990 5,060

10,950 12,380

4,000 5,200 2,880 1,630 300 47 4

3,280 1,070 730

9,850 5,200 2,540 2,100

7,730 3,970 1,040 280 36

8,760 4,190 1,500 420 180

332

Direct Solar Radiation

and minima is quite great; that is the variation of monthly radiation totals may be considerable from year to year. Especially great is the variation of December totals, sometimes exceeding 100 percent relatively to the mean many-year value. A similar conclusion on the variability of monthly radiation totals on a horizontal surface has been reached by Beletzky [I351 at Odessa and by some other authors. The main cause for this variability are the changing conditions of cloudiness and atmospheric transparency. In spite of the considerable variations of monthly radiation totals from year to year, the stability of their mean values is quite high. For illustration consider Table 5.52 which presents computations of monthly totals XS', E = all/;, and

a=

by Gorlenko [136, 1371 (1913-1933 Pavlovsk observations). The value (T here is taken as the stability modulus, and E is the error of the mean total. TABLE 5.52 Variation of Mean Monthly Radiation Totals on a Horizontal Surface. Ajier Gorlenko [136]

Month

Jan. Feb. Mar. Apr. May June July Aug. Sept. Oct. Nov. DX. Yearly

Q, 188 602 2,800 4,198 7,513 7,617 8,408 5,161 2,693 974 162 77 40,393

U

&

45 50 30 25 21 22 20 24 23 26 62 48 8

10 11 7 5 5 5 4 5 5 6 14 10 2

5.10. Variability of Fluxes and Totals of Solar Radiation

333

It is seen from the table that the considerable variation of mean monthly totals on a horizontal surface takes place only in the winter months (November-February). In the other months (March-October) the variation of mean totals is insignificant; for example, o/ZS’ is less than 1 percent. As regards the variability of the mean annual total of radiation on a horizontal surface, it is quite negligible. Tarnizhevsky [1381, using familiar results of mathematical statistics, has analyzed the accuracy of the determination of radiation totals in the general form. With application of the Student distribution, the mean values can be confidentially estimated from the inequality (5.59)

where u is the mean of the general totality with the normal law of distribution of the quantity under consideration (radiation totals), 3, is the standard selection (mean quadratic deviation) to judge on the standard general mean, R is the mean selection from the general totality, n is the number of selection terms (years of observation), and t is the Student distribution parameter P = f ( t , k), where the index k expresses the so-called number of freedom degrees. Setting the reliability a = 1 - P and the index k dependent upon n, it is possible to find t from the corresponding tables, and if the range of variations of the standard totals is known, also to calculate the accuracy of the mean total determination for various observation terms, provided the reliability is known. Tarnizhevsky, using the results of actinometric observations at Pavlovsk, Tashkent, and Odessa, has found that the standards of monthly and annual radiation totals vary within 0.05 to 6.32 cal/cm2. Setting reliability at 90 percent, Tarnizhevsky has calculated Table 5.53, which characterizes the accuracy of the determination of the mean monthly totals of direct, diffuse and global radiation at different standard values and terms of observation. For example, a 15-yr period of observations at Karadag has given the mean annual total of direct solar radiation on a perpendicular surface equal to 129.3 kcal/cm2 and the annual total standard 3.22 kcal/cm2. From Table 5.53 we find that in this case the accuracy of the determination of the mean annual total at the reliability 0.9 equals 1.45 kcal/cm2, or 1.1 percent.

w w

P

TABLE 5.53 Accuracy of the Determination of the Mean Monthly and Annual Radiation Totals, E

Duration of Observation, Yrs

=2

- a (kcal/cm2)at 0 . 9 Reliability. After Tarnizhevsky [138]

Standards, kcal/cma

0.05 0.10 0.25 0.50 0.75 0.00 1.24 1.50 1.75 2.0

2.5

3.0

3.5

4.0

4.5

5.0

6.0

7.0 U

5

2

0.22 0.45 1.12 2.24 3.36 4.48 5.60 6.72 7.84 8.96 11.20 13.44 15.68 17.92 20.16 22.40 26.88 31.36

3

0.08 0.17 0.42 0.84 1.27 1.69 2.11 2.54 2.96 3.38 4.22 5.07 5.92 6.76 7.60 8.45 0.14 11.83

5

0.05 0.10 0.24 0.48 0.71 0.95 1.19 1.42 1.66 1.90 2.38 2.85 3.32 3.80 4.28 4.57 5.70 6.65

10

0.03 0.06 0.14 0.29 0.44 0.58 0.72 0.87 1.02 1.16 1.45 1.74 2.03 2.32 2.61 2.90 3.48 4.06

15

0.02 0.04 0.11 0.22 0.34 0.45 0.56 0.68 0.79 0.90 1.12 1.35 1.58 1.80 2.02 2.25 2.70 3.15

20

0.02 0.04 0.10 0.20 0.29 0.39 0.49 0.58 0.68 0.78 0.98 1.17 1.36 1.56 1.76 1.95 2.34 2.73

30

0.02 0.03 0.08 0.16 0.23 0.31 0.39 0.46 0.54 0.62 0.78 0.93 1.08 1.24 1.40 1.50 1.86 2.17

40

0.01 0.03 0.06 0.13 0.20 0.26 0.32 0.39 0.46 0.52 0.65 0.78 0.91 0.04 1.17 1.30 1.56 1.82

2 tx g,

?it3: EL

5.10. Variability of Fluxes and Totals of Solar Radiation

335

Table 5.54 summarizes Tarnizhevsky’s data on the expedient duration of observations to determine monthly and annual total means with the accuracy of 10 percent and the 0.9 reliability. We can see that long-term observations are required to secure even this low percentage of accuracy. Pivovarova [139], however, has justly noticed that the demands to the relative accuracy of determination of winter totals may be less strict than in the case of the summer months due to the smallness of winter radiation values. The general character of the annual range of monthly totals (one maximum in summer and one minimum in the winter months) is the same at almost all points of the globe, whereas monthly total values differ considerably in dependence upon geographical location and local climatic conditions. It is also interesting to compare the observed monthly totals of radiation on a horizontal surface with the monthly incoming solar radiation under the conditions of an ideal atmosphere and outside the atmosphere. In Fig. 5.35 are given data of Pleshkova (see [140]) for the Tikhaya Bay, Pavlovsk, Tashkent, and Vladivostok. The upper curves characterize the annual range of monthly totals at the above points in the absence of atmosphere; the middle curves determine the monthly variation of the incoming radiation in an ideal atmosphere; the lower curves portray the annual range of the observed monthly totals. It is natural that at southern points (highest solar elevations) the observed radiation totals are closest to the values calculated for the ideal atmospheric conditions. Interesting is the double annual range of monthly totals at Vladivostok. The forming of the second minimum of incoming radiation in summer is caused by the local increase of summer monsoon cloudiness. As seen from Fig. 5.35, at the Tikhaya Bay there is no incoming radiation during the polar night (October-February), and it is at maximum in July (3.2 kcal/cm2 mo). At Pavlovsk, monthly totals on a horizontal surface vary from 0.7 cal/cm2 mo. in December to 8.1 kcal/cm2 mo in July. At Tashkent, the minimal monthly income equals 1.9 kcal/cm2 mo and falls on December, the July maximum being 16.7 kcal/cm2mo. Monthly radiation totals calculated for an ideal atmosphere are comparatively little different from the possible values (maxima of the possible totals are by 5 to 6 kcal/cm2 mo less than the corresponding values for an ideal atmosphere). The relation between the lower mean curves (taking into account the mentioned correction) can therefore approximately characterize the difference between the real and possible totals of solar radiation on a horizontal surface.

W

w

Q\

TABLE 5.54 Terms of Observation Necessary for Determining Mean Monthly and Annual Radiation Totals with Accuracy 10 percent and reliability 0.9. Afrer Tarnizhevsky [138] ~

~~

Radiation

~

~

June

July

Aug.

Sept.

Oct.

Nov.

Dec.

Year

7

5

7

7

3 8

4 6

3 13

8 8

18 14

14 18

3 4

6

4

4

3

3

3

4

9

7

3

19 13

15 8

12

7

13 5 5

18 7 6

15

13

11 3 5

7

23 6 8

15 9 5

15 26

20 21

4 3 3

70

10

12

11

8

6

10

7

57

23

a2

3

Jan.

Feb.

Mar.

Apr.

May

28 15 5

15 15

19 8

7 6

9

10

51 16

36 15

15

38

Tashkent Direct Diffuse Global Pavlovsk Direct Diffuse Global

6

4

0

-

Odessa Direct

337

5.10. Variability of Fluxes and Totals of Solar Radiation

30 -

r

0

,E 2 0 -

5

\

-

8

Y

c 0 ._ '0 ._ 10U

I + !

0-

I

30

0

,E 20 E 0 \ 0

y" c

.-0 '0 lo ._ 'D 0

n

I 1:

:

:

!

I - :

:

:

!

:I

m m m x x

FIG. 5.35 Annual variation of the monthly totals of solar radiation on the horizontal surface in the absence of atmosphere, at the ideal and the real atmospheres for Tikhaya Bay (a), Pavlovsk (b), Tashkent (c) and VIadivostok (6).

5. Seasonal and Annual Totals of Radiative Heat. Similarly with monthly totals, seasonal and annual values are determined in the main by the latitude and the conditions of cloudiness and atmospheric transparency at a particular point of observation. Table 5.55, compiled by Yaroslavtzev [141], gives mean seasonal and annual totals of radiation on perpendicular and horizontal surfaces for Pavlovsk and Tashkent. The first four lines of these data characterize the relative seasonal distribution of radiation totals (seasonal total values in percent relative to the

338

Direct Solar Radiation

TABLE 5.55 Mean Seasonal and Annual Radiation Totals on a Perpendicular ZS and Horizontal ZS' Surfaces at Pavlovsk and Tashkent. After Yaroslavtzev [141]

Pavlovsk (59.7ON)

Tashkent (41.3ON)

Season ZS

Winter Spring Summer

Fall

Year total

5.3 36.5 44.8 13.4 82,623

ZS 2.2 36.4 51.6 9.8 39,758

ZS 10.9 21 .o 40.7 24.4 174,430

ZSI

7.2 25.9 46.0 20.9 101,640

annual total), the lowermost line gives the absolute annual totals (cal/cm2 yr). The income of solar radiation at Tashkent is shown to be much greater than at Pavlovsk, the most pronounced difference being between the totals on a horizontal surface. At both places the maximal income of solar radiation is observed in summer. The same regularity of seasonal radiation distribution holds for the majority of observation points. However, at some places where maxima of cloudiness fall in the summer months, the highest values of radiation shift to spring and fall; for example, at Vladivostok the summer total on a perpendicular surface is 20.6 kcal/cm2, and the spring and fall totals are 30.2 and 33.3 kcal/cm2, respectively. On the basis of observational data of 22 stations located at various points of the Northern Hemisphere, Berland [I421 explored the variation of annual radiation totals on a horizontal surface in dependence upon latitude. Figure 5.36 gives a curve of latitudinal variation of annual radiation totals. As seen from Fig. 5.36, the range of annual totals is from 10 kcal/cm2yr a t v = 80'N lat. to 100 kcal/cm2yr at y~ = 36'N lat. The dispersion of points in relation to the curve is not great, which points out that the dependence upon latitude (that is, upon solar height and daylight duration) is the main factor determining geographical variability of annual radiation totals. Only at low latitudes (v < 4 0') is the observed dispersion of points around the mean curve significant, a result evidently due to the influence of cloudiness on the income of radiation. It has been mentioned above that monthly totals of radiative heat are considerably variable from year to year, and we can justly expect secular variations of annual totals resulting from these monthly fluctuations. Ac-

5.10. Variability of Fluxes and Totals of Solar Radiation

339

cording to actinometric observations by Kalitin [Chapter 3, Ref. 1021 at Pavlovsk from 1913 to 1935, departures of the observed annual totals on a perpendicular surfaces from the mean many-year total vary within - 18 to 13 percent. This range is - 16 to 12 percent for horizontal surfaces. It is interesting that the maximal departures from the mean toward greater and lesser observed totals were recorded in two successive years, 1921 and 1922. In the year of the maximal annual total, the latter was 37 percent of the possible value for perpendicular and 41 percent for horizontal surfaces. The corresponding ratios of the minimal annual totals to possible values are 26 and 31 percent.

+

+

cp

FIG. 5.36

Variation of the annual totals of radiative heat on the horizontal surface in dependence upon latitude.

Less marked departures (in the same years) of annual totals from the mean were obtained by Yaroslavtzev [141] (Tashkent 1926-1945). According to Yaroslavtzev, the variation of annual totals on perpendicular surfaces fluctuates from - 6.6 to 7.6 percent of the mean many-year value. For horizontal surfaces this range is from - 6.8 to 6.2 percent

+

+

340

Direct Solar Radiation

In all cases, therefore secular variations of annual radiation totals at Tashkent do not exceed the limits of & 8 percent of the mean. Approximately the same results are obtained from data of decades of observation at Tbilisi and Irkutsk. The chief cause for the secular variability of annual radiation totals are cloudiness and atmospheric transparency fluctuations. As seen, the variation of annual totals is less than that of monthly and daily totals. The same holds for the mean annual totals. According to Table 5.55, the variability of the mean annual total on a horizontal surface, as taken from data of observations at Pavlovsk, can be neglected. Similar results are available at other points. It should be noted that although annual totals of radiation on horizontal and perpendicular surfaces are considerably variable from year to year, the ratio of these values remains almost stable. According to Yaroslavtzev [141], the value of the x S ' / c S ratio for Tashkent and Pavlovsk has an annual variation, with respect to the mean many-year figure, not greater than f 1 percent. The years 1919 and 1928 were exceptional a t Pavlovsk in that somewhat higher observed fluctuations of c S ' / x S were observed. The low variability of the ratio of annual totals on horizontal surfaces to those on vertical surfaces ensures transcalculation of the considered totals, making use of a definite proportionality coefficient. The numerical value of ZS'/XS will, of course, depend upon latitude increasing equatorward. Figure 5.37 gives a distribution of annual sunshine duration values, obtained by Berland and Danilchenko [1431, which quantitatively describe the main peculiarities of the planetary geographical distribution of incoming radiation. According to Fig. 5.37, the annual sunshine duration over the globe varies from 500 to 4100 h. The sunniest are subtropical deserts and semideserts, especially the Sahara in the Aswan region and the desert in Colorado, U.S.A. On continents, the duration of sunshine always increases offshore. The dullest areas are the western and northern coasts of Eurasia and the western shore of Canada. The South Orkney Islands are the most cloudy spot of the globe, averaging S 5 483 h. Highest of all are antarctic values of incoming radiation. For example, Rusin [144] has estimated maximal flux values on perpendicular surface in Antarctic at about 1.67 cal/cm2 min, which is higher than any recorded value on the globe, including data of high mountainous stations in Central Asia. The same refers to the possible daily totals on a horizontal surface reaching 700 to 740 cal/cm2 on the antarctic coast and 850 to 900 cal/cm2 at the highest peaks of the mainland glacial plateau.

r

n

90

120

160

180

160

120

90

60

w

FIG. 5.37 Geographical distribution of the annual values of sunshine duration (hours).

P

I-.

342

Direct Solar Radiation

Also high are monthly and annual real totals of solar radiation on a horizontal surface, which can be seen from comparison of Table 5.56 [I441 with the above data for temperate and low latitudes. 5.11. Income of Solar Radiation on Slant Surfaces We have considered the cases of incoming radiation on surfaces that are horizontal and perpendicular to the sun's rays. The real underlying surfaces are, however, often nonhorizontal. Of great interest to agrometeorology is the problem of determining the incoming radiation on slopes of various orientations. It is also important for heliotechnique, peat drying methods, and some other applications. 1. General Relations. A great number of investigations have been devoted to calculations of radiation income on slopes. One of the first was made by Smoliakov [145]. Let us consider, following Smoliakov, deduction of the general formula for radiant flux on an arbitrarily oriented slant surface. Denoting by S , the solar radiation flux at the earh's surface on a perpendicular plane at the atmospheric mass m, write the following obvious expression for the radiant flux on the sloping surface: Ssl = S , cos i

(5.60)

where i is the angle of incidence of solar rays on the inclined surface (see Fig. 5.38). It can be shown that cos i is expressed in the following manner: cos i

= cos a

sin h,

+ sin a cos h, cos y

(5.61)

where a is the angle of inclination, ha is the height of the sun, and y = y, - yn where ya, yn are solar azimuths and projections of the normal to the slope on a horizontal surface counted from the meridional plane (azimuths are conventionally assumed to be positive when reckoned clockwise). The azimuth and height of the sun are determined from the following familiar relations : sin ha = sin 'p sin 6 cos yo

=

sin yo =

+ cos 'p cos 6 cos 52

sin h, sin v - sin 6 cos h, cos q~ cos 6 sinQ cos h,

(5.62a) (5.62b) (5.63)

TABLE 5.56 Monthly and Annual Totals (kcal/cm2)of Direct Solar Radiation Incident on a Horizontal Surface in the Antarctic at Mean Cloudiness. After Rusin [144]

Jan.

Feb.

Mar.

Apr.

May

June

July

Aug.

Sept.

Oct.

Nov.

Dec.

Year

Oasis

8.7

6.2

3.0

1.1

0.4

-

-

0.6

2.7

4.8

7.3

9.2

44.0

Mirny

10.6

7.5

4.0

0.9

0.1

-

-

0.8

0.2

5.7

10.3

12.8

54.9

Pionerskaya

13.9

9.1

4.7

1.1

0.0

-

-

0.1

2.9

7.3

11.7

13.7

64.5

Vostok-1

14.9

9.1

3.3

0.4

-

-

-

0.0

1.6

7.4

12.5

15.8

65.0

Komsomolskaya

21.1

12.9

5.7

0.8

-

-

-

0.0

2.7

10.0

17.8

23.7

94.7

VostokaSovetskaya

22.3

14.8

4.2

0.4

-

-

-

-

1.8

8.7

20.0

25.6

97.9

South Pole

15.3

8.8

0.9

-

-

-

-

-

0.2

4.8

12.7

17.5

60.2

Station

Y

L

0

J

zD

*

w

P

w

344

Duet Solar Radiation

where 47 is latitude, 6 is the inclination of the sun, and SZ is the solar hour angle at a particular moment reckoned from the moment of the true noon (Q is assumed to be positive, reckoning clockwise).

FIG. 5.38 The derivation of the formula for the income of solar radiation on a slope.

Taking account of (5.61) to (5.63), instead of (5.60) we now have

S,,

=

+ cos v cos 6 cos y ) + sin a{cos yn[tan q(sin v sin 6 + cos 47 cos 6 c o s 9 ) - sin 6 sin 471 + sin yn cos 6 sinSZ}] (5.64)

S,[cos a(sin 47 sin 6

This formula expresses the general form of the dependence of incoming radiation on the slope of orientation determined by the angles a and y,, for any latitude 47 and at different time of the day (hour angle SZ) or year (solar inclination 6). Let us now turn to consideration of various finite relations following from (5.64). Horizontal Surface. In this case a the below familiar relation : S,

= S,(sin = S,

47 sin 6

sin ha

= 0,

+ cos

and instead of (5.64) we obtain

cos 6 cosSZ)

(5.65)

345

5.11. Income of Solar Radiation on Slant Surfaces

Vertical Surface. Assuming a = n/2, in place of (5.64) find S,

= S,(cos

= S,

y,[tan q(sin q sin 6

sin 6 sec q ]

+ cos q cos 6 c o s 9 )

+ sin yn cos 6 s i n 9 1

cos ha COS(Y,- yn)

(5.66)

If the vertical surface is directed southward (yn = 0), we have Sv,s= S , [tan q(sin cp sin 6

+ cos q cos 6 cos 9 )

- sin 6 sec q ] = S,(sin q sin 6 COSQ = S , cos h, cos yo

- sin

6 cos q) (5.67)

For a vertical surface directed eastward or westward, yn Sv,E,w) = S , cos 6 sinQ

= S,

=

cos h, sin yo

f (n/2), (5.68)

In the case of northern orientation, yn = 180°, Sv,N= S,(sin 6 cos q - cos 6 sin q COSQ)

(5.69)

Taking into consideration (5.67) and (5.68), it is possible to rewrite (5.66) in the following manner:

sv

=

sv,~cos Y n + &,E(w) sin Y n

(5.70)

Sfant Surface. According to (5.65) and (5.66), the formula (5.64) can be transformed as S,, = s h cos a i-S, sin a

or, taking account of (5.70), S,,

=sh

cos a

+ [Sv,scos yn + Sv,E,w,sin y,]

sin a

(5.71)

The first to derive (5.71) was Gordov [146]. This formula shows that the income of radiation on any slope can be easily determined, provided we know the three components that characterize the incoming radiation on the horizontal surface and on the vertical oriented southward or eastward (westward). Using (5.71), it is easy to derive formulas that determine the incoming radiation on slopes oriented in the four cardinal directions. For the south slope, yn = 0, we have S,,,

= s h cos a

+

sin a

(5.72)

346

Direct Solar Radiation

In the case of the eastern (western) slope, yn = &(n/2),

S,,E(W) = Sh cos a For the direction northward, yn (5.69), the following:

S,,,,

=s h

+ S,,E,W) sin a

=

cos a

(5.73)

& n, we have, taking account of

+ S,,

sin a

(5.74)

Diurnal Range of Radiation Income on a Slope. Let us turn to analysis of the daily variation of radiant flux on an arbitrarily oriented slant surface. The formula (5.66) can be abbreviated to the following form: S,l = S,[Al

+ B,

COSQ

+ C, sinQ]

(5.75)

where Al

= cos a

sin 9 sin 6

+ sin a[cos yn(tan 9 sin 6 - sin 6 sec 9)] B, = cos a cos cos 6 + sin a cos yn sin 9 cos 6 Cl = sin a cos 6 sin yn Remember that in the case of a horizontal surface we had an analogous formula of the form Sh = S,[A

+ BCOSQ]

(5.76)

Comparing (5.75) and (5.76), we see that the expression for the daily flux variation on nonhorizontal surfaces acquiers an additional term of the type of C, sinQ, which means that the daily variations of the fluxes S, and S,, are different. Later we shall give some computational results to illustrate these differences.

Duration of Irradiation of Slopes. For practical purposes the problem of duration of irradiation of variously oriented slopes is very important. It is natural that the duration of irradiation of slopes will differ from that in the case of horizontal surfaces, even if the other conditions are equal. We can easily see that the hour angles which determine the moments of the beginning (- Q) and the end (+Q) of irradiation of a slope are in general the roots of an equation following from (5.75): A,

+ Bl cosQ, + C, sinQ = 0

(5.77)

5.11. Income of Solar Radiation on Slant Surfaces

347

2. Computational Results. Since the radiant flux on an arbitrarily oriented slant surface can be comparatively simply and exactly expressed through the flux on a perpendicular surface, there is no need for any special measurement of the incoming radiation on a slant surface. The value of incoming radiation on slopes is usually calculated from data of routine actinometric measurements of radiant flux on a perpendicular surface. Also available are a great number of theoretical calculations of the above value in different conditions of atmospheric transparency. In this case the value of radiant flux on a perpendicular surface was determined by means of calculation from the familiar value of the solar constant and various given values of the atmospheric transparency characteristics (for example, the transparency factor).

Vertical Surface. The most numerous and detailed are theoretical calculations of incoming radiation on vertical surfaces of various orientations, whose practical importance is evident in building problems. Owing to explorations by Weinberg [147], Hamburg [148], Razumov [149], Nicolet and Bossy [150], and others (see, for example, [151-159]), this question has been fully investigated. For illustration let us consider the results of calculations of the daily range of SJS, for vertical surfaces of different orientation in summer and winter at = 60ON. lat. tabulated by Razumov [149]. Values of Sh/S, are given in the third and fourth columns of Table 5.57. It can be seen from Table 5.57 that there is a considerable difference between the daily ranges of Sv/S, values in dependence upon orientation of vertical surfaces. Naturally, in the case of southern walls, the daily range shows maximum at true noon. When southern orientation is eastward or westward, the time of the maximal radiation displaces, with respect to noon, toward earlier (south east walls) or later (south west walls) hours. For example, in the considered case the maximal incoming radiation on a southwestern wall takes place at about 15.00 of true sun time. While in summer the flux of solar radiation on a horizontal surface is maximal in the near-noon hours, in winter the vertical walls of southern orientation at 60' N. lat. receive a greater amount of solar radiation than the horizontal surface, especially at noon. Calculations of Razumov have shown that the same relations hold for 40'N. lat. as well. In the case of west and east walls, the maximum of radiation is still more displaced with respect to noon; for example, in summer a west wall receives the largest amount of radiation at about 1800 of true sun time. In winter the total incoming solar radiation on west and east walls decreases

TABLE 5.57 Daily Range of the Values of the Ratio of Solar Radiant Flcrxes on Surfaces That Are Vertical and Perpendicular to the Rays, SJS,,,, at 600N Iat. After Razumov [149] Slll~,

True Solar Time W

E

9 10 11 12 13 14 15 16 17 18 19 20 21 22

15 14 13 12 11 10 9 8 7

&IS, Summer

5 4

0.8037 0.7880 0.7422 0.6694 0.5744 0.4639 0.3452 0.2265 0.1160

3 2

0.0210

6

Summer (June 21), 6 = 23.5O Winter

0.1133 0.0977 0.0519

S

N

0.5948 0.5676 0.4884 0.3622 0.1977 0.0062

w,

sw

E

SE

O.oo00

0.1993 0.448 0.5964 0.7608

0.2373 0.4585 0.6484 0.7941 0.8857 0.9170 0.8857 0.7941 0.6484

0.0212 0.2335 0.4205 0.5691 0.6698 0.7146 0.7012 0.6306 0.5074 0.3400 0.1396

Winter @ec. 22), S NW NE

s

N

w’

=

23.5O

sw SE

0.3033 0.5154 0.9934 0.9663 0.8870 0.2023 0.4217 0.6218 0.7893 0.9125 0.9832 0.9964

O.oo00 0.2373 0.4585

0.7023 0.8510 0.9517

NW NR

E?a

L’

B B

5.11. Income of Solar Radiation on Slant Surfaces

349

considerably in comparison with the summer radiation. As seen from Table 5.57, on the day of the winter solstice, west and east walls at 60° N. lat. are illuminated during about 2 h only. Walls of northern orientation (north, northwest, norteast) are almost completely without solar irradiation and completely so on the winter solstice day. In the summer months incoming radiation to northeastern and northwestern walls is maximal in the early or late daytime hours, respectively. Vertical walls facing straight north are irradiated twice a day, with a large midday interval when the sun leaves for the south. Mamikonova [160] has investigated the regularities of the variation in daily totals received by vertically oriented walls. Calculations of daily totals were conducted from data of actinometric observations of radiant flux on a perpendicular surface at Kharkov. Figure 5.39 gives a graph of the dependence of the possible daily totals on vertical walls upon their orientation for the days of the year closc to the solstices and equinoxes (according to Mamikonova). The straight lines parallel to the axis of abscissas characterize the daily incoming radiation on a horizontal surface for the corresponding days. In all cases except the summer period the daily radiation on a vertical surface is at maximum for south walls. In summer the maximum falls on eastern and western walls. Likewise for the case of radiant fluxes, the daily totals of radiation on

FIG. 5.39 The possible daily totals of radiative heat received by vertical walls of diferent orientation.

350

Direct Solar Radiation

south vertical surfaces exceed the corresponding totals on horizontal surfaces during the winter, fall, and spring periods. Analogous curves, characterizing the dependence of actual radiation totals received by vertical walls upon their orientation for the same four days of the year, are presented in Fig. 5.40. In the given case our attention is drawn by the asymmetry of the summer curve: The western walls receive less radiation in summer than do the eastern walls. This phenomenon is caused by the increase of cloudiness during the second half of the day in summer. The effect of cloudiness can also explain the considerable decrease of the real totals in comparison with the possible totals during winter and the great difference between the actual radiation totals in spring and fall. Also important is the problem of the duration of irradiation of oriented walls. Interesting results have been obtained by Kastrov [161] and Torletzkaya [162].

240r

Ix I--\

0” 200 0

NE

1

160-

8

z 1200

2

80a a 40-

0

OL FIG. 5.40

N

NE

E

SE

S

SW

W

NW

N

The actual daily totals of radiative heat received by oriented walls.

Table 5.58 gives values of the relative duration of irradiation of south and southeastern (or southwestern) walls under the conditions of a continuously clear sky for the solstice and equinox days at different latitudes. The relative duration of irradiation is understood to be the ratio of the factual duration of irradiation of a given wall s, to the possible duration of sunshine per day, s p . From Sec. 5.9, we have

sp = 2 arc cos(- tan q~tan S ) The s, values for the case of south walls aie determined by the following simple relations :

351

5.11. Income of Solar Radiation on Slant Surfaces

Summer half-year : s, = 2 arc cos(tan 9 tan 6)

Winter half-year : s, = sp = 2 arc cos(- tan 9 tan 6) TABLE 5.58 Relative Duration of Irradiation (Percent)of South and Southeast (Southwest) Walls in the Absence of Cloudiness at Different Latitudes and during Different Periods in the Year. After Kastrov [161]

9 6 23'27'

56'

60'

66' 33'

30'

40'

45'

50'

37 100 100

53 100 100

56 100 100

51

56

55

44

100 100

100 100

100 100

100 100

60 79 94

61 80 98

58 80 100

46 81 100

South wall 23'21' 000 - 23 27

0 100 100

Southeast (southwest) wall 23O21' 000 - 23 27

50 69 14

53 73 79

51

59

76 87

78 90

We see that in winter the value of the relative duration of irradiation of south walls is independent of latitude and always equals 100 percent. The latitudinal dependence of s,/sp in the summer period is quite weak. Practically, the relative duration of irradiation of south walls in the latitudinal limits 40' to 60' may be considered to be independent of latitude in summer as well. All this means that the annual relative duration of irradiation of south walls must be also practically independent of latitude. For southeast and southwest walls, almost always sJsP < 1, with the latitudinal dependence of these values notably developing in winter only. However, since the duration of irradiation during winter is comparatively small, the annual relative duration of irradiation in this case must not be seriously dependent upon latitude. The above results show that theoretical computations based on the use of purely astronomical relations give

352

Direct Solar Radiation

evidence of comparatively small latitudinal differences of the annual relative duration of irradiation of oriented walls in the zone of latitudes 40' to 60'. This fact was empirically stated at an earlier date by Torletzkaya [162]. Using heliograph recordings at various stations located in the latitudinal belt from 43.5' N. lat. to 49' N. lat., Torletzkaya showed that for the particular belt of latitudes, the relative annual duration of irradiation of oriented walls was practically independent of latitude. The means of the considered quantity can be approximately characterized by the following data: Wall Orientation:

N

NE

E

SE

S

SW

W

NW

srlsz, %

4

21

49

78

96

78

50

22

9

Slant Surface. Let us consider the results of calculating incoming direct solar radiation on slopes of different steepness and orientation. Besides the mentioned works, a great many others are devoted to this problem, in particular Gordov [146], Averkiev [163, 1641, Grishcheriko [165, 1661, Kuzmin and Novikova [167]. The above authors have performed calculations of the incoming solar radiation on slopes in various conditions. Gordov [146], for example, has carried out detailed calculations of radiant flux on oriented slopes at 42' N. lat. Grischchenko [165, 1661 has proposed a method for determination of the orientation of slopes from geographical maps and has calculated the incoming radiation on slopes at 44' 40' N. lat. Averkiev [163, 1641, making use of (5.61), has compiled detailed tables and nomograms enabling determination of radiant flux on the surface of a slope in dependence on the components a, h,, and y of (5.61). Consider some of the mentioned results. Table 5.59 gives values of the ratio of radiant fluxes on the surface of a slope and on a horizontal surface, computed by Kuzmin and Novikova [167] for slopes of different steepness and orientation at = 58' N. lat. at true noon. As seen from this table, the incoming solar radiation on south slopes considerably exceeds that on the horizontal surface, and even more on northern slopes. This difference is observed even with slanting slopes as small as GL = 5'. Zakharova [155] has investigated the difference of incoming radiation between the southern and northern slopes of various steepness for the latitudes of 42, 50, 60, and 70'. As shown by Zakharova, the conditions for incoming radiation are not always the most favorable for southern slopes. In the morning and evening hours at all latitudes, especially in the north, the southern slopes are in the losing position as compared with the northern. In the polar conditions of continuous daylight, the northern

TABLE 5.59 Direct SoIar Radiation on Slopes of Different Orientation and Steepness in Portions of the Direct Radiation Incident on the Horizontal Surface for True Solar Noon (q = 580 N. Iat.). After Kuzmin and Novikova [167]

Angle of Inclination of the Slope, deg Date

Solar Horizontal Height at Surface

10

5

Noon, deg

SE, SW

W, E

NE, NW

SE, SW

W, E

20 NE, NW

SE, SW

W, E

NE, NW

N

Mar. 1

24.0

1.00

1.19 1.13 1.00 0.85 0.80 1.38 1.26 0.99 0.71 0.60 1.74 1.49 0.94 9.39 1.17

Mar. 15

29.2

1.00

1.15 1.10 0.99 0.88 0.84 1.30 1.20 0.98 0.76 0.67 1.55 1.37 0.94 0.50 0.32

Apr. 1

37.0

1.00

1.11

Apr. 15

41.8

1.00

1.10 1.07 1.00 0.93 0.90 1.18 1.12 0.99 0.85 0.79 1.22 1.21 0.94 0.67 0.56

May 1

47.0

1.00

1.07 1.05 0.99 0.94 0.91 1.15 1.10 0.99 0.87 0.82 1.26 1.16 0.94 0.71 0.62

0

a

1.08 1.00 0.91 0.88 1.22 1.15 0.99 0.82 0.75 1.39 1.26 0.94 0.62 0.48

w VI w

354

Direct Solar Radiation

slopes experience two maxima of irradiation, at about 04 and 20 h, while the steep southern slopes are exposed to radiation only during the interval from 04-05 to 19-20 h. These circumstances, however, do not change the fact that the total daily, monthly, or annual radiation on the not too steep southern slopes is always maximal. This can be seen from Table 5.60, which summarizes Zakharova's calculations of daily totals of radiative heat for northern and southern slopes of different steepness on the days of solstice and equinox. We see from Table 5.60 that the conditions of the maximal daily radiation on southern slopes are determined by the season and local latitude and, in the long run, by the height of the sun. It is natural in the summer half of the year, especially at low latitudes, that the maximum of radiation will be received by comparatively less steep southern slopes. Steeper slopes will be in worse conditions at that time. In winter, on the contrary, steep southern slopes will receive the maximal amount of radiation, especially at high latitudes. It is interesting to compare the daily incoming radiation on slopes and horizontal surfaces at different latitudes. For example, it is seen from Table 5.60 that a southern slope of 20' steepness at the latitude of Leningrad (rp = 60' N. lat.) in summer receives TABLE 5.60 Daily Totals of Radiative Heat for Southern and Northern Slopes (cal/cm2).Ajier Zakharova [155]

Date

Northern Slopes Southern Slopes Angle of Inclination, deg Horizontal Angle of Inclination, deg Surface 40 30 20 10 10 20 30 40

p = 50°N. lat.

June 22

441

551

630

686

727

745

739

721

618

0 0

101 0

197 289 0 4 7

376 88

443 136

505 193

547 235

577 281

388

488

575

642

709

716

733

721

705

0 0

0 0

91 0

186 0

270 14

346 41

412 67

464 89

503 112

Mar. 22 and Sept. 23 Dec. 22 p = 60°[N. lat.

June 22 Mar. 21 and Sept. 23 Dec. 22

References

355

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Direct Solar Radiation

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86. Murk, H. (1959). A nomogram for computation of certain characteristics of atmospheric transparency. Invest. Atmospheric. Phys. 1. 87. Murk, H. (1959). On the rationality of the Makhotkin turbidity index N. Invest. Atmospheric Phys. 1. 88. Murk, H. (1964). About some properties of the effective transparency coefficient. Invest. Atmospheric Phys. No. 5. 89. Yaroslavtzev, I. N. (1939). On the calculation of the atmospheric turbidity factor. Meteorol. Hydrol. Nos. 7 and 8. 90. Kastrov, V. G. (1928). On the basic actinometric formula. Meteorol. Bull. No. 7. 91. Savinov, S. I. (1929). Apropos of the article by Kastrov 'On the basic actinometric formula.' Meteorol. Bull. No. 7 . 92. Korsak, R. S . (1939). On the influence of fluctuations in atmospheric transparency on the accuracy of the theoretical cadastre of direct solar radiation. Meteorol. Hydrol. No. 3. 93. Yaroslavtzev, I. N. (1932). To the problem of method for reduction of atmospheric transparency coefficients to unit mass. Bull. Perm. Comm. Actinometric (Leningrad) No. 3 (23). 94. Makhotkin, L. G . (1951). Variation of the intensity of nonmonochromatic radiation in a limited interval. Trans. Main Geophys. Obs. No. 26 (88.) 95. Makhotkin, L. G . (1957). Direct radiation and atmospheric transparency. Proc. Acad. Sci. USSR, Ser. Geophys. No. 5. 96. Makhotkin, L. G . (1959). The results of study of direct solar radiation variations. Trans. Main Geophys. Obs. No. 80. 96a. Makhotkin, L. G . (1965). Normal and anomalous variations of atmospheric transparency. Trans. Main Geophys. Obs. No. 169. 97. Hinzpeter, H. (1950). ifber Triibungsbestimmungen in Potsdam in dem Jahren 1946 und 1947. Meteorol. 4, No. 112. 98. Sivkov, S. I. (1940). Some conclusions from actinometric observations at Feodosia and Karadag. Meteorol. Hydrol. No. 10. 99. Makhotkin, L. G . (1960). Atmospheric transparency in Arctic and Antartic. Trans. Main Geophys. Obs. No. 100. 100. Zvereva, S. V. (1961). Atmospheric transparency in Arctic. Trans. Inst. Arct. Antarct. 229. 101. Subbotina, Z. Y. (1960). The transparency coefficient in Antarctic. Trans. Main Geophys. Obs. No. 115. 102. Belinsky, V. A. (1947). An experiment in measuring the radiation balance components in the free atmosphere. Trans. Centr. Aerol. Obs. No. 2. 103. Faraponova, G . P. (1959). Measurements of solar light attenuation in the free atmosphere. Trans. Centr. Aerol. Obs. No. 32. 104. Krug-Pielsticker, U. (1949). Messungen der Sonnenstrahlung bei Flugzeugaufstiegen bis 9 km Hohe. Ber. Deut. Wetterdienstes. No. 8. 105. Kalitin, N. N. (1947). On the attenuation of solar radiation by water vapor and aerosols. Meteorol. Hydrol., Inform. Bull. No. 1. 106. Mamontova, L. I., and Khromov, S. P. (1933). Turbidity factors in different types of tropospheric air masses over Moscow. Meteorol. Z. No. 1. 107. Poliakova, M. N., Sivkov, S. I. and Ternovskaya, K. V. (1935). Actinometric characteristics of tropospheric air masses as observed at Slutzk and Kursk. Geophys. (Moscow) 5, No. 1.

360

Direct Solar Radiation

108. Poliakova, M. N., and Sivkov, S. I. (1935). Variation of atmospheric turbidity associated with the dynamics of air masses. J. Geophys. (Moscow) 5,No. 4. 109. Makarevsky, N. I. (1932). Transmission of direct solar radiation by clouds of the upper layer. J. Geophys. (Moscow)2, No. 1. 110. Kalitin, N. N. (1947). Cloudiness and radiation. Nature No. 3. 111. Cheltzov, N. I. (1952). Study of the reflection, transmission and absorption of radiation by clouds of certain forms. Trans. Centr. Aerol. Obs. No. 8. 112. Neuburger, M. (1949). Reflection, absorption and transmission of insolation by stratus clouds. J. Meteorol. 6, No. 2. 113. Fritz, S., and McDonald, J. H. (1951). Measurement of absorption of solar radiation by clouds. Bull. Am. Meteorol. SOC.32,No. 6. 114. Hewson, E., and Longley, H., (1944). “Meteorology, Theoretical and Applied.” 115. Savinov, S. I. (1931). Relation between cloudiness, sunshine duration and totals of direct and diffuse radiation. Meteorol, Bull. No. 1. 116. Savinov, S. I. (1933). Formulae expressing direct and diffuse radiation in dependence upon the degree of cloudiness. Meteorol. Bull. Nos. 5 and 6. 117. Kopylov, N. M. (1949). Approximate calculations of solar radiation totals. Trans. Main Geophys. Obs. No. 14 (76). 118. Ukraintzev, V. N., and Shepelevsky, A. (1939). On the calculation of direct solar radiation totals from cloudiness. Meteorol. Hydrol. No. 1._ 119. Galperin, B. M. (1949). The radiation balance of the Lower Volga region in the warm period. Trans. Main Geophys. Obs. No. 18 (80). 120. Ukraintzev, V. N. (1939). Approximate calculation of direct and diffuse solar radiation totals. Meteorol. Hydrol. No. 6. 121. Hartmann, W. (1960). Sonnenscheindauer und Bewolkung. Meteorol. Rundschau 13, No. 2. 122. Grunow, I. (1958). ifber die Beziehung zwischen Sonnenscheindauer and Bewolkung. Meteorol. Rundschau 11, No. 4. 123. Sivkov, S. I. (1964). On the calculation of the possible and relative duration of sunshine. Trans. Main Geophys. Obs. No. 160. 124. Holcke, T. (1962). Sonnenscheindauer und Bewolkung. Meteorol. Rundschau, 15, No. 3. 125. Steinhauser, F. (1954). uber die Beziehungen zwischen Sonnenscheinregistrierungen

und Bewolkungsschatzungen und ihre Verwertungsmoglichkeit fur die Berechnung der Sonnenscheindauer aus Bewolkungsbeobachtungen. Wetter Leben 6, NOS. 7 and 9. 126. Nebolsin, S. I. (1949). Climatic outline of the Moscow region. Trans. Centr. Inst. Weather Forecast No. 10 (37). 127. Lileev, M. V. (1947). Radiation characteristics of Sverdlovsk (1940). Trans. Main Geophys. Obs. No. 1 (63). 128. Yaroslavtzev, I. N. (1947). The tension of direct solar radiation in Central Asia. GUGMS, Trans. Local Obs. No. 1. 129. Averkiev, M. S. (1933). The influence of Moscow upon the value of direct solar radiation. Meteorol. Bull. Nos. 8 and 9. 130. Averkiev, M. S. (1947). The radiaction regime in Moscow. Bull. Moscow Univ. No. 8. 131. Shubtzova, V. G. (1949). On the radiation regime of Moscow. Meteorol. Hydrol., Inform. Bull. No. 3.

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132. Kalitin, N. N. (1947). Materials to the investigation of the annual radiative heat income at Pavlovsk. Trans. Main Geophys. Obs. NO. 5 (67). 133. Kalitin, N. N. (1940). Maximum observed totals of radiative heat and their comparison with theoretical data. Rept. Acad. Sci. USSR 30, No. 1. 134. Kalitin, N. N. (1946). Variation in monthly heat totals of solar radiation. Meteorol. Hydrol. Inform. Bull. No. 1. 135. Beletzky, F. A. (1947). Variations of monthly heat totals of direct solar radiation in Odessa. Meteorol. Hydrol., Inform. Bull. NO. 1. 136. Gorlenko, S. M. (1933). To the problem of the stability of the real solar cadastre. Bull. Perm. Comm. Actinometr. (Leningrad) No. 1 (24). 137. Gorlenko, S. M. (1940). The actinometric cadastre of Alma-Ata according to data of 1935-38. Meteorol. Hydrol. NO. 9. 138. Tarnizhevsky, B. V. (1959). 00the accuracy of determination of the mean monthly and annual radiation totals. Trans. Main Geophys. Obs. No. 96. 139. Pivovarova, Z. I. (1963). Direct solar radiation on the USSR territory. Trans. Main Geophys. Obs. No. 139. 140. Kalitin, N. N. (1947). “ The Sun’s Rays.” Acad. Sci. U.S.S.R., Moscow. 141. Yaroslavtzev, I. N. (1948). Heat totals from direct solar radiation a t Tashkent. Trans. Uzbek Geograph. Soc. 2. 142. Berland, T. G. (1949). The radiation and heat balances of continental surfaces without

the tropical latitudes in the northern hemisphere. Trans. Main Geophys. Obs. No. 18 (80). 143. Berland, T. G., and Danilchenko, V. Y.(1961). “The continental Distribution of

Solar Radiation.” Gidrometeoizdat, Leningrad. 144. Rusin, N. P. (1961). “The Meteorological and Radiation Regime in Antarctic.”

Gidrometeoizdat, Leningrad. 145. Smoliakov, P. T. (1929). Study of insolation of terrestrial surfaces. J . Geophys. Meteorol. 6, No. 4. 146. Gordov, A. N. (1938). Calculation of direct solar radiation on different oriented slant surfaces for the latitude 42O. Muter. Agroclimatol. Subtropics USSR, NO. 2. 147. Weinberg, V. B. (1951). “Natural Illumination of Schools.” Building and Architec-

ture Publ. House, Moscow. 148. Hamburg, P. Y. (1951). “Consideration of Solar Radiative Heat.” Building and Ar-

chitecture Publ. House, Moscow. 149. Razumov, I. K. (1934). “On the Solar Radiative Heat Through Glass Surfaces of

Buildings.” W I T 0 of Heating and Ventilation. 150. Nicolet, M., and Bossy, L. (1949). Ensoleillement et orientation en Belgique. 2. Etude pratique des surfaces orientbes. Inst. Roy. Meteorol. Belg., Mem. 32;36 (1950). 151. Averkiev, M. S. (1951). On a certain peculiarity of solar illumination of eastern and western walls of buildings. Meteorol. Hydrol. No. 5. 152. Bogel, A. (1957). Die direkte Sonnenstrahlung auf Westhange. Z. Meteorol. 11,No.3. 153. Bogdanov, P. G. (1958). The solar climate of slopes in south-western Ukraine, the coastal area. Trans. Ukr. Hydrometeorol. Inst. No. 13. 154. Dogniaux, R., (1954). Ensoleillement et orientation en Belgique. V. Etude de I’Cclairement lurnineux naturel. Inst. Roy. Meteorol. Belg. B 12. 155. Zakharova, A. F. (1959). The radiation regime of northern and southern slopes in

dependence upon geographical latitude. Proc. Leningrad Univ., Ser. Geograph. No. 13.

362

Direct Solar Radiation

156. Kartzivadze, A. I. (1959). To the problem of determination of the angle of incidence of the sun’s rays on a slant surface. Bull. Acad. Sci.Geogian SSR 22, No. 1. 157. Riabova, E. P. (1958). Some peculiarities of irradiation of ridged soil surfaces by direct solar radiation. Meteorol. Hydrol. No. 11. 158. Sato, T. (1953). On the problem of the mathematical insolation. J. Meteorol. SOC. Japan 31, No. 1. 159. Philipps, A. L. (1962). Hourly distribution of solar energy availability on surfaces with various orientation. Carribbean J. Sci. 2, No. 2. 160. Mamikonova, S. V. (1956). Insolation of horizontal and vertical surfaces in Kharkov. Meteorol. Hydrol. No. 6. 161. Kastrov, V. G. (1939). To V. V. Torletzkaya’s study ‘On a simplified method for cal-

162.

163. 164. 165. 166. 167.

culation of the duration of illumination by solar rays of walls of various orientation. Meteorol. Hydrol. No. 6. Torletzkaya, V. V. (1939). On a simplified method for calculation of the duration of illumination by solar rays of walls of various orientation. Meteorol. Hydrol. No. 6. Averkiev, M. S. (1939). Auxiliary graphs and tables for calculating the insolation of surface of various orientation. Trans. Moscow Hydrometeorol. Inst. No. 1. Averkiev, M. S. (1933). Insolation of a plane surface inclined to the horizon at the latitude of Nizhny Novgorod. Corky Monthly UEGMC Nos. 11-12. Grishchenko, M. N. (1945). An attempt to estimate the insolation of slopes at the southern Crimean coast. Proc. Acad. Sci. USSR, Ser. Geograph. Geophys. No. 2. Grishchenko, M. N. (1945). On the geomorphological conditions of insolation of slopes. Proc. Acad. Sci. USSR, Ser. Geograph. Geophys. No. 4. Kuzmin, P. P., and Novikova, N. K. (1949). The conditions of snow thaw on slopes. Trans. GGZ No. 16 (70).

DIFFUSE RADIATION OF THE ATMOSPHERE

In the scattering of radiant energy in the atmosphere, the shortwave radiation reaching the earth's surface is not only direct solar but also scattered (diffuse) radiation. Chapter 4 gave the fundamentals of light scattering in the atmosphere. Now we turn to characterization of the observed regularities of the incoming diffuse radiation. It should be noted that the results of measurements of diffuse radiation given in this chapter were in many cases obtained without making corrections for the spectral sensitivity of pyranometers and zonal distribution of diffuse radiation (see Chapter 2). The measured flux values should be therefore considered approximate. 6.1. Energy Distribution in the Spectrum of Diffuse Radiation

1. General Characteristic. As was seen in Chapter 3, the intensity of scattered light significantly depends upon the scattered light wavelength, increasing with the decrease of wavelength in the majority of cases (light scattering on small particles). Therefore it is natural that the energy distribution in the spectrum of diffuse radiation must be different from the corresponding energy distribution in the direct solar radiation spectrum. Table 6.1 presents data of theoretical calculations of the diffuse radiation spectral composition for various scattering media. Spectral fluxes of diffuse radiation are expressed here in 10" cal/cm2 min. The first line of Table 6.1 gives values of spectral radiant fluxes outside the atmosphere in cal/cm2 min according to F. Linke. The next four lines characterize the energy distribution in the spectrum of radiation scattered by 1 cm2 of clean and dry air and also by water droplets of various sizes. 363

TABLE 6.1

Energy Distribution in the Solar Spectrum outside the Atmosphere (lo-' cal/cm' min) and the Spectral Composition of Diffuse Radiation (10-< cal/cm" min)

Son

2.6

11.5

21.8

31.3

35.2

36.0

54.3

62.6

59.7

54.6

48.4

42.9

35.3

27.1

21.0

I cm 8 of clean dry air

4.4

14.4

21.9

23.5

20.8

16.9

17.2

13.7

9.3

4.4

2.8

1.8

o8

O. ,

0.2

100 droplets: r=O.l#

0.05

0.27

0.44

0.54

0.51

0.45

0.52

0.46

0.36

0.25

0.136 0.09

0.046 0.027

0.015

0.14

0.78

1.62

1.78

3.52

3.78

6.41

7.65

7.35

6.67

5.48

3.04

1.36

1.0

4.6

9.2

25 droplets:

r=0.5# 5 droplets: r = 1 I'

12.8

14.6

15.1

23.2

25.6

20.6

14.4

18.6

4.29

20.0

18.0

2.11

15.8

12.4

6.1. Energy Distribution in the Spectrum of Diffuse Radiation

365

As shown by Table 6.1, the distribution of energy in the solar spectrum outside the atmosphere is characterized by the presence of a maximum in the interval 0.46 to 0.48 p. The radiation scattered by dry and clean air (Rayleigh scattering) has its maximum displaced toward shorter wavelengths and in the interval 0.34 to 0.36 p and a second maximum in the range 0.42 to 0.44 p. The diffuse radiation spectral composition is found to be approximately the same (but with a more pronounced maximum in the interval 0.42 to 0.44 p ) also in the case when each cubic centimeter contains 100 water droplets of 0.1-,M radius. A totally different picture is observed with the increase in the size of scatterers. When sizes increase, the diffuse radiation intensity increases noticeably and the energy distribution maximum in the spectrum of diffuse radiation displaces toward longer wavelengths. If the number of droplets is 25 per 1 cm3 and their radius is 0.5 p, the maximum of spectral energy distribution coincides with the corresponding maximum for direct solar radiation outside the atmosphere. At a still more marked accretion of droplets up to 1 p, a secondary maximum of diffuse radiation appears in the interval 0.70 to 0.72 p (a certain “reddening” of diffuse radiation is taking place). From above data of theoretical calculations, it can be concluded that the diffuse radiation of a cloudless sky strongly differs in spectral composition from the direct solar radiation, being far more abundant in shortwave radiation. Just this fact, as has already been mentioned, accounts for the blue color of the sky. As to the spectral composition of an overcast sky, it follows from the consideration of Table 6.1 that in this case the difference between the direct and diffuse radiations must be less than in the event of a clear sky. Although the spectral investigation of diffuse sky radiation has a long history, the available observational data are as yet rather limited. Figure 6.1 presents the curves of diffuse radiation spectral distribution in a clear sky at zenith (curve 1) and at the point of the sky that has minimum luminosity (approximately at an angular distance 90’ from the sun, in the sun’s vertical; curve 2). These data were obtained by Lenz [ l ] with the help of a quartz-optic spectrophotometer at a solar height h, = 41’ on September 13, 1960. The turbidity factor value at the moment of observation, T = 4.5, corresponds to the conditions of a fairly turbid atmosphere. Even at a comparatively low spectral resolving power, the dependence of diffuse radiation intensity upon wavelength is nonmonotonic. A great 2. Observational Results.

366

Diffuse Radiation of the Atmosphere

number of extremes on the curves of Fig. 6.1 is caused, first of all, by the influence of Fraunhofer absorption lines in the solar spectrum outside the atmosphere and of telluric lines belonging primarily to water vapor and

XtC

FIG. 6.1 Spectral distribution of diruse radiation with clear skies at the zenith ( I ) and at a point where the sky luminance is minimal (2).

oxygen. Table 6.2 gives values of wavelength corresponding to the mentioned absorption lines (the last five wavelengths refer to the lines of oxygen and water vapor). TABLE 6.2 Wavelengths of Absorption Lines in the Difuse Radiation Spectrum ( m p ) 328 344 382.5 419 452 517 635 814

332 352 393 423 460 543 656 935

334 359 397 43 1 470 560 687

336.5 361

404 440.5 486 573 718

341.5 373.5 409 447 500 589 762

6.1. Energy Distribution in the Spectrum of Diffuse Radiation

367

From Fig. 6.1 we see that the maxima of diffuse radiation spectral intensity at zenith and at the point of the minimum intensity fall in the given case on the wavelengths 451 and 401 mp, respectively. In Fig. 6.2 are given measurement results of Lenz [ l ] for various points of the solar almucantharat (h, = 40') at angular distances from the sun p = 20°, 50°, 90°, and 180°, obtained at T = 4.5 on Sept. 13, 1960. The maximum of diffuse radiation intensity for all points in seen to fall on the wavelength L = 450 mp. The intensity of diffuse radiation rapidly diminishes with the increase of the azimuth of the considered point relative to the sun. This appears to indicate a considerable elongation of the scattering function over the entire investigated wavelength range.

XI1

FIG. 6.2 Spectral distribution of difuse radiation at diferent points of the solar almucantharat whose azimuth values relative to the sun are given in this figure.

One of the characteristic peculiarities of energy distribution in the diffuse radiation spectrum is a presence of a deep minimum near 430 mp, which can be clearly seen in Figs. 6.1 and 6.2. According to observations of Boiko and Kazachevsky [2], the position of this minimum displaces, depending on the conditions of observation. At the appearance of haze the intensity minimum doubles. The physical nature of the decrease in diffuse radiation intensity near 430 m p remains unknown.

368

Diffuse Radiation of the Atmosphere

The above data have permitted characterization of energy distribution in the spectrum of diffuse radiation from various sections of the sky. For practical reasons the most interesting is the investigation of the spectral composition of diffuse radiation incident on a horizontal surface from the entire celestial hemisphere. Investigations of this kind were first performed by Tikhov [3,4], Krinov and Sharonov [5-81. Contemporary methods for similar measurements have been described in Chapter 2 and the corresponding measurement results will be considered in the next chapter when the global radiation spectral composition will be discussed. 6.2. Angular Distribution of Diffuse Radiation Intensity

Having given an outline of the basic diffuse radiation spectral properties, let us now pass on to characterization of the angular structure of the diffuse radiation field. The qualitative peculiarities of the diffuse radiation angular distribution in a cloudless sky are determined first of all by the influence of two factors: the form of the scattering function and the atmospheric optical thickness in a given direction. Because of the “forward” elongation of the scattering function, an intensity maximum in the vicinity of the sun is always observed. The increase of the optical thickness of the atmosphere at the increasing zenith angle results in the rise of the diffuse radiation intensity horizonward. It must be stressed, however, that the described situation presents an extremely simplified scheme. In reality, the factors whose bearing determines the regularities of the diffuse radiation intensity angular distribution are numerous and interacting in the process of multiple light scattering. The field of diffuse radiation becomes especially complex and nonhomogeneous in the presence of partial cloudiness. Figure 6.3 depicts the intensity distribution of diffuse radiation at a cloudless sky (h, = 39’) according to observations of Kudriavtzeva et al. [9] at Karadag. In the given case the measurements were conducted by means of radiation receivers of two kinds: the Yanishevsky pyranometer and a selenium cell (the angular diameter of the inlet in the tube limiting the instrument’s solid angle was 10’). The zenith radiation intensity was taken for unity. We can thus compare the principles of angular distribution of the energetical (integral) and luminous diffuse radiation intensity. These results give a spectacular characteristic to the main peculiarities in the angular structure of the diffuse radiation field: increase of intensity in the circumsolar zone and toward the horizon, symmetry of angular distribution

6.2. Angular Distribution of Diffuse Radiation Intensity

369

relative to the solar vertical, and minimum intensity in the sun’s vertical at 90’ angular distance from the sun.

FIG. 6.3 Sky distribution of the relative energetical and luminous intensity of diffuse radiation as observed on June 27, 1953 (ha = 39’).

As observed by Yaroslavtzev [lo], this angular distance d varies within fairly wide limits (60 to lOS0), depending upon solar height and radiation wavelength (atmospheric transparency). With the increase of solar altitude, the value d diminishes, slowly as the wavelength shortens. The increase in wavelength raises the value A . Since to the increase of wavelength corresponds (absorption bands not considered) the increase of atmospheric transparency, the latter points out a direct dependence between atmospheric transparency and the angular distance of the region of the radiation intensity minimum from the sun. However, this conclusion contradicts the results of theoretical calculations. From Fig. 6.3 it is seen that although the qualitative peculiarities of the angular distributions of the energetical and luminous diffuse radiation coincide, their quantitative correspondence is not observed. At some points of the sky the relative radiation intensity values differ by several times.

370

Diffuse Radiation of the Atmosphere

Observations show that these differences get less with the increasing solar height; for example, they are insignificant a t h, = 64'. A satisfactory coincidence between the distributions of the energetical and luminous intensity of diffuse radiation is also observed at a solid (or almost solid) cloudiness. To illustrate this conclusion, Fig. 6.4 shows the distribution of energetical and luminous radiation intensity at Karadag with the solar altitude h, = 48' and almost complete (Sc, 9/10) strato-

FIG. 6.4 Sky distribution of the relative energefical and luminous intensity of diffuse radiation from observations on June 14, I953 (ha = 4 8 O , solid cloudiness).

cumulus cloudiness [9]. In this case (transparent cloudiness) the qualitative features of the angular distribution of diffuse radiation remain almost the same as in a cloudless sky. Observations show, though, that at a dense nontrasparent cloudiness, the azimuthal dependence of diffuse radiation intensity is not marked enough and a somewhat monotonous increase of the intensity from the horizon zenithward is observed. A comparative simplicity of the angular distribution of the diffuse radiation in a cloudless sky enables us to inquire into its analytical approximation. E. V. Piaskowska-Fesenkova [Chapter 4, Ref, 391 has shown that

6.2. Angular Distribution of Diffuse Radiation Intensity

371

(the sky sections near the horizon excluded) Z(2) may be presented with 5 to 6 percent accuracy as the product of the function of the azimuth at the sighted point, c(y) and p(8, lo)where 8, C0 are the zenith distances of the point of the sky and the sun: (6.1)

= c(y)p(e, 50)

This simple correlation has enabled E. V. Piaskowska-Fesenkova to solve the problem of determination of the angular distribution of diffuse radiation intensity in a cloudless sky on the basis of data for intensity distribution at only five or six points of the celestial sphere. Numerous theoretical calculations of the diffuse radiation field in a clear sky, performed at the Institute for Atmospheric Physics of the Academy of Sciences of U.S.S.R., have enabled detailed analysis of the dependence of the diffuse radiation intensity on the basic factors. This analysis has confirmed the conclusions deduced from the observed regularities mentioned above. It has been shown by the investigations of Atroshenko, Glazova, Kogan, Koronatov, Kuznetzov, Malkevich, and Feugelson [Chapter 4, Ref. 651 that the most important features of the angular distribution of diffuse radiation intensity are already apparent in the theory of single scattering. In particular, at 8 = CO, the intensity of diffuse radiation at ground level, Z(2)(0, 8, y), equals Z(2)(0, 8, y ) = z*e-T*sec50 sec C0

where

(6.2)

8 =zenith distance of the considered point of the sky to = solar zenith distance z* = optical thickness of the atmosphere in the vertical direction

For instance, it can be seen from (6.2) that an increasing t* results in a maximum of diffuse radiation intensity at a certain z* dependent of solar zenith distance. According to the above investigations, maximum intensity is observed at the following points: For C0

=

{

75O

at

t*

{

0.26 = 0.5

0.87

Piaskowska-Fesenkova [Chapter 4, Ref. 391 has given a detailed treatment to a similar temporal variation of diffuse radiation intensity deter-

3 72

Diffuse Radiation of the Atmosphere

mined by the product ecT* b sec To (it is obvious that at an increase of = sec To , the intensity of diffuse radiation at the given point of the sun's parallel of altitude must have the maximum a t a definite m). The azimuthal dependence of diffuse radiation intensity is determined by the effect of multiple scattering. The latter also greatly influences the intensity variation in dependence upon zenith distance. Calculations show that the intensity of diffuse radiation increases toward the horizon at small z* (high transparency) only. At large t* (low transparency) near the horizon, starting from a certain To value, the decrease in radiation intensity observed is accounted for by a strong attenuation of solar radiation at large t* and Co. It has been mentioned above that the form of the scattering function is an important factor of the angular distribution of radiant intensity. Calculations of the Institute for Atmospheric Physics show that its effect is especially felt at small To. At large solar distances, however, it is smoothed because of the increasing role of the albedo of the underlying surface A on the character of the diffuse radiation angular distribution, and has shown to be generally insignificant. For illustration, Table 6.3 presents data of the Institute for Atmopsheric Physics.

m

TABLE 6.3 Influence of Albedo on the Intensity of Diffuse Radiation. After Atroshenko et al. [Chapter 4, Ref: 651

A

0

0.2

0.6

Co

Co

Number of the Function

e=o

e = 750

e = 00

e = 750

VII VIII

0.6854 0.7822

0.8730 0.9692

0.1155 0.1094

0.8117

VII

0.7295 0.8246

1.0097 1.1022

0.1222 0.1157

0.8323

VIII

VII VIII

0.8354 0.9243

1.3178

0.1381

1.4144

0.1305

0.8819 1.1429

=

30'

=

75O

1 .0768

1 .0966

The considered results are obtained for the two scattering functions VII and VIII. Optical thickness z* = 0.8 and y = 0 (sun's vertical). In the given examples the influence of albedo on diffuse radiation intensity is found to be essential only at large zenith angles of the sun and the

373

6.2. Angular Distribution of Diffuse Radiation Intensity

sighted point, although this is not a typical situation. For instance, in the sun's countervertical (y = 180') at the same initial parameters of computation, = 30° and is about the contribution of reflection is maximal at zenith at 10 to 13 percent This is just the position that is most characteristic (the role of reflection diminishes with the increase of solar zenith distance). For some specific points of the celestial vault and certain atmospheric transparency conditions, the effect of albedo can be exceptionally great. This is shown in Table 6.4, which gives ratios of diffuse radiation intensity at A = 0.8 and A = 0. TABLE 6.4 Values of the Coefficient k = [Z(2~(r)]g=o.8/[Z(2)(r)]g=o. After Atroshenko et al. [Chapter 4, Ref. 651

Ca t*

I

=

Ca

30°

e=oo

e = 15'

e = 15'

y=O'

y = oo

y = 180'

0.2

1.1

1.2

0.60

1.3

2.0

1

I

y=Oo

e = 15' y=Oo

y = 180'

1.3

1.1

1 .o

1.1

3.8

1.3

1.1

1.8

I

e =oo

= 15'

e = 750

It is seen, for example, at z* = 0.6, = 30°, 0 = 7 5 O , and y = 180' that we have k = 3.8; that is, taking account of the reflection changes the intensity of diffuse radiation by almost four times. Averkiev [ I l l has shown that the albedo effect may also be notable in the case of flux of diffuse radiation. Especially interesting is the angular distribution of diffuse radiation intensity in the sun's immediate proximity. Linke and Ulmitz [12] have tabulated (Table 6.5) values of the ratio of diffuse radiation from individual circular zones of the heavenly arch to direct solar radiation at mean atmospheric transparency conditions. The corresponding observations were performed by means of the socalled differential actinometer, which consists of two electric actinometers contacted so that at equal sighting angles of the actinometers the deviation of the connected galvanometer is zero. By changing the sighting angle of one of the actinometers, it is possible to measure the diffuse radiation from various circular zones in the vicinity of the sun. As seen from Table 6.5, the diffuse radiation of the above circumsolar ring zones of the angular halfwidth of somewhat less than 2' is about 1 percent of the direct solar (@

374

Diffuse Radiation of the Atmosphere

TABLE 6.5 Relative Intensity of the Circumsolar Radiation. After Linke and Ulmitz [12]

Zone

I I1 111 IV

Zonal Area, in Quadratic deg

Region of Angular Distances from the Solar Disk Center

17.49 31.92 44.71 56.33

Zonal Half-width

DIS, %

1'47' 1'55' 1'55' 1'50'

0.99 0.71 0.70 1.12

27.5'-4'0.25' 1'43.1'-5'25.6' 3'3.9'-6'50.6' 4'34.4'-8'15'

radiation. Since the area of the zones under consideration increases as they are get farther from the sun, and the ratio DIS does not vary much, it becomes clear that the intensity of circumsolar radiation decreases as it travels from the sun outward. Special investigations of the principles of such a decrease have shown that, in the first approximation, this decrease may be considered exponential. Figure 6.5 presents the results of measurements carried out by Stranz [13], which illustrate the above conclusion. The curves of this figure show the variation of the circumsolar diffuse radiation intensity as it leaves the sun at h, = 18O, and also shows different atmospheric transparency conditions that are characterized by the turbidity factor T values. The value of the circumsolar diffuse radiation intensity at the angular distance from the sun of q~ = 2' is taken to be unity here. As seen from Fig. 6.5, a very

80-

a e - 60-

t

5

Q

a

40-

... '+.

-1

--2

4

I 0"

FIG. 6.5

---___ .. .............--..-

= .-'. ........

20-

2"

I

4" 6" 8" 100 120 ANGLE OF SCATTERING CP

3

14"

Variation of the intensity of the circumsolar diffuse radiation (%) on the angular distance from sun (ha = 18').

(1) T = 7.6; (2) T = 5.2; (3) T = 3.3; (4) T = 2.8.

6.2. Angular Distribution of Diffuse Radiation Intensity

375

rapid decrease in radiant intensity is observed in the immediate proximity of the sun. At long angular distances from the sun (exceeding 6 to So) this decrease slows down. Important in affecting the intensity variation are the atmospheric transparency conditions. In the conditions of a very turbid atmosphere ( T = 7 . 6 ) the decrease of diffuse radiation intensity in dependence on the proximity to the sun is much smoother than in the case of a better transparency (T = 2.8). Measurement of the absolute values of the intensity of circumsolar diffuse radiation shows that they are also considerably dependent on atmospheric transparency conditions. According to Linke and Ulmitz [12], in the first approximation, the intensity of circumsolar diffuse radiation increases proportionally to the product of the turbidity factor and an atmospheric mass m corresponding to the given height of the sun. The above results relate to the conditions of a cloudless sky. It is natural that with a presence of cloudiness in the limits of the circumsolar zone the intensity of diffuse radiation must increase significantly. The presence of cloudiness in the circumsolar zone provokes a notable increase in diffuse radiation, especially in the case of cirrocumulus and altocumulus clouds. In this case the diffuse radiation can increase up to 15 percent of the direct solar radiation. The above dependence is characterized by the case of cumulus cloudiness as presented by N. N. Kalitin [Chapter 3, Ref. 1021 in Table 6.6. The considered observations were conducted at a gradual occultation of the circumsolar zone with ragged but dense enough shreds of cumulus clouds. TABLE 6.6 Influence of CIoudiness on the Value of Circumsolar Diffuse Radiation. After Kalitin [Chapter 4, Ref. 1021

Cloudiness in Instrument's Sighting Field

D, , cal/cm2 rnin

DrllSm, %

0.0185

1.6

0

0.0222

1.9

1 cu

0.0379

3.3

3 cu

0.0498

4.3

5 cu

0.0655

5.8

7 cu

0.0777

6.8

8 Cu

376

Diffuse Radiation of the Atmosphere

As seen from this table, the value of the circumsolar diffuse radiation was increasing approximately in proportion to the degree of cloudiness. The data show that observations over direct solar radiation in the presence of cloudiness near the sun should be performed with great care and special corrections for diffuse radiation should be introduced if there are any clouds in the actinometer’s sighting field.

6.3. Fluxes of Diflluse Radiation 1. Dependence of Difluse Radiation Flux upon Solar Height. We know from both observations and theoretical calculations that the magnitude of the considered flux in a clear sky is dependent first of all on solar height, atmospheric transparency conditions, and the albedo of the underlying surface. Let us therefore consider the effect of these factors on flux values. Figure 6.6 presents the dependence of diffuse radiation flux on a horizontal surface upon the turbidity factor T at various solar heights, as observed by Barashkova [14, 151 at Karadag. It is seen that with the increase of the turbidity factor the diffuse radiation increases considerably, the more as the sun is higher, and that at a stable atmospheric transparency the radiant flux shows a marked increase at the increase of the solar altitude, especially in a very turbid atmosphere. It should be noted that the character of the dependence of diffuse radiation on the turbidity factor may be other than that of Fig. 6.6. For instance, Ross and Avaste [16] observed an increase, not decrease, of the derivative dD/dT with the increase in the turbidity factor at Tartu (Estoina). Similar results have also been observed for a number of other points. Another important factor is the albedo of the underlying surface. If the albedo is great enough, then the radiation reflected from the underlying surface and scattered by the atmosphere in the direction of this surface may cause a significant increase in incoming diffuse radiation. It is natural that this effect will be the greatest in the presence of snow cover. This is confirmed by observations. Table 6.7 gives the results of measurement of diffuse radiation flux at various underlying surface albedos according to data by Zavodchikova and the author [17]. Table 6.7 indicates a notable increase of diffuse radiation flux with the increase of the albedo of the underlying surface. It should be mentioned, however, that the observations were not simultaneously and the quantitative relations that follow from Table 6.7, should therefore be treated as approximate.

377

6.3. Fluxes of Diffuse Radiation

T

FIG. 6.6 Dependence of diffuse radiation upon the furbidify factor for different soIar heights.

TABLE 6.7 Dependence of a Flux of Diffuse Radiation on the AIbedo of the Underlying Surface with a Clear Sky. After Kondratyev and Zavodchikova [17]

Surface

Thin grass Clay soil Snow fall two days prior to the observation Snow of day preceding observation day

Albedo, %

Diffuse Radiation Flux, cal/cm2 min

23.7 27.6 85.7 89

0.085 0.110 0.144 0.181

2. Daily and Annual Variation of Diffuse Radiation Flux in a Cloudless Sky. The flux of diffuse radiation in a clear sky has a simple daily range, with the maximum at about noon caused by the variation of the solar height during the day. This can be seen in Table 6.8 compiled by Galperin [18] from data of observations at various points (for comparison, data on global radiation are also given). In many cases the daily variation of solar radiation flux is asymmetrical with regard to noon. For example, in Vienna the afternoon values are somewhat less than those before noon, which is caused by a higher trans-

378

Diffuse Radiation of the Atmosphere TABLE 6.8 Fluxes of Global and Diffuse Radiation in a Cloudless Sky (in cal/cm2min). After Galperin [18]

Solar Height, deg Stations 5

10

Global Radiation Drifting Chelyuskin Cape Helsinki Pavlovsk Karadag Blue Hill

0.30 (0.06) 0.18 0.08 0.18 0.18 0.09 0.18

Diffuse Radiation Drifting Chelyuskin Cape Helsinki Pavlovsk Wien Pyatigorsk Karadag Nice Tashkent

0.03 0.03 0.03 0.03 0.03

15

20

25

30

40

0.33 0.31 0.30 0.29 0.32 0.31

0.47 0.43 0.42 0.41 0.45 0.44

0.61 0.56 0.55 0.55 0.56

0.74 0.64 0.66 0.69 0.69 0.69

0.86 0.96 0.93 0.93

0.11 0.08 0.07 0.08 0.09 0.10 0.12 0.10 0.08

0.13 0.09 0.08 0.09 0.10 0.11

0.15 0.10 0.09 0.10 0.12 0.12 0.15 0.12 0.09

0.10 0.11 0.14 0.14 0.17 0.13 0.10

-

-

-

0.08 0.05 0.05 0.05 0.05 0.05 0.07

0.10 0.07 0.06 0.07 0.07 0.08

-

0.09 0.06

-

0.05

-

0.55

-

-

-

parency at Vienna in the afernoon (the observations were carried out in the city itself and the higher atmospheric transparency in the afternoon hours was due to an intensive turbulent mixing, which helps to “dissolve” the smoke and dust accumulated over the town during the first half of the day). For the majority of other points, an inverse asymmetry is more characteristic, caused by the decrease of atmospheric transparency in the afternoon. The annual range of diffuse radiation shows a maximum in the summer period when the sun is highest. The character of the annual range is also determined by the peculiarities of the annual variations in atmospheric transparency and the albedo of the underlying surface.

3. The Relation between Fluxes of Direct Solar and Diffuse Radiation in a Cloudless Sky. Since the flux of direct solar radiation on a perpendicular surface, similarly to that of diffuse radiation with a clear sky, is dependent

379

63. Fluxes of DitFuse Radiation

first of all on the height of the sun and the atmospheric transparency conditions, it is interesting and practically important to study the correlation between these quantities. An example of such a correlation is given in Table 6.9, made up from observational data by Kalitin [19] at Pavlovsk. To tabulate these data, Kalitin used recordings at the Yanishevsky pyranograph and at an actinograph over a five-year term of observations. Diffuse radiation flux values were expressed in thousands of calories per square centimeter per minute. TABLE 6.9 Dependence of Valuesof Difuse Radiation Flux on a HorizontalSurface on SolarHeight and Dependence of Direct Solar Radiation Values on a Perpei.dicular Surface. After Kalitin [19]

Solar Height, deg 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 36 38

40 42

44 46 48 50

S,,, , cal/crnamin

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 35

28 50

I5

42 63 88

37 53 47 72 61 54 97 80 68 61 123 103 87 74 125 105 88 149 122 101 144 118 167 136 156 175 196

65 74 85 98 111 125 140 153 168 190 207 225 245

1.0 1.1 1.2 1.3 1.4

73 81 91 101 113 123 135 147 162 175 190 207 223 235 246 260 276

76 83 92 101 109 118 129 139 149 160 173 183 205 216

88 97 105 112 120 127 136 146 162 173

99 105 111 117 131 139 114

Using Table 6.9 it is possible to evaluate approximately the diffuse radiation flux from measurement data of direct solar radiation. As shown by Table 6.9, at a constant solar height (but with a varying atmospheric

380

Diffuse Radiation of the Atmosphere

transparency), an inverse correlation takes place between fluxes of direct solar and diffuse radiation; that is, the greater the direct solar radiation value, the less the diffuse radiation. This happens as the result of the increase (at a constant height of the sun) of the direct solar radiation flux, which means an increase in atmospheric transparency. It must be mentioned that when compiling and using similar tables one should also take into consideration the dependence of diffuse radiation flux on the albedo of the underlying surface. As has already been noted, the variation of albedo at the appearance of snow cover and at thaw greatly affects the magnitude of diffuse radiation flux. 4. Theoretical Calculations of Difuse Radiation in a Cloudless Sky. The comparatively simple principles governing the variation of diffuse radiation fluxes in a clear sky enables us to realize theoretical calculations of these quantities. Numerous calculations of this kind have been performed at the Institute for Atmospheric Physics on the basis of the method of successive approximations (see Sec. 4.6). Makhotkin [20, 211 has shown that at the albedo A = 0, approximate estimates of diffuse radiation fluxes can be obtained from the following formula, derived after integration of (4.82) : socos e, - So cos 0, e-70 % D= 1 ~~z~sec Bo

+

Barashkova [I51 has calculated diffuse radiation flux values from this formula at the solar constant So = 1.88 cal/cm2 min, zo = 0.3, and for various values (Table 6. lo). Barashkova explains the systematic exaggeration of the calculated values by the neglect of the absorption effect in the TABLE 6.10 Measured and Calculated Values of Diffuse Radiation Flux (cal/cmamin). After Barashkova [15]

Observation Measurement

Calculated Values 10

20

30

40

50

60

0.25 0.34 0.50

0.13 0.14 0.12

0.21 0.23 0.18

0.26 0.26 0.21

-

-

0.28 0.24

0.29 0.24

0.33 0.30 0.24

Avg.

3.07

0.12

0.13

0.15

0.16

6.3. Fluxes of Diffuse Radiation

381

deduction of the formula (6.3). As to the qualitative aspects of the variation of diffuse radiation fiux, this formula is quite satisfactory for their description. Regarding the absorption effect, the theory shows that the diffuse radiation flux depends on the ratio of the scattering coefficient to the sum of the coefficients of absorption and scattering. This accounts, in particular, for the above discussed differences in the dependence of radiant flux on the turbidity factor, as shown by observations at different points. At the increase of the role of absorption (increasing factor of humid turbidity), but at a constant residual turbidity, the flux of diffuse radiation decreases, and vice versa. The slowing down of the increase in the flux of diffuse radiation at Karadag, with an increasing turbidity factor (Fig. 6.6), may thus be explained by the fact that at this point the increase of T is accompanied by an increase of the ratio WIT. On the other hand, whenever the increase in turbidity is determined primarily by the increase of radiation attenuation due to scattering, an increase of dDldT must be observed with the increase of T. Besides the approximate theoretical formulas for calculation of fluxes and totals of diffuse radiation, several empirical formulas have been suggested. For instance, Kastrov [22] has obtained a relation

D = cm-b

(6.4)

where c and b are empirical constants. Sivkov [23] has proposed substituting the following formula into (6.4):

D=

~2/sinho

(6.5)

However, all the foregoing formulas are very limited in application. De Bary [24] and Bullrich el al. [25] have performed interesting estimates of contributions to diffuse radiation by multiple molecular scattering, primary aerosol scattering, and aerosol scattering of higher orders at different condition of observation. Their calcualtions show that the influence of aerosol multiple scattering is largely caused by tiny aerosol particles. The importance of this observation becomes evident if we view it from the standpoint of the role of aerosol multiple scattering estimated by making use of calculations related to molecular scattering. 5. Difuse Radiation of a Cloudy Sky. As was shown in Chapter 4, the

appearance of cloudiness brings about a marked increase of diffuse radiation flux. Clouds containing a great number of coarse scatterers in the form

382

Diffuse Radiation of the Atmosphere

of water droplets or ice crystals are powerful centers of radiation scattering. The intensive scattering and considerable absorption of solar radiation by clouds are just the reason for a certain small amount of radiation being transmitted through the upper layer (middle layer at times) clouds. The lower layer cloudiness, as well as the cloudiness of the middle layer (in the majority of cases), is completely opaque to direct solar radiation. Just how much diffuse radiation flux increases at the appearance of cloudiness can be seen in Fig. 6.7, which presents a pyranograph record from observations by Kalitin [Chapter 3, Ref. 1021 at Pavlovsk on two successive days, April 11 and 12, 1934. The first day was cloudless, and the second

: "E

1

"

z + 0 9 n a

n

I

HOUR

FIG. 6.7 Fluxes of difuse radiation for clear and cloudy skies.

was entirely overcast with altostratus clouds (As). It is seen that in the noon hours the diffuse radiation flux on April 12 was 7.6 times that on April 11. It is natural that the magnitude of diffuse radiation flux must be essentially dependent on the amount and form of cloudiness. As in the case of a clear sky, the solar height and the albedo of the underlying surface have important effects on the radiant flux. A detailed investigation of the dependence of diffuse radiation flux upon the degree of cloudiness and solar height for clouds of various forms has been carried out by Gushchina (see Chapter 3, Ref. 1121) on the basis of observational data at Pavlovsk. Even though the obtained values are insufficiently accurate (due to faulty methods of observation), the following data enable a reliable analysis of the basic regularities of the variation in diffuse radiation flux with respect to the degree of cloudiness, solar height, and cloud forms. The curves obtained by Gushchina (Fig. 6.8) characterize the dependence of diffuse radiation flux on the degree of cloudiness and solar height for cirrus (Ci), cirrocumulus (Cc), altocumulus (Ac), and cumulonimbus (Cb) clouds. The considered curves are the isolines of the diffuse radiation flux (the flux values in cal/cm2 min corresponding to different isolines are given on the right side of the figure).

383

6.3. Fluxes of Diffuse Radiation

By analyzing Fig. 6.8 it is possible to draw the following conclusions: In the case of a cloudy sky, as well as in the absence of clouds, the flux of diffuse radiation increases with the increase of solar height, this increase being the greater the higher the degree of cloudiness (in all the considered cases the isopleths get nearer in the region of high values of degree of cloudiness). The regularities of variation of radiant flux in dependence on the degree of cloudiness at a constant solar height differ for different clouds. In the presence of cirrus or altocumulus clouds, the flux increases with the increase of cloudiness. b

col/cm2 min

0270

0.250

0.150

0.050 0 1 2 3 4 5 6 7 8 9 1 0 FORCE ,, CLOUDINESS ml/cm2mln 0250

0.150

0.050

0 1 2 3 4 5 6 7 8 9 1 0 FORCE CLOUDINESS

FIG. 6.8 Efect of cloudiness on fluxes of diffuse radiation.

But cumulonimbus clouds have a different effect. In this case the increase of radiant flux with the increase of cloudiness takes place at solar heights h, < 40' as long as the degree of cloudiness does not exceed force 5 to 6. At a further increase of cloudiness the diffuse radiation flux starts falling. It is easily understandable that the cause for this is the fact that at too high solar altitudes, the increase in the amount of cumulus clouds and their power leads to a very significant attenuation of solar radiation and the resulting decrease in the flux of diffuse radiation. A similar principle of

384

D b e Radiation of the Atmosphere

the variation of diffuse radiation flux in dependence upon the amount of dense clouds of the lower and intermediate layers was observed at other points as well. Consideration of Fig. 6.8 allows an approximation of the diffuse flux value as it increases in passage toward the denser and lower clouds. The passage from cirrus to cirrocumulus and altocumulus clouds is accompanied by an increase of the diffuse radiation flux at a constant solar height. However, at further condensation and an increasing power of clouds, a decrease of the incoming diffuse radiation takes place. This is due, as has already been mentioned, to a marked increase in attenuation (absorption, in particular) of radiation by clouds. In the conditions of variable cloudiness, the daily range of diffuse radiation is complex and irregular, whereas in the presence of a solid cloud cover (when the variation of the flux value is mainly determined by the variation of solar height) there is observed a very simple time-based variability of diffuse radiation flux, with a maximum at about noon. The annual range of diffuse radiation flux in correspondence to a definite moment of the day is largely determined by the yearly variation of solar height and the amount of cloudiness. Usually, maximum annual flux values are observed in the warmer half of the year. This can be seen, for example from data of Table 6.11, compiled by Tzutzkiridze [26] over observations at Tbilisi in 19371947. The given flux values were averaged from the values corresponding to a given moment of time over all days of observations (at different weather conditions). From the given data we see that in many cases, and especially at an uninterrupted cloud cover, the values of diffuse radiation fluxes are quite large. In individual cases they may exceed 1 cal/cm2 min. It is of interest, therefore, to consider maximum flux values observed under different conditions. Table 6.12 presents maximal values of diffuse radiation flux at various cloudiness conditions according to observations by Temnikova [27] at Pyatigorsk with the help of a Yanishevsky pyranometer (March, 1939October, 1940). In all cases of a dull sky the degree of cloudiness was of force 8 to 10. This table shows that the highest maximum flux value (0.98 cal/cm2 min) was observed at a continuous altocumulus and altostratus cloudiness. The least maximum took place in the case of cumulonimbus clouds. Values approximately similar to those of Table 6.12 were obtained by Yaroslavtzev [28] during observations at Tashkent (employing the Yanishevsky pyranometer). According to these observations (1937-1943) the highest maximum flux value was recorded with solid altocumulus clouds, and reached 0.80 cal/cm2 min.

TABLE 6.11 Mean Daily and Annual Variation of Diguse Radiation Flux as Observed at Tbilisi. After Tzutzkiridze [26]

Hours Months 5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

January

-

-

-

0.05

0.11

0.17

0.22

0.24

0.23

0.20

0.13

0.06

-

-

-

0.16

February

-

-

0.02

0.07

0.15

0.22

0.26

0.28

0.27

0.25

0.19

0.12

0.02

-

-

0.17

March

-

0.01

0.06

0.14

0.23

0.31

0.36

0.37

0.37

0.23

0.27

0.19

0.10

0.02

-

0.21

April

-

0.04

0.12

0.19

0.26

0.32

0.35

0.39

0.37

0.32

0.27

0.20

0.12

0.03

-

0.23

May

0.02

0.08

0.15

0.22

0.27

0.31

0.33

0.35

0.34

0.32

0.27

0.22

0.15

0.08

0.02

0.21

June

0.03

0.09

0.16

0.22

0.28

0.30

0.30

0.31

0.30

0.29

0.26

0.21

0.16

0.09

0.03

0.20

July

0.03

0.10

0.17

0.23

0.28

0.32

0.34

0.35

0.34

0.32

0.28

0.23

0.17

0.10

0.03

0.22

August

0.01

0.07

0.15

0.21

0.25

0.28

0.30

0.29

0.29

0.27

0.23

0.19

0.12

0.06

0.01

0.18

September

-

0.03

0.09

0.16

0.20

0.23

0.24

0.25

0.23

0.21

0.18

0.13

0.08

0.02

0.16

October

-

0.01

0.06

0.11

0.15

0.19

0.23

0.25

0.23

0.20

0.15

0.09

0.03

-

November

-

-

0.02

0.07

0.12

0.17

0.20

0.21

0.20

0.17

0.11

0.05

-

-

December

-

-

0.01

0.04

0.10

0.15

0.17

0.18

0.17

0.14

0.09

0.02

-

-

-

Average

0.02

0.05

0.09

0.14

0.20

0.25

0.28

0.29

0.28

0.25

0.20

0.14

0.11

0.06

0.02

0.18

0.14 0.13 0.11

386

Diffuse Radiation of the Atmosphere

TABLE 6.12 Maximum Values of Diffuse Radiation FIux according to Observations at Pyatigorsk. After Temnikova [27]

Atmospheric Mass Cloud Form

Noon 5

4

3

2

1.5

St

0.11

0.18

0.29

0.36

0.45

0.45

Cb

0.09

0.12

0.19

0.21

0.22

-

Ci, Cs

0.18

0.16

0.24

0.38

0.49

0.12

Ac, As

0.18

0.22

0.28

0.52

0.98

0.90

Fog

0.11

0.13

0.19

0.34

-

0.38

Clear

0.13

0.14

0.16

0.20

0.18

0.22

The highest flux values were observed in arctic conditions, where in a number of cases the measured flux magnitudes exceeded 1 cal/cm2 min in spite of comparatively low solar heights. This is explained, as was pointed out by Kalitin [Chapter 3, Ref. 1021, by the presence in most cases of clouds that are good radiation scatterers and by the albedo of snow-covered underlying surface that affected the incoming diffuse radiation.

6. Theoretical Calculation of Difuse Radiation of Clouds.+ The calculation of diffuse radiation of clouds presents a special interest because the presence of cloudiness greatly influences the magnitude of diffuse radiation flux. The theoretical calculation of diffuse radiation of clouds is quite complicated, especially for cumulus cloudiness which has isolated irregular formations and a considerable vertical extension. The problem looks less difficult in the case of stratus clouds, which may be presented in the form of a homogeneous horizontal surface of moderate thickness. The available theoretical calculations are based on just this approach. The most important results have been obtained by Feugelson [29], Korb and Moller [30], and Romanova [31, 321. When calculating the intensity of shortwave diffuse radiation of stratus clouds, Feugelson made use of the general equation of radiative transfer (see Sec. 1.6): E. M. Feugelson is coauthor of this section.

6.3. Fluxes of Diffise Radiation

387

where ZA(z,r ) is the shortwave, direct, and scattered radiation intensity, and aAand kAare coefficients of scattering and absorption relative to unit volume. Since the absorbing media in a cloud are water droplets and water vapor, and the scattering media are water droplets and humid air,

LA = kA,l@l+ kA,Z@Z 8A = a1,zez

+

Or,&

The total scattering function y,(z, rr, r ) must be correspondingly determined from the formula

In the latter relations kA,l and kA,zare mass absorption coefficients of water vapor and liquid water; aA,zand Y ~ are , the ~ mass coefficients of scattering and the scattering function of water droplets; aA,3 and yA,3are the same quantities for air; e3 are densities of water vapor, liquid water, and air in the cloudy medium, respectively. Taking into account the fact that the intensity of direct solar radiation is distributed in such a way that it is zero in all directions except the direction to the sun, instead of (6.6) we have

where IA,D(z,r ) is the intensity of diffuse radiation, n S i is the intensity of direct solar radiation at the upper cloud boundary, 8, is the solar zenith distance, H is the cloud layer thickness, z is a vertical coordinate reckoned from the lower cloud surface, and ro is the direction of the incidence of direct solar radiation. It has been mentioned above that the role of absorption and scattering in shortwave radiation attenuation is not the same. This can also be seen from Table 6.13, which present values of mass and volume coefficients of scattering by water droplets and of absorption by water vapor and liquid

TABLE 6.13 w

7 I e Coefficients of Absorption and Scattering for Water Droplets and Water Vapor. After Feugelson [29]

Band Denomination

AL

k,, cm2/g

l,, km-1

k,, cm-2/g

k,, cm-1

0.7004.719 0.719-0.721 0.7214.740 0.74(M. 790

0.006 0.012 0.016 0.025

0.00017 0.0034 0.00046 0.00072

0.019 0.261 0.019

0.0093 0.13 0.0093

0

0

0.8 p

0.790-0.814 0.8140.816 0.816-0.840 0.840-0.860

0.030 0.030 0.031 0.040

0.00086 0.00086 0.00089 0.0011

0.020 0.479 0.020 0

0.0098 0.23 0.0098 0

eu=

0.860-0.91 5 0.9154.935 0.9254.990 0.990-1.030

0.069 0.147 0.352

0.0020 0.0042 0.011 0.010

0.038 1.153 0.038 0

0.019 0.56 0.019 0

v

1.030-1.112 1.112-1.148 1.148-1.230 1.230-1.240

0.186 0.583 1.100 1.206

0.0053 0.017 0.031 0.034

0.030 0.107 0.030 0

0.015 0.54 0.015 0

W

1.240-1.321 1.321-1.449 1 .449-1.530

1.166 12.7 38.6

0.033 0.36 1.10

0.047 3.157 0.047

0.023 1.55 0.023

9

1.530-1.755 1.755-1.965 1.965-2.190

15.8 47.8 40.7

0.45 1.37 1.16

0.022 4.178 0.022

0.011 2.05 0.011

a

0.400

00 00

u,, cmz/g

5 1600

32

16, 39 28, 26 20, 29

B

+

!!

5 Feugelson’s data

Other authors’ data

6.3. Fluxes of Diffuse Radiation

389

water in dependence on wavelength. Feugelson computed the mass scattering coefficient on the basis of the following formula for the scattering coefficient calculated per single particle :

where a is the droplet’s radius, e = 2na/iZ, and K ( e ) is the effective cross section of scattering. Using this formula it is possible to write an equation for a mass coefficient of scattering by water droplets:

where n(a, z) is a number of droplets of a radius per unit volume at a height z. Curves of size distribution of the number of droplets in a cloud, obtained in explorations of cloud microstructure, do not relate to unit volume but to the number of droplets detected by the instrument. If, however, we assume that the same distribution is valid for each unit volume, then the value n(a, z) can be expressed as

where n(z) is a general number of droplets per unit volume, and p(a, z) is the portion of droplets of a radius according to the distribution curve derived from observations. The total number of droplets per unit volume is expressed by the cloud water content, ez(z), in the following manner:

n(z) = 4 ZG 3

ez(4 00

(6.10)

$p(a, z) da

where p = 1 g/cm3 is water density. On the basis of Eqs. (6.8) to (6.10), we have

(J&Z

=

(6.1 la)

4

-z 3

Jw 0

dp(a, z) da

(6.11b)

390

Diffuse Radiation of the Atmosphere

These formulas were used by Feugelson for the computation of a2 and 8, at e, = 0.2 (see Table 6.13). The mass scattering coefficients of humid air can be calculated on the basis of the molecular scattering theory from the formula a1.,3

32n3 3N@1*

= -(n -

1),

where n is the refraction index of humid air equal to 1.000253 at 1= 0.54p. The function of scattering on air molecules, is described by the Rayleigh formula (6.12) YL.3 = %[I cos2(r, r”

+

The scattering function of cloud droplets is characterized by a strong forward elongation. An example of such a function is given in Fig. 6.9. are negligible in comCalculations show that the values and aL,3yL,3 parison with cl,, and al,,yl,,.

FIG. 6.9 (il

+ iJ2

the scattering function of a polydispersional cloud with a mean droplet radius of 4 p at A = 0.45 p.

= y(v)

391

6.3. Fluxes of Diffuse Radiation

The values of the absorption coefficient of water vapor presented in Table 6.13, are calculated for the density of saturated water vapor at O O C . These values, as are those for k,,,, are insignificant compared with 5,,2; nevertheless they must not be neglected in solving (6.7). Owing to intensive scattering, the path of the light beam inside the cloud turns out to be sufficiently long and the small volume absorption becomes considerable. In Fig. 6.10 are given the albedos of cloud layers of various optical thicknesses (discussed later), computed by Romanova [32], in dependence upon the value ,8 = k/(k 5).

+

A

I

1

0

0. I

1

0.2

P FIG. 6.10 Dependence of doud albedo upon the value 20, tll = 2.5.

B = i/(i+ 3) for

t o=

00,

12,

In the visible spectrum the absorption, as compared with scattering on droplets, may be neglected and the transfer equation (6.7) accordingly simplified. Let us introduce an integral intensity of diffuse radiation:

where 2

=

2 p is the boundary of the diffuse radiation spectrum on the

392

Diffuse Radiation of the Atmosphere

side of long wavelengths. The transfer equation (6.7) will then be of the form

-case - _ _ 31, a2e2

-

’z

72

s

ID@, r’)y(z, r, r’) do‘

+ $ {exp[ - seco,

Jr

02ed z ] }

We now introduce the optical thickness t=

s” 02e2dz

(6.14)

0

Eq. (6.13) thus is reduced to

case‘ I D =at 4n

I ID(~,r ‘ ) y ( t ; r‘, r ) do‘ S‘

+,{exp[-(t*

-

t)sece,]}y(;;ro,r)-

I,

(6.15)

For solution of Eq. (6.15) it is necessary to give form to the scattering function. Feugelson presents the scattering function in the form of a series in Legendre polynomials : N

r ) = X cnPn[cos(r’, r)l

~(r’7

(6.16)

n=O

where 1, are constant coefficients, and Pn is Legendre’s polynomial of the nth order. Table 6.14 displays cn values for the scattering function of a droplet at e = 30. For the boundary conditions, values may be taken o f t h e intensity of diffuse radiation incident on the upper and lower cloud boundaries from outside. Supposing that the incident diffuse radiation at the boundary level is isotropic, and denoting the radiant fluxes at these levels by Fl(0) and F2(t), we write 1 7t (6.17a) I ~ ( oe), = F,(o), o I eI 1 I ~ (e)~= ~ F,(o), , where

to is

n 2-

- <e 5

the full optical thickness of the cloud layer.

(6.17b)

TABLE 6.14 c, Coefficients for (6.16). After Feugelson [29] ~~

n

1

2

3

4

5

6

7

8

9

10

cn

2.48

3.70

4.18

4.60

5.02

5.39

5.89

6.39

6.79

7.23

n cn

n Cll

11 7.47 21 8.59

12 7.75 22 8.82

13 7.85 23 8.93

14 7.96 24 9.13

15 8.07 25 9.31

16 8.08 26 9.45

17 8.19 27 9.68

18 8.28 28 9.77

19 8.33

20 8.53

29

30

10.00

10.11

9

w

2

P

% U

n

31

32

33

34

35

36

37

38

39

40

8

cn

10.27

10.42

10.55

10.66

10.85

10.90

11.16

11.26

11.44

11.62

P

n

41

42

43

44

45

46

47

48

49

50

3 a

Cn

11.64

11.68

11.61

11.33

11.05

10.38

n

51

52

53

54

55

56

Cn

4.56

n

61

Cn

23

3.44 62 0.13

2.62 63 0.03

2.43

64 0.03

2.73

2.73

9.62 57 2.72

8.46 58 2.19

7.32 59 1.41

P

5.90 60 1.05

65 0.04

w

v)

w

394

Diffuse Radiation of the Atmosphere

At the boundary conditions (6.17) and the scattering function (6.16), the solution of (6.15) may be presented in the form N

ZD(r, r ) =

C n=o

A,(t, 0) cos ny

.. .

(6.18)

where y is the azimuth of the r ray. The functions A,(r, 0) are determined from a system of integrodifferential equations, which was solved by Feugelson according to the method of successive approximations. Thanks to a special choice of zero approximation this was found possible with only a few successive approximations. Omitting cumbersome intermediate calculations, we shall give only the final expression for the intensity of diffuse radiation outgoing through the lower boundary of a cloud of moderate thickness ( H > 100 m): (6.19) where

where p = cos0 and pa = cosea, and p , a mean cosine of the angle of scattering, is a parameter depending on the form of the scattering function and the position of the sun and must be determined in the solution of the problem. In Fig. 6.11 are given p values in dependence upon pa for a case of the scattering function corresponding to e = 20; 30, The C/Sp, values are given in Table 6.15. The formula (6.20) does not take account of the fluxes of diffuse radiation incoming at the cloud boundaries (F,(O) = F,(t0)= 0). As seen from (6.20), with clouds of moderate thickness, the diffuse radiation outgoing through the lower cloud surface is independent of azimuth y and is in a linear dependence upon cos 0. The intensity of diffuse radiation is maximal zenithward (0 = 0, p = 1) and decreases toward the horizon as 0 increases. With the increase of cloud thickness the rate of the variations of luminosity from the zenith horizonward decreases. This rate gets higher with an increase of the elongation of the scattering function and the height of the sun over the horizon. The

395

6.3. Fluxes of Diffuse Radiation

PO

FIG. 6.11 Dependence of the mean angle of scattering upon solar position.

luminosity distribution in a cloudy sky is thus radically different from that in a clear sky whose angular luminosity distribution reflects qualitatively the scattering function of the sun's rays. TABLE 6.15 The Parameter CISp,. After Feugelson [29] TO

30

20

30 50 70 30

10

20

30

40

50

1.00 0.81 0.68 0.92

0.70 0.52 0.41 0.58

0.62 0.37 0.29 0.43

0.41 0.30 0.23 0.34

0.34 0.23 0.20 0.28

Figure 6.12 gives theoretical curves characterizing the directional dependence of the intensity of diffuse radiation leaving through the upper cloud surface. In the given case the intensity depends on 8 and y. Within the azimuth variation 0 5 y 5 looo, the reflected light intensity varies in a weak and fluctuating manner depending upon 8,. At y > 100' and especially at y 180' (that is, for antisolar direction) the cloud luminosity always increases toward the horizon, and does so more rapidly as the the sun approaches closer to the horizon.

-

396

Diffuse Radiation of the Atmosphere

a+zL--> 8.200 .-. --_---. L

45

90

135

180

0.6

45

0

90

135

180

FIG. 6.12 Directional distribution of the intensity of diffuse radiation on leaving through the upper cloud surface at t o= 30. (a) 5 = 30'; (b) 5 = 50'; (c) C = 70'.

I 00

300

600

I

9oD

SOLAR ZENITH DISTANCE ,r90

FIG. 6.13 Dependence of cloud albedo upon cloud amount and zenith solar distance for direct solar radiation. (1) spherical function, H = 600 m;(2)nonspherical function, H = 600 m;(3) spherical function, H = 300 m; (4) nonspherical function, H = 300 m.

6.3. Fluxes of Diffuse Radiation

397

In Fig. 6.13 is given a cloud albedo for direct solar radiation in the visible spectrum. It can be seen from this figure that at 8, > 30°, the albedo of weak stratus clouds (z,, 10) shows a notable dependence upon solar zenith distance. In the 10 to 50 range of optical thickness, a rapid increase of albedo with the increase in optical thickness is observed. At zo > 50 this dependence disappears. Figure 6.14 shows the spectral albedo of a thin (z = 6) and a thick (z = 30) cloud at two solar heights, C = 30' and 5 = 60'.

-

4

a

FIG. 6.14 Spectral albedo of cloudr.

The qualitative comparison of the considered results of computation with observational data (see the above material and data on albedo in the following chapter) finds satisfactory agreement. This comparison, however, is difficult to realize because it requires fulfillement of the conditions for a complete agreement of parameters corresponding to the real conditions and those used in calculations. The problem of such comparison is still to be solved. Feugelson, in an earlier work [33], limited the expansion of the scattering function by three terms,

y(r, r') = 1

+

c1 Pl

+

c2 P2

and took ,ii = 4, thus obtaining very simple formulas for computing al-

398

Diffuse Radiation of the Atmosphere

bedo of direct solar (As), diffuse ( A D ) and transmitted (P) radiation: A,=l--

7

2cose,+

l2 1 + ( 1

7

1

+a

-+)To

7

I

I.)+-

1-

A,=

6[1 + ( I

(6.22)

7

P= 6[1 + ( 1

(6.23)

-+)to]

These formulas give less reliable results than those previously presented, since they exaggerate A and lower P, but nevertheless they may be of use for rough calculations. For approximate calculations of monthly totals of diffuse radiation D at any cloudiness conditions, many empirical formulas have been proposed. Savinov, for example, found the following relation :

CD

=

C D(1 - n )

+ knl C Q

(6.24)

P

0

where X o D is the incoming diffuse radiation in a cloudless sky, and the index of the sky’s dullness* is given by n1 =

1-S,+n 2

The first term in (6.24) determines the amount of diffuse radiation from the cloudless part of the sky; the second characterizes the income of radiation scattered by clouds. According to the considered formula, we have in the two finite cases: (a) nl = 0 (clear sky); E o D = C,D. (b) nl = 1 (fully overcast sky); C D

= k&Q.

* n is the degree of cloudiness (in portions of unity), &Q is a possible total of global radiation.

6.3. Fluxes of Diffuse Radiation

399

In accordance with (6.24), even in the case of a overcast sky a certain quantity of the diffuse radiation of the clear sky from the atmosphere above the clouds penetrates through the cloud cover and attains the earth’s surface in the form of a multiply scattered radiation of the clear sky. Introducing a coefficient a = E p D / Z p S ,rewrite (6.24) in the following manner : C D = 2 S’[a(l- n ) kn,(l a ) ] (6.25)

+

+

P

Savinov [34], having determined the coefficients a and k from observations, showed that the results of calculating mean monthly diffuse radiation totals with the help of this formula are in satisfactory agreement with observations. Table 6.16 lays out 01 values obtained by Galperin [35] from measurement data on diffuse radiation by means of the Yanishevsky pyranometer. TABLE 6.16 Annual Range of the a Coefficient at Pavlovsk (1929-1933) and Saratov (1936-1939). After Galperin [35] Stations

Jan. Feb. Mar. Apr. May June July Aug. Sept. Oct. Nov. Dec.

Pavlovsk

33

24

18

13

11

10

11

14

13

19

39

47

Saratov

36

29

22

17

18

17

18

18

17

24

28

38

As seen from this table, the coefficient a has a summer minimum and a winter maximum. The increase of 01 in the winter period is mainly caused by the decrease of Qp and also by the influence of a high snow cover reflectance : the diffuse radiation flux accumulates the radiation, reflected from the snow cover, by secondary scattering. According to Table 6.16, even in the warmer half of the year the diffuse radiation totals in a clear sky are so large that it makes their neglect absolutely impossible. As was noted by Galperin, on individual summer days when the total turbidity factor T = 5.2, the coefficient a had reached values of 45 percent. The problem of the geographical variability of the coefficient a has not been solved completely at present. Further investigations of the Savinov formula (6.25) are necessary in order to compare data from various climatic conditions. The coefficient k values will be given in Chapter 8 in the discussion of the problem of calculating diffuse radiation totals.

400

Diffuse Radiation of the Atmosphere

6.4. The Main Observed Regularities of the Variability of Diffuse Radiation Totals As has been mentioned above, the income of diffuse radiation constitutes a large portion of the direct solar radiation income. In this connection the problem of the radiative heat totals of diffuse radiation becomes particularly interesting. The present section will treat only the cases of heat totals on a horizontal surface. 1. Daily Heat Totals of Dzruse Radiation. It is evident that the mean monthly daily totals must have an annual variation, with a summer maximum resulting from the higher summer altitudes of the sun. This can be seen in Table 6.17 made up by Kalitin [Chapter 3, Ref. 1021 from observational data at various points. The Pavlovsk data (Table 6.16) were obtained from diffuse radiation recordings over 10 years (1931-1940). The radiation receiver used in 19311935 was the Kalitin pyranometer, and in 1936-1940, the Yanishevsky pyranometer. It should be noted that in the final processing of the Pavlovsk data, Kalitin introduced corrections for spectral composition and zonal distribution of diffuse radiation, which considerably changed the initial results by increasing radiant fiux values. The initial means of daily diffuse radiation totals were the following: Jan. Feb. Mar. Apr. May June July 16

44

86

124

137

157

127

Aug. Sept. Oct. Nov. Dec. 119

81

43

16

9

The last column on Table 6.17 gives the mean daily totals for the year. In spite of the fact that the different methods used at different points make the data of Table 6.17 not quite comparable, the table is still representative of the principles of the annual and latitudinal variation in daily diffuse radiation totals. According to the above table, at all points the maxima are observed in summer months. A somewhat peculiar annual range was recorded at Tacubaya (Mexico), where the second half of the year gives higher radiation totals than the first. This is evidently due to greater cloudiness in the second half of the year at that place. The latitudinal variability of daily diffuse radiation totals differs throughout the year. In winter, when in the north the solar heights are small and the duration of daytime is short, the daily totals in the south are much larger than in the north. In summer, thanks to a longer daytime duration, the difference in daily incoming diffuse radiation between the north and the south is smoothed out.

P

P

TABLE 6.17 Mean Daily Heat Totals of Diffuse Radiation (cal/cm2). After KaIitin [Chapter 3, Ref. 1021 Point of Observation

Oct.

Nov.

Dec.

Year

113

55

22

13

107

119

106

103

45

24

81

234

191

135

85

48

35

131

135

202

156

117

135

141

164

Latitude

Jan.

Feb.

Mar.

Apr.

May.

June

July

Aug.

Sept.

Pavlovsk

59'41'

24

63

116

149

179

195

188

164

Benson

51'30'

35

46

75

68

106

119

128

Paris

48'49'

40

67

109

168

215

244

Tacubaya

19'24'

88

82

186

252

268

201

402

Diffuse Radiation of the Atmosphere

In connection with the problem of the dependence of incoming solar radiation upon observational conditions, it is interesting to study maxima and minima in diffuse radiation totals. Tables 6.18 tabulates maximal (Omax) and minimal (Omin)values of mean monthly totals as obtained by Kalitin a t Pavlovsk, taking into account the spectral and angular corrections. The amount of cloudiness on the day of extremum is given as average for three morning, three noon, and three evening hours. For the same intervals the prevailing cloud form was determined. The last column indicates maximum-minimum ratios. It can be seen from Table 6.18 that the annual range of maximal and minimal daily totals is almost the same as their daily means. The highest maximum of daily totals over 10 year (Pavlovsk) was observed on June 4, 1938 and was equal to 372 cal/cm2 day. On that day the sky was completely covered with altostratus, altocumulus, and stratocumulus clouds. The minimum daily totals of 3 cal/cm2 day fell on Jan. 1, 1937 and Dec. 8, 1938 in the presence of a continuous cover of stratus clouds. The limits of variations in daily diffuse radiation totals indicated by Table 6.18 are fairly wide: The maxima are occasionally 20 to 30 times higher than the minima. The most variable is the incoming diffuse radiation during autumnwinter. It has been mentioned above (see Sec. 6.3) that the incoming diffuse radiation is especially large in the Arctic. In the polar day period, daily totals there reach 500 to 600 cal/cm2 day. Daily totals, as well as diffuse radiation flux, are dependent on the elevation of an observational point above sea level. Similarly to diffuse radiant fluxes at a cloudy sky, the daily totals increase with the increasing of the height of the observation point above sea level. This increase is most marked in the summer half of the year when the sun is at its highest. 2. Monthly Heat Totals of Diffuse Radiation. Like the daily totals, the monthly heat totals of diffuse radiation have a simple annual range with a summer maximum. Table 6.19 gives a yearly variation in the monthly diffuse radiative heat means as derived by Yanishevsky [36] from data of observations at Pavlovsk from March, 1938, through September, 1940. It can be seen that the maximum monthly total for the given point was observed in June and was equal to 5.90 kcal/cm2 mo. The December minimum equals 0.37 kcal/cm2 mo. Comparing the data of Table 6.19 with the monthly totals of direct solar radiation on a horizontal surface (Pavlovsk), given in Table 5.51, we see that the incoming diffuse radiation is quite notable and is a fair portion

TABLE 6.18 Maximum and Minimum Values of Daily Heat Totals of Diffuse Radiation (cal/cmz).After Kalitin [Chapter 3, Ref.1021

Maxima Months

Minima Cloudiness

Year

Date

XmaxD

Cloudiness Year

Amount

Form

Date

XminD

Amount

Form

January

1933

31

65

10-10-10

Ci, As, Cs

1937

1

3

10-10-10

St

22

February

1936

28

163

10-10-10

As, Sc

1939

19

22

10-10-10

St

7

March

1931

29

332

10-10-10

Ns

1932

4

35

Cloudless

9

April

1931

11

326

10-10-7

Ci, Sc

1934

1

54

Cloudless

6

May

1936

17

322

10-10-10

Cb, Au

1935

27

63

Cloudless

5

June

1935

30

340

10-10-7

Ci, Cs, Ar, Sc

1931

27

58

04-10

cs

6

July

1938

4

312

10-10-10

As, Ac, Sc

1933

6

63

1-04

Ci, Sc

6

August

1937

3

280

10-7-10

Sc, Cu, Frost

1931

27

44

10-10-10

Nb

6

September

1933

4

237

10-1-10

Ci, Sc, Frost

1935

30

23

10-10-10

St, Frost

10

October

1933

4

189

10-10-8

sc

1937

26

6

10-10-10

St

31

November

1932

3

17

10-10-10

St

1938

28

4

10-10-10

St

18

December

1933

2

38

10-10-10

sc

1939

8

3

10-10-10

St

13

404

Diffuse Radiation of the Atmosphere

of the direct solar radiation. Moreover, in the period from September to March, the diffuse radiation totals exceed those of the direct solar radiation; for example, the December minimum of diffuse radiation is by 5.3 times larger than the corresponding direct solar radiation total. TABLE 6.19 Annual Variationof Mean Monthly Heat Totals of Diruse Radiation at Pavlovsk, 1938-1940. After Yanishevsky [36]

Nov. Dec.

Year

0.64 1.43 3.34 4.37 5.84 5.90 5.61 4.31 3.30 1.52 0.59 0.37

37.21

Jan. Feb. Mar. Apr. May June July

Aug. Sept. Oct.

The importance of incoming diffuse radiation can be clearly noticed even at such a point as Tashkent, where cloudiness is insignificant and the direct solar radiation by far exceeds that of Pavlovsk. This is illustrated by Table 6.20, which presents means of monthly diffuse radiation totals from observations by Yaroslavtzev [28] at Tashkent in 1937-1943. The second line of Table 6.20 gives values of the ratio of diffuse radiation totals to those of global radiation in percent. It is seen that at Tashkent the portion of monthly incoming diffuse radiation is not less than 16 percent of the global incoming radiation. Note here that the presence of monthly maxima of diffuse radiation at this point is caused by an intensive cloudiness in the springtime. From comparison of Tables 6.19 and 6.20 one can see that the total diffuse radiation per year is almost the same for both Pavlovsk and Tashkent; at Pavlovsk, however, a more pronounced annual range of monthly totals is observed. It should be stressed that the comparison of the above data must be considered conditional because Table 6.19 includes corrected pyranometer readings whereas the data of Table 6.20 have been processed without correction. Very interesting results of comparison of monthly diffuse radiation totals at various geographic points have been obtained by Chernigovsky (see [37, 38]), as shown in Table 6.21. It can be seen that at the drifting polar stations in the central Arctic, the summer monthly totals of diffuse radiation are two to three times larger than those at southern latitudes. The Antarctic has no such large amount of incoming diffuse radiation because the lower layer clouds that are favorable for the increase in diffuse radiation are rarer here than in the Arctic. Diffuse radiation makes the main con-

Annual Range of Mean Monthly Diffuse Radiation Totals at Tashkent in 1937-1943. After Yaroslavtzev [28]

:l:D, kcal/cm' mo :l:D/:l:Q, %

Jan.

Feb.

Mar.

Apr.

May

June

2.36

2.67

3.78

4.20

4.41

3.71

59

50

46

33

27

20

July

3.20 16

Aug.

Sept.

2.93

2.71

16

20

Oct.

2.71 30

Nov.

Dec.

Year

2.18

1.87

36.73

41

51

27

6.4. Main Observed Regularities of the Variability of Diffuse Radiation Totals 405

TABLE 6.20

406

Diffuse Radiation of the Atmosphere TABLE 6.21

Monthly Totals of Diffuse Radiation in Central Arctic, Antarctic, and on the Caucasus (kcal/cma).After Chernigovsky [37]

Month

North Pole North Pole North Pole 2,1950-1951

Jan. Feb. Mar. Apr. May June July Aug. Sept. Oct. Nov. DK. Year Total

6,1957

7,1957 -

1.4 5.3 14.0 17.7 14.3 9.5 3.8 0.4

-

-

2.5 2.9 3.8 4.9 6.0 5.8 5.4 4.6 4.3 3.6 2.6 2.1

-

48.5

1.1

-

11.3 13.0 10.6 7.3 2.8 0.6

13.9 11.o 7.0 1.5

-

-

-

-

66.5

Sukhumi 1950-1955

Month

Mirny 1956-1957

July Aug. Sept. Oct. Nov. DK. Jan. Feb. Mar. Apr. May June

0.2 1.o 2.6 5.7 6.8 7.6 6.8 5.5 2.8 0.8 0.4 0.1

Year Total

40.3

tribution to the radiation budget in the Arctic; in the Antarctic it is the direct solar radiation.

3. Annual Heat Totals of Difuse Radiation. The main factors to determine these quantities are the latitude and cloudiness regime of a given point. There is no definite regularity in the latitudinal variation of annual totals, owing to the fact that in the north the cloudiness is usually greater than in the south. The dependence of annual totals of diffuse radiation upon latitude is shown in Table 6.22, compiled by Berland [39]. TABLE 6.22 Annual Heat Totals of Diffuse Radiation at Various Latitudes. After BerIand [39]

Latitude, deg. :

80

75

70

65

60

55

50

45

40

35

D, kcal/cm2yr:

47

41

40

38

37

37

39

43

49

52

We see that within the range from 30° to 35O N. lat, the annual totals vary insignificantly. The latitudinal variation of annual diffuse radiation totals is small, and since the incoming direct solar radiation increases

6.4. Main Observed Regularities of the Variability of Diffuse Radiation Totals 407

southward, it becomes clear that the relative portion of annual diffuse radiation must decrease toward the south. The mean latitudinal distribution of the ratio of the incoming diffuse radiation to that of the global radiation is characterized by a minimum at about 30' (in the areas of least cloudiness in subtropical and tropical latitudes). Farther south, however, the portion of diffuse radiation relative to the global radiation regains its increase. Observations show that annual sums of diffuse radiation may vary considerably from year to year, mainly as a result of the corresponding variation of the cloudiness regime. For instance, according to data by Yaroslavtzev for Tashkent, the mentioned variations in annual totals from year to year are as much as 12 to 16 percent of the many-year mean. And the processing of observational data during IGY and IQSY, performed by Pleshkova [40], gives still higher values of 20 to 27 percent. Interesting calculations of optical properties of haze and clouds in the visible and infrared regions have been performed by Deirmendjian [41] who has proved the necessity of using exact formulas of the Mie scattering theory and the more adequate microphysical aerosol model characteristics for solution of similar problems. These calculations deal with two models of haze and one of cloud that are satisfactorily reflecting the real cumulus cloud microstructure in the layer 230 to 2100 m and also the microstructure of intercontinental and coastal haze. In the mentioned work are also given formulas for calculation of the absorption and scattering volume crass sections and of the scattering matrix components at a given size distribution of aerosol particles. To overcome computational difficulties it is important at this point to choose an analytical approximation of microstructure. The results of calculation of extinction coefficients and albedo (ratio of the coefficient of attenuation due to scattering to the extinction coefficient) relate to the wavelength range 0.45 to 16.6 p and to the above three models of aerosol microstructure. These data are very spectacular in illustrating the great influence of optical properties (refraction index) of aerosol particles, varying in dependence on wavelength and microstructure, upon the extinction coefficients and albedo values. In the case of haze the extinction coefficient is shown mainly to decrease with wavelength. In cloud, however, an attenuation maximum in the near-infrared takes place, partly caused by the increasing absorption coefficient of water. The calculated scattering functions for reciprocally perpendicular polarized components of scattered light show that in all cases is observed a marked anisotropy of scattering (elongation of functions in the direction

408

Dif€use Radiation of the Atmosphere

of the incident light) and a polarization maximum at large scattering angles. Under these conditions the position and magnitude of maximal polarization and the function elongation notably depend upon microstructure and aerosol optical properties, that is, wavelength. Deirmendjian, in particular, has found the presence of a very intensive aureole and rainbow at scattering by a cloud of radiation with 0.45 p wavelength. The peculiarity of scattering on polydispersional aerosol consists in the presence of one chief rainbow, which suppresses those of higher orders that are so clearly developed in light scattering by a single particle. The same refers to the phenomenon of countercorona (glories). Deirmendjian has also made the important conclusion that when studying radiation scattering by aerosol of a wide particle spectrum, it is impossible to find an “equivalent” particle whose scattering might be identical with the scattering by polydispersional particle mixture, since the main effect of “polydispersional” scattering consists in smoothing (suppressing) the extreme peculiarities of the scattering by individual particles, In this connection the introduction of the concepts of the mean modal radius, etc., is useless except perhaps in the case of particles whose dimensions are much less than the wavelength. The forms of the scattering function and of polarization are independent of particle concentration and are determined solely by their size distribution and complex refraction index. The angular scattering characteristics can thus be measured when using optically thin layers for which the effect of multiple scattering is negligible. The problem of determining aerosol microstructure from measurement data on the angular intensity and light polarization distribution in the case of optically thick layers cannot be considered to be solved, although it is clear that such peculiarities, as for example, rainbow and countercorona are certainly informative as regards microstructure. Comparison of the above calculations with observational data shows fairly satisfactory agreement. In particular, it has been shown that in the case of nonabsorbing particles the intensity of light scattered at an angle of about 45’ depends only on the volume concentration of scatterers and not on the aerosol microstructure.

REFERENCES 1. Lenz, K. (1961). ‘‘Uber das Streuspektrulnder Atrnosphare. Optik und Spektroskopie aller Wellenlangen.” Akademie Verlag, Berlin. 2. Boiko, P. N., and Kazachevsky, V. M. (1960). “Observation of the spectral sky brightness by means of the photographic method, Bull. Astrophys. Inst. Acad. Sci. Kazakh SSR, 10.

References

409

3. Tikhov, G. A. (1934). Spectral illumination of a horizontal surface. Coll. Papers Aerophotometry No. 1. 4. Tikhov, G. A., and Drury, V. I. (1934). Spectral properties of illumination of horizontal surfaces in the daytime and in the twilight. Coll. Papers Aerophotometry No. 2. 5. Krinov, E. L., and Sharonov, V. V. (1936). Spectrophotometric investigation of global and diffuse daytime illumination. J. Geophys. 6, Nos. 2 and 3. 6. Sharonov, V. V. (1938). Spectral composition of the daytime sky. Priroda, No. 4. 7. Krinov, E. L. (1942). Spectral sky brightness. Proc. Acad. Sci. USSR, Ser. Geograph. Geophys. No. 4. 8. Krinov, E. L. (1939). Spectral daytime illumination of horizontal surfaces within the infrared region. CON. Works of TzNIZGAZK No. 1. 9. Kondratyev, K. Ya., Kudriavtzeva, L. A., and Manolova, M. P. (1955). Distribution of the energetical and light intensity of diffuse atmospheric radiation over the celestial sphere. Bull. Leningrad Univ. No. 5 . 10. Yaroslavtzev, I. N. (1953). Distribution of brightness over the sky. Proc. Acad. Sci. USSR, Ser. Geophys. No. 1. 11. Averkiev, M. S. (1965). On the influence of albedo of the underlying surface on diffuse radiation. Bull. Moscow Univ. [5] No. 1. 12. Linke, F., and Ulmitz, E. (1940). Messungen der zirkumsolaren Himmelstrahlung. Meteorol, 2. 57. 13. Stranz, D. (1942). Die Himmelstelligkeit in unmittelbarer NHhe der Sonne. Meteorol. 2.59, No. 4. 14. Barashkova, E. P. (1955). Dependence of diffuse radiation on the turbidity factor, Trans. Main Geophys. Obs., No. 46. 15. Barashkova, E. P. (1959). Diffuse radiation at Karadag. Trans. Main Geophys. Obs. No. 80. 16. Ross, Y. K. and Avaste, 0. A. (1959). Diffuse radiation at Tartu. Invest. Atmospheric Phys. No. 1. Tartu. 17. Kondratyev, K. Ya. and Zavodchikova, V. G. (1953). Spatial distribution of diffuse and reflected radiation. Bull. Leningrad Univ. No. 2. 18. Galperin, B. M. (1961). Fluxes of global and diffuse radiation in the Arctic. Trans. Arct. Antarct. Znst. 225. 19. Kalitin, N. N. (1943). Diffuse radiation of a cloudless sky. Rept. Acad. Sci. USSR. 39, No. 8. 20. Makhotkin, L. G. (1953). On the methods for calculation of diffuse illumination at a clear sky. Proc. Acad. Sci. USSR, Ser. Geophys. No. 5 . 21. Makhotkin, L. G. (1959). Regularities of the variation of diffuse radiation at a cloudless sky. Trans. Main Geophys. Obs. No. 80. 22. Kastrov, V. G. (1938). Actinometrical properties of mist. Social. Econ. Cereals NO.3. 23. Sivkov, S. I. (1964). To the methods for computing the possible radiation totals. Trans. Main Geophys. Obs. No. 160. 24. de Bary, E. (1964). Influence of multiple scattering of the intensity and polarization of diffuse sky radiation. Appl. Opt. 3. 25. Bullrich, K., de Bary, E., Danzer, K., Eiden, R. and Ulger, K. (1965). Research on atmospheric optical radiation transmission. Sci. Rept., No. 3, A F 61 (052)-565, Mainz.

410

DiBFuse Radiation of the Atmosphere

26. Tzutzkiridze, Y. A. (1947). Tension of diffuse radiation at Tbilisi. Trans. Local Obs., No. 1. 27. Temnikova, N. S. (1941). Diffuse radiation of the sky in connection with cloud forms. Meteorol. Hydrol. No. 5. 28. Yaroslavtzev, I. N. (1947). Diffuse radiation at Tashkent. Meteorol. Hydrol., Znform. BuIl. No. 6. 29. Feugelson, E. M. (1964). “Radiative Processes in the Stratus Clouds.” Nauka, Moscow. 30. Korb, H., and Moller, F. (1962). Theoretical investigation of energy gain by absorption of solar radiation in clouds. Final Tech. Rept., Contr. DA-91-591-EVC-1612, Munchen. 31. Romanova, L. M. (1962). Solution of the equation of radiative transfer at the scattering function highly differing from the spherical. Opt. Spectry. (USSR)(English Transl.) 13, Nos. 3 and 6. 32. Romanova, L. M. (1963). Radiation field in the plane layers of a turbid medium with a strong unisotropic scattering. Opt. Spectry. (USSR)(English Transl.) 14, No. 2. 33. Feugelson, E. M. (1961). Radiative properties of clouds St. Proc. Acad. Sci. USSR, Ser. Geophys. No. 4. 34. Savinov, S. I. (1933). About the formulae expressing direct and diffuse radiation in dependence upon the degree of cloudiness. Meteorol. Hydrol. Nos. 5 and 6. 35. Galperin, B. M. (1949). To the method for approximate calculation of solar radiation totals. Meteorol. Hydrol., Inform. Bull. No. 4. 36. Yanishevsky, Y. D. (1951). The problem of measurements by means of pyranometers and diffuse radiation at Pavlovsk. Trans. Main Geophys. Obs. No. 25 (88). 37. Chernigovsky, N. T. (1961). Some characteristics of the radiation climate of Central Arctic. Trans. Arct. Antarct. Znst. 229. 38. Gavrilova, M. K. (1963). “The Arctic Radiation Climate.” Gidrometeoizdat, Leningrad. 39. Berland, T. G. (1965). Variability of solar radiation incident on the earth’s surface. Trans. Main Geophys. Obs. No. 119. 40. Pleshkova, T.T. (1965). Diffuse radiation and its anomalies during the IGY and IQSY on the territory of the USSR. Trans. Main. Geophys. Obs. No. 179. 41. Deirmendjian, D. (1964). Scattering and polarization properties of water clouds and hazes in the visible and infrared. Appl. Opt. 3, No. 2.

7 ALBEDO OF THE UNDERLYING SURFACE AND CLOUDS

On reaching the earth's surface the direct solar and diffuse radiation are reflected from it. Reflectivity (albedo) of different kinds of underlying surface essentially differs. As was mentioned in Chapter 2, measurements of the integral albedo employ various instruments, for the most part pyranometers. For spectral measurements a variety of spectrophotometric instruments has been constructed. The determination of albedo values is exceptionally important in the study of radiation balance of underlying surfaces because only by knowing the albedo it is possible to find out the amount of heat obtained in the underlying surface by the absorption of incident radiation. The present chapter treats the problem of albedo of various underlying surfaces, of clouds, and of the planet Earth, which is of much practical importance. 7.1. Spectral Albedo of Natural Underlying Surfaced The investigation of the integral albedo regularities is essential principally for meteorology, but the information on spectral albedo is of wider interest for bioclimatology, biophysiology, aerosurvey, and other investigations. At present, researchers of quite a number of scientific fields work with spectral reflectivity of natural underlying surfaces and various telluric formations. However, the available data on spectral albedo are still scarce and not always reliable enough. Almost all data on spectral albedo of natural underlying surfaces have been obtained either by means of glass filt Z. F.

Mironova is coauthor of this section. 411

412

Albedo of the Underlying Surface and Clouds

ter measurements, which have wide and not sufficiently clear limits of the radiation transmission range, or by measuring spectral brightness of various natural formations in fixed directions with the help of monochromators that use photographic or photoelectric spectrum registration. None of these methods can yield satisfactory results to meet the demands of meteorological applications. Measurements of albedo by means of pyranometers equipped with filters permit only a rough and not quite simple picture of spectral albedo distribution (often filter transmission regions overlap, which makes the unsatisfactory radiation monochromatization even worse). It is true that the recent appearance of interferometric filters with very narrow transmission bands has significantly widened the prospects of filter application, but to date their development is still in the beginning stage. The numerous measurements of spectral brightness of natural formations are quite interesting for meteorology, but in application to the study of the regularities of radiation balance of natural underlying surfaces they are of limited importance. It is known that only in the case of isotropic reflecting surfaces, are the values of spectral brightness (usually determined in relation to perfectly white screen luminosity) and of albedo found to be identical. Almost all natural surfaces, however, are anisotropic radiation reflectors. Therefore, between the results of measurement of spectral luminosity and albedo is observed only a qualitative correspondence, approximately similar to the range of dependence of reflectivity upon wavelength. As regards direct spectrophotometric measurements of albedo, they are few so far. The first to conduct such measurements was the staff of the corresponding department at Leningrad University in 1953-1954. It is a common view that albedo characterizes a given underlying surface. This is valid in relation to the integral albedo in the first approximation. Although it has been stated in the result of many measurements that almost always there is an observed dependence of integral albedo upon solar height, as a rule this dependence is not marked (excepting albedo of water basins). It is also important that the dependence of integral albedo upon solar height is monotonic and fairly stable. This means that to the given kind of surface corresponds a definite curve of the daily albedo range. The matter looks more complex with respect to spectral albedo, which is greatly dependent on conditions of illumination. In the case of spectral albedo the properties of a given underlying surface and the conditions of illumination are often almost equally significant factors in determining albedo values. This makes difficult analysis of spectral albedo measurement

7.1. Spectral Albedo of Natural Underlying Surfaces

413

results. It is natural, howeker, to classify such results first of all in dependence upon the kind of underlying surface following the type of scheme suggested by Krinov [l]. 1. Soils and Bare Surfaces. The main peculiarity of this type of underlying surface is a monotonous increase of albedo with the increasing wavelength in the visible and near-infrared. This conclusion, first obtained from data of spectral luminosity measurements, was later fully confirmed by the results of spectrophotometric albedo measurements. The latter showed also that the variation of spectral albedo in the bared soil in dependence upon illumination conditions is rather weak, not exceeding the range of 5 to 10 percent relative to the mean. Ashburn and Weldon [2] have investigated the spectral albedo of various underlying desert surfaces. To monochromatize radiation (the region 0.40 to 0.65 p), these authors used interferometric filters. Although the obtained albedo values varied within very wide limits (from 3 to 74 percent), in all studied cases (except a basaltic lava surface) an increasing albedo with the increase of wavelength was observed. For this the albedo values at the boundaries of the investigated spectral interval differed by approximately twice. The same authors have also found certain data on the dependence of albedo upon solar height. In all the cases the albedo has been observed to grow as the height of the sun decreases to 10'; at a further decrease of solar height, a decrease of albedo takes place. Most pronounced is such dependence of albedo upon solar height for long waves. It should be noted, however, that the number of observations of the daily spectral albedo range in the considered investigation was quite small, which prevents us from relying on the obtained data about the dependence of albedo upon solar height. The study of spectral reflectivity of trampled earth in the wide spectral range from 0.6 to 1.35 p, carried out by Krassilshchikov et al. [3-61, has shown that the reflectivity growing with an increase of wavelength is observed approximately up to 1.1 p, after which it begins to decrease with the increase in wavelength. Zaytov and Indichenko [7] have found a maximum of spectral reflectivity of various soil types near 1.3 p and also a dependence of reflectivity upon soil surface roughness. Dirmhirn [S-91, investigating the property of light limestone and dark gneiss in the range 0.3 to 2.6 p, has shown that the entire range demonstrates a monotonic increase of reflectivity with an increase in wavelength up to very high values. For instance, the reflectivity of limestone at 1 = 2.6 p is 87 percent, whereas at 1 = 0.5 p, it equals 40 percent.

414

Albedo of the Underlying Surface and Clouds

2. Vegetative Covers. Voluminous literature is dedicated to the problem of spectral reflectivity of vegetative covers and their various elements (leaves, stalks, etc.). The feature peculiar to the spectral albedo range for several kinds of green (rich grass, green tree leaves, etc.) is low albedo value in the visible spectrum, with a small maximum in the interval 0.50 to 0.55 p and a minimum in the chlorophyll absorption band interval near 0.65 p. At wavelengths about 0.70 t o 0.80 p and more is observed a strong increase of albedo, and when the wavelength further increases, a sharp decrease of albedo takes place. Albedo values may vary according to the kind of vegetative cover. In Fig. 7.1, built up from data by Kondratyev er al. [lo-141, are shown spectral albedo characteristics of green vegetation. Differences in values fall chiefly on the interval 0.70 to 0.90 p. According to data of other authors (for example, Pronin [15]) the albedo of the green increases monotonously up to the wavelength il = 1 p. Billings and Morris [16], studying spectral albedo of different types of vegetation, have found high albedo values (about 50 percent) and an almost complete absence of selectivity in the nearinfrared (from 0.7 to 0.8 to 1.1 p). Analogous results have been obtained by Egle [17]. At I > 1.1 p, however, the latter has found a strong decrease

I

400

500

600

700

800

Wavelength

900

1000

*

X,mp

FIG. 7.1 Spectral albedo of various plants. (1) sudangrass (Aug. 1963); (2) maize (Aug. 1963); (3) clover (Aug. 1963); (4) lucerne (June 1961); (5) lucerne (July 1961).

415

7.1. Spectral Albedo of Natural Underlying Surfaces

in reflectivity down to values of the order of 10 to 15 percent, and less at iZ = 2.4 p. Very interesting data on luminosity coefficients of leaves, measured in laboratory conditions, have been derived by Dirmhirn [8, 91 for the infrared up to ;Z = 2.6 p. According to her data, the luminosity coefficient of green leaves has a chief maximum in the interval 0.75 to 1.4 p and then decreases erratically with the increase of wavelength, showing intermediate maxima at about 1.7 p and 2.3 p . The cause for a notable discrepancy between measurement results on spectral reflectivity of the green is difficult to state. It may be the fact that many investigations are conducted in the laboratory and therefore cannot fully correspond to outdoor results. For certain types of plants it has been observed that albedo decreases with increasing wavelength at about 0.8 to 1.0 p. Evidence for this, for example, is given by the fact that, according to data by Matulavicene [18], the vegetative cover albedo in the visible spectrum is by far higher than the integral. A notable decrease of spectral luminosity coefficients for many kinds of vegetation in the range from 0.8-0.9 to 1-2 p have been found by Perevertun [19]. Similar results are shown by Yaroslavtzev [20] and Miller [21]. The latter gives the following schematic table of the dependence of the albedo of coniferous trees upon wavelength : Wavelength, p : 0.35

0.35-0.76

0.76-2.0

2.0-5.0

10

%:

4-8

15

5

3

Albedo,

5-8

A number of investigations have found a strong variability of spectral albedo in dependence upon phase of vegetation for various plants. Such a dependence for a maize field is given in Fig. 7.2 by Kondratyev and MiA

30.

ae*

20

a

10

I

400

500

600

700

000

900

-

k,m p FIG. 7.2 Daily means of spectral albedo for maize. Aug. 1963. (1) ensilage; (2) high stand; (3) yellow.

416

Albedo of the Underlying Surface and Clouds

ronova [14]. It is seen that with the growth of plants (in the transition from ensilage crops to high stalks), the profile of the chlorophyll absorption band changes for a less marked one, the albedo in the 0.45 to 0.68 p interval somewhat increases, and its values in the range of wavelengths 0.75 to 0.85 p decrease significantly. In the case of yellow corn a further increase of albedo in the range 0.55 to 0.73 p takes place and the chlorophyll absorption band disappears. Quite interesting are data by Tihomirov [22, 231 and Artzybashev and Belov [24] which characterize seasonal variation of the spectral luminosity coefficient for deciduous and coniferous foliage in the interval 0.56 to 0.725 p. The factors influencing spectral albedo magnitude are the conditions of illumination, the angle of incidence of direct solar radiation, and the relation between the directed and diffuse radiant fluxes. According to Kondratyev and Mironova [13, 141 and other authors, on cloudless days the diurnal spectral albedo variation for all vegetative surfaces is characterized by the presence of a minimum value at daytime, with the most variation in the longwave spectral region. Figure 7.3 presents a daily range of albedo for thick grass (the albedo value at noon is taken to be 100 percent). With the wavelengths 0.80 and 0.90 p (this value being 140 to 150 percent relative to noon), for 0.50 p

I ' 6

I

8

10

I

12 14 Hour

I

I

16

18

+

FIG. 7.3 Daily variation of albedo relative to noon for thick grass. (1) 500 rnp; (2) 800 mp.

the albedo equals 110 to 120 percent. The foregoing statement suggests a conclusion that the daily variation of the integral vegetation albedo with a noon minimum appears to be explained first of all by the analogous variation of albedo for longwave radiation. The same conclusion follows from observations by Yaroslavtzev [20]. In recent years wide scale aircraft measurements of spectral luminosity

7.1. Spectral Albedo of Natural Underlying Surfaces

417

of different natural formations have been conducted in connection with solving various aerosurvey problems. Numerous results of such investigations in forestry can be found in the works of Artzybashev and Belov [24] and Spurr [25]. These results fully confirm the regularities of the spectral luminosity coefficient variation in the visible range, but give somewhat lowered (relative to the surface measurements) values of spectral luminosity coefficients. The latter is explained by a greater roughness of the forest foliage, by the influence of shade, and by a lesser reflection of light by bare branches. A detailed review of data on albedo in the nearinfrared is given by Avaste [26].

3. Snow Cover. The spectral albedo of snow cover is extraordinarily variable in dependence upon variation of the properties of snow cover and the conditions of illumination. The author and collaborators [10-141 have obtained fully satisfactory data on the spectral albedo of snow. It follows from these data that spectral albedo of the surface of clean snow under a cloudless sky shows a rather weak dependence upon wavelength in the central part of the visible spectrum 0.50 to 0.80 p, decreasing with the increase of wavelength in the range from 0.50 to 0.30 p and from 0.80 to 1 p (Fig. 7.4).

I

Joo

.

.

500

.

.

.

700 A. m p

.

900

.

*

FIG. 7.4 Spectral albedo of the surface of clean snow.

Yaroslavtzev [20] has studied albedo of snow for infrared radiation (the filter isolated the interval 0.8 to -3.5 p). In this case the albedo of snow was considerably less than the integral. This means that a decrease of albedo takes place in the infrared. Such a conclusion is confirmed by the results of laboratory measurements of spectral luminosity coefficients of settled snow, conducted by Dirmhirn [8, 91: The snow brightness coefficient steedily decreases down to A = 2.6 p, where it amounts to only a few percent. It is of interest that the transition from the snow as a good reflector (the visible spectrum) to the snow as a blackbody (the infrared) is very

418

Albedo of the Underlying Surface and Clouds

gradual. The transitional region of the spectrum embraces the interval of wavelength values that are several microns wide. An important factor influencing albedo values is roughness of snow cover. When the latter increases (in otherwise equal conditions), a decrease of albedo for all wavelengths is observed, which fully agrees with the results of theoretical analysis of the effect of surface roughness on albedo, as performed by Shifrin [27]. With a change in surface roughness is also observed a certain variation of the character of the dependence of albedo upon wavelength. The mentioned peculiarities of the dependence of albedo upon roughness is illustrated by the data of Table 7.1. TABLE 7.1 Dependence of Spectral Albedo upon Snow Cover Roughness. After Kondratyev et al. [lo]

Date

Hour

Cloudiness

State of Snow Cover

Wavelength,

400 501 615 742

March 7, 1957

10

0/0

Dry, clean, friable, smooth surface

73

76

73

70

March 17,1957

10

010

Dry, clean, friable knobby surface

65

61

60

60

The strong dependence of the spectral albedo of snow on its state gives basis to suppositions that, by studying albedo variations in different spectral intervals, it is possible to obtain valuable information about variations in the state and physical properties of snow cover. The solution of such an inverse problem, however, is made difficult by the presence of considerable variations of the spectral snow albedo in dependence upon illumination conditions. The appearance of haze or cloudiness (even more so) greatly affect the albedo value. Also necessary is a further development in the theory of snow cover albedo. In this connection may be mentioned important works by Giddings and Chapelle [28]. The daily range of the spectral albedo of snow cover is determined by a complex interaction of two main factors: variation in the state of snow cover and illumination conditions (angular distribution of global radiation intensity). Even with a cloudless sky, both factors show a fluctuating variation with time. Accordingly, a very complex diurnal albedo variation is observed. The diurnal albedo variation for snow cover, as investigated

7.1. Spectral Albedo of Natural Underlying Surfaces

419

in March in bright weather (solar height increasing up to 41°), is characterized by a minimum at about noon. The most marked variations should be connected with solar heights less than 25'. The considerable variations in the daily range of snow albedo, similar to the case of vegetation, related to longer waves should be explained evidently by complex interaction of diffuse and directed radiant fluxes in the process of reflection. In the case of continuous cloudiness of the stratus type the albedo values are practically invariable throughout the day. To conclude what has been said above about spectral albedo of snow cover, it should be noted that in spite of a very strong variability of snow albedo values, the selectivity of radiation reflection by snow is weak as a rule. In this respect the snow cover is directly opposite to vegetation, for which selectivity is essential. Owing to the comparatively low selectivity of radiation reflection by snow, the spectral composition of reflected radiation in this case differs insignificantly from that of incident global radiation. 4. Wafer Basins. The spectral albedo of clean water is practically fully determined by the effect of surface reflection and can be easily calculated from the familiar Fresnel formula. Calculations of this kind for albedo of water in the spectral region 0.214 to 1.256 p for direct solar radiation have been made by Ter-Markariantz [29]. Since the water refraction index in this region is steadily decreasing with an increase of wavelength, the calculations give a steady decrease of albedo with the increase of wavelength, which is more pronounced at great angles of radiation incidence (except very near 90'). However, even at great angles of incidence the variation of albedo in dependence upon wavelength is quite low. For instance, at an 80' angle of incidence the albedo values corresponding to the above spectral limits are 36.8 and 34.4 percent. Analogous results have been obtained by Funk [30]. Czepa [31] has performed calculations of water albedo for global radiation, taking into account the angular distribution of diffuse radiation intensity and the correlation between the direct solar and diffuse radiation. Assuming that the albedo of water for a diffuse cloudless sky radiation equals 7.75 percent and is independent of wavelength, and taking into consideration the relation between the diffuse and direct radiation a t different wavelengths, Czepa derived curves of dependence of albedo for global radiation from a cloudless sky upon wavelengths in the form presented in Fig. 7.5. As seen, the spectral range of albedo is significantly dependent upon solar height. At greater altitudes of the sun the obtained results are analogous to those by Ter-Markariantz [29]; that is, albedo

420

Albedo of the Underlying Surface and Clouds

decreases with an increase of wavelength. The inverse dependence of albedo upon wavelength takes place at low altitudes of the sun. For an altitude equal to 22.5' there is no observed selectivity of reflection.

0 0.4

05

0.6

0.7

P

FIG. 7.5 Dependence of the spectral albedo of water upon solar height for global radiation in a cloudless sky.

It should be noted herewith that since the results referring to great solar heights can be easily explained qualitatively (the contribution of diffuse radiation in this case is small and therefore the deciding role belongs to the dependence of the direct solar radiation albedo upon wavelength), the increase of albedo with an increase of wavelength at low altitudes is not quite clear. From the point of view of Ter-Markariantz's results, a decrease of albedo with the increase in wavelength should be expected in this case. As regards albedo in the presence of solid cloudiness, it may be considered to be practically independent of wavelength. In the real conditions the albedo of water basins is determined by a sum total of the surface reflection and the so-called reverse scattering of radiation. The latter is determined by water turbidity. This is why in real conditions the spectral albedo depends upon the turbidity of water basins; the real observed spectral albedo range differs from the one presented in Fig. 7.5. According to data by Kalitin [32] and Hulburt [33] on the visible spectrum, the albedo is at maximum in the central visible and decreases toward both shorter and longer waves. The same result was obtained by Hulburt from calculations for distilled water. Kondratyev et al. [13, 141 have investigated spectral albedo of water surface (bottom at 60 to 70 cm). For the considered case (Fig. 7.6) are peculiar small albedo values throughout the whole spectrum, the albedo

7.1. Spectral Albedo of Natural Underlying Surfaces

42 1

decreasing toward the shortwave and longwave regions. Interesting aircraft measurements of the spectral sea luminosity in the region 450 to 750 mp have been made by Semenchenko and Spytkin [35].

YO1

/

0 400

-

500

*

=

600

700

800

hmrr

FIG. 7.6 Daily mean spectral albedo for water surface.

5. The Planet Earth. The solar radiation reaching the earth is reflected not so much by the earth’s surface as by the clouds and atmosphere in the result of intensive radiation scattering. We shall therefore give a brief account of these factors in solar radiation reflection. Experimentally, the spectral albedo of the planer Earth is almost unexplored. There are only some theoretical evaluations available. Voluminous literature is devoted to the theoretical investigation of albedo of scattering media. The quantitative calculations of spectral albedo of the clouds, atmosphere, and the earth-atmosphere system are few and approximate. According to Hewson’s [Chapter 4, Ref. 1141 data, the albedo of clouds is practically independent of wavelength up to the values near 1.3 p. In the region of longer waves a fluctuating decrease of albedo takes place, with minima near 1.5 and 2.0 p and maxima at 1.7 and 2.2 p (calculations were made for the interval 0.4 to 2.4 p). Absolute albedo values are considerably dependent upon microphysical characteristics, the vertical cloud power, and solar height. A qualitative presentation of the spectral albedo of the earth as a planet and its components can be given by the results of Fritz [36], tabulated below. Note in Table 7.2 that all albedo values are expressed in relation to the solar constant values in the corresponding spectral intervals. The limits of the spectral regions are defined as il < 0.40 p (ultraviolet) and il > 0.74 ,u (infrared). As seen, the albedo of clouds is almost nonselective. The albedo of the atmosphere rapidly decreases with an increase in wavelength, which is quite natural because molecular scattering is important in the given case. The spectral variation of albedo of the planet Earth is determined by the effect of atmospheric albedo. Analyzing the peculiarities of spectral albedo of various underlying sur-

422

Albedo of the Underlying Surface and Clouds TABLE 7.2 Spectral Albedo of the Planet Earth. After Fritz [36]

Area

Earth’s surface

Ultraviolet Spectrum

Visible Spectrum

Infrared Spectrum

Integral Spectrum

1.1

2.4

2.4

2.3

Clouds

20.0

25.1

22.2

23.3

Atmosphere

28.9

11.5

2.8

9.1

Planet Earth

50.0

39.0

27.4

34.7

faces, it is possible to isolate whole classes and groups of surfaces with the common property of dependence of albedo on wavelength. On the basis of such analysis Krinov [ l ] has worked out a spectrophotometric classification of natural formations, whose main features are followed in the distribution of the material of the current section. The peculiarities of radiation reflection may also be characterized by classifying surfaces according to the type of angular distribution of reflected radiation (isotropic, specular, etc.). For this, it is necessary to have information about indicatrixes of radiation reflection by different surfaces. Unfortunately such information is as yet scarce and superficial. In summarizing what has been said above, it should be stressed once more that the spectral albedo investigations of natural underlying surfaces are at present in the primary stage of development. Essential perfection of measurement methods and prolonged accumulation of experimental material are necessary before the reflective properties of natural underlying surfaces can be sufficiently studied. Particularly interesting is an almost uninvestigated problem of albedo in the infrared. Let us turn now to characteristics of the integral albedo variability in the case of natural surfaces subjected to shortwave radiation.

7.2. Albedo of Various Continental Underlying Surfaces Almost all the performed observations over albedo were made by means of measuring the incident and reflected shortwave radiant fluxes at a height 1 to 2 m above the level of the underlying surface (see [37]). Measurements of this kind allow characterizing the albedo of small underlying sections. During recent years attempts have been made to measure albedo from helicopters and aircraft (see, [37a, 37bl). To some extent these measure-

7.2. Albedo of Various Continental Underlying Surfaces

423

ments obtained direct average albedo values for underlying surfaces. We shall now discuss the characteristics of these surface albedo measurement results. A great variety of different kinds of underlying land surface determines the notable diversity of the observed albedo values. The following sections consider data of albedo observation for different groups of underlying surface.

1. AZbedo of Soil Surface. Table 7.3, compiled from results of different authors (see [38]), gives albedo values for various soils without vegetation. It should be noted that the methods employed in the measurements were not uniform and therefore there is a varying degree of accuracy in the measurements. TABLE 7.3 Albedo of DiHerent Soil Covers ( A l

= 0.3

to 2 p )

Soil

Albedo, %

Black earth, dry Black earth, moist Gray earth, dry Gray earth, moist Blue clay, dry Blue clay, moist Fallow field, dry surface Fallow field, wet surface Ploughed field, moist Surface of clayey desert Yellow sand White sand Gray sand River sand Fine light sand

14 8 25-30 10-12 23 16 8-12 5-7 14 29-3 1 35 3440 18-23 43 37

The results given in Table 7.3 show a great variability of albedo in dependence upon soil quality. It is necessary to mention here that the actual observed values of albedo in various concrete cases are still more variable. Even the albedo of some definite kind of soil may be likely to vary, depending on certain factors. From Table 7.3, for example, it is seen that albedo is essentially dependent upon humidity of the soil. The decrease of soil albedo in the latter case may be explained by the fact that the albedo of wa-

424

Albedo of the Underlying Surface and Clouds

ter (see the next paragraph) is significantly less than the albedo of land surfaces. The decrease of albedo in soil watering largely accounts for a notable increase of the radiation balance of irrigated fields. In Kirillova’s [39] estimation the irrigated surface albedo is 5 to 8 percent less than that of a dry area. Besides humidity, the form of the soil surface is another factor influencing albedo. Skvortzov [40] has investigated the albedo of the same soil under different tillage (lowland light clay earth). The results of his measurements are given in Table 7.4. TABLE 7.4 Albedo of Lowlund Light CIUYEarth. After Skvortzov [40] (dl = 0.3 to 2.0 p )

Surface

Albedo, %

Leveled Covered with dust Covered with a crust from dryiig after watering Covered with small clods Covered with big clods Newly ploughed field

30-31 28 27 25 20 17

As seen from Table 1.4, the albedo shows a significant decrease with the increase of surface roughness. It was pointed out in Chapter 1 that such variation of reflectivity is effected by the relatively more intensive radiation absorption by “cavities” of a rough surface in comparison with a horizontal surface. Also effective in albedo variation is color of the soil. This can be seen from consideration of data (Table 7.3) related to sand. By changing the soil surface color (dyeing), it is possible to change the thermal level of the soil, and permits active interference with the soil thermal regime. 2. Albedo of Vegetative Covers. As in the case of soil albedo, the albedo of vegetation is strongly variable (see [37c, 471). Table 7.5, borrowed from the same sources as Table 7.3, gives albedo values of different vegetative covers (grass, forest). This table gives only approximate albedo values of different vegetative covers. Their actual variety takes place in far wider limits, showing at the same time variations within given types. Observations demonstrate, for example, that albedo of plants varies considerably from phase to phase in the process of growing.

7.2. Albedo of Various Continental Underlying Surfaces

425

TABLE 7.5 Albedo of Different Kinds of Vegetation (AL

= 0.3

Vegetation

to 2 . 0 , ~ ) Albedo, %

Rye and wheat in different phases of vegetation: Summer wheat Winter wheat Winter rye Grass cover: High standing grass Green grass Dry grass wizened in the sun

10-25 16-23 18-23

18-20 26 19

Forest vegetation: Tops of oak Tops of pine Tops of fir

18

14 10

Different small plants: Cotton Lucerne (early florescence) Rice field Lettuce Beet Potatoes Heather wasteland

20-22 23-32 12 22 18 19 10

3. Albedo of Snow and Ice. The albedo of snow is the most variable quantity. Attaining almost 100 percent for dry snow, it go as low as 20 to 30 percent in the case of dirty moist snow. Figure 7.7 gives a curve of snow albedo variation during winter as observed by Kalitin [Chapter 3, Ref. 1021 at Pavlovsk. This figure is very spectacular in illustrating the great variability of the snow albedo value. The albedo fluctuations in the given case result from fresh snowfall, thaws, and gradual pollution of snow. For instance, there were no thaws until March 12, and the albedo was almost constant at 80 percent. After the thaws on March 12-14, 15-17, and 20-23, the albedo lowered to 66 percent. A fresh snowfall on March 24 caused a new increase of albedo up to 80 percent. Still more pronounced were the albedo variations due to similar causes in April.

426

Albedo of the Underlying Surface and Clouds 100-

a

40 -

2025 4

O

ii h '

14

' ii

FEE

22 26 30 16' 24'is' MAR

4

12

8

i'i

16 23 4

14

10' ' 19' 25' APR DAYS OF THE MONTH

9

k

16 22

' 12 ' 19'

i4

MAY

FIG. 7.7 Albedo variation for snow cover.

High albedo values are observed in Antarctic conditions, where its annual mean over the entire Antarctic is about 83 to 84 percent. The same holds for Arctic snow albedo. Observations of Chernigovsky [41] at the station North Pole 4 have found 96 to 98 percent albedos in newly fallen snow. In Table 7.6 are presented observational results of R u s h [42] at Mirny, characterizing the snow albedo variability in the Antarctic conditions, depending upon solar height and cloudiness. TABLE 7.6 Variation of Dependence of Snow Albedo (Percent) upon Solar Height and Cloudiness at Mirny Station. After Rusin [42]

Solar Height, deg Cloudiness

o/o

5

10

15

20

25

30

35

40

45

91

83

81

80

78

76

75

74

74

10/0, all forms of upper

and middle layer clouds 10/10

92

91

84

84

81

75

80

79

90

89

89

89

92

93

92

93

Hence it is seen that the albedo of snow has a notable daily range (these data will be discussed later on). Also significant is the dependence of albedo upon cloudiness: The denser and the lower the clouds are, the more pronounced is the albedo increase, especially at great solar altitudes. A still stronger dependence of albedo on cloudiness conditions was observed at the Oasis station, where the albedo measured under solid cloudiness of the upper and middle layers appeared to be by 50 percent higher, on the average, than under a clear sky.

427

7.2. Albedo of Various Continental Underlying Surfaces

Albedo of ice is less known. In Table 7.7 are given the results of sea ice albedo measurements in the area of the White Sea, performed by Kuzmin [43]. As seen from this table, the albedo of sea ice surface varies from 30 to 40 percent in the given case. TABLE 7.7 Albedo of the Surface of Ice in the White Sea. After Kuzrnin [43]

State of Surface Sea ice, slightly porous, milky blue Thawed ice, strongly porous

Albedo, % 36 41

Ice slightly powdered with snow Ice area under a layer of clean water, 15 to 20 crn thick white porous ice

31

Layer of frozen water on the sea bottom surface

12

26

Considerably higher values of ice albedo were obtained by Koptev [44] from aircraft measurements in the region of the Laptev Sea and the Kara Sea (August-September, 1959). According to these data, the mean albedo of ice under thawed snow is 76 percent. With a decrease of the upper snow, however, the albedo of ice cover becomes notably less. For instance, at the snow spread of 60 percent, the albedo of ice is only about 58 percent. It is natural that the albedo of ice is also dependent upon mass area. Observations of Koptev [44] point out a linear increase of ice albedo with increasing areal dimensions. 4. Daily Range of Albedo. Observations detect daily albedo variation expressed in a dependence of albedo upon solar height. In the majority of cases observed, there was an increase of albedo with the decreasing height of the sun. Figure 7.8 presents isopleths of albedo of the underlying surface, plotted by Yaroslavtzev [20, 451 from data of all-year radiation observations at Tashkent. Inspection of this figure shows that in all months the daily range of the albedo of the underlying surface at Tashkent is characterized by the presence of an albedo minimum in the near-noon hours. The elaborate investigation of the daily range of vegetation albedo carried out by Tooming [47] has shown that the increase of albedo with a decrease of solar height is attenuated: At solar altitudes about 5 to loo, a maximum is observed, after which the albedo considerably decreases. Shifrin [27] and Ross [48, 491 have worked out a theory of radiation re-

428

Albedo of the Underlying Surface and Clouds

flection by vegetative cover, which considers the latter as a layer of turbid medium with given optical properties. Calculations show that such a model of vegetative cover not only enables qualitative description of the regularities of the vegetation albedo daily variation, but also-provided the corresponding parameters are chosen-quantitative agreement of theoretical calculations with the data of observation. It turns out that the main factor in the daily albedo variation is the variation of the relation between direct solar and diffuse radiation. The albedo for diffuse radiation shows a weak dependence upon solar height, varying from 22 to 26 percent. On the contrary, the albedo for direct solar radiation notably increases with a decrease of solar height. This is the cause for the increase of albedo at a decreasing height of the sun in the daily range, whereas the decrease of albedo at very low altitudes of the sun results from the increase of the portion of diffuse radiation, the albedo for which is less than for direct solar radiation.

FIG. 7.8 Zsopleths of the albedo of natural underlying surfaces at Tashkent.

Tooming [47] has found an asymmetry in the daily range of vegetation albedo : The afternoon values exceed the morning values by approximately 10 percent. This appears to be explained by the variation in the spectral reflectivity of leaves during the day. Quite singular is the daily albedo variation ir, the case of snow cover. The observational results of R u s h [42] give evidence of the increase of

7.2. Albedo of Various Continental Underlying Surfaces

429

the albedo of dry clean snow with a decrease of solar height. Different authors obtain analogous results. Certain observations find, however, an inverse daily variation of snow albedo. For example, according to measurements of snow albedo in March-April 1952, in the Leningrad region, carried out by Ter-Markariantz [50], the daily albedo range has a noon maximum and a decrease at the increase of atmospheric mass (decreasing solar height). Observations on March 21, 1952 gave the following results: 2

m A,

%

3

90.5 89.0

4

5

87.2 86.5

It should be noted that the processing of the foregoing observations was performed by improved methods, taking into account the spectral and angular corrections for the incident and reflected radiant fluxes. Similar results, were also obtained by Yaroslavtzev [20]. An absence of the daily snow albedo range was observed. To understand the regularities of the daily range of snow albedo, let us clarify the possible causes for the albedo variation in dependence upon solar height. As was shown in Sec. 7.1, the albedo of different underlying surfaces essentially differs over spectral regions, and consequently the reflection of radiation by surfaces is selective. On the other hand, the spectral composition of direct solar radiation incident on an underlying surface varies at a varying solar height, for when the latter decreases the “reddening” of solar radiation takes place. Taking into consideration these two circumstances, it is possible to conclude that the variation in the spectral composition of the incident direct solar radiation flux in dependence upon solar height may be a cause for the daily albedo variation. Another cause for this variation is the dependence of surface reflectivity upon angle of incidence of radiation. As we found in Sec. 1.4, reflectivity of dielectrics considerably increases with an increase of the angle of incidence. Qualitatively the same effect must take place in the case of natural underlying surface. In this case the influence of surface roughness is also important. Detailed calculations performed by Kastrov [51] for various models of surface roughness show that the influepce of the angular dependence of reflection on the daily albedo range is considerably more than the influence of the spectral composition variations of the incident radiation. It is of importance, however, that (according to calculations by Kondratyev and Ter-Markariantz [50,521) with different relation of the global radiation spectral composition and the spectral albedo both increase and decrease of albedo with a decrease in solar height may be observed. As to

430

Albedo of the Underlying Surface and Clouds

the angular dependence of reflection, it always determines the increase of albedo at an increasing height of the sun. This means that the variability of the snow albedo daily range is evidently caused first of all by the “spectral” factor.

5. Annual Range of Albedo. As the character of the underlying surface at a given point considerably varies during the year, this causes the albedo to have a notable yearly variation. For geographical points in the temperate and northern latitudes, it is characteristic to find a great increase of albedo from the warm half of the year to the cold when snow falls and a snow cover appears. The regularities of the annual albedo range at different geographical points are not yet sufficiently known. The most interesting work in this field has been done by Yaroslavtzev [20] on the basis of allyear actinometric observations at Tashkent. Figure -7.9 presents the curve of the annual variation in monthly albedo means of natural earth cover at Tashkent. We see that even at the latitude of Tashkent, the increase of al-

FIG. 7.9 Annual variation of the albedo of natural underrying surfaces at Tashkent.

bedo in winter, due to the appearance of snow, is quite notable. In individual years, depending upon yearly climatic peculiarities, the annual albedo variation may differ from year to year. Table 7.8 gives data characterizing the annual range of monthly albedo means according to observations by Yaroslavtzev at Tashkent (Central Asia). As seen from Table 7.8, the albedo means for the same month may vary considerably from year to year. For instance, in December this varia-

43 1

7.3. Albedo of Water Basins

TABLE 7.8 Annual Range of Monthly Albedo Means as Observed at Tashkent in Different Years (Percent). After Yaroslavtzev [20]

Month

1945

1946

1947

1948

1949

1950

Jan. Feb. Mar. Apr. May June July Aug. Sept. Oct. Nov. Dec.

46 48 28 17 18 18 21 24 24 22 15 36

34 20 23 25 24 24 24 21 24 22 26 61

47 15 20 23 17 18 16 19 23 19 19 28

26 27 24 18 18 19 22 22 22 24 25 51

50 34 13 19 16 15 15 18 20 19 18 16

55 45 17 17 17 20 21 22 19 19 35 56

tion is from 16 percent in 1949 to 61 percent in 1946, that is, a differential of almost four times. Such pronounced variations of albedo means for winter months appear to be characteristic at southern points where the duration of snow cover shows strong fluctuations in different years. It is natural in intermediate and northern latitudes with stable snow that the year-to-year albedo variation in winter would be less. The climatic peculiarities of individual years affect even the annual albedo means. For example, the average annual albedo at Tashkent varies within 19 to 20 percent. The observational data of Yaroslavtzev in Table 7.8 show that extreme care must be taken in the determination of albedo means used in various climatic calculations of the underlying surface radiation balance, since even average albedo values are variable to some degree.

7.3. Albedo of Water Basins The available data on the albedo of water basins are not numerous. On the other hand, it has been stated that the albedo of smooth transparent sea surfaces (or, in general, water basins) can be theoretically computed with sufficient accuracy. Therefore let us first consider the results of theoretical calculations of water basin albedo and then give data of observations.

432

Albedo of the Underlying Surface and Clouds

1. Albedo of Water Basins for Direct Solar Radiation. Since clean water is a very poor conducting medium, the reflectivity of its surface for unpolarized light can be computed from Fresnel’s formula (see Sec. 1.4): A=-

1 2

sin2(i n) [ sin”i + n) -

+

tan2(i - n) tan2(i n)

+

1

(7.1)

where i is the angle of incidence of radiation and r is the refraction angle. The angles of incidence and refraction of (7.2) are interconnected by the familiar sine law: -sin = ni sin n where n is the refraction index for the medium (for water in the visible region, n = 1.33). Table 7.9 gives the results of albedo calculation for a water surface (n = 1.33) according to (7.1). By using Table 7.9 it is possible to determine the albedo of sea (lake, river) for direct solar radiation of any zenith distance (numerically equal to the angle of incidence) or any height of the sun. Observations over the albedo of a smooth nonturbid sea surface agree satisfactorily with data of theoretical calculations. This can be seen from Fig. 7.10, which presents the results of measurements and calculations of water surface albedo in cloudless weather and in the absence of roughness TABLE 7.9 Dependence of Water Surface Albedo upon Angle of Incidence of Radiation

Angle of Incidence, deg 0

10 20 30

40 50 60 70 80 90

0

1

2

3

4

5

6

7

8

9

2.1 2.1 2.1 2.1 2.1 2.1 2.1 2.1 2.1 2.1 2.1 2.1 2.1 2.1 2.1 2.1 2.1 2.1 2.1 2.1 2.1 2.1 2.1 2.1 2.1 2.2 2.2 2.2 2.2 2.2 2.2 2.3 2.3 2.3 2.3 2.4 2.4 2.4 2.4 2.5 2.5 2.5 2.6 2.7 2.8 2.9 3.0 3.1 3.2 3.4 3.5 3.6 3.8 4.0 4.2 4.4 4.1 5.0 5.4 5.8 6.2 6.6 7.0 7.5 8.2 9.6 10.4 11.4 11.4 12.4 13.6 14.8 16.2 17.8 19.6 21.5 23.8 26.0 28.8 31.4 35.0 38.6 42.8 47.6 52.9 58.6 65.0 12.0 80.6 89.6 100.0

7.3. Albedo of Water Basins

433

[53]. At the same time the data of Fig. 7.10 show that there are certain discrepancies between the measurement and calculation results. For instance, according to the most careful observations by Grishchenko [53], a full coincidence of the measured and calculated values takes place solely at h, = 17O. If h, < 17O, the observed albedo is lower than that calculated. An inverse relation is valid at h, > 17'. This discrepancy is explained by the influence of two factors: (1) sea turbidity (reverse scattering of radiation by sea water thickness); and (2) contribution of the reflected diffuse radiation whose dependence upon solar height is more complex than suggested by the Fresnel formula (8.1). Also important is the fact that the actual sea surface is never ideally smooth.

"1

lOOr

I

\

FIG. 7.10 Albedo of the wafer surface for direrent solar heights in clear weather. (1) from Fresnel's formula; (2) from observations by Angstrom; (3) from Kumin's observations; (4) from data of the Karadag Observatory.

Pyranometric measurements conducted by Ter-Markariantz [54, 551 have shown that the value of the radiant flux scattered backward by the sea is strongly dependent upon water transparency. For instance, at h, = 25' and a transparency in the Secci disk equal to 10 to 15 m, the outgoing radiation scattered by sea is about 0.0015 cal/cm2 min. At the same solar heigh and a transparency of 1 to 2 m, the contribution of back scattering

434

Albedo of the Underlying Surface and Clouds

increases from 0.0055 to 0.075 cal/cm2 min; that is, it is four to five times greater. It is natural that this would appear to denote a notable dependence of sea albedo upon water transparency. Observations show that the portion of reverse scattering in the reflected radiation becomes larger at an increasing solar height. However, the relative contribution of radiation scattered by water thickness to albedo (the relation of radiation scattered by sea to global radiation) is practically independent of solar height, since with the increase of the latter the global radiation notably increases. Thorough experimental investigations of lake albedo, Forsh [56] showed that at solar heights exceeding 20°, the tint (transparency) of water mass does not affect the albedo. At low altitudes of the sun, its effect is quite pronounced.

2. Albedo of Water Basins for Difluse Radiation. It is perfectly evident that the albedo of water basins for diffuse radiation incident on the water surface at various angles from different sky sections must be other than the albedo for direct solar radiation. It is also natural that albedo values for diffuse radiation will be chiefly determined by the regularities of the distribution of diffuse radiation over the sky. Observations of the albedo of the sea for diffuse radiation give evidence of great variability of this quantity. According to data taken by Kuzmin [43] at the White Sea, in the case of solid cloudiness of the lower layer the albedo values for diffuse radiation vary at times within 4 to 18 percent. As average values of the sea albedo for diffuse radiation at solid lower layer cloudiness, Kuzmin gives values from 4.8 to 8.1 percent. Approximately similar results have been derived by Ter-Markariantz [54, 551. Since the available data on the water basin albedo for diffuse radiation are scarce, theoretical calculations of this value are all the more interesting. The first to perform theoretical calculations of diffuse radiation reflection from smooth (polished) surfaces was Gershun [57]. If we assume that the diffuse radiation is isotropic (that is, that its intensity is independent of direction) the albedo of a smooth surface for diffuse radiation, A D , can be determined from the following formula: AB= 2

A ( @ cos 8 sin 8 d8 0

where 8 is the zenith angle, and A ( @ is the albedo value for a ray whose angle of incidence on the reflecting surface equals 8. The A ( 8 ) values may be calculated for different 8 from the Fresnel formula. Gershun has calculated the albedo A(@ from the above formula

435

7.3. Albedo of Water Basins

for substances with different refraction indexes, n, varying from 1 to 2. In the case of water surface (n = 1.33), the albedo value for isotropic diffuse radiation was estimated by Gershun at about 6.5 percent. Later calculations by Lauscher [58, 591 gave values A D = 6.6 percent, which practically coincide with the above result by Gershun. The diffuse radiation is not actually isotropic, especially in the case of partial cloudiness. It is therefore necessary to calculate albedo values of water basins for diffuse radiation, taking into account the observed angular distribution of radiation. This author and Ter-Markariantz [50] have made similar calculations of sea albedo from data of observations over the angular diffuse radiation distribution in the Crimea (Karadag) at noon and with a cloudless sky. The results of these calculations are given in Table 7.10. TABLE 7.10 Sea Albedo Values for Diffuse Radiation in a Cloudless Sky (1951). After Kondratyev and Ter-Markariantz [50] Date of Observation June June June July

28 30 29

4

State of Sky

Albedo, %

Clear Clear Cirrus clouds at horizon Cumulus clouds in West

8.5 8 9.9 11.1

We see in Table 7.10 that the albedo values for diffuse radiation, taking into account the observed angular distribution of the latter, are greater than the corresponding values computed on the assumption that diffuse radiation is isotropic. The discrepancy is especially marked in the case of clouds near the horizon. This can be easily explained. It is known (see Chapter 6) that the intensity of diffuse radiation increases toward the horizon, the more so on the sunny side. At the same time, the angles of incidence of radiation downcoming from the sky sections near the horizon are very significant; that is, the albedo for such radiation is quite great. It is natural therefore that taking account of the actually observed distribution of diffuse radiation will be more likely to lead to greater albedo values for the full flux of diffuse radiation than will the idealized case of isotropy. Furthermore, we also see that variations in the directional distribution

436

Albedo of the Underlying Surface and Clouds

of diffuse radiation affect the albedo values. This confirms the conclusion that the regularities of directional distribution of diffuse radiation are the main factors when determining albedo values for radiant flux. As is known, the diffuse sky radiation is polarized. Calculations by this author and Kudriavtzeva [60] showed that the dependence of albedo for diffuse radiation upon the state of polarization was well marked. For the albedo of global radiation, however, the polarization of diffuse radiation in a cloudless sky is practically unimportant. Calculations for the conditions of continuous low cloudiness show that in this case the albedo usually does not exceed 6 percent. The measured values for the same conditions vary from 5 or 6 to 12 percent, averaging 7 to 8 percent; that is, they exceed the calculated values by 1 to 2 percent. This difference should be attributed to calculations that neglect the effect of reverse scattering and to the assumption that water surface is horizontal.

3. Albedo of Water Basins for Direct Solar Radiation in the Presence of Commotion. The preceding data refer to the albedo of smooth water surfaces for direct solar radiation. It is natural that the presence of roughness must affect albedo values, since in this case the angles of radiation incidence relative to the sea surface undergo changes. Consider the results of approximate calculations of the roughness effect on albedo as performed by this author and Ter-Markariantz [61, 621. To calculate the albedo of rough sea it is necessary t o choose a certain wave profile. We know that in the first approximation the sea-wave shape may be considered cycloidal, and the following calculations are based on this assumption. The principle of the calculation is: Having expressed the angle of incidence of solar radiation on the sea surface through a solar zenith distance, 8 , and an angle of inclination of the wave profile at the given point, /?, we use the Fresnel formula to compute the dependence of albedo upon the horizontal coordinate A ( x ) . By computing the value

we obtain the mean sea surface albedo. Here L is the length of the cycloid section corresponding to the full circumference orbit (wavelength). Similar calculations can also be performed for the case where, in place of a given cycloidal wave profile, experimental data are used to describe statistical characteristics of the form of rough sea surface. As a result of calculations of the dependence A ( x ) and the subsequent averaging, approximate albedo means were obtained (Table 7.11). These

437

7.3. Albedo of Water Basins

data clearly show that the albedo of rough sea differs greatly from that of calm sea. TABLE 7.11 Albedo of Smooth and Rough Sea Surfaces. After Ter-Markariantz and Kondratyev [61,62]

Albedo of Sea, %

Solar Zenith distance, deg

0 30 60

Rough

calm

13.1 3.8 2.4

2.2 6.2

2.1

At small zenith distances of the sun, commotion of the sea provokes a great increase of albedo. For example, at 8 = 0' in the case of calm sea, the albedo is about 2 percent, whereas in the presence of waves, it increases up to 13 percent. Although the latter value is exaggerated as a result of approximated calculations, nevertheless it is illustrative in that the actual roughness is shown to cause a notable increase of albedo at small solar zenith distances. In the case of long zenith distances the albedo of rough sea, on the contrary, is less than for smooth surface. If, for instance, the albedo for calm sea at B0 = 60' is 6.2 percent, as calculated from Fresnel's formula, in the presence of waves it decreases to 2.4 percent at the same zenith distance BO. Such decrease of wave albedo at large O0 is perfectly natural because in this case an important factor appears, namely, crests of waves shading horizontal wave surfaces. The reflection of solar radiation takes place from the steep sections of wave crests, relative to which the angle of radiation incidence is very small. Taking into account the effect of diffuse radiation shows that it somewhat decreases the dependence of albedo upon roughness. Ter-Markariantz [63] and Mullamaa [63a, 63b] have performed theoretical calculations of albedo of rough sea, taking into consideration the statistical regularities determining wave forms. The comparison of the above results with observational data leads to full qualitative agreement. We write the following relation : A,

= A,

- AA

where A , is the albedo in roughness, A , is the albedo at calm. According

438

Albedo of the Underlying Surface and Clouds

to observations by Ter-Markariantz [63, 641, the correction for roughness in the solar height range 10' < h < 60' assumes the below values: ha'

AA, %

10 f10.0

20

30

40

+1.5

-1.0

-1.8

50

60

-2.5

-2.9

These data may be used for introduction of corrections for roughness when theoretically calculating albedo of water basins. 4. Albedo of Water Basins for Global Radiation. The sea albedo values for global radiation are of highest practical importance. It is evident that the albedo for global radiation can be determined by the following relation:

A=

+

Rs iRD R D ~ S' D

+

where Rs and RD are fluxes of direct solar and diffuse radiation reflected from the sea surface, RDg is a radiant flux scattered by water thickness and outgoing through the surface of the water basin, and S' D is the global radiation. The above analysis of the dependence of the albedo components upon various factors shows that the albedo of a water basin for global radiation must be first of all dependent upon the height of the sun, cloudiness conditions, water surface state (roughness), and water transparency. In Fig. 7.1 1 are presented the results of prolonged observations by Grishchenko [53] at the Black Sea (see also [46]) related to conditions of cloudless and little cloudy sky (force 1 to 2, not more). The nonmonotonic character of the dependence of albedo for global radiation upon solar height is interesting: At very low altitudes, we can observe a decrease of albedo due to the increasing contribution of diffuse radiation, for which the albedo at such altitudes is far less than for the global radiation. The effect of roughness is very clear, especially at low solar heights. The comparison of curves 1 and 2 in Fig. 7.1 1, obtained from observations with the Fresnel curve (3) finds that the latter is only schematic (and wrong at low altitudes) in describing the water albedo variations in dependence upon solar height. A considerable dispersion of points in relation to the curves in Fig. 7.11 indicates the influence of sea transparency and other factors and also of measurement errors. As has been stated, the albedo of water basins is much dependent upon cloudiness conditions. In the case of global radiation this conclusion can be illustrated by data of calculations by Ter-Markariantz [63], given in

+

439

7.3. Albedo of Water Basins

ap

4

10

0

20

30

50

40

60

70

ho

FIG. 7.11 Dependence of the sea surface albedo upon solar height.

Fig. 7.12. Hence it follows that at low altitudes of the sun, the water albedo strongly decreases with an increase in cloudiness. At great heights 26 -

-

22

18 -

z 14-

10 -

30 6-

40

-

560-

2 '

70

0

i

6

I

I

I

1

4

8

1

0

DEGREE OF CLOUDINESS

FIG. 7.12

Water albedo at different solar heights z and cloudiness degrees for global radiation.

440

Albedo of the Underlying Surface and Clouds

this dependence becomes inverse and far weaker (in absolute values). For climatological calculations the problem of water albedo means (for a given day or month) presents much interest. Table 7.12 gives the results of calculations of daily albedo means in different cloudiness conditions for different latitudes and months as performed by Ter-Markariantz [63, 651, who has measured the albedo for global radiation at a roughness of force 1 to 3 and water transparency 8 to 12 m in the Secci disk. Such conditions may be considered the most characteristic for the U.S.S.R. seas. It should be noted, however, that the data of Table 7.12 relate to the case of lower layer cloudiness, nontransparent for solar radiation. Analogous calculations have been made by Sivkov [64]. 7.4. Albedo of Clouds

The preceding section has given data characterizing the great scattering power of clouds. Since clouds are intensive radiation scatterers, it is natural that they must possess large albedo. Direct measurements of cloud albedo are possible only during actinometric observations from aircraft or balloons elevated above the upper cloud surface. In recent years quite a number of such measurements have been made; for example, by V.A. Belinsky [Chapter 5, Ref. 1021, N. I. Cheltzov [Chapter 5, Ref. 1111, Koptev [44], M. Neuburger [Chapter 5, Ref. 1121, Fritz [66], Piatovskaya [67], Robinson [68], etc. For measuring fluxes of incident and cloud-reflected radiation, pyranometers are used. One instrument is fixed on the aircraft, with its receiving surface upward ; another has its surface directed downward. Such aircraft measurements have their peculiarities, compared with the usual methods for pyranometric radiation measurements ; for example it becomes important to make allowance for aircraft pitch and yaw, which causes deviations of the pyranometer’s position from the horizontal. Another factor to be considered is variation of temperature and air pressure, which also affect readings. Observations by Cheltzov [Chapter 6, Ref. 1111 in the Moscow and Arkhangelsk regions have found a considerable dependence of the albedo of clouds upon the latter’s vertical power. Figure 7.13 gives the results of flights at Arkhangelsk in 1949-1950. Each point characterizes an albedo mean obtained as the result of averaging over approximately 40 single values. These measurements all were taken under conditions of solid cloudiness. Although the dispersion of the above points is great, nevertheless it is possible to plot smoothed curves that characterize the dependence of different cloud albedos upon thickness. It should be mentioned

441

7.4. Albedo of Clouds

TABLE 7.12 Daily Means of Albedo at Different CIoudiness Degrees. After Ter-Markarianti [63, 651 Cloudiness, in tenths

Jan.

Feb. Mar. Apr.

May June July

9.3 9.2 8.9 8.6 8.0 7.7 7.0

7.7 7.6 7.5 7.5 7.2 7.0 7.0

6.7 6.7 6.7 6.8 6.8 6.8 7.0

5.7 5.8 5.9 6.0 6.4 6.6 7.0

5.8 5.8 5.9 6.1 6.3 6.6 7.0

5.6 5.7 5.8 5.9 6.2 6.5 7.0

5.6 5.7 5.8 6.0 6.3 6.5 7.0

5.6 5.6 5.7 5.9 6.2 6.5 7.0

6.3 6.4 6.4 6.5 6.6 6.8 7.0

0 2 4 6 8 9 10

12.6 12.2 11.7 11.0 9.7 8.6 7.0

9.1 9.0 8.8 8.5 8.0 7.6 7.0

7.5 7.5 7.4 7.4 7.4 7.2 7.0

6.3 6.3 6.4 6.5 6.6 6.8 7.0

6.1 6.1 6.2 6.4 6.5 6.7 7.0

5.7 5.7 5.8 6.0 6.2 6.6 7.0

5.7 5.8 5.9 6.0 6.4 6.6 7.0

6.4 6.4 6.4 6.5 6.7 6.8 7.0

8.1 10.7 11.4 13.3 8.0 10.6 11.1 13.0 8.0 10.2 10.8 12.4 7.9 9.7 90.1 11.6 7.5 8.9 9.1 10.1 7.3 8.1 8.3 8.9 7.0 7.0 7.0 7.0

50' 0 2 4 6 8 9 10

18.0 17.2 16.3 14.9 12.4 10.3 7.0

13.2 13.8 13.2 12.2 10.5 9.1 7.0

9.6 9.5 9.2 8.9 8.3 7.8 7.0

7.7 7.7 7.6 7.5 7.3 7.2 7.0

7.2 7.2 7.1 7.1 7.1 7.0 7.0

6.7 6.7 6.7 6.8 6.8 6.9 7.0

6.6 6.7 6.7 6.8 6.8 6.9 7.0

6.2 6.9 6.9 6.9 7.0 7.0 7.0

8.1 11.0 16.0 22.1 8.0 10.7 15.3 21.4 7.9 10.5 14.7 20.0 7.8 9.9 13.6 18.3 7.5 9.0 11.4 15.0 7.3 8.2 9.7 12.5 7.0 7.0 7.0 7.0

27.0 25.8 24.1 21.5 17.0 13.1 7.0

2.03 19.4 18.3 16.7 13.8 11.2 7.0

13.1 12.8 12.1 11.3 9.9 8.8 7.0

8.9 8.7 8.6 8.3 7.9 7.5 7.0

7.4 7.3 7.3 7.2 7.2 7.2 7.0

7.4 7.3 7.3 7.2 7.2 7.1 7.0

7.7 7.6 7.6 7.5 7.3 7.2 7.0

8.0 10.2 16.8 25.6 27.4 8.0 10.0 16.2 24.8 26.3 7.9 9.7 15.4 23.1 24.6 7.8 9.4 14.2 20.8 22.1 7.6 8.6 11.9 16.6 17.6 7.4 7.9 10.0 12.8 13.7 7.0 7.0 7.0 7.0 7.0

-

27.1 25.8 23.8 20.3 16.1 12.4 7.0

20.3 19.6 18.3 16.7 13.6 11.1 7.0

12.5 10.0 11.7 9.8 11.6 9.5 10.9 9.1 9.7 8 . 5 8.6 7.9 7.0 7.0

8.9 8.8 8.6 8.3 7.9 7.6 7.0

9.2 10.5 15.8 25.6 9.1 10.3 15.3 24.6 8.9 10.0 14.5 23.1 8.6 9.5 13.8 20.8 8.1 8.8 11.6 16.6 7.7 8.0 9.6 13.1 7.0 7.0 7.0 7.0

Aug. Sept. Oct. Nov. Dec.

v = 30' 0 2 4 6 8 9 10

5.8 5.8 5.9 6.0 6.4 6.6 7.0

8.2

8.1 8.0 7.8 7.6 7.4 7.0

9.7 9.5 9.3 8.9 8.3 7.8 7.0

7=400

7

=

v = 60' 0 2 4 6 8 9 10

7 = 70' 0 2 4 6 8 9 10

-

-

24.3 23.4 21.5 18.8 14.6 11.3 7.0

442

Albedo of the Underlying Surface and Clouds 700. t0 -

r

600-

t.e--

I

-

-h --3

4 l

500-

O

I /

D

d /

E

I4 0 0 -

g

300-

5

200.

// //

/

/ /

&’

A

/+

,

/ 4.A

,’,I

100.

4

/

d a

/ ’

BA/da

$7

s

x

s

i

J /

9

/

I

,’ ,,,/

O/

,,‘, .. / ’/ /‘ 4,

/.<<

r

that the above curves were extrapolated to the zero cloud power, after considering that, with the tendency of cloud power toward zero, the cloud albedo must approach the albedo of the underlying surface. As seen, in all cases there is observed a monotonous increase of albedo at the increasing cloud thickness. This process is more rapid at small thicknesses (up to 200 to 300 m) and slows down at a continued increasing power. The cloud form strongly affects albedo; for instance, according to the Arkhangelsk data, the cloudiness of equal thickness (300 m) but of different form was characterized by the following albedo values: Ac at about 73 percent, Sc at 64 percent, mixed Sc-Cu at about 52 percent, Sc with clearings at 46 percent. Analogous flights at Moscow give these values for the same cloud power as: Ac at 71 percent, St at 56 percent, Sc with clearings at 46 percent. We see that the greatest albedo values are observed in the case of stratocumulus cloudiness. The cause for this appears to be that clouds of this form contain a large number of ice crystals whose property produces intensive reverse scattering. The comparison of Cheltzov’s data with other results shows fairly satisfactory agreement. For example, in Fritz’s [66] estimation the albedo of altostratus clouds varies within the limits from 39 to 83 percent. Koptev’s [44] measurement data for the Laptev and Kara Seas (1959) can be seen in Table 7.13. These data illustrate the presence of a notable dependence of cloud albedo upon character of the underlying surface. For example, the albedo in the interlayer between stratus and altocumulus clouds above water is evaluated at twice less than the albedo above ice.

443

7.4. Albedo of Clouds

TABLE 7.13 Albedo of Low Clouds. After Koptev [44]

Cloudiness Cloudiness degree 10 (St, Sc, Ns) Cloudiness (St, Sc, Ns), ice through clearings The same, water through clearings Inside the cloud over ice The same, over water In the interlayer between St and Sc, over ice The same, over water

Albedo, % 58 70 32 65 35 68 33

Number of Cases 65 21 16 7

40 5 5

From what has been said above, it follows that the albedo cloud is strongly dependent upon many factors. Meanwhile, it is often important to know a mean cloud albedo. According to Fritz [66], this mean, computed with allowance for spreading of clouds of different form and capacity, makes about 50 to 55 percent. Robinson [68] gives a somewhat higher value of 60 percent with a variation from 29 to 87 percent. In Sec. 6.3 we considered the results of theoretical computations of stratus cloud albedos by Feugelson [Chap. 6, Ref. 291. Comparing these results with the above observational data, we see that the main qualitative conclusion about the considerable dependence of albedo upon cloud thickness is clearly valid for both cases. The quantitative coincidence is, of course, hard to expect because theoretical calculations can fully correspond to the actual cloud conditions, neither in initial data nor in the assumptions for derivation of formulas. An interesting conclusion on the basis of the theoretical calculations considered above was made about the presence of a marked dependence of thin cloud albedo upon zenith solar distance at 8, > 30'. There are no observational data on this dependence at present. It may be that the dispersion of points in Fig. 7.13 is due somewhat, to this same dependence of albedo upon zenith solar distance. The problem of radiation absorption by clouds is quite interesting. However, this problem has not yet been clarified. For example, according to Cheltzov, the portion of the absorbed radiation is very small, varying from 3 to 7 percent. Measurements by Robinson have led to bigger values, from 13 to 25 percent, and a mean at 22 percent. In Fritz's evaluation the absorption of solar radiation may reach 20 to 30 percent. According

444

Albedo of the Underlying Surface and Clouds

to Monteith [69], the absorption of solar radiation by clouds averages 8 to 9 percent. If we consider the big absorption values to be correct and take into account that absorption takes place mainly in the near-infrared, this means that the albedo of clouds will be comparatively small in the infrared and very high in the visible and ultraviolet. In Robinson’s [68] estimation the mean albedo for the infrared must be about 45 percent and up to 85 percent in the visible and ultraviolet regions. Unfortunately no experimental data on the spectral cloud albedo are available.

7.5. Geographical Distribution of Albedo The values of albedo given above for various surfaces are local; that is, they characterize reflectivity of comparatively small surface areas at a certain moment of time. Meanwhile, for climatological calculations of net radiation, it is important to know average albedos of extended underlying surfaces over prolonged time intervals (month, year). The consideration of albedo values as given in Secs. 7.2 and 7.3 shows that from the standpoint of classification of different territories over albedo values for temperate latitudes, it is necessary first of all to isolate forests, woodless areas, and water basins (lakes and seas). For this, the presence of snow or ice is extremely effective in relation to albedo values. Table 7.14 illustrates characteristic albedos of various underlying land surfaces, obtained by Mukhenberg [70] from generalization of numerous observational data. It is natural that, being averaged (may be considered as monthly means), all these data present only the most characteristic features of different surface albedos. In actual fact, albedo of a given type of underlying surface may vary, depending on various factors, within fairly wide limits. Figure 7.14 presents curves of recurrence of albedo values for different underlying surfaces in different seasons. These data by Barashkova et al. [37a] have been obtained for the U.S.S.R. territory. The first thing that draws attention here is a clear distinction between the recurrence curves related to winter (stable snow cover) and other seasons (excepting semidesert and desert). The comparison of these data with the values of Table 7.14 is very spectacular in showing to what degree the averaged values of Table 7.14 are approximate relative t o actuality. Making use of Table 7.14, Mukhenberg [70] has plotted charts of geographical albedo distribution over the continental Northern Hemisphere for January and July (Figs. 7.15 and 7.16).

445

7.5. Geographical Distribution of Albedo

TABLE 7.14 Albedo of Underlying Land Surfaces. After Muhenberg [70]

Albedo, %

Type of Surface Stable snow cover in high latitudes (above 60') Stable snow cover in temperate latitudes (less than 60') Forest with stable snow cover Unstable snow cover in spring Forest with unstable snow cover in spring Unstable snow cover in fall Forest with unstable snow cover fall Steppe and forest during the period between snow melting and mean daily air temperature from l0'C Tundra in the period between snow thaw and mean daily air temperature from l0'C Tundra, steppe, deciduous forest in the period from the mean temperature l0'C in spring to snowfall Coniferous woods in the period from the mean temperature l0'C in spring to snowfall Forest exfoliating in the dry season; savanna, semidesert in the dry season The same in the rainy season Desert

A1

u 30405060

,

.-2

FIG. 7.14

0

10 20 30 40 50 60 70

80 70

45 38 25 50 30 13 18 18 14 24 18 28

A%

.... ....., . .... 3

---- 4

Recurrence of certain surface albedos: (a) coniferous woods; (b) deciduous woods; (c) wooded steppe and steppe; (6)semidesert and desert. (1) winter; (2) spring; (3) summer; (4) fall.

446

Albedo of the Underlying Surface and Clouds

FIG. 7.15 Albedo of the continental surface of the northern hemisphere, January.

It is natural that in January the zone of largest albedoes (80 percent) occupies the high latitudes where the woods are scarce and the snow cover keeps fresh. To the south the albedo decreases as the forest area expands, and reaches 45 percent in the fully wooded country. The albedo increases again when the steppe appears (70 percent). Still more to the south, the albedo variation is connected with the state of snow cover. In the zone of subtropics and tropics the main role belongs to the presence or absence (desert) of vegetation and also soil humidity. Depending on what January is for a given time (rainy or dry), the albedo varies from 18 to 24 percent. The characteristic albedo for desert is 28 percent (see Table 7.14). Since in July the snow (or ice) is absent everywhere except the polar regions, the geographical variability of albedo in this month is not great:

7.5. Geographical Distribution of Albedo

447

FIG. 7.16 Albedo of the continental surface of the northern hemisphere, July.

The albedo varies from 16 to 20 percent over the vast surface of the Northern Hemisphere. A slight decrease of albedo (15 percent) is observed in the zone of extensive coniferous woods. The maximum albedo takes place in the dry areas and deserts (up to 28 percent). Having classified albedo values over different underlying surfaces, it now becomes necessary to work out a method for averaging albedos over long time intervals. The time relation of sea albedo has been considered in Sec. 7.3. The case of continental underlying surfaces has been treated by Budyko [34] and Berland [Chapter 5, Ref. 1421. The most natural method of calculation of monthly seasonal or yearly albedo means consists in determining average weighted values that correspond to a total of global radiation heat in a time interval during which

448

Albedo of the Underlying Surface and Clouds

some constant albedo value is observed. For example, the seasonal mean may be computed from

where XQ is the global radiation, and A is albedo. The indices 1, 2, 3 determine the months of the season. The formula for computing the annual mean may be presented as

where the indices h, v, e, a mean that the corresponding value refers to winter, spring, summer, or fall. The main features of the annual albedo range for continental surfaces are determined first of all by the appearance of snow or its melting. Thus, the chief factor in the annual mean of albedo a t a point of sufficiently homogeneous territory is the duration of snow cover. Taking a uniform value for the southern European U.S.S.R. territory albedo equal to 18 percent in the warm half-year, Budyko [34] used a formula of the type of (7.4) to calculate the dependence of the annual albedo mean at a given point upon the duration of snow cover. Figure 7.17 gives

DAYS

FIG. 7.17 Dependence of albedo upon duration of snow cover.

References

449

the curve of dependence obtained. As seen from Fig. 7.17, the annual albedo mean increases considerably with an increase of duration of the snow period. Using (7.3) and (7.4), Berland [Chapter 5, Ref. 1421 has performed numerous calculations of yearly and seasonal albedo means for various points on the European U.S.S.R. territory, which enabled plotting of charts of geographical albedo distribution over the mentioned area. Barashkova et al. [37a] have generalized observational data of actinometric stations by plotting analogous charts for all months and the entire U.S.S.R. territory. When analyzing these charts, one’s attention is drawn first of all by a great variety of albedo values due to the variegated surface cover. It is obvious that maximal albedos take place in winter and minimal in summer. It is also obvious that the greatest geographical variability of albedo is observed in winter with a stable snow cover in the north and changing cover conditions in the south. The geographical distribution of albedo in the winter half-year (November-March) is not zonal: In the western U.S.S.R. territory the albedo is 10 to 15 percent less than in the eastern section. In June-September the albedo over the most part of the U.S.S.R. is 15 to 20 percent. Maxima (30 to 35 percent) are observed in central Asia. On the northern European territory of the U.S.S.R. and in the Maritime Province (Far East) the albedo values somewhat exceed 20 percent. On the average, the highest annual values are found in the far north (40 to 50 percent) and the southeastern part of the European territory of the U.S.S.R. (in excess of 30 percent). The latter case may be accounted for by the summer effect of sandy surfaces, whose albedo is comparatively high. REFERENCES 1. Krinov, E. L. (1947). “Spectral Reflectivity of Natural Formations.” Acad. Scienc. U.S.S.R. Press, Moscow. 2. Ashburn, E. V., and Weldon, R. G. (1956). Spectral diffuse reflectance of desert surface. J. Opt. SOC.An. 46, No. 8. 3. Krassilshchikov, L. B. and Golikova, 0. I. (1960). A photometric installation for measuring spectral brightness coefficients. Trans. Main Geophys. Obs. No. 100. 4. Krassilshchikov, L. B., and Tzarevskaya, A. A. (1960). An installation for measuring reflection functions in the region 0.6-2.5 p . Trans. Main Geophys. Obs. NO. 100. 5. Krassilshchikov, L. B., and Piatovskaya, N. P. (1957). Spectral reflection functions of some surfaces in natural illumination on a dull day. Trans. Main Geophys. O h . , No. 68. 6. Krassilshchikov, L. B., Golikova, 0. I., and Novoseltzev, E. P. (1957). Photoelectric measurements of relative spectral brightness coefficients. Trans. Main Geophys. Obs. No. 68.

450

Albedo of the Underlying Surface and Clouds

7. Zaytov, I. R. and Indichenko, I. G. (1958). On spectral reflectance of certain types of soil. Bull. High Schools, Geod. Aerosurvey No. 1. 8. Dirmhirn, I. (1957). Zur spektralen Verteilung der Reflexion natiirlicher Medien. Wetter Leben 9, Nos. 3-5. 9. Dirmhirn, I. (1964). “Das Strahlungsfeld im Lebensraum.” Akad. Verlagsges., Frankfurt. 10. Kondratyev, K. Ya., Mironova, Z. F., and Dayeva, L. V. (1960). Spectral albedo of snow and natural covers. Coll. Papers ICY, Leningrad Univ. 11. Kondratyev, K. Ya. (1960). Spectral albedo of natural underlying surfaces. Meteorol. Hydrol. No. 5. 12. Kondratyev, K. Ya., Mironova, Z. F., Otto, A. N., and Popova, L. V. (1962). Spectral albedo of some underlying surfaces in the visible and near infrared regions. Sci. Bull. Geol. Geograph. Inst. Acad. Sci. Lithuanian SSR 13, No. 1. 13. Kondratyev, K. Ya., Mironova, Z. F., and Otto, A. N. (1965). Spectral albedo of natural underlying surfaces. Probl. Atmospheric Phys., Leningrad Univ. No. 3. 14. Kondratyev, K. Ya., Mironova, Z. F., and Otto A. N. (1964). Spectral albedo of natural surfaces. Pure Appl. Geophys. 59. 15. Pronin, A. K. (1949). Study of vegetation by means of aerosurvey in various spectral zones. Trans. Lab. Aerornethods Acad. Sci. USSR 1. 16. Billings, W. D., and Morris, R. J. (1951). Reflection of visible and infrared radiation from leaves of different ecological groups. Am. J. Botany 38. 17. Egle, K. (1937). Zur Kenntnis des Lichtfelders der Pflanze und der Blattfarbstoffe. Planta 26, No. 4. 18. Matulavicene, V. I. (1957). Albedo of some underlying surfaces in the Lithuanian SSR. Sci. Bull. Geol. Geograph. Inst. Acad. Sci. Lithuanian SSR 5. 19. Perevertun, M. P. (1957). Spectral reflectance of certain plants in the range 6501,200 mp. Trans. Asrobotan. Sect. Acad. Sci. Kazahk SSR 5. 20. Yaroslavtzev, I. N. (1952) Albedo of natural soil cover at Tashkent. Proc. Acad. Sci. USSR, Ser. Geophys. No. 1. 21. Miller, D. H. (1955). “Snow Cover and Climate in the Sierra Nevada, California.” Univ. of California Press, Berkeley, California. 22. Tihomirov, V. S. (1950). Spectral reflectance of relict plants. Proc. Acad. Sci. Kazakh SSR, Ser. Astrobotan. Nos. 1 and 2. 23. Tihomirov, V. S. (1953). On the amplitude of seasonal variations of certain spectral reflective properties of plants. Trans. Astrobotan. Sect. Acad. Sci. Kazakh SSR 1. 24. Artzybashev, E. S., and Belov, V. F. (1958). Reflectance of wood species. Trans. Lab. Aerornethods Acad. Sci. USSR 6 . 25. Spurr, S. H. (1948). “Aerial Photographs in Forestry.” New York. 26. Avaste, 0. A. (1963). Spectral albedo of the main types of underlying surfaces in the near infrared. Invest. Atmospheric Phys. No. 4. Tartu. 27. Shifrin, K. S. (1953). To the theory of albedo. Trans. Main Geophys. Obs. No. 39. 28. Giddings, J. C., and Chapelle, E. L. (1961). Diffusion theory applied to radiant energy distribution and albedo of snow. J. Geophys. Res. 66, No. 1. 29. Ter-Markariantz, N. E. (1957). Dependence of the reflection coefficient of water on the incident light wave length. Trans. Main Geophys. Obs. No. 68. 30. Funk, J. P. (1957). Der Einfluss der Polarization auf die Reflexion an Wasserflachen. Arch. Meteorol., Geophys. Bioklimatol. A10, No. 213.

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31. Czepa, 0. (1954). uber die spektrale Reflexion der Globalstrahlung an WasserEichen. Acta Hydrophsica 1, No. 4. 32. Kalitin, N. N. (1937). On the study of spectral albedo of water surface. Meteurul. Hydrol. No 9. 33. Hulburt, E. 0. (1945). Optics of distilled and natural water. J. Opt. SUC.Am. 58, No. 11. 34. Budyko, M. I. (1956). “The Heat Balance of the Earth’s Surface.” Gidrometeoizdat,

Leningrad. 35. Semenchenko, I. V., and Spytkin, A. V. (1961). Measurement of spectral sea brightness from aircraft. Oceanol. 1, No. 5. 36. Fritz, S. (1949). The albedo of the planet Earth and of clouds. J. Meteurul. 6, No. 4. 37. Bibliography on Albedo. (1957). Meteorol. Abstr. Bibliog. 8, No. 7 . 37a. Barashkova, E. P., Gayevsky, V. L., Dyachenko, L. N., Lugina, K. M., and Pivovarova, Z. I. (1961). “The Radiation Regime of the USSR Territory.” Gidrome-

teoizdat, Leningrad. 37b. Kirillova, T. V., and Malevsky-Malevich, S. P. (1964). On measurement of the sea albedo from helicopter. Trans. Main Geophys. Obs. No. 150. 37c. Beliayeva, I. L. Rachkulik, V. I., and Sitnikova, M. V. (1965). Connection of the

brightness coefficient of the system soil-vegetation with the quantity of vegetative mass. Meteorol. Hydrol. No. 8. 38. Kondratyev, K. Ya. (1965). “Actinometry.” Gidrometeoizdat, Leningrad. 39. Kirillova, T. V., and Nesina, L. V. (1955). Influence of irrigation on the variation of the heat balance components on a wheat field. Trans. Main Geophys. Obs. No. 53 (115).

40. Skvortzov, A. A. (1928). To the problem of the climate in oasis and desert and some peculiarities of their heat balance. Trans. Agr. Meteorol. 20. 41. Chernigovsky, N. T. (1959). On the radiative properties of snow in Central Arctic. Trans. Elbrus Expedition, Vol. 1. Nalchik. 42. Rusin, N. P. (1959). Radiation balance of snow cover in the Antarctic. Trans. Main GeophyA. Obs. No. 96. 43. Kuzmin, P. P. (1957). “Physical Properties of Snow Cover.” Gidrometeoizdat, Leningrad. 44. Koptev, A. P. (1961). Albedo of cloud, water and snow and ice. Trans. Arct. Anturct. Inst. 229. 45. Yaroslavtzev, I. N. (1948). Reflected radiation from natural surfaces at Tashkent. Proc. Acad. Sci. USSR, Ser. Geograph. Geophys. No. 6. 46. Preobrazhensky, L. Y. (1964). Measurement of albedo of sea. Trans. Main Geophys. Obs. No. 150. 47. Tooming, H. (1960). Daily and seasonal variations of albedo of some natural un-

derlying surfaces in the Estonian SSR. Invest. Atmospheric Phys. IPhA Acad. Sci. Est. SSR, Tartu, No. 2. 48. Ross, Y. K. (1962). To the theory of albedo of vegetation. Sci. Bull. Geol. Geograph. Zst. Acad. Sci. Lithuanian SSR 13. 49. Ross, Y. K., and Nilson, T. (1963). To the theory of the radiation regime of vegetative cover. Invest. Atmospheric Phys., ZPhA Acad. Sci. Est. SSR, Tartu. No. 4. 50. Kondratyev, K. Ya., and Ter-Markariantz, N. E. (1953). On the daily variation of albedo. Meteorol. Hydrol. No. 6.

452

Albedo of the Underlying Surface and Clouds

51. Kastrov, V. G. (1955). To the problem of the daily range of the albedo of the earth’s surface. Trans. Centr. Aerol. Obs. No. 14. 52. Kondratyev, K. Ya., and Ter-Markariantz, N. E. (1953). To the methods for processing observations over albedo. Meteorol. Hydrol. No. 7. 53. Grishchenko, D. L. (1959). Dependence of the albedo of the sea upon solar height and roughness of the sea surface. Trans. Main Geophys. Obs. No. 80. 54. Ter-Markariantz, N. E. (1957). On reflection of radiation by the rough sea. Trans. Main Geophys. Obs. No. 68. 55. Ter-Markariantz, N. E. (1957). Return scattering of radiation by the sea. Trans. Main Geophys. Obs. No. 68. 56. Forsch, L. F. (1954). Reflection of solar radiation from water surfaces of lakes. Trans. Lab. Lakes Acad. Sci. USSR 3. 57. Gershun, A. A. (1928). To the problem of diffuse light transmission. Trans. Opt. Znst. (Leningrad) 6, No. 38. 58. Lauscher, F. (1955). Optik der Gewasser. Zn “Handbuch der Geophysik,” Vol. VIII. Springer, Berlin. 59. Lauscher, F. (1952). Sonnen- und Himmelstrahlung in Meer und in Gewassern. Part I. Theorie der Strahlungsreflexion. Arch. Meteorol., Geophys. Bioklimatol. E4, No. 2. 60. Kondratyev, K. Ya., and Kudriavtzeva, L. A. (1955). To the problem of albedo of water basins. Meteorol. Hydrol. No. 3. 61. Kondratyev, K. Ya., and Ter-Markariantz, N. E. (1956). On the reflection of radiation by the sea. Proc. Leningrad Univ., Ser. Phys. No. 9. 62. Kondratyev, K. Ya., and Ter-Markariantz, N. E. (1953). Albedo of the rough sea. Meteorol. Hydrol. No. 8. 63. Ter-Markariantz, N. E. (1959). On calculation of albedo of water surfaces. Trans. Main Geophys. Obs. No. 80. 63a. Mullamaa, Y. R. (1964). Reflection of direct radiation from sea surface. Proc. Acad. Sci. USSR, Ser. Geophys. No. 8. 63b. Mullamaa, Y. R. (1964). “Atlas of the Optical Characteristics of the Rough Sea Surface.” Tartu, Inst. Atmospheric Phys. Acad. Sci. Est. SSR. 64. Sivkov, S. I. (1952). Geographical distribution of the effective albedo values of water surface. Bull. Geograph. Soc. 84, No. 2. 65. Ter-Markariantz, N. E. (1960). Mean daily albedo values. Trans. Main Geophys. Obs. No. 100. 66. Fritz, S. (1950). Measurement of the albedo of clouds. Bull. Am. Meteorol. SOC. 31, No. 1. 67. Piatovskaya, N. P. (1960). Measurement of albedo from aircraft. Trans. Main Geophys. Obs. No. 109. 68. Robinson, G. D. (1958). Some observations from aircraft of surface albedo and the albedo and absorption of cloud, Arch. Meteorol., Geophys. Bioklimatol. B9, No. 1. 69. Monteith, J. L. (1959). The reflection of short-wave radiation by vegetation. Quart. J. Roy. Meteorol. SOC.85, No. 366. 70. Mukhenberg, V. V. (1963). Albedo of underlying surfaces on the USSR territory. Trans. Main Geophys. Obs. No. 139. 71. Sitnikova, M. V. (1964). The results of measurement of the albedo of various underlying surfaces. Trans. Hydrometeorol. Znst. Central Asia No. 18.

GLOBAL RADIATION

Chapters 5 and 6 have dealt with details of the problem of direct solar and diffuse radiation. In real conditions, the total income of shortwave radiation is composed of two parts, that of direct solar and of diffuse radiation. The total sum of fluxes (or totals) of direct solar and diffuse radiation is called global radiation flux (or total). Only in the case of solid cloudiness (or partial, when the sun is behind a nontransparent cloud) does the global radiation have one diffuse component. The main regularities of variation in the global radiation are presented in the following sections.

8.1. Energy Distribution in the Spectrum of Global Radiation A great number of experiments to determine the global radiation spectral composition have been performed by Tikhov [Chapter 6, Refs. 3, 41, Krinov [Chapter 6, Refs. 5 7 1 , Sharonov [Chapter 6, Refs. 5 , 6 ] ,and others. Tikhov was the first to show experimentally that the spectral composition of global radiation received by a horizontal surface is practically independent of solar height and consequently remains constant throughout the day. This conclusion was confirmed later on by numerous observations of Krinov, Sharonov, and others. Figure 8.1 gives a mean curve of global radiation spectral distribution (interrupted curve) as obtained by Herman [I] from averaging over 14 serial observations of a cloudless sky. The solid curve characterizes the diffuse radiation spectral composition of a clear sky. When plotting these curves the radiant flux at 1 = 500 mp was taken to be unity. According to Fig. 8.1, the spectral distribution of global radiation has a chief maximum in the interval 480 to 500 mp and very weak secondary extremes. 453

454

Global Radiation

300400500

600

Wavelength

FIG. 8.1

700 A, m p

800

Energy distribution in the spectra of globaI ( I ) and difluse (2) radiation at a clear sky.

The interrupted curve of Fig. 8.1 is characteristic of the relative energy distribution in the spectrum of global radiation. Nevertheless it can be used for the computation of the global radiation spectral distribution in absolute units. As will be shown later, the global radiation in a cloudless sky may be approximately considered dependent only upon solar height and atmospheric transparency. Therefore it is natural that spectral fluxes of global radiation will also depend first of all upon solar height. Since, on the other hand, the spectral composition of global radiation is independent of solar height, it becomes clear that the dependence on solar height for a spectral flux of global radiation will be uniform throughout the spectrum. To obtain the value of a spectral radiant flux at a given solar height in absolute units, it is therefore sufficient t o multiply the corresponding ordinate of the interrupted curve (Fig. 8.1) by a certain coefficient independent of wavelength. Such coefficients, f(h,), computed by Gotz and Schonmann [2] have the following values:

h,,

f(h,):

60' 176

55' 166

50' 154

45' 140

40' 126

35' 111

30' 94

25' 77

Multiplying all the ordinates of the interrupted curve by the valuef(h,) corresponding to the given solar height h, , obtain a curve of global radiation spectral distribution in absolute units W/cm2 l mp. It should be noted, however, that this method is only approximate. The investigation of the global radiation spectral composition of a partially cloudy sky, performed by Gotz and Schonmann, has led to the conclusion that in this case the energy distribution in the global radiation spectrum is highly variable and depends first of all on whether the sun is obliterated by clouds. On the whole, though, the spectral distribution of global radiation is in this case almost the same as for a clear sky. This can be seen in Fig. 8.2, which presents a solid curve obtained on Aug. 31, 1942, at

8.1. Energy Distribution in the Spectrum of Global Radiation

455

h, = 40.3' and cirrus clouds (Ci) of force 3, and a dashed curve plotted from observational data on Aug. 25, 1942, at ha = 48.2' and cumulus clouds (Cu) of force 3. In the case of cirrus clouds occulting the sun, a certain bluing of global radiation was observed, due to the attenuation of radiation of the red spectrum by a cloud film.

-ac c

a

.t c

-

1.0

L

-ca0.5 P

e

W

0 300

400

500

600 700 Wavelength x,mp

000

FIG. 8.2 Energy distribution in the spectrum of global radiation in diferent cIoudiness conditions. (1) Ci force 3, Aug. 1942, h, = 40.3O; (2) Cu force 3, Aug. 1942, h, = 48.2'.

Condit and Grum [3] have realized measurements of the spectral energy distribution of global radiation in the ultraviolet and visible ranges at various solar heights and cloudiness conditions for inclined surfaces. The measurements were conducted at Rochester, N. Y., in 1962. The purpose was to study radiant fluxes on a surface inclined at 15' relative to the vertical with azimuths (relative to the sun) equal to ,'O 30°, 45', and 180', and also on vertical surfaces and surfaces perpendicular to the sun's rays. In measurements of the spectral radiation compositon, a modified Beckman spectroreflectometer DK-2R was used. A tungsten bulb served for standard reference, the directly measured value being the ratio of signal from the sky and the standard source. The stability of the photometric scale was checked by a special secondary standard with the same energy distribution in the emission spectrum as that of the standard source. The accuracy of measurement was about & 1.5 percent and independent of wavelength, since the slit width was automatically adjusted to secure a constant exit signal on the photomultiplier. The scanning interval for wavelengths from 330 to 700 m p was 105 sec. The spectral resolution did not deviate more than 5 mp. Characterization of the atmospheric turbidity was obtained by conducting measurements of the light polarization from a point on the sky where the polarization was at maximum. The cloudiness conditions were checked by photographing the celestial hemisphere at

456

Global Radiation

1-min intervals. Data of Condit and Grum show a considerable decrease in values of global radiation fluxes on the surface inclined at 15’ to the vertical and directed toward the sun (in the range near 400 to 500 mp) in the transition from clear sky conditions to the appearance of haze. However, the radiation in the mentioned wavelength range became more intensive again in the presence of light (semitransparent) clouds. At a dense, solid cloudiness the radiation of the range 400 to 500 mp is slightly less intensive than for a clear sky. When examining data on the variability of the global radiation spectral composition and of the color temperature of a cloudless sky in dependence on solar height, taken from June 26 to July 3, it was found that the global radiation at low altitudes of the sun on June 26 was more “blue” than on Aug. 3. This was puzzling because on Aug. 3 the atmosphere was more transparent. The considered results stress the effect of solar height on the variability of the global radiation spectral composition. Such variability was practically unobservable in the presence of solid dense cloudiness. Also notable was the variation of the global radiation spectral composition in dependence upon orientation of surfaces relative to the sun at different solar heights (clear sky). References [4-61 give a detailed treatment of the spectral composition of global and diffuse radiation in the visible and near-infrared regions. A series of similar investigations have also been made by Kondratyev et al. [7-91, whose data confirm the conclusion about the approximate constancy of the global radiation spectral composition of a clear sky throughout the day in spite of the considerable deviations observed. An attempt at comparison of the results of measurement with computational data for the Rayleigh atmosphere has shown that even in the mountainous conditions the agreement is unsatisfactory. The main result of experiments in the global radiation spectral composition is the conclusion about the approximate daily constancy of the considered distribution in the interval 0.35 to 0.80 p in a clear (or partially occulted) sky. The causes for the lack of dependence of the global radiation spectral composition upon solar height are evident. When the solar height decreases, the radiation gradually, loses more and more blue and violet rays, which results in the reddening of solar radiation. At the same time, however, there is an increase of the relative portion of diffuse light that is rich in blue-violet rays. This increase practically compensates for the attenuation of the radiation in the blue-violet spectrum. Thus the global radiation spectral composition does not undergo any significant variation at varying heights of the sun.

8.2. Fluxes of Global Radiation

457

Theoretical calculations by Rautian [ l l ] and Schulze [12] quantitatively confirm, in agreement with the above experimental data, the fact of approximate constancy of the spectral distribution of global radiation during the day. It should be mentioned here that the absence of notable dependence of the global radiation spectral composition upon solar height is observed at medium heights only. If h, < 10 to 15, and the solar height decreases further, a certain bluing of the global radiation takes place. 8.2. Fluxes of Global Radiation

The chief principles of variation in the flux of global radiation are determined by the complex influence of the above-considered factors, which account for the more significant variations in fluxes of direct solar and global radiation. These factors are solar height (or atmospheric mass in the direction toward the sun), atmospheric transparency conditions, degree of cloudiness, duration of sunshine, and similar influences. Let us now turn to the roles of these factors, analyzing the observed regularities in the variation of global radiation fluxes. Note here that throughout the current chapter (except Sec. 84) we shall speak about radiant flux on a horizontal surface. 1. Dependence of Global Radiation Flux upon Cloudiness Degree, Solar Height, and Atmospheric Transparency Conditions. The degree of cloudiness and the solar altitude (or atmospheric mass in the direction toward the sun) are of prime importance in determining variation of the global radiation flux. In the presence of cloudiness, the radiant flux may both increase and decrease. If the cloudiness is partial and the sun is not obliterated, then in this case the flux of global radiation is greater, as a rule, than in a clear sky (fluxes compared at the same height of the sun), In the case of continuous cloudiness the incoming global radiation is always less than that from a cloudless sky. Table 8.1 is very spectacular in illustrating the dependence of global radiation flux upon cloudiness. The observations were conducted by Nebolsin [14] at various degrees of cloudiness in summer. The considered data of Table 8.1 confirm the above conclusions on the dependence of global radiation flux upon cloudiness. Analogous results were obtained by Nebolsin and other authors for different seasons. The averaging of the results of observations of global radiation in various cloudiness conditions over a long term shows that cloudiness always decreases the income of global radiation. That is why, in particular, the de-

458

Global Radiation TABLE 8.1

Dependence of Global Radiation Flux upon Cloudiness as Observed during the Summer Months (cal/cm2min) in the Moscow Region. After Nebolsin 1141

Mass

8 5 4 3 2 1.5 Noon

Cloudless

0.16 0.18 0.23 0.35 0.60 0.97 1.12

Sun and Clouds

0.14 0.26 0.25 0.41 0.57 0.88 1.15

Sun Transparent through Clouds

Sun behind Clouds

Ci

St

High

Low

0.07 0.20 0.22 0.35 0.52 0.78 0.92

0.05 0.06 0.11 0.18 0.41 0.55 0.41

0.08 0.11 0.14 0.19

0.09 0.11 0.16 0.18 0.29 0.35 0.39

0.44 0.71 0.63

Continuous Cloudiness

0.05 0.06 0.08 0.15 0.14 0.28 0.37

pendence between the mean incoming global radiation and the degree of cloudiness is always inverse. We shall consider this problem in more detail further on. It is natural that incoming global radiation is strongly influenced not only by the amount but also by the form of clouds. To characterize the latter effect, Table 8.2 gives the results of observations by Kalitin [15] of direct solar, diffuse, and global radiation at continuous cloudiness of the following forms: Ci, Cs, Ac, As, St fr, Sc, St. The considered data were obtained by processing five-year (1936-1940) actinograph and pyranograph recording at Pavlovsk. As seen from Table 8.2, the decrease of global radiation flux as compared with the conditions of a clear sky is observed with all cloud types. It is most pronounced in the presence of low clouds, since in this case the clouds are fully nontransparent for direct radiation and the flux of diffiuse radiation turns out to be rather small (at low solar heights even less than at a clear sky). The least decrease in global radiation flux is observed at the upperlayer cloudiness, which is semitransparent to direct solar radiation. However, even in this case, the relative decrease of global radiation flux at low heights of the sun in comparison with the corresponding value for a cloudless sky is quite noticeable. To illustrate the decrease of global radiation flux with solid cloudiness, Table 8.3 gives values of the relative (in percent) decrease for various cloud forms and solar heights, computed by Kalitin [15] from data of Table 8.2. It is seen from Table 8.3 that in the

459

8.2. Fluxes of Global Radiation

TABLE 8.2 Dependence of Fluxes of Direct Solar, Diffuse and Global Radiation on a Horizontal Surface at Solid Cloudiness upon Cloud Form (cal/cm2min). After Kalitin [15]

Cloud Form

Cloudless Ci cs Ac As Stfr ScorSt

Solar Height, deg 5

7.5

10

15

20

25

30

40

50

0.03

0.06

0.00 0.00 0.00 0.00

0.00 0.00 0.00 0.00 0.00 0.00

0.09 0.00

0.13 0.00

0.22 0.04

0.00

0.46 0.21 0.09

0.59 0.32 0.18

0.00

0.00

0.84 0.60 0.42 0.12

0.00 0.00 0.00

0.00 0.00 0.00

0.33 0.11 0.01 0.00

0.01

0.05

1.10 0.90 0.70 0.31 0.01

0.00 0.00

0.01

0.05

0.00

0.00

0.03

0.04

0.05

0.07

0.06 0.06 0.05 0.02 0.01 0.02

0.12 0.29 0.40

0.09

2.5

0.00 0.00

Cloudless Ci cs Ac As Stfr sc St

0.02 0.03 0.04

Cloudless Ci cs Ac As Stfr sc St

0.05 0.03 0.04 0.04 0.03 0.01

0.94

0.03 0.01 0.00

0.01

0.00

0.01

0.05

0.06 0.06 0.05

0.02 0.01 0.02

0.00

0.00 0.00 0.00 0.00

0.00 0.00

0.00

0.00 0.00 0.00 0.00

0.09 0.09 0.07 0.03 0.02 0.02

0.05 0.09 0.11 0.11 0.09 0.03 0.04 0.04

0.07 0.13 0.15 0.16 0.13 0.08 0.06

0.08 0.16 0.19 0.22 0.17 0.08 0.13 0.08

0.09 0.19 0.23 0.26 0.20 0.13 0.17 0.10

0.10 0.22 0.27 0.31 0.24 0.16 0.20 0.13

0.11 0.26 0.34 0.39 0.30 0.21 0.27 0.16

0.35 0.24 0.33 0.19

0.13 0.07 0.09 0.09 0.07 0.02 0.02 0.02

0.18 0.09 0.11 0.11 0.09 0.03 0.04 0.04

0.29 0.17 0.15 0.16 0.13 0.05 0.08 0.06

0.41 0.27 0.20 0.22 0.17 0.08 0.13 0.08

0.55

0.69 0.54

0.96 0.86 0.76 0.51 0.35 0.22 0.27 0.16

1.22 1.19 1.10 0.75 0.45 0.29 0.35 0.19

0.00 0.00

0.05

0.40 0.32 0.26 0.20 0.13 0.17 0.10

0.45

0.31 0.25 0.16 0.20 0.13

0.44

overwhelming majority of cases, the decrease of global radiation at solid cloudiness is quite great. The data of Table 8.2 permit analyzing the relation between the direct solar and diffuse components of the global radiation at different cloudiness degrees and solar heights. We see that in the case of a fully overcast sky the direct solar radiation can reach the earth's surface only when semitransparent upper layer clouds are present and the height of the sun is not too low. With the fully overcast sky, as a rule, only diffuse radiation is observed at ground level. In a clear sky the portion of direct solar radiation first increases rapidly, and then slowly as the sun rises.

460

Global Radiation

TABLE 8.3 Relative Decrease of Global Radiation Flux with Solid Cloudiness of Different Form and at Different Solar Heights in Comparison with the Corresponding Values for a Clear Sky (Percent). After Kalitin [15]

Solar Height, deg Cloud Form

Ci cs Ac As St fr sc St

5

10

20

30

40

50

44 33 33 44 78 89 78

50 39 39 50 83 78 78

34 51 46 59 80 68 80

22 35 55 64 77 71 81

10 20 46 63 77 72 83

2 10 38 63 76 73 84

From Tables 8.1 and 8.3 it can be concluded that with both clear and overcast sky the flux of global radiation is essentially dependent upon solar height. As shown in Chapters 5 and 6, the fluxes of direct solar and diffuse radiation on horizontal surfaces increase with an increasing solar height. Therefore it is can be expected that the same dependence will be observed for the global radiation. We shall consider later the manner in which this dependence can be presented analytically for the case of a clear sky. Note here that in the first approximation these dependents are proportional. At uninterrupted cloudiness, the dependence between global radiation flux and solar height appears to be far more complex. According to Table 8.3, the relative decrease of the global radiation flux in comparsion with the corresponding value for a clear sky is slightly dependent on solar height only in the case of the lower layer clouds (St, Sc, St fr). This means that solely in the presence of uninterrupted cloudiness of the lower layer does the global radiation flux vary approximately in proportion to solar height, since such proportionality takes place for global radiation flux from a cloudless sky. On the other hand, in the case of the upper and middle layers cloudiness, observations find a notable dependence of the relative decrease of global radiation flux upon solar height. Consequently the dependence of the global radiation flux at solid upper and middle cloudiness must be nonlinear. The validity of the above conclusions becomes tangible if the data of Table 8.3 are used for graphic presentation of the dependence of global radiation flux in a completely obliterated sky upon solar height.

8.2. Fluxes of Global Radiation

46 1

Although cloudiness and solar height are the dominant factors for global radiation, observations for the clear sky find a marked dependence of global radiation upon atmospheric transparency. This can be seen in Fig. 8.3, which shows dependences of global radiation flux upon turbidity factor at different atmospheric masses, as derived by Barashkova [I61 from data of observations at Karadag. It is evident that with a decrease of atmospheric transparency (increasing turbidity factor), a notable decrease of global radiation takes place, especially at high elevations of the sun.

0

0.6

I 0

2

3

4

T

FIG. 8.3 Dependence of global radiation upon the turbidity factor as observed at Karadag.

Chapter 6 presented data that characterized the dependence of diffuse radiation flux upon underlying surface albedo. Obviously a similar dependence must be observed in the case of global radiation. However, at present there are no sufficient quantitative data available to substantiate this statement. Observations show that, other conditions being equal, the global radiation increases (by 10 to 15 percent) when snow cover appears. 2. Daily and Annual Range of Global Radiation. It is natural that the simplest daily and yearly variation in global radiation flux must be observed with a completely clear sky, since in such a case the variation is mainly determined by solar height variations. According to Kalitin [15], from observations made at Pavlovsk, the daily and annual range of hourly totals of global radiation in a cloudless sky is determined quite simply and is characterized by the presence of one noon maximum (summer maximum, correspondingly). More comoplex, however, is the calculation of the annual range of hourly radiation totals: A certain decrease of incoming radiation is observed in June, which appears to be caused by the decreasing atmospheric transparency. Secondary maxima in the incoming global radiation take place in the fore- and afternoon hours in December. Their

462

Global Radiation

appearance can also be explained by the decrease of atmospheric transparency. Simple enough regularities of the daily and annual ranges of global radiation flux are also found from averaging observational data over all days, that is, with average cloudiness conditions. Observations by Kalitin [I51 at Pavlovsk (summer) have shown that the incoming global radiation at noon hours considerably exceeds that of the afternoon time. Also marked is the dominance of the incoming global radiation in the first half-year over that in the other half-year. These peculiarities of the daily and yearly range of global radiation result from a somewhat lower cloudiness in the morning hours (correspondingly, in the first half-year) as compared with the afternoon hours (the second half-year). Moreover, the absolute values of hourly global radiation totals in the summer noon hours for the average cloudiness conditions turn out to be considerably less than in the case for a clear sky. Figure 8.4 gives isopleths of hourly global radiation totals, plotted by Berland [Chapter 5, Ref. 1431 for Dixon (island) and Leopoldville. These data characterize the peculiarities of the daily and annual global radia(a)

(b)

J

I 1 0 12 14 16 I

20

FIG. 8.4 Daily and annual range of global radiation (cal/crn2hour) for Dixon (a) and Leopoldville (6).

tion variation in the polar latitudes and in the equatorial zone. It is shown that the isopleths plotted from Dixon data are strongly elongated in the horizontal direction, which is due to the all-day incoming solar radiation during the polar day and its absence during the polar night. On the contrary, the Leopoldville isopleth elongation is vertical, which reflects a rel-

8.2. Fluxes of Global Radiation

463

atively even distribution of incoming global radiation throughout the years. The symmetrical zones of maximal values of global radiation refer to the equinoctial periods. The daily global radiation range in the equatorial zone is characterized by the prevalence of maxima in the afternoon hours, caused by the daily cloudiness variation. The detailed discussion of the daily global radiation range in the principal climatic zones can be found in Berland's work [16a]. At temperate latitudes, observers also find an inverse regularity, that is, the daily global radiation maximum falls during the afternoon hours. According to Birukova [17], such a phenomenon resulting from the morning predominance of cloudiness can be observed, for example, at Riga in summer.

3. Theoretical Calculations of Global Radiations. The problem of calculating global radiation fluxes is solved simply for the case of a clear sky. Let us write the transfer equations (4.82) derived in Chapter 4: dF1 - dF, - E~ sec 8,F2 - ~ E F , dt dt Add the following boundary conditions to (8.1):

F,

=

socos 8,

Fi = AF,

at

t = to

at z = O

(8.2) (8.3)

where So is the solar constant, to = J," a(z) dz is the atmospheric optical thickness in the vertical direction, and A is the earth's surface albedo. Remember that (8.1) relates to the conditions of a purely scattering atmosphere and, strictly speaking, describe the transfer of monochromatic radiation only. Simple calculations lead to the following results of integration of (8.1) at the boundary conditions (8.2) and (8.3) for a downcoming global radiation flux F,:

For a flux of global radiation at the level of the earth's surface we have

(t = 0),

464

Global Radiation

Assuming A = 0, instead of (8.5) we have

As shown in Chapter 4, the asymmetry of the mean atmospheric scattering function is quite notable. It may be assumed that, in the order of value, E~ = 10-l. Since t N 0.3, this means that el-c 21 0.03. Therefore, at not too large sec 8, (in any case at sec 8, < 5), the following expansion into a series is valid: ee1~0(sec%-2) N 1 elto(sec 0, - 1)

+

Using this approximate presentation, instead of (8.6) we find

Although this formula is extremely simplified, its comparison with observational data on the daily range of the integral global radiation realized by Berland [Chapter 5, Ref. 1431 has found satisfactory agreement. Table 8.4 gives the coefficient f = values calculated by Berland [18] from data on the dependence of possible monthly global radiation totals upon latitude as found by him [Chapter 5, Ref. 1421. Berland pointed out that the use of the coefficients of Table 8.4 enables satisfactory description of the daily global radiation range in a cloudless sky. Such a result is quite normal when related to the fact that Eq. (8.7) is identical with the Kastrov formula (5.25), which describes the daily variation of direct solar radiation. From Table 8.4 it is seen that the coefficients are small in winter. The solar height being sufficiently great, in this case the second term in the denominator of (8.7) may be neglected. Strictly (zo = 0) this corresponds to the case of the absence of atmosphere and cannot be connected with the description of the daily global radiation range at ground level. However, as already mentioned, observations sometimes give a direct proportionality between global radiation flux and solar height. For instance, according to Cheltzov [191, from Arkhangelsk observations, the following empirical formula is valid:

F2(0) = 0.02552 h,

(8.8)

where h, is expressed in degrees, and F,(O) is in cal/cm2 min. A relation of this type is an extremely simplified formula and can be derived from (8.7).

TABLE 8.4 Annual Range of the Coefficientsf. After Berland [18] 00

P

Latitude, deg

Jan.

Feb.

Mar.

Apr.

May

June

July

Aug.

Sept.

Oct.

Nov.

Dec.

3

30

0.030

0.060

0.080

0.140

0.170

0.190

0.190

0.165

0.120

0.090

0.070

0.070

9

40

0.030

0.045

0.085

0.120

0.165

0.185

0.190

0.170

0.130

0.100

0.075

0.070

b 5d

V

50

0.055

0.065

0.085

0.110

0.150

0.175

0.180

0.175

0.140

0.100

0.085

0.070 0 3

60

0.045

0.070

0.090

0.110

0.140

0.160

0.165

0.165

0.155

0.090

0.055

0.035

466

Global Radiation

It is obvious that such a relation does not have sufficient generality. However, by making use of (8.7) one can derive formulas for computation of the possible daily global radiation totals. In the solution of this problem a method suggested by Makhotkin can be used for the computation of possible daily totals of direct solar radiation (Sec. 5.9). The theoretical description of the global radiation variability with partial cloudiness presents great difficulties due to the complex accounting for the angular cloudiness distribution over the celestial sphere and for cloud optical properties. The problem of the incoming global radiation in the presence of uninterrupted clouds can be solved in a much simpler way. Berland and Novoseltzev [20] suggested that in this case the same system of equations and boundary conditions should be used, as considered above. However, for the coefficients mJ1 and m 2 r 2 , these authors took mlrl

=

ZF

m 2 r 2= iiiF - pemaz where p

=

if - mar,)

e-maTo

where iii, are certain constant (mean) values of the coefficients m and r, m, = secO,, and is the coefficient r value in the direction toward the sun. The use of the above expressions for the coefficients leads to the following result of the integration in the transfer equations :

r,

where Ei(x) is the integral exponential function. Let us now present the connection between radiant fluxes in the presence of cloudiness ( F 2 9 and in a cloudless sky (F2) in the following form:

FAn) = F2(1 - cLnL - c,nM - tun,)

(8.10)

where n,, n*M,nu are the cloudiness degrees of the lower, middle, and upper layers, respectively. The coefficients c can be obtained from the relation

(8.1 1 )

8.2. FImw of Global Radiation

467

where F(l) is the global radiation at a given uninterrupted cloud layer, which can be calculated from (8.9). Equation (8.7) is proposed for the computation of F2. Applying the described method for global radiation calculation, Berland and Novoseltzev [20] found satisfactory agreement of the results of computation with observational data. For example, calculations give c, 21 0.8 for the lower-layer clouds of about 800-m thickness (characteristic for summer on the European territory of the U.S.S.R.).According to summer observations by Kalitin [15], c, = 0.8 as well. The coefficient c, averages 0.5. For the coefficient c,, a considerable dependence on solar height is observed: c, = 0.2

c,, = 0.4

> 20° at h, < 20° at h,

The preceding theoretical formulas are important in that they enable analysis of the dependence of the c coefficients upon various factors. For example, it was shown, that, in accordance with observational data, the c values are practically independent of underlying surface albedo in the summer time, but are considerably varying in winter. The coefficient c, and c, values are chiefly determined by cloud thickness. In climatological calculations of global radiation totals are widely spread numerous empirical formulas. We shall mention here the most common of them. The Savinov-Angstrom formula is most often used : (8.12) where Q, Qp are the actual and the possible totals of global radiation, and k = Ql/Qp is an empirical coefficient characterizing shortwave radiation trasmission by clouds. The n1 value can be generally determined as a sky clearness parameter, n, = (1 - S1 n)/2. In the most used variant of (8.12), however, n, = n is taken (where n is a mean cloudiness degree in portions of unity), since in this case data on the duration of sunshine are not needed. In a number of investigations, n, = 1 - s1 was also used. Table 8.5 gives the coefficient k values for different latitudes q~ in the case where n, = n, obtained by Berland [Chapter 5, Ref. 1431. These data were used to compute the geographical distribution of global radiation totals.

+

468

Global Radiation

TABLE 8.5 Mean LatitudinaI Variation of the Coefficient k. After Berland [Chapter 5, Ref. 1431

75 0.55

70 0.50

65 0.45

60 0.40

0.38

35 0.32

30 0.32

25 0.32

20 0.33

0.33

55

15

50 0.36

45 0.34

40 0.33

10 0.34

5 0.34

0 0.35

Albrecht [21] has shown that the coefficient can be calculated from the relation (8.13) where h,, is mean maximal solar height for a given month. The functions f(n/nL) and q@max) are of the following form:

f

(5)

= 0.615

n + 0.157 nL

(8.14)

with a constant value of 1.243 for n/nL > 4, and

~ ( h , , ~=) 0.50

+ 0.14 tanh ( hmax 17034.5'

(8.15)

As seen, Eqs. (8.13) and (8.15) allow computation of the coefficient k from data on cloudiness and maximal solar height. Averkiev [22, 231 has found out that this method for computing k and Q yields lowered radiation totals in the winter season, since no account is taken of the effect of the underlying surface albedo. The latter effect can be considered if we introduce a factor GI = 1/(1 - A y ) , where y 2: 0.2 0.5 n into Eq. (8.12). In a number of investigations attempts have been made to differentiate the influence of different layers clouds on the incoming global radiation by taking into consideration the effect of the underlying surface albedo. Averkiev, for example, has proposed the formula

+

(8.16) where kL ,y are empirical coefficients. The value y characterizes the portion of the reflected radiation returned to the earth from the atmosphere. Aver-

469

8.2. Fluxes of Global Radiation

kiev has shown that y can be calculated from the formula = 0.2

+ 0.5 ("'2")

(8.17)

By analyzing observational data on the European territory of the U.S.S.R. Averkiev showed that the formula (8.16) was more exact than (8.12) at n, = n. However, it should be noted that (8.16) is at the same time more complex than (8.12) and requires more information about cloudiness. From the available experimental results, it may be concluded that the dependence of global radiation upon cloudiness degree is nonlinear. Following this, Berland [24] proposed a formula for computation of global radiation totals : (8.18) Q = QJl - (a bn)nl

+

where a and b are constant. The coefficient b can be taken as constant and equal to 0.38. The coefficient a values for different latitudes are tabulated below : QP:

a : @: a :

0 0.38 45 0.38

5 0.40 50 0.40

10 0.40 55 0.41

15 0.39 60 0.36

20 0.37 65 0.25

25 0.35 70 0.18

30 0.36 75 0.16

35 0.38 80 0.15

40 0.38 85 0.14

Berland has shown that the use of the linear dependence of global radiation on cloudiness leads to the lowering of computational values in summer and to their exaggeration in winter. The most simple formula for computation of daily and monthly global radiation totals was suggested by Ukraintzev [25]: Q=ms+n

(8.19)

where s is the duration of sunshine in hours and m, n are empirical coefficients. The comparison of the results of calculations according to (8.19) with observational data shows that the coefficients m and n are dependent on solar height. In this connection Sivkw [26, 271 has proposed the following formula, instead of (8.19), for computing monthly totals of global radiation : (8.20) Q = 0 . 0 0 4 9 ( ~ ~ ~ )10.5(sin ~ . ~ ~ h,)2.1

+

where s,

is the duration of sunshine over a month, h,,,,

is the noon al-

470

Global Radiation

titude of the sun on the 15th of the given month. This formula allows determining monthly totals for the latitudes 35 to 65'N, with an error not more than 10l percent.

8.3. The Main Observed Regularities in the Variability of Global Radiation Totals Having learned the observed principles of variation of global radiation flux and the methods for computing fluxes and totals of global radiation, let us turn now to consideration of the results of measurements and calculations of global radiation totals.

1. Daily Totals of Global Radiation. From the results presented in Sec. 8.2, it is clear that the values of daily global radiation totals must be determined mainly by the degree of cloudiness and the height of the sun within the given 24 h, and it is obvious, therefore, that daily totals for a cloudless sky (the possible totals) must have a simple annual range with a summer maximum. Table 8.6 summarizes Berland's [28; Chapter 6, Ref. 1431 data for the latitudinal variation of the possible daily totals of global radiation for the whole year in months (in the 1956 International Pyrheliometric Scale). The peculiarities of the annual range of global radiation at different latitudes are quite evident here. As seen, the simple range of the high, intermediate, and low latitudes (summer maximum) is not kept in the equatorial zone : the lower atmospheric transparency in summer and little variation of solar height throughout the year lead to the presence of maximum incoming radiation in winter. The influence of cloudiness more or less changes the character of the annual variation in daily radiation totals. In the majority of cases the annual variation of daily means of global radiation totals for average cloudiness conditions has no qualitative difference with that of such totals for a clear sky. This can be seen, for example, in Fig. 8.5 (see [Chapter 5, Ref. 1431) presenting curves of the annual range of the daily possible (QJ and actual Q,) global radiation totals averaged over a month, and also of the real totals of diffuse radiation (D,). As seen, the points situated outside the tropics in the Northern Hemisphere are characterized by a maximum of global radiation in June. The most marked amplitude of the annual range takes place at high latitudes (the island Dixon). In the Southern Hemisphere (Windhook, Mirny) this maximum falls in December (summer of the Southern Hemisphere). An interesting transitional regime of the annual variation in global radiation is

8.3. Main Observed Regularities in Variability of Global Radiation Totals 471

TABLE 8.6 Possible Global Radiation Q, (cal/crnaday). After Berland [28; Chapter 5, Ref. 1431 Latitude, Jan. deg North 90 85 80 75 70 65 60 55 50 45

Feb. Mar. Apr.

May June July

Aug.

Sept. Oct. Nov. Dec.

0

0

35 30 25 20 15 10 5 0

0 0 0 1 24 58 102 159 220 290 352 410 463 511 555 595 635 666

0 0 18 51 92 142 204 270 340 402 460 509 552 590 624 650 671 688

4 24 69 132 198 264 325 384 438 389 538 580 613 640 663 68 1 695 704 707

328 336 354 385 430 478 526 569 608 642 668 689 703 710 710 705 698 688 672

720 716 706 690 675 672 684 707 729 746 759 764 763 754 740 72 1 696 667 635

856 846 828 805 774 751 753 768 780 787 790 788 780 768 750 724 692 656 618

780 77 1 754 727 700 692 703 722 742 75 1 772 775 771 760 743 721 694 662 628

424 430 439 455 480 613 550 590 628 662 678 706 716 719 716 709 698 680 660

78 100 140 191 248 311 371 425 474 519 559 596 628 653 673 688 698 701 698

0 0 15 44 90 142 200 257 318 377 433 483 530 572 608 636 661 681 696

0 0 0 16 45 85 133 190 256 318 375 430 482 530 572 610 643 672

0 0 8 37 70 131 193 260 320 378 43 1 484 530 575 618 656

South 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90

695 722 744 762 776 787 792 792 789 779 762 743 735 742 763 792 811 820

704 715 722 726 726 718 701 680 654 622 586 548 505 469 442 420 408 404

704 694 679 660 639 61 1 578 540 499 454 406 353 298 240 187 140 96 56

654 631 601 566 527 489 447 40 1 353 302 245 184 127 74 31 0 0 0

602 567 527 485 439 392 343 287 233 178 125 79 36 8 0 0 0 0

580 535 49 1 442 395 348 297 24 1 180 125 75 32 3 0 0 0 0 0

590 550 507 464 416 366 316 265 206 150 98 52 15 1 0 0 0 0

634 602 568 531 49 1 447 398 350 297 24 1 180 124 75 32 4 0 0 0

687 670 648 622 519 558 522 482 439 393 340 280 220 165 115 69 30 2

704 705 700 690 679 662 642 616 584 547 507 464 416 375 343 318 303 296

698 717 734 746 755 760 760 752 738 720 702 690 685 688 70 1 721 736 742

693 726 753 774 793 810 822 830 832 824 811 804 807 820 838 856 874 886

40

0

0 0 0

472

Global Radiation

i'\

,' .............

\.

\.-.-

....

... 1

-

Q,

-.-__.Q

1

1

1

1

1

1

1

1

,

,

,

......... D

FIG. 8.5 Annual variation of daily values of the possible Q, and real Q totals of global and diffuse radiation D averaged over the month. (a) Dixon; (b) Moscow; (c) Vladivostok; (d) Leopoldville; (e) Windhook; (f) Mirny.

observed near the equator (Leopoldville), where there are two maxima of spring and fall at the maximal noon solar heights. An anomalous yearly variation of global radiation is observed at Vladivostok, with a summer secondary minimum due to a considerable increase of cloudiness at this time of year. Figure 8.6, plotted by Berland [Chapter 5, Ref. 1431, gives observational data characterizing the latitudinal range of daily global and diffuse radiation totals averaged over a year. Also given are the corresponding curves for the possible global radiation ((3,) totals and for the incoming solar radiation outside the atmosphere (Qoa). Examination of Fig. 8.6 shows that in the range of latitudes from the North Pole to 55' N. lat., the incoming global radiation varies comparatively little. To the south of 55' N. lat. the totals of global radiation rapidly increase, attaining maxima in the zone of tropics and subtropics where the annual means for individual points are the most variable due to the presence

8.3. Main Observed Regularities in Variability of Global Radiation Totals 473

N 80.

60.

40.

20.

0

20.

40.

60.

80.S

Q

FIG. 8.6 Latitudinal variation of daily totals of global and difuse radiation averaged over the year.

of deserts and monsoon areas. In the vicinity of the equator a decrease in the incoming global radiation, due to the effect of cloudiness, is observed. The amplitude of the latitudinal variation in the annual means of global radiation totals within the year is 450 cal/cm2day for the entire globe. It should be noted here that, as seen in Fig. 8.6, the latitudinal variation of diffuse radiation totals is rather slight. Comparing the incoming solar radiation outside the atmosphere and the possible global radiation totals, we see that about 20 percent of energy is lost in the scattering and absorption of solar radiation by the atmosphere. The presence of cloudiness decreases the incoming shortwave radiation additionally by 20 to 30 percent. Thus, during the year, the earth’s surface receives on the average only 50 to 60 percent of solar radiation from outside the atmosphere. Section 5.10 discussed the problem of variability of radiation totals from year to year. We shall add here only a notion on the variability of incoming global radiation as being less than that of direct solar radiation (see [Chapter 6 , Ref. 391). For example, the deviation of the direct incoming solar radiation from the many-year mean at Vladivostok is f 27 percent. Since these fluctuations of direct solar radiation are somewhat smoothed by the oppositely directed variability of diffuse radiation, the monthly mean of

474

Global Radiation

global radiation shown there has only f 12 percent deviation. For Tash8 and & 5 percent. According to Berland kent the respective figures are [Chapter 5, Ref. 1431, the greatest variability of monthly global radiation values is observed at high and intermediate latitudes, the least in the tropics and subtropics.

2. Monthly Global Radiation Totals. Similar to the daily mean, the monthly mean of global radiation has an annual range greatly dependent upon peculiarities of the annual variation of average monthly cloudiness. Since in the majority of cases the minimum of cloudiness within the year falls during the warmer period, the yearly variation of monthly totals of global radiation with a summer maximum is the most representative. This conclusion is illustrated by Fig. 8.7, which gives isopleths of the ratio of monthly totals of global radiation to yearly totals; Qmo/Qyr,depending on latitude and season as plotted by Berland [29]. This ratio is expressed in percent.

80 9 700 al

.-c c 60-

50-

FIG. 8.7 Zsopleths of the ratio Qmo/Qyrin dependence upon Iatitude and season (percent).

As seen from Fig. 8.7, the maximum in the annual range of relative monthly totals at all the considered latitudes occurs in June, the annual range being the more pronounced as the observation point moves farther north. For instance, in the far north during the polar night the incoming global radiation is completely absent, and in the summer time the monthly totals reach 20 to 25 percent of the annual total. Observations show that in southern U.S.S.R., the maximum in monthly totals is often shifted to

8.3. Main Observed Regularities in Variability of Global Radiation Totals 475

July, and to May in the monsoon areas. In isolated years both at intermediate latitudes and in the north, the maximum incoming radiation can be observed in July and in May. The minimum of global radiation almost always occurs in December. We now consider data of observations characterizing relations among monthly totals of direct solar, diffuse, and global radiation. Table 8.7 presents the annual variation of monthly means of direct solar, diffuse, and global radiation from observational data of Galperin [30] at Saratov (the Volga region) in 1935-1937. The observations were conducted with the help of the Yanishevsky pyranometer and the Kalitin pyranometer. As shown by Table 8.7, even at the latitude of Saratov, which has a considerable number of bright days, the incoming diffuse radiation constitutes a large portion relative to the incoming global radiation. This again gives evidence of the importance of diffuse radiation (see Chapter 6 ) . As we found before, the incoming global radiation in the presence of cloudiness is less than at a cloudless sky. The actual monthly global radiation totals are thus much smaller than the possible values. Table 8.8 gives an annual range of average monthly real and possible totals and their ratio, based on Galperin’s data. Table 8.8 shows that in winter the actual monthly totals at Pavlovsk were over 200 percent less than the possible values. Galperin’s data for Saratov give a somewhat less pronounced difference. It was stated in the preceding section that the main factor determining the relation between actual and possible radiation totals was the degree of cloudiness. The latter factor, that is the heavier cloudiness at Pavlovsk compared with that at Saratov, accounts for the actual incoming global radiation at Pavlovsk showing more difference from the possible than at Saratov. It is evident that monthly totals of global radiation must be considerably dependent on latitude. In analyzing observational data, one’s attention is drawn by a contrary range of monthly totals in dependence upon latitude (40 to 80’ N. lat.) in the warm and in the cold half-years. In summer the relative incoming global radiation increases northward, whereas in winter this trend reverses. This is a natural consequence of the above-mentioned fact about the main portion of the yearly incoming global radiation falling at high latitudes during summer while the winter radiation constitutes a small part of the annual value. At low latitudes the distribution of global radiation over the year is more even. Numerous observations point out at an exceptionally great role of global radiation totals in the Arctic and Antarctic. In Rusin’s [3 I ] estimation the summer shortwave radiation in central Arctic reaches 30 kcal/cm2 mo,

TABLE 8.7 Annual Range of Average Monthly Totals of Direct Solar, Diffuse, and Global Radiation at Saratov (kcallcm2mo). After Galperin 1301 Jan.

Feb.

Mar.

Apr.

May

June

July

Aug.

Sept.

Oct.

Nov.

Dec.

0.81

1.45

3.80

7.96

9.96

10.42

10.62

7.24

5.93

3.33

1.23

0.57

?

U

CS’

c

m hi

ED

2.04

2.81

3.95

4.10

5.56

5.76

5.28

5.37

3.63

2.63

1.66

1.36

! a i 6’

2.85

CQ ED CS’ C D

+

3

%

72

4.26

66

7.75

51

12.06

15.52

16.18

15.90

12.61

34

36

36

33

43

9.56

38

5.96

44

2.89

57

1.93

71

Annual Range of Average Monthly Actual and Possible Global Radiation Totals ot Pa)'!Q)'sk (kealfem' min rna). Afler Galperin [30} Nov.

Dec.

Qmo,~

1.5

3.9

9.3

14.2

19.8

21.2

22.2

17.3

to.7

5.7

1.8

0.7

Qmo.~

0.7

2.0

4.9

8.8

13.1

15.8

12.4

to.7

5.9

2.7

0.8

0.3

W In

62

62

66

75

56

55

47

44

43

3

53

td

51

In

47

In

Qmo,~

I-

Qmo,~

8

Oct.

W N

Sept.

I-

Aug.

W

July

W N

June

m

May

In

Apr.

In

Mar.

3

Feb.

d

Jan.

8.3. Main Observed Regularities in Variability of Global Radiation Totals

TABLE 8.8

477

478

Global Radiation

that is, more than the corresponding equatorial value. Even when averaged over the year, the global radiation is approximately the same as at the equator, the annual total amounting to 120 to 130 kcal/cm2. The annual mean of incoming global radiation in the Arctic is 30 to 40 percent less than in the Antarctic, where in central areas elevated at 3 to 3.5 km above sea level, the global radiation is approximately twice that of the central Arctic regions. According to Gavrilova [32], the mean annual Arctic incoming global radiation decreases with latitude. For instance, in the western part of the Soviet Arctic, the annual totals decrease from about 70 kcal/cm2 at the polar circle to less than 60 kcal/cm2 at 80' N. lat. A somewhat higher incoming global radiation is observed in the Canadian Arctic: about 85 kcal/cm2 at the Arctic Circle and more than 70 kcal/cm2 at 80' N. lat. In summer (May-August), however, the monthly totals in the Arctic are larger than in the temperate latitudes (by about 1 to 3 kcal/cm2 mo in the Soviet Arctic and by 3 to 4 kcal/cm2 mo in the North American Arctic), with the greatest incoming radiation being observed in the central part of the Arctic basin. The difference in shortwave incoming radiation between the two polar areas is accounted for by the antipodal role of diffuse radiation whose contribution to the global radiation in the Arctic amounts to about 74 percent, whereas in the Antarctic, the leading component of the global radiation is direct solar radiation (60 to 80 percent). 3. Seasonal and Annual Totals of Global Radiation. It is perfectly evident that seasonal and annual totals of global radiation, similarly to the daily and monthly totals, are determined first of all by cloudiness and solar height within the season or year at a given geographical point. Hence it follows that the seasonal totals must be maximal in spring and summer and minimal in fall and winter. It is also obvious that the seasonal and annual totals must be considerably dependent on latitude. Table 8.9 presents values of the seasonal and annual totals of direct solar, diffuse, and global radiation obtained by Kalitin [I51 from observational data taken at different points of the U.S.S.R. Table 8.9 clearly shows the difference between the spring and summer and the fall and winter totals of global radiation. Also evident is the fact that the farther to the south, the less this difference becomes. Both seasonal and annual totals generally increase southward though not in proportional increments. It is easily understandable that the main cause for breaking the smooth dependence of incoming global radiation on latitude are the peculiarities of the cloudiness regime at a given geographical point.

8.3. Main Observed Regularities in Variability of Global Radiation Totals 479

TABLE 8.9 Seasonal and Annual Totals of Direct Solar, Diffirse, and Global Radiation (kcal/cmz> at Different Points of the U.S.S.R. After Kalitin [15]

Point

Tikhaya Bay

Latitude Longitude Radiation Winter Spring Summer Fall

80'19

52'48'

CS' CD

ZQ Tiksi Bay

71'35'

128'56'

CS' CD

CQ Yakutsk

62'01'

129'43'

ZS' CD

CQ Pavlovsk

59'41'

30'29'

CS' CD

CQ Novisibirsk

55'02'

82'54'

CS' CD

CQ Minsk

53'54'

26'33'

Cs' CD

CQ Voron ezh

5 1'40'

39'13'

CS' CD

CQ Evpatoriya

45'09'

33'15'

CS' CD

CQ Karadag

41'54'

35'12'

CS' ZD

ZQ Kislovodsk

43'54'

42'42'

CS' CD

CQ

Year

0 0 0

15 22

8 23 31

1 1 2

16 39 55

0 1 1

12 18 30

14 19 33

1 4 5

27 42 69

2 2 4

20 12 32

30 14

5

44

4 9

57 32 89

1 2 3

14 11 25

22 13 35

4 4 8

41 30 71

3 4 7

18 10 28

28 10 38

0 5 11

55 29 34

2 4 6

16 13 29

22 15 37

6 6 12

46 38 84

2 5 7

16 14 30

27 15 42

8 7 15

53 41 94

2 6 8

22 15 37

38 14 52

15 77 9 4 4 24 121

5 6 11

23 12 35

37 12 49

15 7 22

80 37 117

7 10 17

17 18 35

27 17 44

15 8 23

66 53 119

7

480

Global Radiation

Table 8.9 also enables characterization of the relation between seasonal and annual totals of direct solar, diffuse, and global radiation at different latitudes. We see that in winter, at all points except Yakutsk, the diffuse radiation prevails over the direct solar radiation. For such northern points as Tiksi Bay and Tikhaya Bay, the incoming diffuse radiation at any time of the year, and consequently over the year as a whole, considerably exceeds the incoming direct solar radiation (excepting Tikhaya Bay in fall). However, not only in the north but also in the south the incoming diffuse radiation constitutes a large portion of the integral radiation income. For instance, at Kislovodsk, the portion of diffuse radiation in the integral yearly total of global radiation amounts to 44.5 percent. The least relative incoming diffuse radiation, equal to 24 percent, was observed in summer at Karadag, where the average annual value is 32 percent. It is interesting that at Yakutsk the portion of diffuse radiation over the year averages 36 percent. Yakutsk, thus, in spite of its comparatively northern situation, is as sunny as Karadag at the southern coast of the Crimea. Yakutsk, however, presents an exceptional case. As a rule, the portion of diffuse radiation in the total incoming radiation increases with an increase of latitude. The data given above show the dependence of average daily and monthly global radiation totals on latitude. The latitudinal dependence of seasonal and annual totals is similar. 4. Geographical Distribution of Seasonal and Annual Totals of Global Radiation. The investigation of the geographical distribution of seasonal and annual global radiation totals was made possible comparatively recently when the available observational data became sufficient for solving this problem. Prolonged and reliable terms of actinometric observations were conducted on the U.S.S.R. territory. The first chart of geographical distribution of annual global radiation totals over the European territory of the U.S.S.R. was plotted by Kalitin [33, 341. In the result of manyyear investigations and on the basis of generalizing the available worldwide actinometric data, Berland [28; Chapter 5, Ref. 1431 has plotted world charts of geographical distribution of monthly (for all months) and annual global radiation totals. Figure 8.8 presents a chart of geographical distribution of daily global radiation totals averaged over the year on continental territories. An analogous chart has been plotted by Landsberg [35]. In Berland’s map we see that the mean annual totals all over the globe vary from 150 to 600 kcal/cm2 day. At high and intermediate latitudes the distribution of global radiation is close to the zonal. At tropical latitudes the zonality is broken, for in the vicinity of the equator a decrease of incoming

8.3. Main Observed Regularities in Variability of Global Radiation Totals 481

482

Global Radiation

global radiation takes place as a result of increasing cloudiness. The maximal annual income of global radiation is observed in the high-pressure belts of the Northern and Southern Hemispheres. The culminant value is recorded in the Sahara at the middle Nile area. For the geographical distribution of global radiation in winter, it is characteristic to find a rapid decrease of incoming global radiation from the intermediate toward the high latitudes, with this distribution being mainly zonal. The summer period is characterized by low gradiants of global radiation over vast areas. During the polar day the income of global radiation in the polar regions by far exceeds that at the equator. Note, for example, that in Rusin’s [31] estimation the mean daily totals in the Antarctic may reach 1000 cal/cm2 day. Barashkova et al. [Chapter 7, Ref. 37b], using observational data only, have carefully studied the regularity of global radiation geographical distribution on the U.S.S.R. territory. Figure 8.9 gives their map of geographical distribution of annual radiation totals. It can be seen that the distribution of global radiation is clearly nonzonal: The western regions receive less radiation than the eastern, especially on the European territory of the U.S.S.R. The annual totals vary from 60 t o 70 kcal/cm2yr in the north to 150 kcal/cm2 yr in southern Central Asia. The comparison with the possible totals shows that the actual values constitute the least portion of the possible (52 to 69 percent) in northwestern parts of the European U.S.S.R. territory, and the greatest portion (75 to 88 percent) in Central Asia. A considerable number of investigations is devoted to the regularities of global radiation distribution for individual countries and territories (for example, see [36-571 and [58]). 5. Totals of Global Radiation Absorbed by the Underlying Surface. The preceding section has characterized the incoming direct solar, diffuse, and global radiation on an underlying surface. It is clear that a large portion of the radiation that reaches the underlying surface is reflected from it. Therefore the values of the radiation absorbed by natural underlying surfaces considerably differ from the incoming radiation values. In the preceding chapter data were given for the underlying surface albedo, from which it is possible to judge the magnitude of the difference between received and absorbed radiation. Now we shall consider only the problem of geographical distribution of seasonal and annual totals of the absorbed global radiation. Figure 8.10 presents the geographical distribution of annual totals of the absorbed radiation, obtained from the data of Fig. 8.9. The comparison

I

60

80

100

120 ~~~

FIG. 8.9 Geographical distribution of annual global radiation totals (kcal/cma)over the URSS territory.

~~~

FIG. 8.10 Geographical distribution of annual totals of the absorbed global radiation (kcal/crn2) over the USSR territory.

8.4. Incoming Shortwave Radiation on Oriented Slant Surfaces

485

of both figures shows a similarity of the geographical distributions of annual totals of the global and absorbed radiation. The latter values vary from 40 kcal/cm2 yr in the north to 120 kcal/cm2 yr in Central Asia. Analysis of observational data shows that the greatest contribution to the annual totals of the absorbed radiation occurs in June, and the least in December. It is clear that the similarity of the fields of the global and absorbed radiation may not be found in isolated months and seasons in individual geographical areas. For example, it has been mentioned above that exceptionally high values of global radiation are observed in the Antarctic in summer. The large albedo of the underlying surface there, however, means that the portion of the absorbed radiation relative to the global is only 10 to 20 percent. That is why the monthly totals of the absorbed radiation do not exceed 4 to 5 kcal/cm2 and annual totals are no more than 16 to 20 kcal/cm2.

8.4. Incoming Shortwave Radiation on Oriented Slant Surfaced The problem of incoming shortwave (diffuse or global) radiation to slopes is of great interest in connection with the solution of various problems in such areas as agriculture, heliotechnique, and building. Consider now, following the results obtained by Kondratyev et al. [58-661, the main principles controlling the incoming diffuse and global radiation to slopes. The incoming radiation falling on oriented slant surfaces depends upon many factors such as the angle of inclination and the azimuth of the surface, solar height above the horizon, atmospheric transparency, albedo of the underlying surface, and obliteration of the horizon. To calculate fluxes of diffuse or global radiation on slopes in each concrete case is rather complex. The problem is simplified if the relative flux values are determined, that is, if the ratio of the flux on a slope to the flux on a horizontal surface and the fluxes on the horizontal surface are known. The relative flux values depend mainly on the surface orientation and solar height. The dependence on albedo and other factors is considerably decreasing. For steep slopes and at high albedo values, however, the reflected radiation may be quite important in the global radiation flux. Figures 8.11 and 8.12 present the measured dependence of the ratio of diffuse and reflected radiation on a slope (DSl rsl) to the flux of diffuse radiation on a horizontal surface (Dhor)upon angle of inclination and orien-

+

t M. P. Fedorova is coauthor of this section.

486

Global Radiation

tation of the surface from clear sky. The observations were conducted in the presence of snow cover, with an albedo at 60 percent. The interrupted curves show the dependence of the relative fluxes on angle of inclination and surface azimuth for summer at the same solar heights and an albedo value of 20

Q

FIG. 8.11 Dependence of the value of the relative diffuse radiation flux upon angle of inclination cx and surface orientation. (1) Leningrad region, Aug. 1958, clear sky, h, = 26O, A = 0.59; (2) Karadag (Crimea), July 1956, clear sky, h, = 26O, A = 0.20.

percent (the lettering N, S, E, and W denotes the points of the compass). From the comparison of the curves it is seen that the relative fluxes of the diffuse and reflected radiation on slopes considerably increase with an increase of albedo, which is especially marked in the case of steep slopes.

8.4. Incoming Shortwave Radiation on Oriented Slant Surfaces

487

The comparison of Figs. 8.11 and 8.12 also shows that the relative fluxes of the diffuse and reflected radiation increase as the elevation of the sun above the horizon increases. This fact was noted earlier by Bogdanovich [67], who had measured the illumination of oriented vertical surfaces by scattered light.

0

FIG. 8.12 Dependence of the value of the relative diffuse radiation flux upon the angle of inclination a and surface orientation. (1) Leningrad region, Aug. 1958, clear sky, h, = 34O, A = 0.59; (2) Karadag, July 1956, clear sky, h, = 34O, A = 0.20.

In the presence of snow cover almost all slopes receive more diffuse radiation than do the horizontal surface. The dependence of the ratio (DSl rsl)/Dhorupon angle of inclination and surface azimuth at continuous cloudiness and in the presence of snow has the same character as for the summer conditions of solid cloudiness. The incoming radiation on slant surfaces is practically independent of surface azimuth. Figure 8.13 gives the results of measurements of fluxes of diffuse and reflected radiation on slopes at uninterrupted cloudiness and a surface albedo of about 70 percent. The dashes denote the curves for the summer conditions (the albedo about 20 percent) at the same solar height. The radiation reflected

+

488

Global Radiation

by snow notably increases the relative flux values. With an increase of the angle of inclination the reflected radiant flux increases, nearly compensating for the decrease of the diffuse radiant flux, so that the ratio is almost independent of inclination and approaches 100 percent.

I

z

rk n

Q

FIG. 8.13 Dependence of the relative difuse radiation flux value upon angle of inclination (Y and surface orientation at a fully overcast sky. (1) Leningrad region, March 1958, Sc force 10, h, = 17O,. A = 0.71; (2) Leningrad region, Oct. 1954, Sc force 10, h, = 16O, A = 0.20.

TiF-

he FIG. 8.14a Values of the relative difuse radiation fluxes from semicircular sky zones for surfaces of direrent steepness directed sunwards. Zone 0-27.5'. Solid curves, solar semicircle; dashed curves, antisolar semicircle.

8.4. Incoming Shortwave Radiation on Oriented Slant Surfaces

489

When investigating the radiation regime of slopes it is of interest to determine fluxes of diffuse radiation from separate sky sections. Such data rnable evaluation of the influence of obliteration of the horizon on the eadiation regime of slopes. Using measurement data of the angular distribution of diffuse radiant intensity, Fedorova [66] has computed fluxes of diffuse radiation on slant surfaces of different orientation from six semicircular sky sections. The circular zones were limited by parallels of altitude 0 to 27.5', 27.5 to 52.5', and 52.5 to 77.5'. Each zone was cut into two parts by a plane perpendicular to the solar vertical plane. Thus diffuse radiation fluxes on the given surface were determined from the circumsolar semicircular zones and from the semicircular zones opposite the sun. In Figs. 8.14(aYby c) and 8.15(a, b, c) are presented dependences of the relative diffuse radiation flux values (the ratio of the flux from an isolated

t

ae

blb

43 FIG. 8.14b Zone 27.5-52.5'.

section to the integral flux from the entire sky) on solar height for each of the semicircular zones. Such curves are plotted for surfaces with angles of inclination lo', 20°, 30°, 50°, 70°, and 90' oriented on the azimuths ' 0 and 180' relative to the solar azimuth, and for a horizontal surface (a = 0'). These graphical results of calculations show that in the majority of cases

490

Global Radiation

FIG. 8 . 1 4 ~ Zone 52.5-77.5'.

I

FIG. 8.15a Values of the relative diffuse radiation fluxes from semicircular diflerent steepness directed opposite to the sun. Zone 0-27.5'.

sky zones of

8.4. Incoming Shortwave Radiation on Oriented Slant Surfaces

"e FIG. 8.15b

Zone 27.5-52.5O.

FIG. 8.15~ Zone 52.5-77.5'.

49 1

492

Global Radiation

it is essentially necessary, when evaluating the radiation regime of slopes, to take into consideration the obliteration of the horizon on the sunny side. To estimate the role of the reflected radiation, fluxes of reflected radiation on slopes were computed and their portion in the integral flux of diffuse and reflected radiation on a slope (in percent) was determined. The reflected radiation fluxes (rsl) were computed from the formula for isotropic radiation : . a rS1= rhor sin22 where rho, is the flux of reflected radiation on a horizontal surface directed downward and a: is the surface inclination angle. As has been stated, the snow surface reflection is essentially anisotropic. Thus, calculations for the isotropic approximation enable only rough evaluation of the portion of the reflected radiation in the radiant flux on a slant surface. In the case of surfaces directed sunward such calculation will give lowered results, owing to the specular character of reflection; and in the case of surfaces oriented in the opposite direction, the results will be exaggerated. From the consideration of Table 8.10 (see [64]) it is possible to estimate the difference between precise and approximate calculations. This table TABLE 8.10 Relative Values of Reflected Radiation Fluxes on Slant Surfaces (rsI/rhor,%) Calculated from the Exact and Approximate Formulas for the Conditions of a Clear Sky and in the Presence of Snow Cover (Melting Granular Snow, A = 0.50) and h, = 30' (March 22, 1956). After Kondratyev and Fedorova 1641 I

Azimuth y (Reckoned from the Solar Azimuth), deg

Angle of Inclination, deg

0

30 50 70 90

~

90

180

270

6.0

rSl (Y - = sin2x 100% 2 rhor 6.1

13.0 (95.5)

5.2 (-22.4)

4.6 (-31.4)

(- -10.5)

35.0 (96.0)

16.8

13.8 (-22.9)

18.5 (3.4)

17.9

(- 6.2)

60.0 (82.6)

25.8 (-21.6)

34.6 (5.2)

32.9

(- 3.6)

83.0 (66.0)

40.0 (-20.0)

53.6 (7.2)

50.0

(- 4.0)

31.7 48.0

493

8.4. Incoming Shortwave Radiation on Oriented Slant Surfaces

gives values of the relative reflected radiation fluxes on slopes (rsl/rhor,%) calculated from the measured angular distribution of the reflected radiation intensity from the surface of snow. The rightmost column gives such values for the isotropic approximation. The figures in brackets give the percentage of discrepancy between the approximate and exact values. The calculations show first that the incoming reflected radiation to slopes is essentially dependent on their orientation. Second, they show that for steep slopes directed sunward the isotropic approximation gives greatly lowered values of the relative reflected radiation fluxes. For slopes with azimuths 90' and 270' relative to the solar azimuth, the differences are not great. For slopes opposite to the sun direction, the isotropic approximation gives notably exaggerated values. The results of approximate calculations of the portion of the reflected radiation in the diffuse (in percent) for the winter and summer conditions are presented in Table 8.11. As seen from this table, in the presence of snow cover the reflected radiation flux on steep slopes is larger than the diffuse radiation flux. For the solution of a number of practical problems it is interesting to estimate the influence of surface albedo on the relative values of global TABLE 8.11 Values of the Ratio of the Reflected Radiation Flux to the Flux of the Reflected and Diruse Radiation on Slant Surfaces (rsl/Dsl rsl, %) in a Cloudless Sky; Diferent Albedo Values and h, = 34'

+

Angle of Inclination, deg

N

S

W

E

A = 0.60 15 30 50 70 90

7.3 26.8 54.7 81.4 100.0

5.2 15.8 30.0 41.1 51 .O

6.3 21.8 46.5 63.5 76.5

5.9 19.4 37.2 57.0 63.4

A = 0.20 15 30 50 70 90

1.9 8.2 20.2 37.0 43.0

1.4 4.9 11.4 19.2 29.2

1.6 6.1 15.4 26.7 40.6

1.7 6.5 16.4 28.2 43.0

Directional Exposure

494

Global Radiation

radiation fluxes on slopes. Figure 8.16 gives curves of the dependence of the relative global radiation fluxes (Ssl DS1 rsl)/Shor Dhor) on angle of inclination and surface orientation from a clear sky and with surface albedo of 0.45 (melting granular snow). The dashed curves are for the summer conditions with surface albedo of 0.20. With an increase of albedo the relative global radiation flux values for steep slopes directed sunward increase considerably. For slanting surfaces (a < 30') of all orientations and for the slopes oriented opposite t o the sun, the value of the relative global radiation flux does not show significant variation.

+

I

3

0

5

I

0

+

7

I

0

+

9

I

0

a

FIG. 8.16 Dependence of the relative global radiation flux value upon the angle of inclination a and surface azimuth y (azimuth reckoned from the direction sunwards). (1) Leningrad region, March 1959, clear sky, h, = 2S0, A = 0.45 (melting granular snow); (2) Karadag, July 1956, clear sky, h, = 2S0, A = 0.20.

Figure 8.17 presents the dependence of the relative global radiation fluxes on angle of inclination and surface azimuth for the solar height h, = 30°, obtained from data of terminal measurements (solid curves) and from recordings (interrupted curves). Also given is the dependence of the relative fluxes for the summer conditions at the surface albedo of 0.20. The curves show an increase of the relative flux values with an increase of albedo. Thus the relative global radiation flux values in certain cases are essentially dependent on surface albedo. It is obvious that the albedo affects the relative global radiation fluxes to a lesser degree than do the relative diffuse radiation fluxes.

8.4. Incoming Shortwave Radiation on Oriented Slant Surfaces

495

Table 8.12 summarizes the results of calculations of the portion of the reflected radiation in the global radiation flux on slopes for the summer and winter conditions with a clear sky. Since the albedo of the underlying surface increases from 0.20 to 0.60, the portion of the reflected radiation in the global radiation for the corresponding surfaces increases two to three times. However, in the flux of global radiation on slant surfaces oriented toward the sun, the reflected radiation portion is insignificant; even at the albedo 0.60, it amounts to less than 20 percent. It should be noted, though, that the exact calculation of the reflected radiation portion may be estimated

FIG. 8.17 Dependence of the relative global radiation flux value upon the angle of inclination (r and surface azimuth y. (1) Karadag, July 1956, clear sky, h, = 30°, A = 0.20; (2) Leningrad region, Aug. 1950, clear sky, h, = 30°, A = 0.20; (3) Leningrad region, March 1960, clear sky, h, = 30°, A = 0.77.

at twice that; that is, in reality this portion for steep slopes directed sunward (a > 70') at A = 0.60 may exceed 30 percent. The reflected radiation is quite important to the incoming global radiation on surfaces oriented opposite to the sun direction, and in certain cases it constituted the main portion of the radiant flux to a slope. The relative values of global radiation fluxes on surfaces inclined at from ' 0 to 90' and oriented on the azimuths O', 90' and 180' relative to the sun's azimuth at the solar height from 20' to 70' are given in Figs. 8.18. through 8.20. The curves are plotted from summer pyranometric measure-

TABLE 8.12 Values of the Ratio (Percent) of the Reflected Radiation FIux to the FIux of Global Radiation on Slant Surfaces at Different Surface Albedo Values

ha a, deg

12'. A = 0.65 (0 WSW)

Azimutha, deg/Orientation

I N 15 30 50 70 90

h,

=

2.4 14.8 29.8 39.0 59.0

h,

W

S

=

0.8 2.3 5.0 8.4 12.9

0.6 1.7 3.4 5.5 8.0

loo, A

= 0.46

90'

180'

E 3.7 14.8 32.4 52.2 68.3

= 0.60 (0 S) Azimuth, deg/Orientation

N

S

W

E

1.9 20.5 49.0 80.6 91.0

0.8 2.7 6.2 10.9 17.4

1.2 5.7 21.5 53.3 68.2

1.0 3.8 10.5 20.9 39.0

h, = 28O, A = 0.40

y,

= 261°,

l

N

S

W

E

1.6 13.9 50.8 74.2 99.0

0.8 2.6 6.2 11.0 17.4

1.1 5.3 19.7 65.6 75.0

1.0 3.7 10.2 20.7 39.2

A = 0.20, h, = 28'

y , = 1020, h, = 740, A = 0.20

a, deg

' 0 15 30 50 70 90

0.4 1.1 2.3 3.8 5.7

0.7 2.7 8.0 13.3 22.7

2.0 8.6 22.7 36.6 54.4

Oo

90°

180°

N

S

W

E

N

S

W

E

0.5 1.6 3.7 6.4 10.0

0.7 2.7 7.8 18.0 37.6

1.4 16.4 38.0 53.0 67.0

0.3 1.5 4.8 13.4 37.5

0.3 1.5 4.7 12.2 33.4

0.2 0.8 1.9 3.5 6.0

0.6 6.7 17.5 27.9 43.0

0.4 1.8 7.2 30.8 47.0

0.3 1.3 3.8 8.8 19.2

0.5 5.2 19.2 32.8 41.6

0.3 0.9 2.2 4.3 7.4

Note: The surface azimuth in degrees was reckoned from the solar azimuth.

o\

h , = 35O, A

28O, A = 0.64 (0 SSE)

=

Azimuth, deg/Orientation

l

eJ

9

13 P

8.4. Incoming Shortwave Radiation on Oriented Slant Surfaces

497

ment data for a cloudless sky and a surface albedo of about 20 percent. These curves make it possible to determine the incoming global radiation on surfaces of a given orientation for similar conditions, provided the global radiation fluxes on the horizontal surface are known. The data of the

FIG. 8.18 Dependence of the relative globaI radiation flux value upon solar height surfaces of different steepness with the azimuth y = '0 (in relation to the sun).

for

above mentioned measurements and recordings have been used to analyze the daily variation of the global radiation fluxes on different slopes oriented to the four points of the compass. The curves of the daily range of global radiation fluxes for various surfaces, plotted from data of registration, are given in Figs. 8.21 through 8.23. As seen from these figures, the differences in variation between the slope curves and the horizontal surface curves are quite pronounced for

498

Global Radiation

FIG.

both slanting and steep surfaces. At 60' N. lat., in the presence of snow cover, all southern surfaces receive much more global radiation during the day than does the horizontal surface. Interesting data on the daily range of global radiation fluxes on slopes

ho FIG. 8.20 Dependence of the relative global radiation flux value upon solar height for surfaces of diferent steepness with the azimuth y = 180' in relation to the solar azimuth (a is the angle of inclination of the slope).

8.4. Incoming Shortwave Radiation on Oriented Slant Surfaces

499

Houa FIG. 8.21 Daily variation of global radiation fluxes for slopes of 15' steepness oriented on the cardinal points. Leningrad region, March 1960, clear sky, A = 0.77. S, south slope; E, east slope; N , north slope; Hor, horizontal surface.

are described by Mukhenberg [67a]. By recording the daily variation of global radiation on different surfaces at a fully overcast sky it was shown that all surfaces of equal inclination receive an almost equal amount of radiation per day. At partial cloudiness the incoming radiation on slopes depends not only upon cloudiness, its degree, and character, but also and

HOUR

FIG. 8.22 Daily variation of global radiation fluxes for slopes of 30' steepness oriented on fhe carPdinal points. Leningrad region, March 1959, clear sky, A Fig. 8.21.)

= 0.45.

(Key as in

500

Global Radiation

FIG. 8.23 Daily variation of global radiation fluxes for slopes of 75O steepness oriented on the cardinal points. Leningrad region, December 1959, clear sky, A = 0.97. (Key as in Fig. 8.21.)

essentially upon distribution of clouds over the sky. The daily totals of global radiation were determined and their relative values for different surfaces calculated by planimetric measurements of the curves for the daily global radiation variation. The results of such calculations for the case of four clear days in the presence of snow are given in Table 8.13. Table 8.14 presents similar results for four fully cloudy days. Table 8.13 shows that the relative global radiation totals for a clear sky vary noticeably from day to day. The given months are December, March and April, so that the variation in the relative totals in Table 8.14 could be due not only to different snow albedo values but also to a variation in the angle of incidence of the sun’s rays on the slant surfaces. It may be noted that in the case of northern slopes, the relative totals first decrease with an increase of the angle of inclination, and then increase (for steep slopes). In the case of eastern slopes the considered totals do not show any significant variation in dependence upon angle of inclination. For southern slopes, observations indicate an increase of the relative totals with an increasing steepness. With a completely overcast sky (Table 8.14) the differences in relative

8.4. Incoming Shortwave Radiation on Oriented Slant Surfaces

50 1

TABLE 8.13 Daily Totals of Global Radiation (cal/cm2)from a Clear Sky and in the Presence of Snow Cover

Dec. 11, 1959, A = 0.97 y = 108.2

March 22, 1960, A = 0.75 y = 316.7

March 26, 1960, A = 0.70 y = 384.8

Apr. 4, 1960,

A

= 0.08

y = 340.8

N

15 30 45 60 75 90

77.7 59.2 21.7 -

0.72 0.55 0.20 -

212.5 41.0 66.6 100.0 131.6

0.67 0.13 0.21 0.32 0.42

260.0 100.0 102.0 116.1 133.0

0.67 0.26 0.26 0.30 0.35

312.8 149.0 126.5 130.8 169.4

0.92 0.44 0.37 0.41 0.50

E

15 30 45 60 75 90

43.8 62.0 62.5 -

0.40 0.57 0.58 -

369.0 288.1 408.0 400.8 357.7

1.17 0.91 1.29 1.26 1.13

381.8 347.5 380.0 415.5 328.8

0.99 0.90 0.99 1.08 0.99

404.0 350.4 476.0 373.7 356.9

1.19 1.03 1.40 1.09 1.05

15 30 45 60 75 90

201.3 218.0 256.8 -

1.86 2.02 2.37 -

489.0 611.9 715.0 -

1.54 1.93 2.96 -

1.27

-

-

489.7 611.0 704.0 -

1.59 1.83 -

538.5 555.7 543.0 726.3 550.3

1.58 1.63 1.59 2.14 1.62

W 60 90

127.0

1.17 -

361.2 424.0

1.14 1.34

357.5 429.0

0.93 1.12

481.0 545.0

1.41 1.60

S

Note:

&

-

=

-

-

total on slope. on horizontal.

x h o r = total

totals between different days are not great. A decrease of the relative totals at the increasing angle of inclination is observed. For a more detailed treatment of the dependence of the relative daily totals of global radiation upon surface orientation, underlying surface albedo, and cloudiness, the available experimental material is not sufficient. The measurements and calculations considered above for diffuse and global radiation fluxes on slant surfaces show that their relative values are essentially dependent on albedo of the underlying surface and may vary

502

Global Radiation

TABLE 8.14 Daily Totals of Global Radiation (cal/cm2)from an Overcast Sky and in the Presence of

Nov. 28, 1959, l o p 0 St, A = 0.80 y = 19.3

Dec. 12, 1959, l o p 0 St, A = 0.80 y = 19.7

Dec. 18, 1959, 10/10 St, A = 0.90 y = 18.6

March 15, 1960, l o p 0 St, A = 0.75 y = 79.9

I

N

45 60 75

16.3 12.4 12.8

0.84 0.64 0.66

16.7 12.7 13.1

0.85 0.64 0.66

15.7 12.0 12.4

0.84 0.64 0.67

67.5 52.3 66.3

0.85 0.65 0.83

E

45 60 75

21.2 16.9 16.3

1.10 0.88 0.84

21.7 17.3 16.7

1.10 0.88 0.85

20.4 16.3 15.7

1.10 0.88 0.84

82.6 70.0 67.5

1.04 0.88 0.85

s

45 60 75

19.4 19.1 15.4

1.00 0.99 0.80

19.8 19.5 15.8

1.00 0.99 0.80

18.7 18.4 14.9

1.00 0.99 0.80

80.3 79.0 63.8

1.00 0.99 0.80

W 60

15.7

0.81

16.1

0.82

15.2

0.82

65.0

0.81

noticeably for the summer and winter conditions. Therefore, to determine such relative flux values on oriented slopes in the presence of snow, it is necessary to plot empirical curves of the dependence of these values upon angle of inclination, surface azimuth, and solar height, which requires further accumulation of observational data.

8.5. Income of Global Radiation under Vegetative Covers The investigation of the regularities of incoming global radiation under vegetative covers presents a great interest in its effect on plant life. Radiant energy is a most important factor of the environment in determining vegetative life activity. Omitting the problems of phytophysiology, we shall give only the results of some observations concerning the effect of global radiation upon various vegetative covers. 1. Grass. It is natural that the shading of soil by vegetation causes a considerable decrease of incoming global radiation as compared with that received by bare surface. This can be seen, for example, in Fig. 8.24, which presents curves that characterize the penetration of radiation into various

8.5. Income of Global Radiation under Vegetative Covers

503

40

:

0

-

1111111111

20 40 60 80 0 20 40 60 80 100 0 20 40 60 80 100 120 140 160 cm cm cm

FIG. 8.24 Penetration of radiation inside the grass stand (in percent of the incident radiation). (1) soya; (2) oat; (3) Jerusalem artichoke; (4) potato; ( 5 ) millet; (6) sunflower; (7) beet; ( 8 ) safflower; (9) horse beans.

agricultural plants as observed by Kudriavtzeva [68]. On the axis of ordinates are plotted value of the ratio of radiant fluxes on the horizontal surface at some depth within the herbage to those above the plants' surface; the axis of abscissas gives the distance in centimeters from the plant's surface to the receiving surface of an instrument probe inside the plant. The maximal abscissa of each curve corresponds to the height of the herbage. Observations were conducted by means of a thermoelectric pyranometer. The sky was dull (absence of direct solar radiation). As seen from Fig. 8.24, the regularities of the decrease in global radiation as it penetrates into the grass differ for different vegetative covers. In the case of soya, potato, and beet, observation finds a rapid decrease of incoming radiation with increasing depth of penetration. For the herbage of Jerusalem artichoke and common sunflower, a slow decrease of incoming radiation in the same process is characteristic. Also notable is the fact that the portion of radiation reaching the soil surface considerably depends on type of grass. In the case of soya, potato, millet, and Jerusalem arti-

504

Global Radiation

choke, this portion constitutes 10 percent of the incident radiation, whereas in the case of oats, it increases up to 40 percent. Lopukhin [69-711 has carefully studied the main regularities of global radiation distribution within cotton and lucerne, the penetration of direct solar and diffuse radiation inside the mass of vegetation, and the absorption by cotton, which is very important for understanding the life activity of plants. Figure 8.25 presents a variation in daily totals of the net radiation components for cotton during the period of vegetation, according to Lopukhin’s pyranometric measurements. Curve 1 characterizes the variation of

c ) -

7

FIG. 8.25

8

--

A

9

Variation of daily totals of the net radiation componentsfor cotton during the vegetation period.

global radiation daily totals and curve 2, the variation of daily totals of radiation reflected by a cotton field. Curve 3 determines the daily total range of global radiation absorbed by the soil under the cotton cover. The latter values were determined as a difference between the incoming solar radiation on the soil and the global radiation reflected by the soil surface. The ordinates of curve 4 are the differences between the curves 2 and 3,

8.5. Income of Global Radiation under Vegetative Covers

505

and characterize the variation of daily radiation totals absorbed by the cotton. The hatched area of Fig. 8.24 determines the integral amount of radiation absorbed by the cotton mass during the period of vegetation. The figures 7, 8, and 9 in the lower part of the graph denote respectively the vegetation periods : budding, florescence, ripeness. The data of Fig. 8.24 make possible a careful analysis of the variation in the cotton radiation regime components during the entire vegetative period. As seen from this figure, the value of radiation absorbed by the sheltered soil, and consequently the penetration of global radiation inside the vegetative cover, smoothly decrease as the cotton grows, whereas the values of the radiation absorbed by cotton show a variation approximately parallel to the incoming global radiation. However, the maximum of the radiation absorbed by the cotton growth is notably displaced in relation to the maximum of the incoming global radiation. Note here that the minimum of curve 4, occurring on the first days of August, is due to accidental circumstances (the plants were infested by mite). Under normal conditions such a minimum would not be present; a maximum of absorbed radiation is observed during florescence. Analyzing the relation between curves 3 and 4, we see that in the primary period of vegetation the absorption of radiation by soil prevails over the radiation absorption by cotton, and vice versa, for beginning with the period of flowering, cotton becomes the main absorbent. For this reason phytoclimate of the cotton field is divided into two periods, before and after florescence. As main quantitative relations characterizing the cotton radiation regime, the mean obtained by Lopukhin [70] from averaging two-year observational data may be given. The incident global radiation reflection by the cotton cover and the sheltered soil averages 20 percent. The other 80 percent are absorbed by the underlying surface, in the proportion of 43 percent by the soil surface and 37 percent by the vegetation. As was shown by Lopukhin, the chief portion of radiant energy absorbed by cotton is lost in evaporation, and only a comparatively small part of radiant energy is realized in the process of photosynthesis. According to Geodakian [72], the coefficient of use of solar radiant energy in photosynthesis varies within 1 to 2.5 percent. It is obvious that the main reason for the decrease of incoming radiation in the sheltered soil is the developing vegetation shade. Special measurements of shade degrees were carried out by Lopukhin [70] and have shown that during the period of vegetation, the degree of shade increases from several hundreds in the period of budding to 0.7-0.8 at the end of flore-

506

Global Radiation

scence and the beginning of ripeness. The considered values were determined by counting the shaded marks out of the general number of 50 marks equally spaced (at about 5 cm) on a white lath placed under the cotton. Such measurements were conducted at ten different radially directed orientations of the lath. The degree of shade can be found as a ratio of the average number of shaded marks to the total marks. The investigation of global radiation penetration into maize, realized by Tooming [73] (see also Shablovskaya [73a] and Yefimova [73b]), has shown that in the beginning of the vegetation period the main bulk of global radiation (78 percent) is absorbed by soil, 15 percent is reflected, and only 7 percent is absorbed by maize. At the end of the vegetation period the herbage albedo increases up to 24 percent, the portion of the global radiation absorbed by plants to 68 percent, and only 8 percent of the radiation is absorbed by soil. In studying the peculiarities of phytoclimates it is of interest to explore not only the integral but also the spectral characteristics of the net radiation components. In this respect observations by Lopukhin [70] for cotton and lucerne fields deserve attention, and Table 8.15 summarizes his relative TABLE 8.15 Relative Values of Radiation Reaching the Shaded Soil and Absorbedby Cotton over Different Spectral Intervals (Percent in Relation to the Global Radiation Flux above the Vegetative Cover over the Corresponding Spectral Intervals). After Lopukhin 1701

Budding

Net Radiation Components

Radiation absorbed by cotton Integral Blue (0.469 p ) Green (0.532 y) Red (0.642 p ) Infrared (1.040 p ) Radiation reaching soil Integral Blue (0.469 p ) Green (0.532 p ) Red (0.642 y) Infrared (1,040 p )

Ripeness

6/21

7/1

7/15

8/1

8/15

9/1

9/15

15 16 8 18 -

12 19 9 20 3

28 28 22 28 11

46 64 59 64 22

54 72 68 69 26

58 75 72 74 30

55 76 71 74 27

48 69 63 66 24

47 65 60 62 23

93 83 91 78

84 80 89 76 88

64 71 75 70 72

44

33 25 26 28 46

30 22 22 22 38

31 22 23 23 41

40 29 32 31 49

43 34 36 35 59

I

Florescence

35 38 34

55

11/1 11/15

8.5. Income of Global Radiation under Vegetative Covers

507

values of radiant fluxes (in different spectral intervals) reaching the soil and absorbed by cotton during the periods of vegetation. From the consideration of Table 8.15 it follows that cotton is clearly selective as regards radiation transmission and absorption, a selectivity that varies within the entire period of vegetation. For example, the relation between the absorbed radiation of the green and of the red spectral intervals is thus considerably variable. In the beginning of the budding period, the cotton absorptivity in relation to “red” radiation is more than twice that for green rays, whereas by the end of the period of ripeness, the respective values become practically equal. In general, the selectivity of radiation absorption by cotton gradually decreases with its development, and only the absorptivity for infrared rays remains (about three times less than the absorptivity for the other spectral intervals even at the end of the period of ripeness). Somewhat less evident is the selectivity of radiation transmission by cotton. It is essential to note, however, that in this case there is no observed decrease of selectivity in the process of cotton growing. It is quite evident that the selectivity of radiation transmission and absorption by cotton is largely due to the selectivity of the optical properties of its foliage and branches. In this connection the problem of radiative properties of cotton, and of other leaves in general, is quite interesting. Let us now consider this problem. 2. Reflection, Transmission, and Absorption of Solar Radiation by Leaves ofPZants. Observations show that leaves of plants are, as a rule, semitransparent. Therefore the radiant flux incident on their surface is not only partly reflected by it and partly absorbed by leaf thickness, but is also transmitted through the leaves. There have been made numerous determinations of the values of reflection, transmission, and absorption of shortwave radiant energy (direct solar and diffuse radiation) by leaves. Different measurement methods were used in similar investigations. Makarevsky [74], for example, determined the reflectivity values of leaves from data of albedo measurements at 3-m2 surface covered with fresh plucked leaves. The values of radiation transmission by leaves were determined from pyranometer recordings, with the leaf freely exposed and covered by a glass cover (the leaves were always placed on the glass cover with their upper surface outward). Having measured the reflectivity ( A ) and transmissivity ( P ) of leaves, it is possible to find their absorptance a as a residual term in the relation A P a = 1. Kalitin [75] has worked out methods for measuring radiation reflection and transmission by separate leaves with the help of a

+ +

508

Global Radiation

modified Yanishevsky pyranometer in which a selenium photocell serves as a radiation receiver for the same purposes. Lopukhin [70, 76, 771 used an Ulbricht sphere. To characterize the results of measurements of reflection, transmission, and absorption of the integral global radiation flux by leaves of different plants, observational data by Makarevsky [74] are given in Table 8.16. As seen from this table, in all the considered cases the values of reflection and transmission differ but comparatively slightly, whereas the absorptivity of leaves exceeds their reflectance and transmittance by about two times. The data of Table 8.16 give evidence of the fact that the radiative properties of leaves depend on their development and lifetime and also on the color and state of their surface. For instance, the variation of the radiation transmission by cabbage leaves, due to the above factors, reaches 70 percent. The transmissivity of a dark red cabbage leaf is only 18 percent, but that of a thin white leaf is 31 percent. TABLE 8.16 Reflection, Transmission, and Absorption of the Integral Solar Radiation Flux (Percent) Plants. After Makarevsky [74] by Leaves of Di’erent

Plant

Elder Jasmin Cabbage Cabbage Cabbage Cabbage Cabbage Cabbage Cabbage Maple Maple Maple Sheep sorrel Burdock Coltsfoot Coltsfoot Cucumber Rhubarb Sugar beet

Reflection

22 24 25 23 27 30 21 24 25 21 23 23 24 24 23 22 24

Transmission Absorption

27 29 24 21

51 47 51 56

-

-

31 24 25 18 29 34 38 28 24 28 21 28 24 28

46 55 47 41 35 49 53 48 49 52 54 48

Leaf Characteristics Dark green Green Green, thin, new Green Light green White, thin White Yellow-red Dark red Green Yellow-green Yellow-brown Green Green Green Green Green Green Green

8.5. Income of Global Radiation under Vegetative Covers

509

Observations by Kalitin [75] and the data of Table 8.16 (see the measurement for maple leaves) show that autumn yellow leaves are essentially different from summer green leaves in their radiative properties. The reflectivity and transmissivity of the latter are comparatively small, but the absorptivity is considerable. Conversely, the yellow leaves in fall are poor absorbents but are intensive reflectors and transmitters. A peculiar feature of certain southern plants with dense leaves is a low transmissivity of leaves; for example, magnolia leaves transmit only 4 percent of the incident radiation. The leaf moisture content greatly affects its radiative properties. Leaves with a big store of moisture rather weakly reflect and transmit, but are powerful radiation absorbents. At shortage of moisture their reflectivity and transmissivity notably increase, and the absorptivity decreases accordingly. The determination of the values of reflection, transmission, and absorption of radiation by leaves of plants in different spectral intervals has revealed the selectivity of leaf radiative properties. Data on spectral albedo of vegetation have already been given in the preceding chapter. A detailed investigation of the spectral radiative properties of leaves and other parts of plants has been performed by Lopukhin [70], who used glass filters for isolating radiation of different spectral intervals. Table 8.17 summarizes some of the measurement results for cotton leaves (standard 0 to 460) that are turned with their upper surfaces sunward. The data in Table 8.17 show a presence of considerable selectivity of cotton-leaf spectral radiative properties and also its dependence upon vegetation phase (leaf surface state). Lopukhin divides all the studied leaves into three groups according to the character of spectral absorptivity. To the first group belong green leaves whose characteristic is most intensive absorption of blue (Aeff = 0.469 p ) and violet (Aeff = 0.392 p ) rays and a weak absorption of infrared (Aeff = 1.040 p ) rays. Yellow and dark reddish leaves that differ from the green by a more smooth decrease of absorptivity at an increasing wavelength constitute the second group. The yellow leaf absorptivity in the visible spectrum is much lower than that of the green leaf. The red leaf possesses a higher absorptivity of yellow-green rays. Spotted leaves are included in the third, group. Their absorptivity differs comparatively little from that of the yellow leaves and is characterized by a smooth decrease with an increase of wavelength. A characteristic peculiar to all measurement data on absorptivity as described above is the considerable decrease of absorptivity in the red and infrared spectral regions. The same results have been obtained by Lopukhin from measurement data on the reflectivity of green lucerne leaves and

TABLE 8.17 Spectral Radiarive Properties of Cotton (Leaf Characteristics). After Lopukhin [70]

Green new, 2 days

Green cotyledonous

Reflection 0.392 0.469 0.532 0.642 0.737 1.040

s 0

Full Ripeness

Flowering

3efore Flowering

Green, 3d from above on the main stalk

Red’ 3d from above

Yellow, 13-14th from above

Yellow, 14th Spotted yellow from above with green and red, green and red 14th from above spots No. 2 No. 1

No. 4a

No. 1

No. 7

No. 6

No. 4

No. 4

5.4 9.9 16.6 12.2 31.7 50.6

4.4 12.1 21.2 13.5 26.2 47.7

4.8 8.8 13.1 10.2 21.9 38.3

5.5 9.6 14.7 11.3 26.7 47.3

7.4 8.3 9.7 13.0 29.4 49.6

8.2 13.2 33.6 36.4 44.2 48.9

11.6 17.2 33.7 39.1 43.8 48.5

0.0 0.3

0.0 2.3 18.1 12.2 26.9 49.3

0.0 0.7 7.7 4.0 15.7 37.5

0.0 0.6 4.5 4.8 21.6 36.0

0.0 1.8 19.8 22.0 27.8 36.6

0.0 1.9 19.9 19.4 28.0 38.2

1.2 14.5 17.6 27.0 39.2

95.2 78.9 68.8 77.6 51.2 12.4

94.5 89.7 77.6 84.7 57.6 15.2

92.6 91.1 85.8 82.2 49.0 14.4

91.8 85.0 46.6 41.6 28.0 14.5

88.4 80.9 46.4 41.5 28.2 13.3

86.0 85.6 61.9 50.0 30.2 14.2

Transmission 0.392 0.469 0.532 0.642 0.737 1.040

2.5 15.9 38.1

0.0 0.2 7.9 1.8 14.0 31.7

Absorption 0.392 0.469 0.532 0.642 0.737 1.040

94.6 89.8 77.9 85.3 52.4 11.3

95.6 87.7 70.9 84.9 59.8 20.6

5.5

Note: No. 4, etc., represent series.

14.0 13.2 23.6 32.4 32.8

9

E

8.5. Income of Global Radiation under Vegetative Covers

511

red poppy petals. On the other hand, in all cases observed there was an increase of the reflectivity and transmissivity of leaves with an increase in wavelength. In investigating various problems of phytophysiology it is important to compute theoretically the value of the shortwave radiation absorbed by leaves as derived from the usual actinometric measurement data. It is clear that the solution of this problem is reduced to calculating the absorption of global radiation by a slant surface of arbitrary orientation. Therefore, for determination of the global radiation absorbed by leaves, it is possible to apply the methods for computing direct incoming solar radiation on slopes of arbitrary orientation as described in Chapter 5. For this purpose Lopukhin [77] worked out a much simpler semiempirical method, which was then used to determine the global radiation absorbed by cotton leaves. To compute the incoming diffuse radiation on a leaf’s surface inclined at an angle a to the horizon, Lopukhin proposed using the following formula : 4, = kk19 (8.21) where k = qa/qo,k = q,/q, q is the incoming diffuse radiation on a horizontal surface placed above the vegetative cover, and q, is the incoming diffuse radiation on the leaf‘s horizontal surface (q f q,, for the leaf‘s surface may be shaded by the foliage). Observations show that the coefficient k may be presented as a function only of the angle of inclination a and the leaf’s elevation above ground level 1. The value k, should be determined from direct measurement data obtained as close as possible to the investigated object. The direct incoming solar radiation on the leaf’s surface can be calculated from the formulas derived in Sec. 5.11. Thus the formula for computing the global radiation absorbed by the whole leaf’s surface has been proposed by Lopukhin to be (8.22) where S , is the flux of direct solar radiation on a surface perpendicular to the sun’s rays; i is the angle of incidence of the sun’s rays on the leaf‘s surface (sin i can be found from the formulas of Sec. 5.7); N is the degree of shade on the leaf’s surface, determined visually; a,, a, are the leaf’s absorptivity for direct solar and diffuse radiation, respectively ; and q-. is the incoming diffuse radiation on the lower surface of the leaf. When making use of (8.22) it should be borne in mind that the leaf’s orientation and its degree of shading may vary during the day. Therefore it is necessary either to substitute into (8.22) some average i and N

512

Global Radiation

values or to take account of the variation of i and N within the day. Calculations of the radiation absorbed by cotton leaves, as performed by Lopukhin with the help of (8.22), have shown that the absorbed radiation values vary from leaf to leaf. As a rule, the lower leaves absorb less radiation than the upper, but in individual cases the absorption by both types is quite comparable. Great differences are also observed in the daily range of absorption by different leaves. For example, the maximum of the absorbed radiation for some leaves takes place at 0700, whereas for others it occurs at 1800 of the true solar time.

3. Global Incoming Radiation Income below the Forest Cover. Radiant energy is a most important factor of the environment in determining the development and growth of forest. For this reason the problem of incoming radiation below the forest cover has been given much attention. It is obvious that the amount of incoming global radiation to the forest must be considerably less than to an open area. The shading of soil by trees causes a notable decrease of the duration of irradiation and consequently of the quantity of incoming radiation. However, it is evident that the amount of global radiation received by the forest must be strongly dependent on the types of trees, for these determine the conditions of the surface shading under the forest cover. To illustrate this statement, Table 8.18 summarizes observations of the conditions found with different forest cover types, performed by Sakharov [78] with the help of a Yanishevsky pyranometer. TABLE 8.18 Dependence of Mean Incoming Global Radiation in the Forest upon Forest Type (Percent of Global Radiation in Open Place). After Sakharov [78] Type of Forest Pinetum cladinoso-muscosum Pinetum sphagnosum Pinetum racciniosum Pinetum moliniosum Quercetum tiliosum Pinetum tiliosum Pinetum carylosum, with a second layer of fir Pinetum myrtillosum with a second layer of fir Alnetum filipendulosum Pinetum hylocomiosum with a second layer of fir Piceetum myrtilloso-oxalidosum

Global Radiation, % 39.8 36.1 24.9 16.2 14.8 11.2 10.7 9.5 7.8 6.5 5.2

513

8.5. Income of Global Radiation under Vegetative Covers

In the majority of cases a slight cloudiness (force 1 t o 3), Ac, As, and Cu, was observed. The flux of global radiation on a horizontal surface in the forest is expressed in percent relative to the corresponding radiant flux in an open area. The values in Table 8.18 are average values. In isolated cases the incoming global radiation below the forest cover of a certain type may fluctuate. Table 8.19 presents Sakharov’s [78] observed minimal and maximal global radiation flux values under different forest covers and the corresponding means in an open area under the conditions of a cloudless sky. TABLE 8.19 Variation of Incoming Global Radiation (callem? min) under Forest Cover in Isolated Cases. After Sakharov [78]

Under Forest Cover Type of Forest

1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.

Pineturn cladinoso-rnuscosum Pinetum racciniosum Pinetum sphagnosurn Pinetum carylosum Trernuleturn tiliosum Pinetum moliniosum Pinetum rnyrtillosum Querceturn tiliosum Piceetum myrtilloso-oxalidosum Alnetum filipendulosum Pinetum hylocomiosum

Open Space

1.17

1.13 0.97

1.13 1.37 1.34 1.17 1.02 0.90 1.11 0.97

Range

Amplitude

0.09-1.19 0.07-1.05 0.10-1.01 0.014.68 0.014.46

1.10 0.98 0.91 0.67 0.45 0.38 0.37 0.35 0.34 0.28 0.12

0.06-0.44 0.0 0 . 41 0.06-0.41 0.02-0.36 0.00.32 0.03-0.15

From Tables 8.18 and 8.19 it follows that the most favorable conditions for global radiation penetration beneath the surface cover are observed for the forest types 1, 2, 3, 6, 8, as listed in Table 8.19. For these types of forest the mean of the relative incoming radiation and the minimal flux values in all cases exceed 0.05 cal/cm2 min and are therefore greatest. The other types of forests, with a thicker upper layer (fir, alder) or a presence of the second layer (types 4, 7, 1 1 , and 5) are characterized by a less penetration of global radiation beneath the forest cover. Quite important to the characterization of forest phytoclimate is the problem of direct incoming solar radiation on the surface of the soil (or grass) under the forest cover. Sakharov’s observations have shown that in the majority of cases the direct solar radiation reaches the underlying surface

514

Global Radiation

beneath the forest cover during a part of the day. For instance, on Aug. 3, 1939, the direct solar rays were observed to hit the pyranometer’s receiving surface for all types of forest except type 4. The values of direct solar radiation under the forest cover and in an open area certainly are not equal. As has been shown by Ivanov [79], the solar radiant flux value at the level of an underlying surface beneath the forest cover (on the “solar spot”) is inversely proportional to the square of the distance from the clearing in the foliage to the “spot” on the soil. This results from the “solar spot’s’’ being a picture of the clearing on the “screen” formed by the underlying surface beneath the forest cover. According to Pivovarova and Guliayev [80], the global radiation under young birch trees during the period of full foliage constitutes 5 to 8 percent and after the leaves fall off, this increases to 20 percent in relation to the global radiation beneath the forest cover. The data of Table 8.20 show that even in the case of one certain forest type, the global radiation fluctuations beneath the tops of trees may be quite notable. Such fluctuations are due to outside factors (solar height, atmospheric transparency conditions, cloudiness, wind shaking the tops of trees) as well as to the character of the forest itself. Viewed from this point, the most essential forest characteristics are its composition in type, age, congestion of the tree tops, number of trunks per unit area, and presence or absence of a young stand. TABLE 8.20 Mean Incoming Global Radiation in Forest Clearings and Sheltered Areas (Percent of Amount in an Open Area). After Zakharova [82] Surface

Global Radiation, %

Northern outskirts Middle clearing Southern outskirts Forest at 10 m from outskirts Forest at 30 m from outskirts

12.9 9.0 7.9 6.0

17.4

The above-considered observational data characterize the influence of various factors on the incoming global radiation found beneath the forest cover at ground level. Let us now clarify the problem of global radiation variation at different heights under the forest cover, as observed by Kuzmin

8.5. Income of Global Radiation under Vegetative Covers

515

[8 11. Kuzmin constructed special suspended devices enabling elevation of pyranometers, which served as radiation receivers to any height beneath the tops of trees. The measurements were conducted at two wooded areas with different estimation characteristics. The most significant factors were the following: area 1 was wooded with type 9 trees (stand density, 0.30) and pine (stand density, 0.20), whose height averaged 20 m. The main types were intermixed with birches and alder undergrowth. The total stand density averaged 0.50. The forest stand density is taken as the ratio of the total sum of the tree cross sections at a given area to the total of cross sections of the standard wood thickness of the same area. Area 2 consisted of type trees (stand density, 0.55) of about 20-m height, fresh fir growth (stand density, 0.24) and a small amount of rowan tree (mountain ash) and alder offsprings. The total stand density of the area was 0.79 on the average. Figure 8.25 presents Kuzmin’s curves of the global radiation variation (in percent of the incoming radiation at the forest outskirts) at different heights under the trees of area 1, with a cloudless sky. One’s attention is immediately drawn to the sharp contrast in amounts of incoming ra-

HEIGHT ABOVE SOIL

FIG. 8.26

Variation in the global radiation income at different heights under the forest cover, Area I .

diation between the upper and lower sheltered zones. In the upper zone rapid decrease of radiation is observed as it approaches the soil surface, whereas the lower zone radiation is almost independent of height (below 5 m). The cause for this is as follows: The instrument elevated into the upper zone is affected by a rapid increase in shade due to the closing of the tops of trees, while in the lower zone the degree of shade at a horizontal surface is practically constant because of the absence of undergrowth. As seen from Fig. 8.26, the character of global radiation variation at different

516

Global Radiation

underforest heights is strongly dependent upon solar height. With a decrease in solar height the curves of global radiation variation lose their steepness and the difference between the upper and lower sheltered space is considerably equalized. The results obtained by Kuzmin for area 2 with young fir undergrowth were different. In that case (see Fig. 8.27) the rapid decrease of global radiation was observed down to the soil surface without separate upper and lower zone characteristics. At low solar heights, similarly to the preceding case, observations showed a marked attenuation of global radiation and also a more smooth and slow variation of global radiation in dependence upon elevation above ground level. 100

80

s-

60

0 .c

.-

40

[L

20 0 5

10 Height above soil

FIG. 8.27

15

Variation of the global radiation income at different heights under the forest cover, Area 2.

Observations of the incoming global radiation at different shaded heights with a dull sky gave results qualitatively similar to the ones obtained with a cloudless sky. Area 1 continued isolating the two zones, the upper ( 5 to 15 m) where the global radiation was variable with height approximately according to linear law, and the lower (below 5 m) which is characteristic of an insignificant variability of global radiation in dependence on height. At area 2 the variation of radiation with height is approximately linear throughout the entire thickness to the soil surface. It is worth noting that the curves characterizing the global radiation variation at different heights under the forest cover in the case of an overcast sky are far more stable than the corresponding curves obtained from observations with a clear sky. In connection with the investigation of the forest radiation climate, it is practically important to study the regularities of incoming global radiation inside the wooded areas and at their outskirts. Obviously the incoming

8.6. Penetration of Radiant Energy into Water, Ice, and Snow

517

global radiation on open surfaces must be larger than that under the forest cover. As illustration, Table 8.20 gives Zakharova’s [82] results of pyranometric measurements carried out in the summer of 1950 in Moldavia (southwestern territory of the U.S.S.R.) under the forest cover (doublelayer growth of beech, hornbeam, and oak with a total closeness of tree tops of 0.6 to 0.8 and a height of 10 to 16 m) and in a forest clearing of 5 x 30 m elongated from west to east. As seen from Table 8.20, the global radiation in the middle of the clearing turned out to be twice as much as in the forest, and in the outskirts, it was three times the forest figure. This gives evidence of the conditions of irradiation being more favorable on open surfaces than in the sheltered areas. Analogous results for field-protection forest barriers have been obtained by Golubova [83], who has also shown that the amount of incoming global radiation inside the field-protection forest barrier decreases at the same rate as inside the forest.

8.6. Penetration of Radiant Energy into Water, Ice, and Snow When incident on the surface of such media as water or ice, the direct solar and diffuse radiation partly penetrate these media. Depending on water or ice transparency, the radiant energy may be transmitted in the given media to different depths. Even such a seemingly nontransparent medium as snow may receive considerable direct solar and diffuse radiation. Some data characterizing the penetration of direct solar and diffuse radiation into water, ice, and snow are given below. 1. Water. The radiative transfer in water is a very complicated problem of water optics, in which connection it has been given much attention in a number of theoretical and experimental investigations. ,We shall consider below only the most essential results of experimental investigations in the transmission of solar radiation by water. Section 3.2 gave data characterizing the absorption of solar radiation by clean water. However, even in clean water the attenuation of shortwave radiation is due not only to absorption but also to scattering. For natural water basins turbid with foreign admixtures, the role of radiation attenuation due to scattering becomes still more important. Table 8.21 presents Hulburt’s [Chapter 7, Ref. 331 laboratory measurement data on the coefficients of total attenuation uL, scattering oA,and absorption kA for distilled water and sea water containing plankton (see also [83a, 83bl). The total attenuation coefficients were determined from the attenuation

518

Global Radiation TABLE 8.21

Coefficients of Total Attenuation, Scattering, and Absorption for DistiIled and Sea Water (cm-'). After Hulburt [ Chapter I , 33 ] ~~

Distilled Water

Sea Water

1, P a,

0.40 0.41 0.42 0.43 0.44 0.45 0.46 0.47 0.48 0.49 0.50 0.51 0.52 0.53 0.54 0.55 0.56 0.57 0.58 0.59 0.60 0.61 0.62 0.63 0.64 0.65 0.66 0.67 0.68 0.69 0.70

8.0 7.0 6.1 5.3 4.6 4.0 3.6 3.6 3.65 3.7 3.8 3.9 4.0 4.2 4.4 4.7 5.3 6.6 8.4 12.0 19.7 24.3 26.5 28.0 29.2 30.8 33.5 37.5 40.6 46.7 57.6

0,

k,

3.57 3.23 2.95 2.69 2.45 2.25 2.04 1.89 1.72 1.59 1.47 1.35 1.25 1.17 1.09 1.oo 0.932 0.868 0.810 0.756 0.708 0.662 0.618 0.581 0.535 0.507 0.483 0.455 0.429 0.404 0.380

4.4 3.8 3.1 2.6 2.1 1.7 1.6 1.7 1.8 2.1 2.3 2.6 2.8 3.0 3.3 3.7 4.4 5.7 8.6 11.2 19.0 23.6 25.9 27.4 28.7 30.3 33.0 37.0 40.2 46.3 57.2

80.5 x 10-4 70.5 62.8 55.6 51.2 47.5 44.7 41.9 38.8 36.8 35.1 33.7 33.1 32.3 32.3 32.3 33.1 36.5 42.9 46.5 47.6 48.8 50.0 51.8 54.2 56.3 48.9 63 74

62.5 x 10-4 52.5 44.8 37.6 33.2 29.5 26.7 23.9 20.8 18.8 17.1 15.7 15.1 14.3 14.3 15.3 15.1 18.5 24.9 28.5 29.6 30.8 32.0 33.8 36.2 38.3 40.9 45 56

Plankton, k,

60 x 10-4 49 43 36 31 28 25 22 18 16 14 13 12 11 10 9 8 7 6 5 4 4 4 4 4 2 1 0 0

of radiation by water as it filled a glass tube over a 364-cm path, with spectral resolution being realized by means of a glass spectrograph. The scattering coefficients uAfor distilled water were found from measurement of the integral light-scattering coefficient u of a tungsten incandescent lamp,

8.6. Penetration of Radiant Energy into Water, Ice, and Snow

519

on the assumption that the 0).values are inversely proportional to the fourth power of wavelength (the light scattered by distilled water had a blue tint, The light scattered by which suggested the validity of the relation 0 2 k4). sea water containing plankton was of whitish hue; therefore it was assumed that in the given case the scattering coefficient was independent of wavelength cm-l. and equaled the integral scattering coefficient 0 = 1.8 x The absorption coefficients k, were computed as a difference ka = a, - 0,. All coefficients aA, ka, and o1 were computed per unit path length in water and therefore expressed in cm-l. The rightmost column of Table 8.21 gives values of the absorption coefficient by plankton contained in sea water, which were determined from the difference in absorption coefficients between natural and distilled water. As seen from Table 8.21, the attenuation of radiation by sea water is considerably more intensive than by distilled water. It should be noted here that the distilled water in the given case was not quite clean optically, but contained a certain amount of minute particulate admixture. Deserving attention are the different dependences of the total attenuation coefficients upon wavelength in the case of sea and distilled water. In the former case a peculiar attenuation minimum (transmission maximum) for yellow-green light is observed. The rightmost column of Table 8.21 shows that the absorption of radiation, especially shortwave, by plankton greatly affects radiation attenuation. In general, the role of the absorption of radiation by water and its admixtures appeared to be quite important. Data of numerous investigations give evidence of the fact that the presence of suspended particulate matter in sea water is most essentially effective with regard to its transparency. Ivanov’s [84,85] laboratory measurements have shown, for example, that the light attenuation coefficient Ua is directly proportional to the weight content of the suspended matter B(a, = cB), with the proportionality coefficient c in such a linear relation depending solely on particle dimensions and wavelength of light. At an increase of the diameter d of the particles from 0.3 to 1.5 p, the proportionality coefficient c increases, reaching maximum at d = 1.5 p, after which it decreases steadily with a further increase in diameter. The dependence of c on light wavelength becomes evident at small diameters of the scattering particles only (the most intensive is the attenuation of shortwave radiation), whereas at diameters exceeding 3 p the dependence loses significance. The latter results are quite understandable from the viewpoint of the conclusions regarding the principles of light scattering on small and large particles (see Chapter 3). Skopintzev and Ivanov [86-881 and other authors have stated that silt

-

520

Global Radiation

combinations brought to the sea with river water considerably affect the sea transparency. For instance, as has been estimated by Skopintzev [86], the attenuation coefficient aL at il= '0.436 p results in 2.8 x cm-' for a water sample from the Sea of Azov and 9.2 x cm-l for a sample from the Baltic Sea (the Gulf of Riga). In both cases the samples were cleaned of the suspended particles by filtration; thus the difference between the attenuation coefficients resulted only from the different contents of silt in the water. It should be noted that the increase of light attenuation in the presence of fine soil particles dissolved in sea water is due mainly to an increase in radiation absorption. The close connection between the transparency of sea water and its content of suspended silt combinations enabled working out methods for determining the amount and weight concentration of suspended particles and organic matter, using the measurements of the attenuation of light by sea water which had been realized by Skopintzev and Ivanov [86-881 and Jerlov [89, 901. Interesting laboratory investigations of the regularities of the attenuation of solar radiation by greatly turbid media (of milk or rosin) with small absorption coefficients but marked scattering factors have been carried out by Timofeeva [91, 92, 92al. Using a lens photometer with a selenium photocell that has a small aperture angle (up to 0.39'), Timofeeva measured the intensity of light at various distances and in different directions from the surface of turbid media. Figure 8.27 gives Timofeeva's curves of the dependence of the relative radiant intensity I/& upon depth I for different q~ angles between the photometer optical axis (sighting beam) and the direction of incidence of solar radiation on the turbid medium surface (the case q~ = ' 0 corresponds to the direction on the light source). As seen from Fig. 8.27, the attenuation of direct solar radiation to a certain depth in the direction q~ = 0 ' is determined by the exponential law I

=

(8.23)

where a is the total attenuation coefficient practically equal to the scattering coefficient, since in the considered case the absorption factor is small. Observations show that the formula (8.23) is valid for such depth 1 to which corresponds the optical thickness a1 equal to 6 to 8. At still lower depths the attenuation slackens, but at a certain 1 > I,, when the light becomes completely scattered, it is again subordinate to the exponential law of the form 1' =

0I e-Q'(I-lo)

(8.24)

8.6. Penetration of Radiant Energy into Water, Ice, and Snow

52 1

where a‘ is the coefficient of attenuation (scattering) of the finitely scattered light, I’ is the intensity of light at a depth Z, at which the light is fully scattered, and I‘ is the intensity of light at a depth 1. It is of interest that, following observations, a conclusion has been reached to the effect that transition to the exponential law of attenuation at great optical thicknesses is in full agreement with Ambartzumian’s [93] analogous theoretical analysis. Comprehensive explanation of the considered results within the limits of approximate theory of multiple light scattering is given by Rosenberg [94] and Lenoble [95, 961. Important contributions to the solution of the problem of radiation scattering in the sea have been made by Ivanoff [96a] and Duntley [96b]. It has been found by Timofeeva [91] that there exists the following dependence* between the values a and a’, where A is a certain constant. The conclusion likewise confirms the analogous deduction following from theoretical works by Ambartzumian [93] and Ovchinsky [97]. The variation of light intensity in dependence on depth is equal in all directions 0’ < v < 120O. At small depths the intensity of light increases with an increase of Z, and on reaching maximum at a certain I, , begins decreasing slowly. The existence of such intensity maxima of the “side” light was first predicted theoretically by Shuleikin [98]. Qualitatively, the presence of maxima can be explained by the increase of the “side” light because of the strong scattering of direct light at small depths. As seen from Fig. 8.27, the intensity ,,,Z increases reaching maxima displace as the angle q~ increases, with first culmination at a certain q~ and then starts to decrease. Timofeeva observed that the depths of the intensity maxima of “side” light decreased with an increasing turbidity of the scattering medium. According to theoretical calculations of Shuleikin, performed on the bases of radiation attenuation must be steadily increasing with an due to scattering only, the value I, increase of v. The presence of the function Zmax(v) maximum is explained by Timofeeva as being due to the growing importance of radiation attenuation at large because of absorption. Timofeeva’s observations show that the attenuation of the directed light is largely caused by scattering, and that the finitely scattered light is chiefly attenuated as a result of absorption. Even in the case of media with a very small absorption coefficient, as investigated by Timofeeva, the portion of the absorbed light of finite scattering reaches 60 percent. According to the data of Fig. 8.28, the light intensity distribution over

* The

following dependence takes place: a‘

=

-v‘/aA.

522

Global Radiation f-

I(

I6

t

0

s 10-

10-

to'

I

10

I -

20

FIG. 8.28 ReIative radiant intensity at different depths under the surface of a turbid medium.

various directions is characterized by considerable asymmetry at small depths and by insignificant asymmetry and constancy of the intensity function form for the finitely scattered light at great depths. Figure 8.29 gives a spectacular presentation of the intensity functions for different depths I in a turbid medium with the attenuation factor a = 1.9 cm-I. We see that

8.6. Penetration of Radiant Energy into Water, Ice, and Snow

523

even at I > 3 cm, the direct light becomes very weak and the function asymmetry sharply decreases. At the same depths, corresponding to large optical thicknesses (finitely scattered light), there is established a directional intensity distribution independent of depth, with somewhat greater light intensities toward the surface of the turbid medium.

x 10 1-20

3

FIG. 8.29 Light scattering functions in a turbid medium.

Timofeeva's experiments made it possible to model the main regularities of radiative transfer in the sea. It is perfectly clear that in natural conditions the process of radiative transfer is more complex and is subject to the influence of numerous factors that are hard to account for; for example, the conditions of irradiation of the sea surface, surface state, and sea turbidity. However, it appears that Timofeeva's observations in natural conditions (the Black Sea) are in quite satisfactory agreement with data of laboratory experiments. It was stated, in particular, that in natural conditions the intensity maximum for angles 0" < ~1 < 120" was observed at a certain distance of the refracted sun's ray from the sea surface to the photometer. The extremal character of the dependence Imax(y)was also confirmed, and it was found that at I >,,I , radiant intensity varies according to the exponential law. All this confirms that the laboratory results are qualitatively correct in describing the main regularities of solar radiation attenuation in the sea. Investigations of Timofeeva enabled study of the main principles of the

524

Global Radiation

directed radiation attenuation in the sea. It is easily understandable that the law of attenuation for hemispherical radiant flux must be, generally speaking, different from that for the directed radiation. Observations by Gershun [99] showed that the luminous radiant flux downcoming on the horizontal surface (illumination of the horizontal surface from above) decreases with the sea depth, according to the exponential law, in very wide limits of depth variation from zero to 100 m. In Kirillova and Burig’s [loo] observations at Sevan (lake in N. Armenian S.S.R.) the attenuation of global radiation flux was best described not by a simple exponent but by the following expression

Q

=

(&+-

Qo{~xP[

(8.25)

az)z]}

The results of measurements at I-m depth give a, = 0.250 and a, = 0.036 m-l. Table 8.22 summarize Kirillova’s values Q/Qo (percent) from data of observations made in the central part of Sevan lake from a cutter (the depth of the lake is 60 m) and from a raft at the depth of 15 m. Comparison of data on the attenuation factors of the directed and diffuse radiation shows that the latter coefficients are much less than the former. TABLE 8.22 Global Radiation Attenuation (Percent) as Observed at Sevan Lake. Afrer Kirillova [loo] Depth, rn Month

0.5

1.0

2.0

3.0

5.0

8.0

10.0

15.0

20.0

0.49 0.48 0.41

0.38 0.43 0.36

0.27 0.34 0.27

0.22 0.28 0.23

0.12 0.19 0.15

0.06 0.12 0.10

0.04 0.09 0.07

0.01 0.05 0.04

0.00 0.03 0.01

0.45

0.36

0.21

0.16

0.10

0.06

0.04

-

-

Cutter: May July Sept. Raft : July

Mokievsky [1011, having realized pyranometric measurements with glass filters, investigated the spectral transparency of water basins. Table 8.23 presents data for different parts of lake Ladoga (NW. U.S.S.R.). The trasparency of water was about 2.5 to 3.5 m in the Secci disk.

8.6. Penetration of Radiant Energy into Water, Ice, and Snow

525

Table 8.23 characterizes the relative energy distribution in the spectrum of radiation penetrating the lake depths (for each spectral interval in percent relative to the whole radiation of the wavelengths 360 to 680 p). The last column indicates the relative (in percent, to incoming global radiation on the water surface) fluxes of the integral global radiation. The presence TABLE 8.23 Variation of the Radiation Spectral Composition with Height under the Water at Lake Ladoga. Ajier Mokievsky [loll

Date, Station

July 22, 1959, Yakimvara creek, buoy

Aug. 4, 1956, point VI-A

Sept. 9, 1959, Motornoye village, regime vertical

Spectral Intervals, mp

Depth

m

Integral 36M50 45C.530 530430 630 630-680

0.1 0.5 1.o 1.5 2.0

41 .O 37.1 36.6 23.5 33.4

17.2 11.4 4.1 20.6 2.1

12.6 20.5 24.8 18.3 31.2

0.1 0.5 1.o 1.5 2.0

29.7 29.7 33.8 22.2

11.1 4.4 5.0 12.8

25.0 31.7 40.2 18.1

0.1 0.5 1 .o 1.5 2.0

46.8 42.2 37.5 33.4 33.4

8.2 8.1 5.1 14.6 10.2

15.8 22.6 24.1 17.8 31 .O

12.7 16.6 21.1 31.1

580

16.5 14.4 13.4 6.5

33.3 34.2 34.2

70.3 30.0 15.5 8.6 5.6 55.8 24.8 11.8 6.6 3.7

-

31.0 46.9

8.7 12.9 21.3 21.0 16.4

20.5 14.2 12.0 13.2 9.0

62.6 32.0 17.1 10.2 6.4

of considerable variation in the spectral radiation composition with depths is quite spectacular. As seen, the Ladoga waters transmit chiefly radiation of the yellow-red spectral region. This is, of course, a peculiarity of the given lake, which contains mechanical admixtures of yellow and brown colors. Reservoirs with different optical properties will otherwise transform the shortwave radiation spectral composition.

526

Global Radiation

2. Ice. It is understandable that the transparency of ice, whose optical nonhomogeneity is greater than that of water, must be considerably lower than the water transparency. Experimental investigations of ice transparency undertaken by Kalitin [Chapter 3, Ref. 1021 are very convincing in confirming this conclusion. Kalitin, using a highly sensitive thermoelectrical pyranometer, made observations over the transparency of ice samples whose outward surface was perpendicular to the incidence of direct solar radiation. According to Kalitin’s data, the values of direct solar radiation transmission by ice vary with ice thickness (2 to 50 cm) from 73 to 3 percent. The transmission of radiation by ice of this thickness is considerably dependent upon its optical inhomogeneity. For instance, the transmission of radiation by a layer of ice 35 cm thick varies in dependence upon its inhomogeneity (cracks, air bubbles) within 33 to 52 percent. A sample of 50-cm thickness with a completely nonhomogeneous (turbid) upper layer 16 cm thick transmits only 3 percent of the direct solar radiation. In the case of diffuse radiation the transparency of ice appears to be considerably higher, reaching 83 to 84 percent for a layer of 4 cm thick and 58 to 60 percent for a thickness of 35 cm. The latter higher transparency in the case of diffuse radiation is explained by a lesser influence (less than for direct solar radiation) of specular radiation reflection from different strata within the ice. The ice transparency is especially high in the ultraviolet spectrum; for example, in the spectral interval 0.332 to 0.476 p, the transmission values of direct solar radiation on a layer of ice 107 cm thick will vary from 46 percent (at il= 0.332 p ) to 55 percent (at il = 0.476 p). An ice rock of 10-cm thickness transmits in this range about 97 percent of the incident radiation. Kalitin’s measurements with ice samples whose surface had been slightly melted for higher transparency generally gave somewhat exaggerated transparency values in comparison with those made for actual conditions. In this connection investigations of ice transparency in the real conditions are of great interest, such as those by Trofimov [102], Pisiakova [103], Lubimova [103a], Ambach [103b, 103~1.Pisiakova realized parallel measurements of global radiation fluxes above and under the ice at a lake in the Leningrad region. The latter measurements were conducted by means of an underwater Yanishevsky pyranometer ; in the measurements above the ice, the common Yanishevsky pyranometer was used. The observations were carried out in the noon hours. In the beginning (Apr. 12, 1941) the ice, of 61-cm thickness, was covered by 4 cm of snow, and the transmission of global radiation equaled 1 percent only. By the end of the obser-

527

8.6. Penetration of Radiant Energy into Water, Ice, and Snow

vations (Apr. 29, 1941) the snow was completely melted and in the thaw the ice thinned down to 32 cm with a gray and wet surface. Under these conditions the transmission of global radiation by the ice increased up to 27 percent. Polli [I041 has determined the total radiation attenuation coefficients in the visible spectrum for ice and water. On the average, the attenuation coefficient of ice appeared to be ten times that of water, but this result may be doubtful at a lower ice transparency. Table 8.24 presents measurement data on the absorption of shortwave radiation at different thicknesses of glacial ice, obtained by Karol [I051 when observing the Fedchenko glacier (the Pamir). It is seen from this table that the absorbed radiation distribution over the ice thickness is not homogeneous, which is due to a strong radiation absorption in the uppermost layer as well as to the different structure of the ice throughout the levels. Practically all global radiation is absorbed by the top 42-cm layer. TABLE 8.24 Absorption of Solar Radiation by Glacial Ice (cal/cm2min). After Karol [lo51

Depth,

cm

0

>42 >82 >112 0-42 42-82 82-1 12

Radiant Flux Penetrating Ice Thickness, D'

Radiant Flux Leaving Ice Thickness,

1.070 0.081 0.047 0.038 0.972 0.030 0.030

0.098 0.058 0.017

D"

0.015

0.023 0.023 0.023

Absorbed Radiation, Q = D' - D"

0.972 0.023 0.030 0.023 0.949 0.007 0.007

3. Snow. Snow cover is still less transparent for global radiation than ice. The radiation penetrating into the snow cover is practically all absorbed by a layer whose thickness does not exceed several tens of centimeters. Observations show that the attenuation of global (or direct solar only) radiation in dependence upon snow depth can be approximately described by the exponential law. According to Sulakvelidze [106], (8.26) where Qd , Q, are downward and upward shortwave radiation fluxes at the

528

Global Radiation

level of 8 cm, e is the snow thickness (g/cmz), and k is the absorption coefficient of snow (cmz/g). Observations have shown that the absorption coefficient for the integral shortwave radiation is a function of snow density:

k

=

1.31 - 1.lle

(8.27)

Values of the absorption coefficient obtained by Sulakvelidze and Okujava [lo61 for different spectral regions are given in Table 8.25. This table shows that the dependence of the absorption coefficient upon wavelength is variable. The largest coefficients are observed in the green-blue spectral region, which accounts for the greenish-blue luminosity of clean snow. The atTABLE 8.25 Spectral Dependence of rhe Absorption Coefficients of Snow. After Sulakvelidze [lo61

Means 0.10

0.19

0.39

0.59

0.65

1.20 1.34 2.00 1.04 1.70

1.11 1.22 1.73 1 .oo 1.50

0.94 1.01 1.23

0.79 0.80 0.82 0.76 0.81

0.79 0.80 0.82 0.72 0.80

0.80

1.16

0.97 1.07 1.39 0.90 1.25

tenuation of radiation by snow is considerably dependent upon the latter’s state and structure. Pyranometric observations, for example, give evidence of a notable increase of the attenuation coefficient of wet snow as compared to that of dry snow. Kuzmin [I071 points out that the mentioned increase of the coefficient of radiation attenuation by snow is mainly due to two factors : (1) Increase of the snow absorptivity in the filling of air bubbles by water, which is a strong radiation absorbent. (2) Change in the snow structure (recrystallization) at the beginning of thaw. The latter factor is especially important.

The exponentially decreasing dependence of global radiation flux upon depth under the snow cover and the considerable attenuation coefficient values cause a rapid decrease in the flux of global radiation at the increasing depth inside the snow thickness. This can be seen in Table 8.26, which char-

8.6. Penetration of Radiant Energy into Water, Ice, and Snow

529

acterizes the absorption of global radiation by snow at different attenuation coefficients over the levels of penetration. As shown by this table, the 5-cm layer of snow absorbs from 34 to 88 percent of all radiation penetrating inside the snow thickness in dependence upon its transparency. At a path of 10 cm the portion of the absorbed radiation varies from 56 to 98 percent; that is, in the most unfavorable conditions the global radiation practically does not fall below 10 cm. TABLE 8.26 Absorption of Global Radiation by Snow at Different Values of the Attenuation Coefficient through the LeveIs of Penetration. After Kuzmin [lo71 Dz, cm

a x loa cm-l

0.83 0.94 1.05 1.17 1.28 1.51 1.75 1.99 2.23 2.88 3.57 4.33

0-5

5-10

10-15

34.4 37.6 41 .O 44.2 47.2 53.0 58.2 62.9 67.2 76.3 83.2 88.4

22.4 23.5 24.2 24.6 24.9 24.9 24.3 23.4 22.0 18.1 14.0 10.3

14.9 14.6 14.2 13.8 13.2 11.7 10.2 8.6 7.3 4.3 2.3 1.2

15-20 20-30 9.7 9.2 8.5 7.7 6.9 5.5 4.2 3.2 2.3 1.0 0.4 0.1

10.7 9.2 7.9 6.7 5.6 3.8 2.6 1.6 1.1 0.3 0.1

30-40 40-50 4.6 3.6 2.8 2.1 1.6 0.9 0.4 0.3 0.1

2.0 1.4 1.0 0.6 0.4 0.2 0.1

50-60

60

0.9 0.5 0.3 0.2 0.1

0.7 0.4 0.1 0.1 0.1

However, even in the most favorable conditions, all radiation is actually absorbed in the layer of 60-cm thickness. At still thicker snow covers (over 60 cm) the solar radiation does not reach the soil surface whatever the snow transparency may be. It should be noted that the determination of the mean attenuation coefficient for the entire snow cover as a whole is not quite justified, since generally, the snow cover consists of nonhomogeneous layers. It is evident, therefore, and also because of the selectivity of radiation absorption by snow, that the mean attenuation coefficient must be dependent upon snow thickness. That is why the determination of attenuation coefficients for isolated snow layers, sufficiently homogeneous in their physical properties, is not too complicated. Such a problem, though, presents greater difficul-

530

Global Radiation TABLE 8.27 The Radiation Regime of Snow Cover. Afier Kalitin [Chapter 3, 102 ]

Daily Radiation Totals Direct solar radiation, cal/cma Diffuse radiation, cal/cmz Global radiation Albedo of the snow surface, % Radiation penetrating inside snow cover, cal/cm2 Thickness of snow cover, cm Transmission of radiation by snow, % Radiation reaching soil surface, cal/cm2 Radiation absorbed by snow, caI/cm2

Jan. 26

40 20 60 78 13

15

5.6 0.73 12.27

Feb. 14 99 34 133 70 40 21 2 0.80 39.20

Mar. 19 213 54 327 70 98 52 0.9 0.88 97.12

ties, which has led to a sparsity of information on the variability of transparency characteristics inside the snow thickness. For description of the radiation regime of the snow cover as a whole, let us consider Table 8.27 in which are given daily totals of the incident and absorbed radiation (cal/cm2)from Kalitin’s work [Chapter 3, Ref. 1021. As seen from the table, in the considered cases the amount of global radiation reaching the soil surface is quite small. At lesser thicknesses than in the given cases the portion of radiation falling on the soil surface may be far greater. In Kalitin’s estimation the daily radiation total on a soil surface covered by an 8-cm snow is 26.5 cal/cm2 for the values of incoming radiation and albedo, as of March 19. The investigation of global radiation penetration inside the snow cover is of great importance in studying the problem of the thermal regime of snow and soil, and particularly in calculating snow thaw. In some cases, owing to intensive radiation absorption inside the snow cover, snow thaw at negative air temperatures may be observed. As the result of solar radiation absorption in the thin surface layer of snow, there takes place the forming of the so-called solar crust and water film. The latter is a transparent ice film that serves as the snow “greenhouse” glass, where the heat accumulation leads to snow thaw at negative air temperatures. For instance, according to observations by Kuzmin [lo71 in the spring of 1952 (Apr. 3-1 1) in the Stone Steppe (southern Ukraine), the main bulk of snow melted at air temperatures from 0’ to -2.2OC.

53 1

8.7. Illumination

8.7. Illumination

The investigation of the regime of natural illumination is of much interest and especially so for various phototechnical applications. Since in the first approximation the illumination and irradiation (that is, global radiation) are connected by a linear dependence, it is possible to use the results of global radiation measurements for rough characterization of the principles of natural illumination, also using the qualitative peculiarities of the regime of illumination dependence upon solar height, cloudiness, albedo of the underlying surface, and other factors. However, the quantitative results will appear unreliable because the so-called light equivalent, a value numerically equal to the number of lumens per cal/cm2min, is extremely variable, when calculated for different observational conditions. Calculations of the author and Pevzner [lo81 have given the following values for direct solar radiation at different atmospheric masses : m=O

k

=

6.58

1 (clean atrn) 5.62

m=

5 (clean atm)

k =

6.42

1 (turbid atm) 4.14

1.4

2

5.46

5.82

4

7.01

5 (turbid atm) 1.89

As seen, even at the same mass but different atmospheric transparency, the light equivalent values are different. In the considered case at m = 1, this difference is about 35 percent. For diffuse radiation in the preceding evaluation, k = 4.70 for a clear sky and k = 6.74 for a dull sky (in both cases the height of the sun is 24'). For global radiation, k = 3.64. The latter case is the most favorable, for as we have seen in Sec. 8.1, the spectral composition of global radiation remains practically constant in a wide range of wavelengths and solar heights. According to Yanishevsky's data [Chapter 2, Ref. 11, the light equivalent for the diffuse and global radiation is about 6 to 8 phot/cal/cm2 min and for the direct solar radiation increases with solar height from 1 at h, of the order of several degrees to 6 phot/cal/cm2 min at the height about 40 to 50'. The preceding examples show that the quantitative characteristics of the regime of illumination require special observations by means of radiation receivers with the same spectral sensitivity as the eye. Measurements of illumination have a very long history, but as Yanishesvky showed, in the majority of cases their results are not sufficiently reliable because of the imperfection of the applied measurement methods.

532

Global Radiation

It is evident that the qualitative principles of the variability of illumination in dependence upon solar height, cloudiness conditions, albedo of an underlying surface, and other factors are the same as those for the variability of global and diffuse radiation. Likewise, the main factors determining the illumination are cloudiness and solar height. As shown by Barteneva and Guseva [109], the values of global and diffuse illumination at different geographical latitudes under the same conditions of observation, as regards height of the sun and the shape and quality of cloudiness, practically coincide and are not a function of latitude. Drummond [110], analyzing the results of measurements of illumination at Pretoria (Union of South Africa), Kew (suburb of London), Tashkent, Wien, and Washington, has come to the same conclusion, which is also held valid by Shifrin and Guseva [ I l l ] . Tables 13 and 14 of the Appendix present data (see [112]) on natural illumination for various conditions of observation with and without snow cover. These tables permit approximate estimates of illumination, provided the conditions of cloudiness and solar height are known. By analyzing the given data it is possible to identify the main factors on which the illumination depends. REFERENCES 1. Herman, R. (1 947). Integrierende Strahlungsmessungen mit Hilfe von Monochromatoren. Optik 2, Nos. 5/6. 2. Gotz, F. W. P., and Schonmann, E. (1948). Die spektrale Energieverteilung von Himmels- und Sonnenstrahlung. Helv. Phys. Acta 21, Part 2. 3. Condit, H. R., and Grum, F. (1964). Spectral energy distribution of daylight. J. Opt. SOC.Am. 54, No. 7. 4. Judd, D. B., MacAdam, D. L., and Wyszecki, G. (1964). Spectral distribution of typical daylight as a function of correlated color temperature. J. Opt. SOC. Am. 64, No. 8. 5. Das, S. R., and Sastri, V. D. P. (1965). Spectral distribution and color of tropical daylight. J . Opt. SOC.Am. 55, No. 3. 6. Kushpil, V. I. (1965). Surface measurements of the angular distribution and spectral composition of diffuse sky radiation. Trans. Main Geophys. Obs. NO. 170. 7. Kondratyev, K. Ya., Burgova, M. P., Badinov, I. Y., and Mironova, Z. F. (1961). Instruments for investigating the spectral light regime Proc. 2nd All-Union Conf. Light Climate, Moscow, Gosstroiizdat. 8. Kondratyev, K. Ya, and Burgova, M. P. (1963). Investigation of the spectral composition of shortwave radiation. Proc. AZZ-Union Sci. Mefeorol. Con$ 1961. VOl. VI. 9. Kondratyev, K. Ya, Burgova, M. P., Grishechkin, V. S., and Mikhailov, V. V. (1965). Investigation of spectral distribution of shortwave radiation. Proc. Acad. Sci. USSR, Ser. Phys. Atmosphere Ocean No. 9.

References

533

10. Koltzov, V. V., and Lazarev, D. N. (1964). Daily variation in the spectral composition of natural radiation in the region 0.3-0.6 p . Zn “Actinometry and Atmospheric Optics.” Nauka, Moscow. 11. Rautian, G. N. (1932). Spectral composition of daylight. Proc. 1st All-Union ConJ Nat. Zllum., 1930, No. 3. 12. Schulze, R. (1951). Zur Abhangigkeit des Globalstrahlenspektrums von der Sonnenhohe. Ann. Meteorol. 4, Nos. 1-6. 13. Brooks, F. A. (1955). Notes on spectral composition of solar radiation and its measurements. In “Solar Energy Research” (F. Daniels and J. A. Duffie, eds.), Madison. 14. Nebolsin, S. I. (1949). Climatic features of the Moscow region. Trans. Central Znst. Weather Pred. No. 10 (37). 15. Kalitin, N. N. (1949). Global radiation at Pavlovsk. Trans. Main Geophys. Obs. No. 19 (81). 16. Barashkova, E. P. (1959). Some regularities in the regime of global radiation. Trans. Main Geophys. Obs. No. 80. 16a. Berland, T. G. (1965). Daily range of solar radiation in the main climatic zones of the globe. Trans. Main Geophys. Obs. No. 179. 17. Birukova, L. A. (1956). On the methods for climatic calculation of daily variation in global and absorbed radiation. Trans. Main Geophys. Obs. No. 66. 18. Berland, M. E. (1956). “Prediction and Regulation of the Thermal Regime of the Boundary Atmospheric Layer.” Gidrometeoizdat, Leningrad. 19. Cheltzov, N. I. (1952). Investigation of reflection, transmission and absorption of solar radiation by clouds of certain forms. Trans. Centr. Aerol. Obs. No. 8. 20. Berland, M. E., and Novoseltzev, E. P. (1962). To the theory of dependence of global radiation upon cloudiness. Sci. Bull. Geol. Geograph. Znst. Acad. SCI.Lithuanian SSR 13, No. 1. 21. Albrecht, F. (1955). Methods of computing global radiation. Geofis. Pura Appl. 32. 22. Averkiev, M. S. (1961). Precise method for computation of global radiation. Bull. Moscow Univ. No. 1. 23. Averkiev, M. S. (1962). On the universal formula for computing global radiation. Meteorol. Hydrol. No. 2. 24. Berland, T. G. (1960). Methods for climatological computations of global radiation. Meteorol. Hydrol. No. 6. 25. Ukraintzev, V. N. (1939). Cloudiness and sunshine. Meteorol. Hydrol. No. 6. 26. Sivkov, S. I. (1964). To the methods of computing the possible radiation totals. Trans. Main Geophys. Obs. No. 160. 27. Sivkov, S. I. (1964). On the computation of the possible and relative duration of sunshine. Trans. Main Geophys. Obs. No. 160. 28. Berland, T. G. (1962). Geographical principles of the solar radiation regime. Proc. All-Union Sci. Meteorol. Con$ 1961. Vol. IV. 29. Berland, T. G. (1948). The radiation and heat balance of the European territory of USSR. Trans. Main Geophys. Obs. No. lo., 30. Galperin, B. M. (1938). Solar radiation at Saratov. Socialistic Econ. Cereals No. 6. 31. Rusin, N. P. (1960). Global radiation in the Antarctic. Trans. Main Geophys. Obs. No. 115. 32. Gavrilova, M. K. (1959). Global radiation in the USSR Arctic and other parts. Trans. Arct. Antarct. Znst. 217. 33. Kalitin, N. N. (1945). Totals of solar radiant heat on the USSR territory. Nature No. 2.

534

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34. Kalitin, N. N. (1947). Development in actinometric investigation for the past 30 years in USSR. Proc. Acad. Sci. USSR, Ser. Geograph. Geophys. No. 5. 35. Landsberg, H. E. (1961). Solar radiation of the Earth‘s surface. Solar Radiation 5, No. 3. 36. Drummond, A. J., and Vowinckel, E. (1957). The distribution of solar radiation through Southern Africa. J. Meteorol. 14, No. 4. 37. Kruglova, A. I., and Shliakhov, V. I. (1962). Global radiation of the Antarctic oceanic waters and of some parts of the Pacific and Atlantic Oceans. Trans. Centr. Aerol. Obs. No. 45. 38. Dogniaux, R. (1954). Etude du climat de la radiation en Belgique. Imt. Roy. Meteorol. Belg. B18. 39. Wexler, H. (1956). Variations in insolation, general circulation and climate. Tellus, 8, No. 4. 40. Black, J. N. (1956). The distribution of solar radiation over the earth’s surface. Arch. Meteorol. Geophys. Bioklimatol. B7, No. 2 . 41. Black, J. N. (1960). A contribution to the radiation climatology of Northern Europe. Arch. Meteorol. Geophys. Bioklimatol. B10, No. 2. 42. Burdecki, F. (1958). Remarks on the distribution of solar radiation over the surface of the earth. Arch. Meteorol. Geophys. Bioklimatol. B8, Nos. 314. 43. Berg, H. (1949). Die Globalstrahlung in Westdeutschland. Ann. Meteorol. 2. 44. Ashbel, D. (1961). “New World Maps of Global Solar Radiation During ICY.” Hebrew University, Israel. 45. Elnesr, M. K. (1956). Contribution a E t u d e de I’energie solaire en Egypte. J. Sci. Meteorol. 8, No. 32. 46. Schulze, R. (1963). Zur Strahlungsklima der Erde. Arch. Meteorol. Geophys. Bioklimatol. B12, No. 2. 47. Tzutzkiridze, Y. A. (1960). The solar cadastre of the territory of Armenia. Trans. Tbilisi Hydrometeorol. Inst. No. 7. 48. Fritz, S . (1949). Solar radiation during cloudless days. Heating Ventilating No. 1. 49. Fritz, S., and MacDonald, T. H. (1949). Average solar radiation in the United States. Heating Ventilating No. 7. 50. Hand, J. F. (1953). Distribution of solar energy over the United States. Heating Ventilating No. 7. 51. Takacz, L. (1958). Normalwarte der Globalstrahlung in Budapest. Idojaras 62, No. 2. 52. Sitnikova, M. V. (1963). Monthly totals of global and diffuse radiation on the territory of Soviet Central Asia. Trans. Centr. Asia Hydrorneteorol. Inst. No. 31 (16). 53. Sekihara, K., and Kano, M. (1957). On the distribution and variation of solar radiation in Japan. Paper Meteorol. Geophys. (Tokyo) 8, No. 2. 54. Mackiewicz, M. (1953). Spatial distribution of insolation in Poland. Przeglad Meteorol. Hydrol. 6, Nos. 112. 55. Mani, A., Chacko, O., and Venkateshwaren, S. P. (1962). Measurements of the total radiation from the sun and sky in India during the ICY. Indian J. Meteorol. Geophys. 13, No. 3. 56. Lindholm, F. (1955). Sunshine and cloudiness in Sweden, 1901-1930. Geograph. Ann. 37, Nos. 1/2. 57. Bibliography on Global Radiation (1954). Meteorol. Abstr. Eibliogr. 5, No. 7. 58. Kondratyev, K. Ya., and Manolova, M. P. (1955). To the problem of the income of diffuse and global radiation on slant surfaces. Meteorol. Hydrol. No. 6.

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59. Kondratyev, K. Ya., and Manolova, M. P. (1956). Income of diffuse radiation on slant surfaces at a cloudless and overcast sky. Proc. Leningrad Univ., Ser. Phys. No. 9. 60. Kondratyev, K. Ya., and Manolova, M. P. (1958). Daily variation and daily totals of diffuse and global radiation on slopes of various types. Bull. Leningrad Univ. No. 4. 61. Kondratyev, K. Ya., and Manolova, M. P. (1958). Radiation balance of slopes. Bull. Leningrad Univ. No. 10. 62. Kondratyev K. Ya., and Manolova, M. P. (1960). Income of diffuse and global radiation on slant surfaces in the presence of snow cover. Bull. Leningrad Univ. No. 16. 63. Kondratyev, K. Ya., and Manolova, M. P. (1960). Radiation balance of slopes. Solar Energy 4, No. 1. 64. Kondratyev, K. Ya., and Fedorova, M. P. (1962). Income of shortwave radiation on slopes of various orientation in the presence of snow cover. Sci. Bull. Geol. Geograph. Znst. Acad. Sci. Lithuanian SSR, 8, No. 1. 65. Kondratyev, K. Ya., and Fedorova, M. P. (1964). On the effect of cloudiness and the occulted horizon on radiation income to slant surfaces. Zn “Actinometry and Atmospheric Optics.” Nauka, Moscow. 66. Fedorova, M. P. (1965). Diffuse radiant fluxes from particular sky sections to slant surfaces. Probl. Attnospheric Phys. No. 3. 67. Bogdanovich, G. P. (1934). Illumination by diffuse light of vertical surfaces of various orientation at a clear sky. J. Geophys. 4, No. 3. 67a. Mukhenberg, V. V. (1965). Some peculiarities of the income of solar radiation on slant surfaces. Trans. Main Geophys. Obs. No. 179. 68. Kudriavtzeva, L. A. (1940). Reflection, absorption and penetration of solar radiation concerning agricultural herbaceous plants. Rept. Agr. Acad. No. 2. 69. Lopukhin, E. A. (1950). To the problem of the radiation regime of cotton. Proc. Acad. Sci. USSR, Ser. Geograph. Geophys. No. 3. 70. Lopukhin, E. A. (1951). An approximate method for consideration of global radiation distribution through cotton. Trans. Tashkent Geophys. Obs. No. 5. 71. Lopukhin, E. A. (1949). To the problem of the role of radiation in cotton growing. Trans. Tashkent Geophys. Obs. No. 3. 72. Geodakian, 0. A. (1951). Balance of solar radiation incident on plants. Rept. Acad. Sci. Armenian SSR 19, No. 2. 73. Tooming, H. (1959). Some problems of distribution of global radiation within vegetative cover. Invest. Atmospheric Phys. No. 1. 73a. Shablovskaya, V. A. (1963). On the distribution of solar radiation through corn crops. Meteorol. and Hydrol. No. 11. 73b. Yefimova, N. A. (1965). Some peculiarities of the meteorological regime inside the vegetative cover of winter-crop wheat and corn. Trans. Main Geophys. Obs. No. 179. 74. Makarevsky, N. I. (1938). On the reflection, transmission and absorption of solar radiation by leaves of plants. Trans. Lab. Light Phys. Phys. Agron. Znst. 75. Kalitin, N. N. (1941). To the study of the radiative properties of leaves of plants. Extl. Botany. [4] No. 5. 76. Lopukhin, E. A. (1948). On the spectral absorption of radiation by cotton. Rept. Acad. Sci. Uzbek SSR No. 9. 77. Lopukhin, E. A. (1951). A method for consideration of short-wave radiation absorbed by cotton leaves. Proc. Acad. Sci. Uzbek SSR No. 4.

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78. Sakharov, M. I. (1940). Radiation and albedo in forest. Mefeorol.Hydrol. Nos. 5 and 6. 79. Ivanov, L. A. (1946). “Light and Moisture in the Life of our Wood Species.” Acad. Sc. U.S.S.R., Moscow. 80. Pivovarova, Z. I., and Guliayev, B. I. (1958). Actinometric observations in woods. Trans. Main Geophys. Obs. No. 86. 81. Kuzmin, P. P. (1949). Radiation balance of forest bodies during thaw. Trans. Hydromefeorol. Znst. No. 16 (70). 82. Zakharova, A. F. (1951). On certain problems of illumination in connection with the tea cultivation under the trees. Bull. Leningrad Univ. No. 9. 83. Golubova, T. A. (1952). On the radiation regime within field-protection forest barrier. Trans. Main Geophys. Obs. No. 36 (98). 83a. Tyler, J. E., Richardson, W. H., and Holmes, R. W. (1959). Method for obtaining the optical properties of large bodies of water. J . Geophys. Res. 64, No. 6. 83b. Pelevin, V. N. (1965). On the variation of the true coefficient of light absorption in the sea. Proc. Acad. Sci. USSR, Ser. Phys. Atmosphere Ocean 1, No. 5. 84. Ivanov, K. I. (1950). On the dependence between the content of suspended particles and the light attenuation coefficient in sea water in the presence of colored humic combinations. Trans. State Oceanograph. Inst. No. 15 (27). 85. Ivanov, K. I. (1950). Variation of the light attenuation coefficient in dependence on the variation in the diameters of suspended particles in water. Rept. Acad. Sci. USSR, 14, No. 5. 86. Skopintzev, B. A. (1950). Organic matter in natural waters (water humus). Trans. State Oceanograph. Znst. No. 17 (29). 87. Skopintzev, B. A. (1947). On the coagulation of humus matter in the river outflow to the sea water. Proc. Acad. Sci. USSR, Ser. Geograph. Geophys. No. 1 . 88. Skopintzev, B. A., and Ivanov, K. I. (1952). Application of photometric measurements to determination of suspended and colored huminous combinations in sea water. Trans. State Oceanograph. Znst. No. 22 (34). 89. Jerlov, N. G. (1951). Optical measurement of particle distribution in the sea. Tellus 3, No. 3. 90. Jerlov, N. G. (1955). Factors influencing the transparency of the Baltic waters. Goteborg Vetenskaps- Vitterhettssamhalles handl. B6, No. 14. 91. Timofeeva, V. A. (1953). Complex light scattering in turbid media. Trans. Sea Hydrophys. Znst. 3. 92. Timofeeva, V. A. (1951). Brightness distribution in strongly scattering media. Rept. Acad. Sci. USSR 76, No. 5. 92a. Timofeeva, V. A. (1957). Propagation of light in the sea. Trans. Sea Hydrophys. Znst. 11. 93. Ambartzumian, V. A. (1942). A new method for calculation of light scattering in a turbid medium. Proc. Acad. Sci. USSR, Ser. Geograph. Geophys. No. 3. 94. Rosenberg, G. V. (1958). The light regime in the depth of a scattering medium and spectroscopy of dispersed matter. Rept. Acad. Sci. 122, No. 2. 95. Lenoble, J. (1956). Application de la mCthode de Chandrasekhar a l’ktude du rayonnement diffuse dans le brouillard et dans la mer. Rev. Opt. 35, No. 1. 96. Lenoble, J. (1956). Etude de la pCnCtration de l’ultraviolet dans la mer. Ann. Geophys. 12, No. 1. 96a. Ivanoff, A. (1956). Etude de pCnCtration de la lumikre dans la mer. Ann. Geophys. 12, No. 1.

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96b. Duntley, S. Q. (1963). Light in the sea. J. Opt. Soc. Am. 53, No. 2. 97. Ovchinsky, B. V. (1943). Application of the equation of radiative transfer to some problems of sea optics. Rept. Acad. Sci. USSR 39, No. 3. 98. Shuleikin, V. V. (1953). “The Physics of the Sea.” Acad. Sci. U.S.S.R. Press, Moscow. 99. Berezkin, V. A., Gershun, A. A., and Yanishevsky, Y. D. (1940). “Transparency and Color of the Sea.” Naval Academy Press. 100. Kirillova, T. V., and Burig, R. F. (1958). On the results of measurement of subaqueous radiation. Trans. Main Geophys. Obs. No. 78. 101. Mokievsky, K. A. (1962). Some results of investigation of the spectral composition of solar radiation penetrating into water. Sci. Bull. Geol. Geograph. Inst. Acad. Sci. Lithuanian SSR 13, No. 1. 102. Trofimov, A. V. (1935). Illumination in the upper sea horizon under the water and under the ice. J . Geophys. 5, No. 4. 103. Pisiakova, N. M. (1947). Observation of the penetration of solar radiation through ice. Meteorol. Hydrol., Inform. B d l . No. 6. 103a. Lubimova, K. S. (1962). Some peculiarities of extinction of solar radiation in the thickness of ice. Proc. Acad. Sci. USSR, Ser. Geophys. No. 5. 103b. Ambach, W. (1959). Ein Beitrag zur Kenntnis der Lichtstrahlung im Gletschereis. Arch. Meteorol., Geophys. Bioklimatol. B9, Nos. 314. 103c. Ambach, W., and Mocker, H. (1959). Messungen der Strahlungsextinktion mittels eines kugelformigen Empfangers in der oberflachehhahen Eisschicht eines Gletschers und im Altschnee. Arch. Meteorol. Geophys. Bioklimatol. B10, No. 1. 104. Polli, S. (1950). Penetrazione delle radiazioni luminose nel ghiaccio e nella neve. Ann. Geofis. (Rome) 3. 105. Karol, B. P. (1962). On the radiative properties of glacial ice. Sci. Bull. Geol. Geograph. Inst. Acad. Sci. Lithuanian SSR 13, No. 1. 106. Sulakvelidze, G. K., and Okujava, A. M. (1959). Snow and its properties. Proc. Elbrus Compl. Exped. 1. 107. Kuzmin, P. P. (1947). Absorption of solar radiation by snow cover. Meteorol. Hydrol., Inform. Bull. No. 5. 108. Kondratyev, K. Ya., and Pevzner, S. I. (1954). On the correlation between the energetical and luminous radiant fluxes. Sci. Bull. Leningrad Univ. No. 32. 109. Barteneva, 0. D., and Guseva, L. N. (1957). The regime of natural illumination in dependence on meteorological conditions. Trans. Main Geophys. Obs. No. 68 (130). 110. Drummond, A. J. (1958). Notes on the measurement of natural illumination. Arch. Meteorol., Ceophys. Bioklimatol. B9, No. 2. 111. Shifrin, K. S., and Guseva, L. N. (1957). Forecasting of natural illumination. Proc. Acad. Sci. USSR, Ser. Geophys. No. 6. 112. Sharonov, V. V. (1945). “Tables for Calculating Natural Illumination and Visibility.” Acad. Sci. U.S.S.R. Press, Moscow.

THERMAL RADIATION OF THE ATMOSPHERE

9.1. The Phenomenological Theory of the Transfer of Thermal Radiation in the Atmosphere Let us turn to the general solution of the problem of thermal radiative transfer in the atmosphere. Generally speaking, this problem should be solved by making use of a complete system of hydrodynamic and radiation equations. The first to formulate such a system was Kuznetzov [l, 21. Kuznetzov succeeded in reaching a number of interesting conclusions on the values of the fluxes of radiant energy and on the vertical temperature distribution in the atmosphere, on the assumption of radiative equilibrium. The mentioned works are particularly interesting because they contain a most complete and exact investigation of radiative equilibrium. The solution of the complete system of hydrodynamic and radiation equations involves considerable difficulties. Although they can be largely overcome, nevertheless it should be stated that the calculation of the transfer of radiant energy from the complete system of equations as mentioned above appears at present to be inexpedient. Moreover, solution of a great number of the most important problems can be accomplished readily by calculating the radiative transfer at a given temperature and density distribution of the radiation absorbent. This makes it unnecessary to consider the complete set of equations. For this reason we shall treat the problem of the calculation of radiative transfer on the basis of the phenomenological transfer equations, with a given distribution of temperature and density of the absorbing substance. In this case the transfer equations (1.83) are linear inhomogeneous differential equations of the first order relative to 538

9.1. Phenomenological Theory of the Transfer of Thermal Radiation

539

the unknown values of the longwave radiation intensities G,(z, 0) and U,(Z, 0). 1. Transfer of Monochromatic Radiation. The solution of the problem of monochromatic radiation transfer reduces first of all to the integration of the transfer equations (1.83). By integrating in this equation, we derive the general formulas for the intensity of monochromatic thermal radiation. Performing subsequent integration over all directions, making use of the relation (1.1 l), we obtain expressions for fluxes of monochromatic thermal radiation. There is no need to dwell on the integration of the simple equations (1.83). The result of the integration follows:

The constants C , and C, are often determined from the following boundary conditions : GA(~ 0)Y= 0,

Ua(0, 0)

Ea(T(0))

(9.2)

The first of these boundary conditions formulates the evident fact that at the upper atmospheric boundary, the intensity of downward thermal atmospheric radiation is zero. The second boundary condition is usually determined on the assumption that the intensity of upward thermal radiation at the earth's surface is equal either to blackbody radiation at the temperature of the surface (as in (9.2)) or to the blackbody radiation corrected by a certain factor almost equal to unity. The latter is justified by some authors on the supposition that the earth's surface can be considered a gray emitter whose radiation therefore may be taken equal to blackbody radiation, excepting a constant coefficient close in value to unity. However, this representation of the boundary condition in question seems illogical. More successful is the following boundary condition : Ua(0,0)

=

Ea(Te(0))

540

Thermal Radiation of the Atmosphere

where T,is the effective surface radiation temperature. If we assume that the earth’s surface is a gray radiator and emits, for example, 6 percent of the blackbody radiation at the surface temperature, according to Kirchhoff’s law it follows that the earth’s absorptivity should be 6. Thus, of all global atmospheric thermal radiation Go incident on its surface, the earth absorbs a fraction equal to 6G,. Since what remains of Gocan only be reflected then in general the considered boundary condition should be written as

It was pointed out in Chapter 1 that the value 6 is not so close to unity that the second term can be ignored. Note here that to simplify the matter we speak about the total flux of thermal radiation ;however, it is evident that these considerations are equally valid for monochromatic fluxes and radiant intensities. Calculations of Kostianoy [2a] show that the correction for the reflected radiation in computing radiant fluxes through a free atmosphere is quite small. Let us return to (9.1) and determine the constants. For simplicity we restrict ourselves to the determination of the constants from the condition (9.2), with the intention of giving the results of the integration of the transfer equation by making use of (9.3). With the boundary conditions (9.2) satisfied, Eqs. (9.1) take the following form:

(9.4a)

(9.4b)

As seen, these expressions contain only the integration over z. To calculate the monochromatic radiant fluxes it is necessary, as has already been pointed out, to perform an additional integration over 19 and ~ 1 , taking account of (1.11). Let us integrate both expressions (9.4) over 8 and ~ 1 ,taking into consid-

9.1. Phenomenological Theory of the Transfer of Thermal Radiation

541

eration the fact that the downward and upward Auxes G,(z) and U,(z) can be determined through the intensities by the following formulas : G,(z) =

12” In‘’ dp,

8 ) sin 8 cos 8 dp,

G,(z,

0

(9.5a)

Taking account of (1.12), we now have

Ba = ZEA We introduce a new variable sec 8 of (9.7) become rn(z, 5 ) =

(9.7d) = t.

1

Then the first two expressions

t-2 e-ra(z*e)t dt

(9.8a)

542

Thermal Radiation of the Atmosphere

Thus, the expression (9.6) can now be presented, taking into account (6.7c), as

Thus, by integrating the intensity equations over 0 and 9,we obtain expressions for the monochromatic thermal radiation fluxes. As seen from (9.10), the fluxes of monochromatic radiation are expressed through such integrals, which can be only calculated numerically or graphically unless certain simplifying assumptions are introduced.

2. Transfer of Nonmonochromatic Radiation. In order to obtain expressions for the full fluxes of thermal radiation it is necessary to integrate over all wavelengths from il = 0 t o il = 00. Thus we finally obtain

The integration over all wavelengths involves particular difficulties due to the very complex structure of the infrared absorption spectra of the polyatomic gases contained in the atmosphere. Lebedinsky [3], investigating the fluxes of thermal radiation in the atmosphere, applied a method similar to that suggested by Ambartzumian for estimating the influence of selective absorption on radiative equilibrium in stellar atmospheres. This method avoids the difficulty encountered in the integration over wavelengths. First carry out certain transformations of the expressions (9. lo), taking into account the identity

9.1. Phenomenological Theory of the Transfer of Thermal Radiation

543

and using the familiar denomination

It should be noted that the substitution of the variable w for the variable rA involves the assumption about independence of the absorption coefficient k Afor altitude. This restriction can be easily removed only when all absorption coefficients are equally dependent on altitude. Actually, if k,(z) = k,(O)f(z), it is possible to introduce an “effective” content of the absorbent

w* =

f(C>e(Od5

thus extending the above consideration to the case of the absorption coefficient’s being dependent on altitude. To simplify exposition let us perform the mentioned transformations on (9.10a) only. We have GAw) = - JWW B%&)42G(kA(PW

som

W))l

(9.12)

Ji

where w, = e(t) dC, and p = e([) d5 is the variable of integration. Now integrate both parts of (9.12) over all wavelengths satisfying the inequality

k

< kA < k

.f dk

and thereby introduce the following designations :

where the integration extends over the mentioned wavelength region. The second of the above relations is essential to Lebedinsky’s method, since it enables derivation of final results in a very simple form. On the other hand, it should be stressed that in actual fact f = f ( k , T ) , and therefore the assumption that f =f ( k ) is considerably limiting in the general treatment. Instead of (9.12) we obtain

544

Thermal Radiation of the Atmosphere

By integrating the above relation over all k values from 0 to 00, we derive the following formula for the total downward flux at the level z : G ( w )= -

The expression in square brackets is the integral transmission function for diffuse radiation (see Eq. (1.30)). We may therefore write

G(w)= -

(9.13)

For the particular case z = 0 ( w = 0), we obtain for the downward atmospheric radiation : GO

-

I,

zu,

(9.14)

BdpF(Pu)

The formula for the upward flux of thermal radation is similarly obtained:

When using the exact boundary condition (9.3), the expression for the upward flux becomes

- (1

-

6)

rrn

BdPF(p

0

+ w)

(9.16)

It is easily understandable that the individual terms of this formula have a simple physical meaning: The first term describes that portion of the radiation emitted by the earth's surface which reaches the considered level ; the second characterizes the emission of the atmospheric layer between the given level and the earth's surface; and the third represents that portion of the downward atmospheric radiation reflected by the earth's surface which reaches the considered level. In the particular case z = 0 (w = 0) for the upward flux at the earth's surface, we have

Uo = 6Bo- (1 - 6)

1"""BdP,(p)

(9.17)

For practical convenience let us transform the general equations (9.13) and (9.16) by partial integration. After performing these elementary

9.1. Phenomenological Theory of the Transfer of Thermal Radiation

545

transformations we obtain G ( w ) = B(w) - B(w,)PF(w,

+ U(W) =

JWm W

- W)

(9.18)

B(w) -

+ (1 (9.19) Consideration of the general equations for the thermal radiation fluxes shows that the calculation of the fluxes at a given distribution of temperature and density of the absorbent with height can be easily performed, provided the integral transmission function for diffuse radiation is known. For this reason, as has already been mentioned, the determination of the integral transmission function is the fundamental problem in the theory of thermal radiative transfer. Analysis of (9.18) and (9.19) shows, furthermore, that these equations are very convenient in plotting nomograms. Actually, any of the terms of (9.18) and (9.19), can be presented in the form of the integral sPF dB, and the sum of the terms can be determined as a similar integral over a closed contour. This means that the radiant fluxes can be calculated as areas in the coordinate system (PF , B). The possibility of this simple graphical interpretation of (9.18) and (9.19) has been extensively used by different authors to plot the so-called radiation charts (nomograms) as a means of computing atmospheric radiant fluxes. Radiation charts will be considered in detail in Sec. 9.3. The integral transmission function for diffuse radiation PF(w) cannot at present be theoretically calculated. The use of (9.18) and (9.19) thus only allows working out semiempirical methods for calculating the thermal radiant fluxes, on the assumption of the transmission function being known empirically and by using the above phenomenological theory. Since, however, the transmission function has not yet been reliably determined, it is of interest to develop purely theoretical methods for computing radiant fluxes based on the use of the theoretical line or band transmission functions considered in Chapter 3. It should be noted that this problem, even in a sufficientlygeneral treatment, is extremely complex, but a practical computational method has not so far been developed.

546

Thermal Radiation of the Atmosphere

9.2. Approximate Transfer Equations and Their Use for Calculating Thermal Radiative Transfer in the Atmosphere In Chapter 3 it was stated that the exact account of the diffusivity of radiation can be replaced with sufficient accuracy by a simpler approximation. By using this approximation we exclude the necessity of performing one of the three above-mentioned quadratures, namely, the integration over all angles 8. In this case the transfer equations for the monochromatic fluxes coincide with (1.83) for the radiant intensities at 8 = 0, provided E, is replaced by B,. It should only be borne in mind that in using (1.83) for the calculation of the radiant fluxes, either the absorption coefficient k , or the density of the absorbent must be multiplied by 1.66. Let us now turn to the derivation of the general formulas for the radiant fluxes, based on the use of the approximate transfer equations as was suggested by this author (see [4, 51). We have the following approximate equations of the transfer of monochromatic radiation :

(9.20b) Remember, in addition to what has been said, that in deriving (9.20) it is assumed that the atmosphere is in the state of local thermodynamic equilibrium. It should also be noted that although (9.20) assumes the presence in the atmosphere of only one absorbing component, actually two components should be considered-water vapor and carbon dioxidebecause of the scheme of compensation of the absorption coefficient or mass (density) of water vapor presented in Sec. 3.3. Thus, despite the fact that the value denotes the density of atmospheric water vapor, (9.20) should account not only for the absorption (emission) by water vapor but also by carbon dioxide. Equations (9.20) enable calculation of the monochromatic radiant fluxes in the atmosphere at a given vertical distribution of temperature and density of water vapor. However, we are interested either in the full radiant fluxes or in the fluxes in isolated spectral regions of finite width. Both fluxes can be obtained from the integration of the expressions for the monochromatic fluxes over the corresponding spectral region. As has already been mentioned, such integration cannot yet be performed directly because of the great complexity of the infrared absorption spectrum of the atmosphere.

9.2. Approximate Transfer Equations

547

The method of the use of the selectivity of absorption suggested by Ambartzumian successfully overcomes this difficulty. In this method it is sufficient to have only a given absorption function. It has been shown above that the infrared absorption spectrum of the atmosphere can be divided into a number of intervals within each of which the absorption can be satisfactorily described by the exponential absorption function. It is easily understandable that by transforming the transfer equations in accordance with the earlier considered selective schematization of the absorption spectrum, and by integrating the transformed equations, we obtain formulas analogous to (9.18) through (9.19), on the condition that the absorption in the individual spectral intervals is described by the exponential absorption function. Let us now pass on to the transformation of (9.20), following the technique of this author. Summing up (9.20) over all wavelengths from 0 to 00 (practically necessary only over the finite wavelength interval), we obtain

a *

8.2

m

m

i.=O

?.=O

c G1 = c k,.ewG1 - x k1eulB1

,+o

(9.21a) (9.21b)

Note here that the sums in (9.21) should be practically considered as the sums over a very narrow, but still of some finite width, spectral intervals. Now combine the terms of these sums into n such selections from the wavelengths so as to make the absorption coefficient for radiation within each selection a little different from the others. Let the contribution of each selected interval to blackbody radiation be

Bj

=

2 B, = pjB 1

where B = aT4. Then, for each of the obtained intervals, determine the mean absorption coefficients (similarly to the method used in Chapter 3) that can be carried out in front of the summation sign for each selection. Denoting the radiant fluxes corresponding to the individual selected intervals by the subscript j , we have

2

= kjew(Gj - pjB)

9 aZ = kje,(pjB -

(9.22a) (9.22b)

548

Thermal Radiation of the Atmosphere

Perform the integration at the following boundary conditions :

z

=

(9.23a)

Gj 10

z=m:

0:

Uj

6pjB

+ (1 - 6)Gj

(9.23b)

-

where j = 1, 2, . n, and 6 is the relative emissivity (absorptivity) of the earth's surface, which, as usual, is considered to be a gray radiator. Introducing the independent variable

in place of (9.22) we have (9.24a)

PjB) "j - - k j ( p j B - U j ) dW

( j = 1,2,

. - -n )

(9.24b)

The boundary conditions (9.23) are transformed as follows: w

=

w,:

w

= 0:

Gj

(9.25a)

=0

Uj = 6pjB

+ (1 - 6)Gj

j

=

1, 2,

--- n

(9.25b)

The integration of (9.24) with the boundary conditions (9.25) leads to the following results:

(9.26b)

Uj(w) = pjB(w) -

On the basis of these formulas the expressions for the integral radiant fluxes are obtained by summing up over all the selected intervals: n

G(w) =

2

Gj(w>

(9.27a)

j=1

(9.27b)

549

9.2. Approximate Transfer Equations

The whole problem finally reduces to the calculation of the integrals in (9.26), which can be easily performed by using the familiar approximate methods for calculating integrals. If the cases of various schematic models of atmospheric stratification are considered, the results can be presented analytically. Particularly simple expressions are obtained if it is assumed that vertical temperature gradient in the atmosphere is constant and that the absolute humidity exponentially decreases with height. It is easy in this case to derive the formula for the effective radiation of the underlying surface :

6

-

dB

pi e-lijww [Ei(kiw,) - ( C

+ In kp,)]

(9.28)

where T, is the air temperature at the level of the tropopause z = H ; A(w,) is the integral absorption function for the entire atmospheric thickness; /3 is the coefficient in the relation ew = ewoe-@characterizing the decrease of absolute humidity with height: 1 a / d z I is the mean for the troposphere, with absolute vertical gradient of value B = crT4; Ei(x) is the integrated exponential function; and C is the Euler constant. It is easy to see that the second term in this formula is corrective. This is clear from the form of the bracketed expression itself. Numerical calculation shows also that it is sufficient to make Y = 2. With this value, p1 = 0.25, p z = 0.1 1, and the corresponding absorption coefficients (diffusivity taken into account) have the value 0.166 and 0.8 cm2/g. As seen, the schematization of the infrared absorption spectrum used in calculating the effective radiation differs numerically from that applied in calculating the fluxes of radiant energy. This difference can be found in the results of the calculations themselves: Since in the problem of radiative transfer it is most essential to describe the absorption in the region of maximal transparency, greater detail was given to the absorption spectrum in the region of small k values and the region of large k was completely excluded in calculating the effective radiation. Neglecting the second term of (9.28) and using the approximate relation B,, - BfI I %/dz I H , we obtain the following simplified formula for calculating the effective radiation :

-

(9.29)

550

Thermal Radiation of the Atmosphere

where C , is a certain corrective coefficient compensating for the neglect of the second term in (9.28). This coefficient can be evaluated theoretically. Let us use the term “normal” for an atmosphere whose vertical temperature gradient from z = 0 t o z = H i s constant and for which the distribution of absolute humidity is described by an exponentially decreasing function of height. From (9.29) it can be seen that in the case of the normal atmosphere, the effective radiation of the underlying surface is determined by five physical quantities: the temperature at the earth’s surface (since Bo = aTO4),the vertical temperature gradient (since the value dB/dz is determined first of all by the value of the vertical temperature gradient y = dT/dz), the total content of water vapor in the atmosphere (woo), the emissivity of the underlying surface (d), and the altitude of the tropopause. As to the dependence of Fo upon the altitude of the tropopause, it is not essential for a given geographical point because the mean H value for a given point is usually not so variable as the other factors. In calculating the effective radiation, however, the form of the absorption function and the values of its numerical parameters are important. Comparison of the theoretical formula (9.29) with the empirical formulas for calculating the effective radiation enables quantitative analysis of the applicability range of the latter. This problem will be dealt with later. In the immediate proximity to the earth’s surface (in the surface atmospheric layer), the observed vertical temperature gradients are usually greater than in a free atmosphere. The vertical temperature profile for this case can be schematized as follows:

where a is constant, E~ is the so-called thermal roughness, and h is the height of the surface atmospheric layer. The positive sign refers to the inversional stratification of the atmosphere and the negative sign to the unequilibrated. Simple calculations show in the considered case that instead of (9.29) the following equation is obtained: Fo

=

[

6 Bo

+ Cl

1z11 ~

H [l - A(w-)]

9.3. Radiation Charts

551

where T&) is the mean value of T3 for the surface atmospheric layer. The positive sign refers to the case of superadiabatic and the negative sign to the inversional vertical temperature gradients. It is quite evident that the second term in (9.30) characterizes the effect of the surface atmospheric layer stratification on the effective radiation. In deriving Eqs. (9.26) through (9.30), we ignored the dependence of radiation absorption on pressure and temperature. In the first approximation this dependence is easily taken into account by introducing the effective content of water vapor, w*, in place of the real content w : (9.31) Usually it is sufficient to take account of the dependence of radiation absorption on pressure only, in which case the usual assumption is f(z)

= d p ( z ) / p , , , where p o is the standard pressure at the earth’s surface (1000 mb), and p ( z ) is the pressure at the level z. 9.3. Radiation Charts The approximate analytical formulas derived in the preceding section are useful in spectacular analysis of the regularities of radiative transfer or in approximate calculations short of data of aerological sounding. Exact calculations of radiant fluxes with the use of aerological data should be performed by means of graphical calculation of the integrals in (9.18) through (9.19), or by use of electronic computers. In the present section we shall consider the so-called radiation charts intended for the above graphical calculation. In the past 20 years a number of different charts has been suggested by such authors as Moller [6, 7, 7a], Dmitriyev [8, 91, Elsasser [lo, Ila], Yamamoto [12], Robinson [13], and others. We shall limit our discussion to two charts, of which the Shekhter chart (see [14, 151) is the most common in the U.S.S.R., and the chart of this author and Nijlisk [Chapter 3, Ref. 11 I ] is the most precise. It should be noted that Kondratyev and Nijlisk [16] and Shekhter [17] have made comparisons of different charts.

1. The Shekhter Chart. The Shekhter chart is based on the general thermal radiation equations derived in the result of the integration of the transfer equations according to the method of Ambartzumian and Lebedinsky. It is easy to see that Eqs. (9.18) and (9.19) for radiant fluxes can be transformed so as to make their right-hand sides represent the integral over a closed

552

Thermal Radiation of the Atmosphere

contour. In fact, instead of (9.18) we can write

=

$ PF(u)dB

(9.32)

where u = ,u - w is the water vapor content measured from the considered level, and u, = w, - w is the total water vapor content above this level. As seen from (9.32), the downward atmospheric radiation at a given level is numerically equal to the area inside the closed contour plotted in the coordinate system (PF, B). Similarly, the upward flux of thermal radiation can be determined. Accordingly, the chart for the calculation of radiant fluxes is a rectangular nomogram whose abscissas are proportional to 0T4 and whose ordinates represent the flux transmission function PF. In practice the axis of abscissas is usually marked with temperature values instead of 0T4. Similarly, the axis of ordinates represents water vapor contents, not P F . The flux transmission function used by Shekhter was described earlier (see Chapter 3). To account for the absorption by carbon dioxide, Shekhter used a technique analogous to that suggested by this author (see Sec. 3.3). In order to compensate the effect of pressure on radiation absorption, the water vapor content is used as the “effective” content described by the formu 1a (9.33) where q is the specific humidity, and e is the density of air. Further on we shall give an example of a calculation made with the Shekhter chart. Shekhter first plotted a radiation chart designed for calculating thermal radiant fluxes. This chart was later modified to allow not only calculation of hemispherical radiant fluxes but also of fluxes within the solid angle enclosed by any two parallels of altitude on the celestial sphere. It can be shown that the downward atmospheric radiation G(w, 6 ) for a spherical belt between the horizon and the parallels of altitude corresponding to the zenith angle 6 is described by the following equation: (9.34)

9.3. Radiation Charts

553

We see in the case, 0 = 0, that this equation gives a relation identical with (9.32) for the downward hemispherical flux of atmospheric radiation. It is obvious that the downward atmospheric radiation for the hemispherical zone within the two arbitrary parallels of altitude can be determined as a difference ebtween two quantities calculated from (9.34). Consideration of (9.34) shows that in plotting the chart for an arbitrary solid angle, the scale of the ordinate axis (which is marked with the values P,(u/cos 0) cos20) must be more complex than in the case of hemispherical fluxes. The Shekhter chart is given in Fig. 9.1, which plots the isolines of constant temperature (vertical lines) and of constant absorbing mass (horizontal lines). On the right-hand side of the diagram is given an auxiliary graph for calculating radiant fluxes within any arbitrary solid angle. Consider first the calculation of the downward and upward hemispherical fluxes

.

FIG. 9.1 Shekhter’s radiation chart ( w , g m ern-=;unit area equals 0.002 cal-

min-I).

554

Thermal Radiation of the Atmosphere

at a certain level in the atmosphere. Let the air temperature at this level be T I . First of all it is necessary to express the temperature variation above and below the given level as a function of the effective absorbing mass measured upward and downward from this level. The effective absorbing mass can be calculated either from a special auxiliary chart or by calculating it layer by layer on the basis of (9.33) and the data of aerological sounding, for which it is necessary to know the variation of specific humidity with height. When the values of the function T(w) for certain considered levels above and below the given level are obtained, they are entered on the chart to plot the corresponding curves T(w) for the atmospheric layers above and below the given level. Let BC and BE (Fig. 9.2) be these curves. The point C of the curve BC obviously corresponds to the highest point of the sounding. Since aerological soundings do not always penetrate great

FIG. 9.2

The use of the Shekhfer radiation chart.

heights, the question arises whether it is possible to extrapolate the curve BC to a height above which the water vapor content is negligible. To realize such extrapolation, the upper right-hand corner of the Shekhter chart (Fig. 9.1) displays an auxiliary table where H is the highest point reached in the sounding, w and w, are the absorbing masses for the whole atmosphere and for the layer of H thickness, and t and tH are the air temperatures at the height 8 km and at H. The table was calculated on the assumption that all water vapor is concentrated in the lower 8 km of the atmosphere. It is intended for the extrapolation of the curve T ( w ) up to the level z = 8 km in calculating the downward atmospheric radiation at the surface level, with aerological data covering the heights H < 8 km.

9.3. Radiation Charts

555

However, even in this case, such extrapolation should be considered approximate. If the point C corresponds to a sufficiently great height (of the order of 8 to 10 km), there is no need to extrapolate the BC curve, and starting from the point C, this curve must be prolonged over the isoline corresponding to the point C of the absorbing mass, and extended to the right-hand vertical coordinate, which is the isoline T = 0. After this, the contour must be closed by adding straight lines DA and AB. The area BCDAB will give the downcoming flux of atmospheric radiation at the given level. The scale in the upper right-hand corner of the chart (Fig. 9.1) is used to obtain the radiant flux in absolute units from the measured area. The upward flux can be calculated similarly. The point E of the curve BE corresponds to the surface level. Since the radiation of the earth’s surface can be identified with the radiation of an isothermal water vapor of infinite thickness, the curve BE must be extended along the isotherm T = To (Tois the earth’s surface temperature) to the point F corresponding to w = 00. Close the contour by adding the straight lines FG and GA. The area BEFGAB will determine the upward flux at the considered level. The difference in area between BEFGAB and BCDAB characterizes the effective radiation of the earth’s surface in the case where the latter radiates as a blackbody. Since the relative emissivity (absorptivity, accordingly) of the earth’s surface is actually less than unity, then at a known 6 it is possible to introduce a correction for the “nonblackness” of the earth’s surface by multiplying the above effective radiation by 6. The calculation of the radiant fluxes for the spherical zone between the horizon and an arbitrary parallel of altitude, whose angular height relative to the horizon is h, shows general similarity with the calculation of the hemispherical fluxes. The difference lies in the technique of determining the absorbing masses. If, for example, h = 50°, then the vertical straight isoline of 50” in the auxiliary chart of Fig. 9.1 will serve as the scale of the absorbing masses. The curved isolines of the auxiliary chart connect the points corresponding to equal w values indicated on the right-hand vertical scale of the radiation chart. As an example let us consider the calculation of the atmospheric emission for the hemisphere and for a spherical zone at h = 50’ (Fig. 9.3). In this chart the area ADBCDA determines the hemispherical flux and the area EOGKME is the downward flux from the spherical zone between the horizon and the parallel of altitude h = 50’. The difference between the areas AOBCDA and EOGKME characterizes the downward atmospheric flux within the vertical cone with apex angle 80’.

556

Thermal Radiation of the Atmosphere

FIG. 9.3 Illustration of the

use of the Shekhter chart.

2. The Chart of Kondratyev and Nijlisk. This chart takes account of the effect of the three main atmospheric components (water vapor, carbon dioxide, and ozone) on radiative transfer by using the most reliable quantitative characteristics of the absorption of longwave radiation. The symbolism PF(w, u, m) denotes the atmospheric transmission function obtained by taking into account the water vapor (w),carbon dioxide (u), and ozone (m).The results of the calculation show that this function can become

M w , u, m) = 0.01 [ P F ( W , 2.4)

+ d&(w,

m)l

(9.35)

where PF(w,u ) is the transmission function, taking into account the absorption by water vapor and carbon dioxide; and dPF(w,m ) is a corrective term for taking account of ozone. The values PF(w,u ) and dPF(w,m), expressed in thousandths of unity, are given in Tables 2 and 3 of the Appendix. By using these tables to determine the transmission functions for different atmospheric levels, it is subsequently possible to calculate the radiant fluxes according to the same method suggested by the Shekhter chart. More detail of the calculation of the transmission function is given in Nijlisk [18]. Figure 9.4 presents the form of the new radiation chart. As before, the temperatures are plotted on the abscissa axis and the transmission functions

9.3. Radiation Charts

557

are on the ordinate axis. Thus, in this case, we enter on the chart the functional T(P,) and not T(w), as with the Shekhter chart. The other computational techniques are the same.

a

FIG. 9.4 A new radiation chart.

Analogous methods of calculation can be used to compute spectral fluxes of radiation. For example, Bolle [I91 has applied the method of radiation charts in computing the thermal atmospheric radiation in the region of isolated spectral lines of water vapor. During recent years work has been done to develop numerical methods for calculating thermal radiation with the help of electronic computers (see [20-221). Kagan [21], for example, has worked out a detailed method for numerical calculations of the integrated longwave radiation fluxes, and this method has used the basis of the calculation of the transmission function derived by Kondratyev and Nijlisk"23, 241. Davis [22] has proposed a technique enabling calculation of the spectral radiant fluxes for spectral intervals of 25-cm-l width and subsequent calculation of the integrated fluxes by summing up the values corresponding to different spectral intervals.

558

Thermal Radiation of the Atmosphere

9.4. Radiative Heat Transfer in Clouds All the results considered above relate to the transfer of thermal radiation in cloudless atmospheric layers. In this case the possibility of ignoring the scattering of radiation is an essential contribution toward simplifying the solution of the problem. The situation is different in the presence of a cloud layer. The dimensions of cloud water droplets are comparable with the wavelengths of thermal radiation. In this case, therefore, it it is quite important to take account of the scattering. The exact solution of the problem of radiative heat transfer in clouds can be obtained only by using the exact transfer equation (1.80). This method was used by Feugelson [Chapter 6, Ref. 91, Yamamoto, and others [25], Shifrin [26] points out that in the considered case the absorption coefficients, like the scattering coefficients, can be correctly determined only by applying the diffraction theory of the radiative properties of cloud droplets. It appears, for example, that by virtue of the diffraction effect, the radiation by a droplet in the region of certain wavelengths exceeds the black radiation at the temperature of the droplet. Shifrin’s calculations of scattering and absorption coefficients make possible an adequate solution of the problem of radiative heat transfer in a monodispersional cloud containing water droplets of some definite radius. The main result of calculations by Feugelson and Shifrin concerning thick stratus clouds is the conclusion that a cloud is “active” with respect to thermal radiation only near its edges, that is, in the boundary layers. The radiant flux penetrating the cloud is completely and immediately absorbed at a distance of only a few tens of meters. This absorption is considerably dependent upon wavelength: It is large in the region of strong absorption where the depth of penetration is only a few meters, and small in the region of weak absorption, with the depth of penetration increasing to 50 to 100 m. The effective radiant flux differs from zero solely within the boundary layers. Inside the cloud it equals zero; and here the upward and downward fluxes are practically identical with the blackbody radiation at the temperature of the corresponding level. The measurements of Gayevsky [27] fully confirm these conclusions. Since the radiating boundary layers of a cloud are comparatively thin, this means that in the first approximation the thermal radiation leaving through the cloud surface can be identified with the blackbody emission at the temperature of the cloud boundary. The latter fact allows use of the above-mentioned simple boundary condition at the cloud surface, reduced on the assumption that this surface is a perfect black radiator. Marshunova

9.5. Effective Radiation of the Underlying Surface

559

[28], however, has shown that this assumption may be considered valid only for warm clouds. Analysis of measurement data on the downward atmospheric radiation through a solid cloudiness and negative temperatures at the lower cloud surface in the Arctic has shown that in this case the relative emissivity of clouds is far less than unity, especially at -6 to - 10°C. This appears to be accounted for by a rapid decrease in the cloud water content and the formation of ice. The mentioned results concern the cases of stratus clouds with sufficiently great optical thickness. It is perfectly evident that they are not representative of thin clouds or of high clouds, which are even thinner. It is clear, in particular, that the radiation leaving the boundaries of the upper layer clouds can no longer be identified with blackbody radiation. This conclusion is supported by the measurement data of Kuhn 1291; for example, in the case of cirrostratus clouds with an altocumulus underlayer, the emissivity turned out to be 0.75. In the presence of high cirrostratus clouds, the emissivity was 0.59. The cirrus emissivity varied from 0.10 to 0.75. In the presence of high cirrostratus clouds, the emissivity was 0.59. The cirrus emissivity varied from 0.10 to 0.75. Recently the investigation of the optical properties of clouds and aerosol layers in the infrared spectrum has been developed further (see [30-351). The result of these investigations will be partly discussed in Chapter 11.

9.5. Effective Radiation of the Underlying Surface and the Downward Atmospheric Radiation One of the most important practical applications of the theory of radiative heat transfer in the atmosphere is the calculation of the effective radiation of the underlying surface and of the downward atmospheric radiation. Modern experimental data on the effective radiation of the underlying surface are rather numerous, but still do not allow solving many important problems of climatology and meteorology. In the climatological investigations of the net radiation, or the heat balance of the underlying surface, all data on the effective radiation are obtained from calculations. For this reason we shall devote special attention to the calculation of the effective radiation of the underlying surface and of the atmospheric emission. 1. Regularities Observed in the Variation of the Eflective Radiation and Downward Atmospheric Radiation. At present the available observational data on the effective radiation and downward atmospheric radiation are

560

Thermal Radiation of the Atmosphere

rather numerous, but are not reliable. As has already been mentioned in Chapter 2, all methods for measuring radiant fluxes, especially during the day time, are imperfect. The measurement errors are usually 10 to 15 percent and still greater in day time when the measurements are indirect. It should also be noted that in many cases the literature gives values of effective radiation of the blackened receiving surface of the instruments, whereas we are interested in the true effective radiation at the ground. Obviously the two values cannot be identical. Thus, the results of the measurements of the considered values given in this section should be treated as approximate. Cloudless Sky. The simplest regularities in the variation of the effective radiation and downward atmospheric radiation are observed with clear skies. It has been stated that the effective radiation has a simple 24-h range, with a maximum at noon and a minimum just before sunrise. More numerous are the investigations of the variation of the effective radiation during the night, since at present only the nighttime radiation can be directly measured with sufficient accuracy. The results of observations show that, as a rule, a monotonic decrease in the effective radiation takes place during the night from the moment of sunset until the moment of sunrise. This can be seen, for example, in Table 9.1, compiled by Chumakova [36] from data of observations by means of the Savinov pyrgeometer at Karadag in 1938. Analogous results were obtained by Pivovarova [37] and other authors. According to Czepa [38], the effective radiation decreases during the night by 10 to 15 percent on the average. The data of Table 9.1 are the values TABLE 9.1 Variation of the Effective Radiation on Clear Nights according to Karadag Observations of I938 (cal/crn2rnin). After Churnakova [36]

True Solar Time Month

May June July August Average

19-20

20-21

21-22

22-23

23-24

24-1

1-2

0.141 0.131 0.139 0.133 0.136

0.139 0.131 0.136 0.130 0.134

0.136 0.126 0.130 0.128 0.130

0.134 0.123 0.133 0.125 0.129

0.134 0.122 0.129 0.123 0.127

0.130 0.120 0.131 0.120 0.125

0.128 0.116 0.132 0.113 0.121

9.5. Effective Radiation of the Underlying Surface

56 1

of the effective radiation of the instrument's blackened receiving surface. Comparison of their range with the variation in the instrument's temperature reveals rather high correlation. This result confirms the fact that the main cause for night variation of the effective radiation is the variation of the temperature of the radiating surface. A somewhat more complex variation of the effective radiation during the night was observed by Yaroslavtzev [39] and Beletzky [40]. According to Yaroslavtzev, the nightime range of the effective radiation at Tashkent, measured with the help of the Angstrom pyrgeometer, averages two maxima. The first, the evening maximum, approximately coincides with the end of the astronomical twilight (h, = -16.5'); the second, the morning maximum, is observed at the beginning of the civil twilight (h, = 5.5'). Two maxima in the nocturnal variation of the effective radiation were also observed by Beletzky at Odessa. The unmonotonic variation of the nocturnal effective radiation was likewise revealed in the theoretical calculations performed by Yegorenkova and Galperin [41] from data of frequent balloon soundings entered in the Shekhter radiation chart. The diurnal range of the effective radiation has been studied mainly by the indirect determination of these values from measurement data on the net radiation and the global radiation. The observations by Pivovarova [31], Eisenstadt and Zuyev [42] and others revealed that the daily range of the effective radiation was slightly asymmetrical relative to noon: The maximal values were observed between 12 and 14 h local time. The highest maximum of 0.44 cal/cm2 min was observed by Eisenstadt and Zuyev in a clearing among desert halaxylon. We see that the effective radiation is dependent on a number of factors, such as the temperature of the underlying surface and the air, and the total water vapor content. The most important of these that determines the daily cycle of the effective radiation is, as was shown by Berland [43], the difference in temperature between the underlying surface and the air at the level of the meteorological observation shelter. The variation of the downward atmospheric radiation during the day is likewise simple. Figure 9.5 presents a curve of the daily variation of radiation quantity as observed by this author and Holm [44] with the help of a special radiometer designed for direct measurement of the atmospheric emission during the day. The two upper curves of Fig. 9.5 characterize the variation of temperature and absolute air humidity at 2-m elevation. As can be seen, the downward atmospheric radiation increases during the first half-day with clear skies. At this time a good correlation is observed between this quantity and the temperature and humidity of the air. The

562

Thermal Radiation of the Atmosphere a q/m3

- 70

20 -0.6

- 65

1816-

- 60

05, 14-

- 55

1210-0.4

-. -. 2 --- 3 1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

7 8 9 10 I1 12 13 14 15 16 17 18 19 20 21 22 2324 HOUR

FIG. 9.5 Daily variation of the atmospheric downward radiation ( I ) , air temperature (2), and the absolute air humidity (3) at the 2 rn elevation according to observations at Koltushi (Leningrad region) on May 15, 1957.

maximum of the downward atmospheric radiation is very “flat”; The decrease occurs only in the evening hours. Other observations and calculations also show that an increase in the downward atmospheric radiation is usually observed in the morning, with a decrease in the afternoon. The maxima are observed some time after noon. The direct measurements of the downward atmospheric radiation realized by Gayevsky [45] and by Paulsen and Torheim [46] show that in winter (with snow cover) this quantity is practically invariable over 24 h. To illustrate the character of the 24-h variation of the downward atmospheric radiation, the desert observational data by Eisenstadt and Zuyev [42] in Table 9.2 can be used. TABLE 9.2 Variation of Radiation Observed under Desert Conditions. After Eisenstadt and Zuyev [42]

Time (h) 0-1 1-2 2-3 3-4 4-5 5-6 6-7 7-8 8-9 9-10 Go(cal/cmzrnin) 0.50 0.50 0.50 0.50 0.50 0.49 0.44 0.47 0.52 0.58 Time (h) 10-11 11-12 12-13 13-14 14-15 15-16 16-17 17-18 Go(cal/cm2min) 0.62 0.64 0.64 0.64 0.63 0.62 0.58 0.56 18-19 19-20 20-21 21-22 22-23 23-24 Time (h) Go(cal/cm2min) 0.56 0.54 0.53 0.53 0.53 0.51

9.5. Effective Radiation of the Underlying SllrEace

563

We see that, according to the above authors, the maximal emission Values are 0.64 cal/cm2 min and the amplitude of variation is 0.2 cal/cm2min. min. Somewhat higher maximal values were obtained by Kirillova and Koocherov [47] from observations over a cotton field near Tashkent in July 1952. It appears that the values of the order of 0.65 to 0.70 cal/cm2min are to be considered as the highest maxima. Analysis of observational data on the effective radiation reveals an annual variation. This can be seen from Table 9.3, which presents the annual range of the daily totals of effective radiation at various points as summarized by Evfimov [48, 491. Although at present these data should be regarded not quite reliable, they are sufficiently correct in the qualitative characteristic of the daily variation of effective radiation. The mean daily totals of the effective radiation vary in the considered points from 128 to 252 cal/cm2day. The character of this variation differs, depending on the location. For instance, at Irkutsk it is simple, and has a maximum in April and a minimum in January. The other points show a more complex variation with several extremes. It will be explained further on that the main causes for the annual range in the effective radiation from a clear sky are the variations in temperature stratification and in the total water vapor content in the atmosphere. Table 9.4 gives data characterizing the annual variation of mean monthly totals of the effective radiation for the same points of Table 9.3 and also for Tashkent and Poona (India). In the considered cases the monthly totals vary from 3.8 to 10 kcal/cm2 mo, with higher values at southern points. An increase in the effective radiation southward is observed for the annual totals as well, although nonmonotonic. The main cause for this increase appears to be the increase in temperature. The maxima in the annual range of the atmospheric emission from a clear sky are usually observed in summer and the minima are in winter.

Cloudy Sky. In the presence of cloudiness the effective radiation and downward atmospheric radiation values undergo considerable changes ; The former decrease and the latter increase, which is accounted for by the fact that clouds are a powerful source of thermal radiation. The variation of the effective radiation in dependence upon cloud amount can be seen from Table 9.5, compiled by Beletzky [40] on the basis of data obtained at Odessa by means of the Aganin condensation pyrgeograph. Since the condensation pyrgeograph is a very primitive instrument, the

TABLE 9.3 Annual Variation of Daily Totals of the Effective Radiation at Dgerent Points with a Cloudless Sky (cal/cm2day). Afrer Evfimov [48, 491

Point

Geographical Coordinates

Jan.

Feb.

Mar.

Apr.

May.

June

July

Aug.

Sept.

Oct.

Nov.

Dec.

3

e

E

: 5

Yakutsk

62'01'

129'43'

180

158

181

180

207

192

190

160

160

192

128

122

P

Pavlovsk

59'41'

30'29'

144

148

180

219

209

202

202

184

171

168

174

163

0

Irktusk

52'16' 104'19'

140

156

189

215

189

167

166

163

158

147

143

124

Karadag

44'54'

235

233

242

230

204

202

222

209

219

252

248

228

s ;P

Q 35'12'

2

565

9.5. Effective Radiation of the Underlying Surface TABLE 9.4

Annual Variation of Mean Monthly Totals of Efective Radiation with Clear Skies (kcalJcm2mo). After Evfmov [48, 491

Place Yakutsk Pavlovsk Irkutsk Karzdag Tashkent

Poona

Jan. Feb. Mar. Apr. May June July Aug. Sept. Oct. Nov. Dec. Year 5.6 4.4

5.6

5.4

6.4

5.8

5.9

5.0

4.8

6 . 0 3.8

3.8

61.5

4.5

4.1

5 . 6 6.8

6.7

6.1

6.5

5.9

5.2

5.2

5.2

5.0

66.8

4.3

4.4

5.9

6.7

5.9

5.0 5.1

5 . 1 4.7

59.8

6.3

6.5

7.5

6.9

6.3 6.1

6.9

4.7

5.2 6.2

6.4

6 . 8 7.3

9.5

8.8 10.0 9 . 4

7.8

7.6

4.6

4.3

3.8

6.6 7.8

7.4

7.1

82.9

7.0 7.1

6.9

6.3

5.8

5.3

75.1

6.9

6.2

7.5

8 . 4 9.0 98.0

6.5 6.9

obtained numerical values (Table 9.5) should be consdiered as only approximate. The causes for this variation will be treated later on. Note here that, according to Yaroslavtzev [39] and Sauberer [50], the effective radiation at a cloudless sky exceeds by 16 to 18 percent, on the average, that with an overcast sky. In practice it is difficult to take account of all the variety of factors determining the variation of the effective radiation in the conditions of cloudiness. At the same time, observations give evidence that the total cloud amount is the main factor of these. In a number of investigations attempts have been made at empirical determination of the dependence of the effective radiation upon cloud amount, and findings indicate that this dependence is nonlinear. For example, from observations conducted at Karadag, Chumakova [5 1] obtained the following relation between the effective radiation F,,n and the total cloud amount, expressed in tenths: = Fo(l-

0.024n - 0.004n2)

(9.36)

where F, is the effective radiation of a cloudless sky. Observations of Barashkova [52] at the same place produced a simpler dependence (n in portions of unity): = Fo(1

- 0.70n2)

(9.37)

However, Barashkova concluded that in particular cases the influence of cloudiness on the effective radiation is essentially dependent on the temperature and humidity of the air. This finds expression in the fact that the relation F,,/F, has a marked annual range.

TABLE 9.5 Influence of Cloudiness on Efective Radiation. After Beletzky [40] Cloud Amount, in tenths

Effective Radiation, cal/cm2 min

Cloud Amount, in tenths

Cirrus

Effective Radiation, cal/cm2 min

Cloud Amount, in tenths

Altostratus

10 10 10 10 10 10 10 10 10 10 10 9 8 7 6 5 5 5 7 7 7 8

0.150 0.114 0.116 0.122 0.134 0.128 0.130 0.118 0.136 0.120 0.126 0.143 0.146 0.137 0.136 0.131 0.136 0.128 0.140 0.142 0.138 0.139

10 10

Cirrostratus 0.120 0.114

10 10 9 4 8 10 10 10 10 10 10

0.078 0.094 0.092 0.147 0.045 0.078 0.083 0.073 0.052 0.042 0.010 Stratus

10 10 10 10 10 10 10 10 10 10 10 10 10

0.013 0.110 0.023 0.033 0.043 0.047 0.015 0.018 0.023 0.015 0.010 0.005 0.000

Effective Radiation, cal/cm2 min

Cloud Amount, in tenths

Cirrostratus 10 10 10 10 10

0.104 0.109 0.118 0.116 0.098

Altocumulus 10 10 10 9 9 6 6

0.065 0.061 0.069 0.051 0.084 0.123 0.123

Effective Radiation, cal/cm2 min

Stratocumulus 10 10 10 10 10

0.040 0.029 0.010 0.018

Nimbostratus 10 0.019 10 0.020 10 0.018

Fractostratus

Stratus 10

0.005

10 10 10 10

0.023 0.005 0.000 0.008

567

9.5. Effective Radiation of the Underlying Surface

A different dependence, Fo,,(n), was found by Sauberer [50].According to this author, Fo,n = FO

-13.8

+ d566.4 - 0.01n2 - 0.224n

(9.38)

10

The atmospheric emission is similarly a nonlinear function of the total cloud amount. Thus, according to Boltz [53],

Go,, = Go(l

+ kn2.5)

(9.39)

where k' is a certain empirical coefficient, and n is the cloud amount expressed in portions of unity. Boltz stated that satisfactory results can also be obtained by using the simpler formula (9.40) Go,n= Go(l k'n2)

+

Analogous relations were obtained by Kreitz [54] and Sauberer [ 5 5 ] . Table 9.6 gives the coefficient k values determined by Boltz for different cloud types ( z is the number of measurements, and N is the number of days on which observations were conducted). Thus, according t o the data of Boltz, on the average the emission of a cloudy sky exceeds that of a cloudless sky by 22 percent. The greatest increase in emission is observed for the case of stratus clouds, and the least, with cirrus. According to Kreitz, the value k' averages 0.27. Barashkova [52] points out that the dependence of downward atmospheric radiation upon cloud amount differs at different values of absolute air humidity (at great humidity a linear dependence takes place). TABLE 9.6 Dependence of Atmospheric Emission on Type of Cloud at n Type of Cloud

Cirrus Cirrostratus Altocumulus Altostratus Cumulus Stratus Avg.

=

10 Tenths. After Boltz [53]

k'

Z

N

0.04 0.08 0.17 0.20 0.20 0.24 0.22

302 59 140 43 82 59 271 1

53 23 63 20 31 15 253

568

Thermal Radiation of the Atmosphere

The diurnal and annual variations of effective radiation and downward atmospheric radiation with cloudy skies is generally far more complicated than with clear skies. Obviously both quantities are particularly variable with variable cloudiness. The order of values and the character of the annual variation in effective radiation under average cloudiness conditions can be seen from the example of the observed monthly totals presented in Table 9.7 by Barashkova. Here the data on the effective radiation during the day were obtained from the difference between the total and the shortwave net radiations. For comparison, the monthly totals calculated by Evfimov (lettered E ) are also presented. Comparing these values we see that the observational data are essentially corrective in relation to the calculated results. As a rule, the Evfimov totals are excessive for winter and underestimated for summer. Examination of Table 9.7 shows that at all points the effective radiation has a marked annual range, with a summer maximum and a winter minimum. The analysis of observational data carried out by Barashkova demonstrated that the highest radiation values are always observed in the summer daytime and the least during winter nights. I t should be noted that the annual range of the nighttime values is very weakly expressed and does not exceed 0.04 to 0.05 cal/cm2 min. Much more marked are the annual variations in the daily effective radiation values. At Karadag, for example, the effective radiation measured at 12.30 h showed an amplitude of the annual range equal to 0.22 cal/cm2 min. The diurnal amplitude here inTABLE 9.7 Annual Range of Monthly Totals of Effective Radiation under Average Cloudiness Conditions (cal/cm2min). After Barashkova 1521 Point

Jan. Feb. Mar. Apr. May June July Aug. Sept. Oct. Nov. Dec. Year

Yakutsk(E)

4.2 3 . 7 4 . 6 3 . 9 3 . 8 3 . 9 3 . 5 3 . 3 3 . 4 3 . 3 2 . 7 2 . 7 43.0 0 . 9 1.1 1 . 9 2 . 8 4 . 7 5 . 2 5 . 6 4 . 4 3 . 3 2.0 1 . 3 0 . 8 34.0

Voyeikovo (Leningrad region) 1 . 5 2.1 2 . 6 3.1 4 . 2 4 . 5 4.2 3 . 3 2 . 8 2 . 9 1 . 6 1.0 32.8 Derkul (CentralAsia) 1 . 5 1 . 3 2 . 3 3 . 6 5 . 1 5 . 7 6 . 0 5 . 6 4 . 3 3 . 2 2 . 2 1 . 7 42.5 Karadag(E)

3.7 3.7 5 . 0 4 . 6 4 . 9 4 . 9 6 . 0 5 . 3 5 . 3 5 . 1 4 . 2 3 . 9 56.6 2 . 4 2 . 2 3 . 7 4 . 5 6 . 7 6 . 8 6 . 9 7.1 6 . 1 4 . 2 2 . 7 2 . 6 55.9

569

9.5. Effective Radiation of the Underlying Surface

creases from 0.01 to 0.02 cal/cm2 min in winter to 0.17 cal/cm2 min in summer. The calculations of the geographical distribution of effective radiation for the earth’s surface, performed by Budyko et a1 [56], showed that the geographical variability of the effective radiation is rather weak. The highest annual totals (80 kcal/cm2 yr) were observed in tropical deserts. Near the equator, both on land and at sea, the effective radiation is lower and amounts to about 30 kcal/cm2 yr. With an increase in latitude it increases t o 40-50 kcal/cm2 yr at 60’.

Eflectof Dry Haze, Smoke, and Fog. Dry haze or fog formed at the earth’s surface can affect the effective radiation and downward atmospheric radiation to a considerable degree. A marked variation of the effective radiation is also observed when the earth’s surface is enveloped with a layer of smoke. The importance of haze provoking a decrease in the effective radiation from a clear sky can be seen from Table 9.8, compiled by Evfimov [49] for Yakutsk. TABLE 9.8 Effective Radiation with a Clear Sky and Dry Haze. After Evfmov [49]

Without Haze

With Haze

Month

June

July August

No. of Cases

cal/min2cm

No. of Cases

cal/cm2min

29 29

0.139 0.139 0.119

6 13 14

0.096 0.116 0.087

45

In June the decrease of the effective radiation at the appearance of haze was 31 percent. In individual cases an even greater decrease of the effective radiation was observed when the formation of haze was accompanied by fog. A considerable variation of effective radiation is also observed in the event of fog without haze. In the same sense is smoke also effective. In some authors’ estimation the decrease of effective radiation by the smoke present in the air near the earth’s surface can vary from 15 to 20 to 70 to 80 percent. Berland and Krasikov [57] obtained a reduction of effective radiation by creating a smoke screen that varied from 0.009 to 0.021 cal/cm2 min, that is, from 12 to 27 percent of the effective radiation in a smoke-free atmosphere. The effect of smoke

570

Thermal Radiation of the Atmosphere

screening can be utilized as a protective cover of plants against frost, since the artificial reduction of the effective radiation slows down the fall of night temperature and can often prevent ground frost. In the U.S.S.R.this method is rather common as a preventive against frost and also as an agent to reduce the danger of afterheating due to solar radiation in the cases when freezing could not be prevented.

2. Empirical Formulas for Calculating Eflective Radiation and Atmospheric Emission. The available observational data do not allow any adequate characterization of the regularities of the spatial and temporal variability of effective radiation and downward atmospheric radiation. To determine these quantities, various methods of calculation are therefore used. In practice, empirical formulas are commonly applied, of which those most frequently used are the Angstrom and Brunt formulas. Let us consider these formulas, first for a clear and then for a cloudy sky. Clear Sky. Analysis of observational results shows that with a clear sky there is a rather close correlation between the effective radiation or downward atmospheric radiation and the air temperature and water vapor pressure at a height of 1.5 to 2 m. This can be seen, for example, from Fig. 9.5, borrowed Berland and Berland's work [58], which presents the dependence of averaged F,/60T4 values upon the water vapor pressure as found by different investigators. In all cases the qualitative character of this dependence is the same. The considerable systematic difference between the curves of Fig. 9.6 is largely due to the nonuniform methods of measurement. The recent measurements (see curve 7) give much lower values of effective radiation than before. The above correlation makes it possible to determine the effective radiation from a clear sky by means of the following general equation: F, = aT"f(e)

(9.41)

where f(e) is a function of the water vapor pressure. The downward atmospheric radiation can be similarly expressed. The empirical formulas of Angstrom and Brunt have the same structure as (9.41). The Angstrom formula for the effective radiation with clear skies is a follows:

Fv = oT4(A

+ Be-Ce)

(9.42)

where o is the radiation constant, T and e are the temperature and pressure of water vapor (in millimeters) at 1.5 to 2-m elevation from the earth's

9.5. Effective Radiation of the Underlying Surface

57 1

e,m rn

FXG. 9.6 The eflective radiation according to observations. (1) Angstrom (1916, California); (2) Angstrom (1933, Sweden); (3) Robitsch (1934, Germany); (4) Rahman (1935, India); (5) Evfimov (1938, Pavlovsk); (6) Chumakova (1947, Karadag); (7) Boltz and Falkenberg (1948, East Germany); (8) as in Table 9.10.

surface, and A, B, C are empirical constants. An analogous equation for the emission of a clear sky can be presented in the form

Go = r3T4(A1 - Be-ce)

(9.43)

where A, = 1 - A, whose validity is understandable in view of (9.3). The empirical constants of the Angstrom’s equation have been determined from numerous measurements of effective radiation and atmospheric emission and also by means of theoretical calculations. Table 9.9 briefly TABLE 9.9 Constants in Angstrom’s Formula

A

B

C

Year

Author and Method

0.21 0.25 0.194 0.23 0.22 0.200 0.180 0.21

0.26 0.32 0.236 0.28 0.148 0.181 0.250 0.174

0.069 0.069 0.069 0.075 0.068 0.070 0.126 0.055

1916 1929 1933 1935 1940 1947 1949 1959

Angstrom, experiment Angstrom, experiment hnstrom; experiment Raman; experiment Phillips; theoretical Chumakova; experiment Boltz, Falkenberg; experiment Knepple; experiment

572

Thermal Radiation of the Atmosphere

summarizes the determined constants. They are seen to be highly variable. This can be accounted for inadequate consideration of all physical factors influencing the effective radiation and downward atmospheric radiation and also by the difference in the applied measurement methods. The following constants are the most frequently used in climatological calculations : A = 0.194, B = 0.236, and C = 0.069. The values of Boltz and Falkenberg (see the last line of Table 9.9) should be considered as the most reliable, for they were obtained by using the vibration pyrgeometer, a modernized instrument. The empirical formula for calculating effective radiation proposed by Brunt is Fo = aT4(a - b

2/e)

(9.44)

where a, b are empirical constants. For atmospheric emision, instead of (9.43) we have

Go = oT4(a,

+b6 )

(9.45)

where a, = 1 - a. The constants in Brunt’s formula, obtained by different authors, are as variable as those of Angstrom’s. This can be seen from Table 9.10. The application of the Angstrom and Brunt formulas with suitably determined TABLE 9.10 Constants in Brunt’s Formula Obtained from Measurements a 0.45 0.66 0.42 0.53 0.57 0.376 0.448 0.39 0.355 0.34 0.305-0.395 0.47

b 0.056 0.127 0.051 0.061 0.095 0.043 0.064 0.058 0.055 0.039 0.040-0.078 0.065

Year

Author

1940 1926 1933 1935 1920 1947 1946 1952 1957 1957 1961 1961

Brunt Robitsch Angstrom Raman and Desay Asklef Chumakova Lutherstein and Chudnovsky Berland and Berland De Coster and Schiiepp Goss and Brooks Marshunova Montheit

573

9.5. Effective Radiation of the Underlying Surface

coefficients yields practically equal results. However, it is evident that Brunt’s formula, since it contains two instead of three arbitrary parameters, deserves preference over Angstrom’s. The empirical formulas were later generalized by taking into account the nonblackness of the earth‘s surface and a sharp variation of the nearsurface temperature with height, which can be considered as a temperature “jump.” By taking account of these two factors, the general equation (9.41) can be presented as

+ (To4- T ) ] = 6uTy(e) + 6 AFo

Fo = 6cr[Ty(e)

(9.46)

where 6 is the relative emissivity (absorptivity) of the earth’s surface, and To is the temperature of the earth‘s surface. Table 9.1 1 gives the corrections AF = 0(TO4- T 4 ) calculated by Berland and Berland [58] for different air temperatures T and A T = To - T. Further on in the current section will be shown that the introduction of corrections in (9.46) is well founded.

Ato to

15 0 15 30

2

4

6

8

10

0.01 0.01 0.02 0.02

0.02 0.03 0.03

0.03 0.04

0.04

0.05

0.05

0.05

0.06 0.07

0.04

0.05

0.06 0.07

0.08

In recent years a great number of new modifications of the Angstrom and Brunt equations have appeared (see, for example, [46, 59-68]). It has been suggested that the empirical formulas for effective radiation and downward atmospheric radiation be substituted by graphical and tabular calculations based on the use of the dependence of these quantities only on temperature and absolute (or specific) humidity near the earth‘s surface. Kovaleva [69, 701 plotted a radiation chart for this purpose. The radiation charts described in Sec. 9.3 (see, for example, [71] are also used.

574

Thermal Radiation of the Atmosphere

Berland and Berland [58] compiled a table of the values of the effective radiation as a function of air temperature and water vapor pressure. (Table 9.12) whose data are more or less satisfactory in agreeing with the available observational results. They are not sufficiently correct, as was shown by Efimova [72] at small values of water vapor pressure. According to Efimova [72], analysis of the measurement data obtained by means of the Yanishevsky pyrgeometer during the IGY at 24 Soviet stations leads to the following analytical approximation of the dependence of monthly means upon air temperature and humidity: Fo = ~ T ~ ( 0 . 2 54 0.0066e)

(9.47)

where the water vapor pressure e is expressed in millimeters and Fo in cal/cm2 min. Marshunova [28] showed that the data of Table 9.12 were exaggerating the effective radiation in the Arctic conditions by 30 to 40 percent. TABLE 9.12 Effective Radiation at Diferent Values of Air Temperature and Water Vapor Pressure. After the Berlands 1581

I

e, mm

c

to,

1 -20 -15 -10 - 5 0 5

10 15 20 25 30

0.11 0.12 0.13 0.14 0.15 0.16 0.17

2

0.12 0.13 0.14 0.15 0.16 0.17

3

4

0.12 0.13 0.14 0.15 0.16 0.17 0.17

5

0.12 0.13 0.14 0.15 0.16 0.17

6

0.13 0.14 0.15 0.16 0.17 0.18

7

8

1

0

0.12 0.13 0.12 0.11 0.14 0.13 0.12 0.15 0.14 0.13 0.16 0.15 0.14 0.17 0.16 0.15

1

0.11 0.12 0.13 0.14

2

1

5

0.10 0.11 0.12 0.10 0.13 0.11

Cloudy Sky. In reality the sky is almost always cloudy, in which connection the problem of the influence of cloudiness on the effective radiation or downward atmospheric radiation is of much importance. When using the empirical equations for the calculation of the considered

9.5. Effective Radiation of the Underlying Surface

575

quantities, it is general practice t o introduce corrections for cloudiness with the help of the relation (9.48)

Fo,n= F,(1 - cn)

where c is an empirical constant, and n is cloud amount. Since the coefficient c depends on the height and density of clouds some authors replace the averaged c with three different coefficients c z , c m , and c,, which characterize the effect of clouds of the lower, middle, and upper levels, respectively. In this case (9.48) becomes

+ cmnm + cun,)l

Fo,n = Fo[1 - ( C Z ~ Z

(9.49)

where nl ,nm ,and nu are the respective values of the low, middle, and upper cloud amount. It should be noted that the low clouds include Ns, St, Sc, Cu and Cb; middle clouds include Ac and As; and upper clouds, Ci, Cs, Cc. As a rule, the value F, in (9.49) is determined from the temperature and humidity of the air at the height 1.5 to 2 m. Therefore, in the presence of a considerable difference in temperature between the soil and the air, it is necessary to correct (9.49) for the temperature “jump” by adding the term 6a(TO4- T 4 ) to the right-hand side. Formulas for the calculation of downward atmospheric radiation from a cloudy sky can be written similarly to (9.48) and (9.49) as

Go,, = Go(l - c’n)

or Go,n = Go11

+ (cz‘nz +

Cm‘nm

(9.50)

+ cu‘fldl

(9.51)

The coefficients characterizing the effect of cloudiness on the downward atmospheric radiation are denoted in (9.50) and (9.51) by primed letters, since their values are obviously different from the corresponding coefficients in (9.48) and (9.49). Taking into account that F0,%= 6(Bo- Go,,) and Fo = 6(Bo - Go),and substituting these expressions for Fo,fiand Fo into (9.48), will obtain the following relation between the coefficients c and c‘: c‘ = C(% BO

-1)

(9.52)

It can be seen from this formula that the values of the coefficients c and c‘ are very much different. Usually, Go is 0.7 to 0.8 B,. Thus the coefficient

576

-

Thermal Radiation of the Atmosphere

+

c’ (0.25 0.40)~.The numerical relation between the coefficients c and c‘ is dependent on atmospheric stratification because (9.52), which deter-

mines the connection between these quantities, contains the downward atmospheric radiation Go. The formula for conversion from c’ to c is obtained by analogy with (9.52) and is c = c’(T 6B - 1) (9.53) The value 0.76 is often used as the mean coefficient c. The coefficients q , c, , and c, , as proposed by different authors, are summarized in Table 9.13, compiled by Galperin [73]. TABLE 9.13 Empirical Coefficients Characterizing the Influence of Cloudiness on Efective Radiation. After Galperin [73]

Investigator

Station

Defant Angstrom Asklef Dorno Evfimov Evfimov Evfimov Evfimov Lutherstein

Stockholm Stockholm Uppsala Davos Tikhaya Bay Schmidt Cape Pavlovsk Tashkent Tashkent

Ca

0.20 0.31,Cs 0.20 0.20 0.22 0.16 0.1, Ci 0.2-0.3, Cs and Cc

cm

0.77,As 0.63,As 0.59 0.57 0.52 0.50 0.4, Ac and Ac trans. 0.6, Ac and Ac op

CZ

0.86 0.90, Ns 0.83,Ns 0.85,Sc 0.84 0.81 0.76 0.67 0.16,Sc trans. 0.8, St and Ns

0.75-0.85,Cu 0.95-1.0,Cb

We see that the coefficient c, averages 0.2; the value c, varies from 0.5 to 0.6, and cz varies over a wider range from 0.6 to 1.0. Since the dependence of the coefficients c upon type of cloud turns out to be significant, it becomes important to take differentiated account of the dependence of effective radiation and atmospheric emission upon cloud amount over separate levels of cloudiness. The introduction of the coefficients c or c’ for the total cloud amount may involve considerable calculational errors.

577

9.5. Effective Radiation of the Underlying Surface

Table 9.14 gives the values of the coefficients cz , c , , c, as determined by Berland and Berland [58]. The average coefficient for all cloudiness is calculated from

In magnitude, the coefficient E is close to c m . Calculations show that the cloud amount of the upper layer, n u , has little effect upon E. The simplified (9.48) has been obtained for such cases when the cloud amount is not differentiated over the levels of cloudiness. TABLE 9.14 Mean Empirical Coefficients Characterizing the Influence of Cloudiness on Effective Radiation. After the BerIands [ 5 8 ]

Latitude, deg Over 60 60-50

50-40

Half-Year

cz

Cm

ca

E

Cold Warm

0.90 0.86

0.77

0.28

0.82

0.72

0.27

0.80

Cold Warm

0.86

0.74

0.27

0.77

0.80

0.67

0.24

0.70

0.82

0.69 0.65

0.24

0.71

0.19

0.69

Cold Warm

0.78

Berland and Berland [58] suggest the calculation of the effective radiation in real cloud conditions from the clear sky values (Table 9.12), with a subsequent cloud correction by means of (9.48) or (9.49), using the coefficients given in Table 9.14. In the presence of a temperature jump near the underlying surface, it is possible to introduce a corresponding correction from Table 9.1 1. Efimova [72] corrected the data of Table 9.1 1 (see (9.47)) and proposed the below formula for calculating mean monthly totals of effective radiation (kcal/cm2 mo) :

F,

= 6aT4(11.7 - 0.40e)(l -

cn)

+ 46aT3(T0 - 7')

(9.54)

where e is the mean monthly pressure of water vapor in millibars, and T is the mean monthly air temperature (OK), 6 = 0.95. In the preceding section it was stated that the dependence of effective radiation or downward atmospheric radiation upon cloud amount was

578

Thermal Radiation of the Atmosphere

observed to be nonlinear. Therefore the empirical formulas considered above, which are intended to take account of the cloud effect, are incorrect, strictly speaking. Budyko et al. [56] calculated the coefficient c in the following equation for effective radiation: FO,%= Fo(1 - cn2)

+ 460T3(TO- T )

(9.55)

The latitudinal variations of this coefficient, as investigated by Budyko, Berland, and Zubenok, are given as Latitude, deg: C:

Latitude, deg: C:

75 0.82

70 0.80

65 0.78

60 0.76

55 0.74

50 0.72

45 0.70

40 0.68

35 0.65

30 0.63

25 0.61

20 0.59

15 0.57

10 0.55

5 0.52

0 0.50

According to Kirillova [74, 751 the atmospheric emission can be calculated from the equation

Go,, = Go(l

+ c’nl.*)

(9.55a)

Although the dependence of effective radiation and downward atmospheric radiation on cloud amount is nonlinear, calculations show that by using the linear dependence with the appropriate empirical coefficients, it is possible to obtain practically identical results, especially in calculating radiation totals over longer periods such as a month or a year. A more serious problem arises from the fact that the coefficients c are strongly variable from month to month. Barashkova [52] calculated that the monthly means of the ratio Fo,JF0 can vary by two to three times in comparison with the annual mean. The use of mean annual c as given above should therefore be considered as a rough approximation. Note here that when calculating nonaveraged values (as in the case of the daily variation), accounting for the nonlinear relation between the effective radiation and downward atmospheric radiation and the degree of cloudiness can become quite important. In this case the use of empirical formulas, in general, has no sound basis, since such formulas characterize only the statistical connection between the effective radiation and downward atmospheric radiation and the values of temperature, water vapor pressure, and cloud amount. All the considered equations express the effective radiation from a cloudy sky on the basis of measurements made for clear sky radiation. It is ob-

9.5. Effective Radiation of the Underlying Surface

579

vious, however, that this method of calculating the effective radiation (or atmospheric emission) is, generally speaking, arbitrary. In this connection the empirical equation for calculating atmospheric emission presents some interest [16]. It is

where T is the air temperature a t a height of 2 m ; no and nz are the total cloud amount and the lower level cloudiness, respectively; cz and c, c, are the empirical coefficients describing the effect of cloudiness on the emission; and A is a coefficient of emission with cloudless skies. All empirical formulas can be used only for the calculation of average values because, with a clear sky, only average values Go/oT4and FoIaT4 can be considered unique functions of the water vapor pressure. Individual values of these ratios at the same water vapor pressure can vary over a wide range. This can be seen in Fig. 9.7, which presents the dependence of Go/aT4(in percent) on the water vapor pressure (in millimeters) as observed by Schmidt [77] at Warnemunde (DDR) with the help of a vibration pyrgeometer. Open circles indicate the relative values of atmospheric emission observed under cloudless conditions, and dots, represent the same values for a cloudy sky. The interrupted curve characterizes the dependence of G0/oTo4on e, calculated from the Angstrom formula with the coefcients given by Boltz and Falkenberg: A, = 0.820, B = 0.250, C = 0.126. The solid curve represents this relationship with the use of Angstrom coefficients: A, = 0.806, B = 0.236, C = 0.069. As seen, the dispersion of the points around the curves calculated from the Angstrom formula is quite great. It is also found that the use of Angstrom’s coefficients systematically underestimates the relative atmospheric emission. The curve of Boltz and Falkenberg is in better agreement with observations. The statistical nature of the empirical formulas is also confirmed by the fact that only when considering the averaged values can the effective radiation or downward atmospheric radiation from a cloudy sky be sensibly expressed in terms of the clear sky values. It is obvious that for any isolated case this relationship is meaningless. It has been shown above that the radiative heat fluxes in the atmosphere are considerably dependent on atmospheric stratification, that is, on the nature of the vertical distribution of temperature and humidity. Hence it follows that neither the effective radiation nor downward atmospheric radiation can be a unique function of either the air temperature and water vapor pressure near the earth’s surface, or of the cloud amount. The cal-

+

580

Thermal Radiation of the Atmosphere

culation of the thermal radiant fluxes, taking into account the atmospheric stratification, can be realized on the basis of the approximate analytical and graphical methods described in the beginning of this chapter.

-

0

0

- 2 .

-

.

s

.

95-

.

-

90

..

--

0

-

0

0 0

O m

CS

DO.

...

.cs

/

/

70

1

/ /

.

/

0 7

e,mm

FIG. 9.7 Dependence of the relative atmospheric downward radiation on the pressure of water vapor.

I, clear sky; 11, with cloud.

3. Comparison of Empirical and Theoretical Formulas for Calculating Eflective Radiation. We considered earlier the empirical and theoretical equations for the calculation of the effective radiation of the underlying surface under cloudless conditions. Now compare, following this author’s work [78], these formulas and attempt to perform a theoretical analysis of the empirical equations and a subsequent determination of the limits of their applicability.

9.5. Effective Radiation of the Underlying Surface

58 1

Let us use the following common version of the Angstrom equation as an empirical example :

Fe = 0.95[0T4(0.194

+ (0.236 x

+

10-0.069e)) a(TO4- T411 (9.57)

The symbols in this formula are the same as those used earlier. The approximate theoretical formula for calculating the effective radiadiation, obtained by taking account of the surface layer, reads as (see (9.30))

Remember that the positive sign in this equation corresponds to the case of the superadiabatic temperature gradients, and the negative sign to the case of inversions. Both (9.57) and (9.58) consist of two terms whose role is perfectly different. The first term of both equations represents the effective radiation in the conditions of a “normal” atmosphere (the vertical temperature gradient is constant from the underlying surface to the tropopause). The second terms give a correction corresponding to the rapid change in temperature (temperature discontinuity) which takes place at the dividing barrier earthatmosphere. The absorption function can be roughly presented as

1 - A(w,)

-“

0.25e-kiWm

(9.59)

The accuracy of this approximation directly depends on the value w, (that is, on the atmospheric moisture content). By taking into account (9.59), now compare the first terms in (9.57) and (9.58). As can be seen from the theoretical formula, the main physical quantities that determine the effective radiation are the temperature and emissivity of the earth’s surface, the vertical temperature gradient in the atmosphere, and the total content of atmospheric water vapor. The value of the effective radiation is also dependent on the form of the absorption function. Let us now analyze (9.57) in order to estimate its accuracy in taking account of all these factors. One can first come to the conclusion that the Angstrom formula takes a correct account of the effect of the emissivity of the earth’s surface. The value 6 = 0.95 is not, of course, universal. Although the first term of Angstrom’s equation contains the air temperature and not the temperature

582

Thermal Radiation of the Atmosphere

of the soil, it should be noted that this fact is not essential, since To in the theoretical formula represents the soil temperature in the “normal” atmosphere when To T with a high degree of accuracy. As seen from (9.57), it does not contain any values characterizing the thermal atmospheric stratification, although it is clear beforehand that the effective radiation must be dependent upon this factor. Let us clarify at what value of the vertical temperature gradient the angstrom formula can be expected to yield best results. Determine this value from the equality of F, and Ft , assuming the following values for the parameters contained in (9.57) and (9.58): e = 7 mm, w, = 1.5 g/cm2, H = 10 km, To = T = 290°K, and c1 = 0.6. The calculation shows that the obtained optimal value of the vertical temperature gradient 7 = 5.S0/km. At y < 7, the formula (9.57) will exaggerate, and at y > y it will underestimate, the effective radiation. This evaluation is practically constant for the range of the water vapor masses of the order 1 to 2 g/cm2. It must be mentioned that Evfimov [48], investigating the angstrom formula by means of statistical analysis of observational data, concluded that p = 6O/km. Thus Evfimov’s estimate practically coincides with the theoretical value obtained above. Consider now what schematization of the thermal radiation absorption spectrum in the atmosphere corresponds to the Angstrom formula. Note first that in the case when the vertical distribution of the absolute humidity is described by a simple exponential function, the atmospheric water vapor mass w is proportional to the absolute humidity at the earth’s surface, e. Taking this into account, (9.57) and (9.58) can be rewritten, omitting the second terms :

-

F, = 0.95aT4(0.194

+ 0.236e-0.8w-)

(9.60)

(9.6 1 )

Comparing (9.60) and (9.61) we find the absorption function that corresponds to Angstrom’s formula as follows :

1 - A(w,)

=

0.194

+ 0.236e-0.8wm

where the denominator must be calculated at the values of To and H used

9.5. Effective Radiation of the Underlying Surface

above, and

p

=

583

5.8'/km. The calculation leads to the following expression:

1 - A(w,)

= 0.138

+ 0.169e-0.8w~

(9.62)

Thus the Angstrom formula corresponds to such a schematization when in a certain spectral interval involving 13.8 percent of the total flux of the blackbody radiation at the temperature of the earth's surface there is a complete transparency, while in another interval including 16.9 percent of the total radiation, the absorption is described by the exponential function and the absorption coefficient equals 0.8 cm2/g. It is clear that such absorption function does not correspond to reality. In the infrared spectrum of water vapor absorption there is no region of complete transparency but a region of weak absorption, which is quite important in the calculation of the effective radiation. This region of weak absorption, including about 25 percent of the total blackbody radiation, is represented in the theoretical equation (9.58). It is obvious that a similar comparison of the theoretical formula with the Brunt equation will lead to the same qualitative conclusion except for the exponential absorption function in this case being substituted by the absorption function in the form according to the square root law. Let us pass now to the analysis of the second terms in (9.57) and (9.58) whose physical meaning was explained in the beginning of the current chapter. Rewrite both terms as follows : 4

Ft

=

f 6a0T;~

x [Ei(-kjew&) - E ~ ( - I ~ ~ Q ~ , E(9.63) ~)]

j= 1

AF,

= 0.95

x 4 d 3 AT

(9.64)

where AT = To - T. The equations (9.63) and (9.64) are completely similar in structure if we assume that the value A

$ a 2 [Ei(-kje,b) - Ei(-kje,,,~,)] = AT j= 1

is equivalent to the temperature discontinuity at the barrier earth-atmosphere. Determine this value numerically at the following parameters : ew0= 7.10-6 g/cm3, E~ = cm, a = 0.226, h = 50 m, k j , according to the data of Chapter 3. The assumed values a and correspond to the difference To - Th = 4'. The calculation gives the value sought, AT = 3.8'. This value practically

584

Thermal Radiation of the Atmosphere

coincides (the permissible error being about I ") with the temperature difference soil-shelter used in (9.57), which gives evidence of the fact that taking account of the dependence of the effective radiation upon the temperature discontinuity in the empirical equation (9.57) has a sound physical basis. Furthermore, it can also be concluded that in the theoretical formula (9.58), the effect of the temperature jump can be accounted for in a simpler way, as in (9.57). The obtained result can be grounded analytically as well. Making use of the expansion into a series at low argument values for the integral exponential functions contained in (9.63), we obtain

h AT? aln-=To-Th "0

However, the temperature difference To - Th (with an accuracy of about 1") coincides with the temperature difference soil-shelter, since the main part of the temperature change takes place in a thin air layer of several tens of centimeters thick adjacent to the earth's surface. Thus, the analysis of the Angstrom empirical formula (9.57) for a clear sky shows, that it is correct in its accounting of the factors that determine the effective radiation magnitude, such as the emissivity of the soil and the temperature jump soil-air. At the same time this formula is inadequate with respect to the influence of radiation absorption, which is quite important in this case, and fails to take into consideration the thermal stratification of the atmosphere In general, the Angstrom equation at w 1 to 2 g/cm2 yields satisfactory results only for the case when the vertical temperature gradient y ,- 6"/km. What has been said gives preference to the theoretical formula (9.58) as being more adequate in representing the true aspects of the phenomenon considered.

-

9.6. Angular Distribution of Intensity of Effective Radiation and Atmospheric Emission over the Celestial Sphere The description of the field of the thermal atmospheric radiation usually reduces to the examination of radiant fluxes. Meanwhile the complete quantitative characteristic of the radiation field can be obtained only with the known spectral composition and the state of polarization of the radiation and the angular distribution of radiant intensity. Since the thermal radiation is unpolarized, there is no problem of determining the state of polarization in this case. For many practical purposes, neither is it necessary to determine the spectral composition of the thermal radiation, so that our

9.6. Angular Distribution of Intensity of Effective Radiation

585

task reduces to the investigation of the integral radiant intensity. It should be stressed, however, that such limitation is not permissible for all cases because the problem of the energy distribution in the spectrum of thermal atmospheric radiation is undoubtedly actual. The present section aims at exposition of the theoretical and experimental results on the intensity distribution of the effective radiation of the earth’s surface and of the atmospheric emission over the celestial sphere in different directions with respect to the vertical, as treated in the work of this author and Yelovskihk [Chapter 1, Ref. 4; 791. This problem, besides being of scientific interest, also holds considerable practical interest for studying the net radiation of plane nonhorizontal or horizontal surfaces screened by various obstacles, and of surfaces with complex configurations. 1. Efective Radiation. Although quite a number of experimental investigations is dedicated to the distribution of the effective radiation over the sky, all are of a chance nature and (which is even more essential) have been performed without parallel aerological soundings. The latter fact minimizes the validity of such experiments because it is impossible to use their results in checking the correctness of the theoretical conclusions. Until recently only the empirical formula of Linke has been used for calculating the intensity of effective radiation over different directions. Such calculations can also be performed with the help of radiation charts. However, the use of radiation charts for this purpose demands great eflort and cannot secure the immediate analysis of those factors that influence the effective radiation intensity. The theoretical solution of the problem of the intensity distribution of effective radiation over the sky was first given in the works by this author [SO-821. Lonnquist’s theoretical investigations [83, 841 were published at a later date. Let us consider how the distribution in question can be obtained theoretically. It will be shown later that the theory of this problem divides into two parts corresponding to the cases of the clear and cloudy sky. We examine both cases separately.

Clear Sky. Let us use (9.4a) to calculate the flux of the thermal atmospheric radiation in the direction of the zenith within the solid angle corresponding to a conic apex angle of 28. For the transition from intensity to flux we have the following general equation: (9.65)

586

Thermal Radiation of the Atmosphere

where Gi,e(z) designates the flux at the height within the given solid angle. Using (9.4), instead of (9.65) we obtain

-k, sec 6

1'" e(C)

d S ] } d6 dq

(9.66)

It is easily seen that (9,66) can be rewritten as

where E,(x) = Jy t-ne-zt dt are certain transcendental functions (in this case n = 2). Introducing the new variable t = sec8, we obtain

=

1,""

t-z{exp[ -tk,

lq e(S)

d S ] } dt

(9.68)

Thus, in order to justify (9.67), it is necessary to prove the validity of the following identity :

Jet W

t-2 e-r,xt dt

= cos

0

IW t r 2 e-rA 1

= cos

0 E2(z,sec 0)

where

Introducing the variable u

=

-zAt, we have

8t

dt

587

9.6. Angular Distribution of Intensity of Effective Radiation

After integration by parts, we obtain

J=co~O[e-~~~~~+t~secOEi(-t~secO)] where Ei(-x) =::J t-'8 dt is the integral exponential function. The functions Ez and Ei are, however, connected by the relation E2(x)= e-z

+ x Ei(-x)

Thus, J = cos 0 E 2 ( t ,sec 0 ) as required. Now perform some transformations of (9.67). First substitute the variable w = J: e(5) d5 for z and rewrite (9.67) for the ground level (z = 0, and correspondingly w = 0) as follows:

Let us integrate the last formula by parts, taking into account the following identity:

Considering E3(0)= 1/ 2 , we have G,,e(O) = B,(T(O)) (1

- c0s2 0 ) - Ba(T(wm))[2E3(kiwJ

- COS' 0 - 2E,(kaw, sec 0 ) ww

+J0

dB

-

2E3(kl,p)- cos2 0 2E3(kfi sec 0)] d p

(9.69)

The function P,,,(w) = 2E3(kaw) is a function of monochromatic flux transmission. Therefore (9.69) can be rewritten as

(9.70) Integrate the last expression over all wavelengths from 0 to co according to the method suggested by Ambartzumian and applied to the earth's atmosphere by Lebedinsky (see Sec. 9.1). For this purpose first integrate

588

Thermal Radiation of the Atmosphere

(9.70) over all wavelengths satisfying the following inequality : k

< k A< k

+ ak

(9.71)

and obtain

wm

dB

Pk,+)

- cos2 0 Pk,&u

sec @ ) ] f ( k dp )

(9.72)

These designations mean

The function f ( k ) in (9.72) is determined from the relation

where the integration extends over all wavelengths satisfying the inequality (9.71). Further integration of (9.72) over all k from 0 to 00 (that is, over the whole spectrum) gives the following expression for the full (integrated) flux of thermal atmospheric radiation within the considered solid angle : G(0, 0 ) =

p)Gk,@(0)dk 0

= B(T(0)) sin2 0

- B(T(w,)) [PF(w,) - cos2PF(w, sec @ ) I (9.73)

sow

where PF(w) = PkSF(wlf(k) dk is the integrated transmission function for the radiant flux. Assuming that Kirchhoff's law is fulfilled at ground level, we write U(0,O) = 6B(To)sin2 0

+ (1 - 6)G(O, 0 )

(9.74)

9.6. Angular Distribution of Intensity of Effective Radiation

589

where U(0,O) is the upward long-wave radiation flux within the given solid angle (at z = 0), B(To)sin2 0 is the blackbody radiation at the temperature of the earth’s surface To in the same solid angle, and 6 is the relative emissivity (absorptivity) of the earth’s surface. On the basis of (9.74) we obtain for the effective radiation of the earth’s surface : F(0, 0)= U(0, 0)- G(0, 0) = 6[B(To) sin2 0

- G(0, O)]

or, using (9.73), F(0, 0 )= 6B(T,) [PF(wm) - cos2 0 Pp(wmsec O)] ww

-6

lo

dB

[PF(p)- cos2 0 Pp(p sec O)] dp

(9.75)

where the notation B(T(w,)) = B(T,) has been introduced for brevity. By taking 0 = 4 2 , we obtain for the effective flux Fo of a hemisphere: Fo = F(0,

+) (9.76)

Making use of (9.75) and (9.76) we have

[

B(T,)PF(wm sec 0)-

F(0,O) = Fo (1 -cos2 0 )

B(T,)PF(%J -

rw

1

(dB/dp)P,(p-sec 0) dp

yw(dB/dp)P,(p)dp 0

(9.77)

It is easy to see that if the integration over 8 is omitted, the following expression is obtained, connecting the intensities of effective radiation in the direction of the zenith angle cfe) and in the direction of the zenith

UO): E(T,)PJ(wWsec 0) fe

=fo E(Tw)P,(w,)

where E = (I/n)B,PJw) transmission function.

=

-

rm

(dE/dp)P& sec 0) dp

s’”” (dE/dp)P,Cu)dp

(9.78)

0

some-kwf(k)dk

is the integrated intensity

590

Thermal Radiation of the Atmosphere

Equations (9.77) and (9.78) are exact expressions for the distribution of the flux and intensity of effective radiation over the clear sky. The structure of these formulas is such that they are the best fit for graphical calculations with the help of the coordinate system (PF ,B) or (PJ, E). However, as has already been mentioned, the calculation from radiation charts is fairly complicated and exclusive of the immediate analysis of the various factors influencing the intensity or flux of the effective radiation. On the other hand, the application of (9.77) and (9.78) for numerical calculations also demands prolonged efforts. In this connection our task consists in simplifying the above formulas in order to obtain approximate theoretical equations suitable for practical calculations. To accomplish this, we may use the approximate method for calculating the integrals of (9.77) and (9.78) suggested by this author. It was shown in Sec. 9.2 of the current chapter that the expressions in the numerator and denominator of (9.77) can be presented in the form

where c is a constant, H is the height of the tropopause, and I dB/dz I is the mean vertical gradient of the function B = aT4 in the troposphere. For the calculation of the fluxes of effective radiation or atmospheric emission, the transmission function PF(ww)is determined as follows :

PF(w,)

+ 0.1 le4.8wm

= 0.25e-0.166ww

(9.80)

Making use of (9.79) and (9.80), instead of (9.77) we obtain

Equation (9.78) can be transformed in a similar way except that in this case it is necessary to use the intensity transmission function. All this considered, instead of (9.78) we obtain

Calculations show that for approximate estimations of the relative in-

9.6. Angular Distribution of Intensity of Effective Radiation

591

tensity of effective radiation, this formula can be substituted by the simpler expression F~ =foe-kw,(sec e-1) (9.83) where k is a numerical coefficient representing an average absorption coefficient for the spectral interval, including the transparency window and the adjacent regions of weak absorption. It can be seen that the use of (9.79) results in derivation of fairly simple approximate equations for the distribution of the flux and intensity of effective radiation over the sky. The main physical meaning of these results consists in that the relative fluxes F(0, O)/F(O) and intensities f ( 0 ) / f o of effective radiation are functions only of the zenith distance and the total water vapor content in an atmospheric column of unit section. According to (9.82), the ratio of the intensities of effective radiation in the direction of the zenith angle 0 and the direction of the zenith is identical with the ratio of the transmission functions for the corresponding directions. It is easily understandable that the last result is natural for the isothermal atmosphere. In fact, taking account of Kirchhoff's law, for the intensity of atmospheric emission in the direction of the zenith angle B(ge) and the direction of the zenith we can write ge

U

=

[1 - P,(w, sec e ) ] -T4 n

Hence (9.84) For the effective radiation intensity we have

fe=6

(-

U

7c

T4- go

=

6PJ(w, sec 0)

U

-T4 76

Thus we obtain the following expression, identical with (9.82): (9.85) The above examination of Eq. (9.82) shows that, following the law of

592

Thermal Radiation of the Atmosphere

distribution of the relative intensity of effective radiation, the real atmosphere can be considered as “quasi-isothermal,” but this should be accepted with caution because the conclusion is not valid for all cases, since (9.79) satisfies only the condition when there is a decrease of temperature with height in the atmosphere. For inversions, this relation is incorrect and must be replaced by a more general one. Thus the applicability of (9.81) through (9.83) is restricted to the case where temperature decreases with height. The invalidity of these formulas for inversion conditions is seen already in their examination. With a sufficiently intense inversion and large 8, the inequality fe < 0 can take place, and consequently &Ifo < 0, while from (9.82) and (9.83) it follows that f& > 0 always. Very simple formulas for the relative intensity of effective radiation for the case of surface inversions of any intensity can be obtained by means of the approximate method of calculating the effective radiation in inversion conditions, suggested by this author and Vasilyeva [85]. In this case, for the effective radiation intensity directed zenithward we have

This the temperature at the upper boundary of the where Eh = (o/n)Th4, inversion, AT = Th - To is the temperature “intensity” of inversion, and a, and b, are constants. The expression for& can be presented by analogy with (9.86) as

For the relative intensity of effective radiation we now have (9.87) where a = aJ6, b = bl/6. In the finite case, AT+ 0, this equation becomes identical with (9.85). Let us conclude at this point the theoretical examination of the distribution of the intensity and flux of effective radiation over the clear sky and pass on to the analysis of observational results. This author and Yelovskikh [Chapter 1, Ref, 4; 791 observed the effective radiation intensity distribution over the sky during cloudless nights of June-July, 1953, at Voyeikovo (Leningrad region). The measurements were conducted by means of two radiometers whose receiving surface, placed at the bottom of a thick copper cylinder, is a thermopile of 20 manganin-

9.6. Angular Distribution of Intensity of Effective Radiation

593

constantan couples. The receiving inlet is a slot with aperture angle of 15' width. Inside both instruments there are four diaphragms. One of the instruments was covered by a KRS-5 filter fixed at 53 mm from the receiving surface. The thermocurrent was measured with the help of a mirror galvanometer with small inside resistance (20 Q) and sensitivity A. The use of two instruments with and without filter was meant to check the effect of wind on the readings of the unprotected instrument. It was shown that the deviations of the exposed instrument were unstable at the wind speed of 4 to 5 m/sec and more. The filtered instrument gave stable deviations at any wind speed. The relative values of the effective radiation intensity measured by both instruments were identical except in the case of strong wind (the corresponding sets of unstable readings from the exposed instrument were neglected). The evaluated relative error in the measurement of the effective radiation intensity was for both instruments about *3 percent. The above theory is based on assumption of the horizontal homogeneity of the atmosphere. This accounts for the fact that the effective radiation intensity described by the theoretical formulas is dependent on the zenith distance and independent of the azimuth of the sky section considered. Since the atmosphere is not horizontally homogeneous, it is essential to state whether the actual inhomogeneity of the cloudless atmosphere affects the radiant intensity. For this purpose the radiant intensity was measured for the zenith angles 20°, 40°, 70°, 80°, and different azimuths (with 50 to 70' intervals over the azimuth). The measurements showed that the azimuthal variation deviates slightly from the limits of the measurement errors. However, it is still small enough to be negligible. Figure 9.8 gives the results of 18 sets of measurements for the relative intensity of effective radiation by means of a filtered instrument. Usually the instruments were directed eastward or toward northeast. The readings were taken every loo of the zenith distance and at 8 = 55'. The curves of this figure were obtained by calculating, with the help of (9.82), two extreme values of the total water vapor content in the atmosphere: w, = 1.57 g/cm2 and w, = 3.75 g/cm2 corresponding to the engaged 18 series of observations. The total water vapor content in a column of atmosphere of unit section was calculated from the results of aerological soundings up to the heights of about 25 to 30 km. Since the time of soundings and observations was seldom simultaneous, the data of aerological soundings had to be interpolated. As seen from Fig. 9.8, the finite curves are fairly satisfactory in limiting the region of the measured values of the effective radiation intensity. An

594

Thermal Radiation of the Atmosphere

8,deg

FIG. 9.8 Relative intensity of the effective radiation according f o 18 sets of measurements with a filtered instrument. (1) observational results; (2) results of calculations from Eq. (9.82) at w, = 1.57 g/cma (upper curve) and w, = 3.75 g/cm2 (lower curve).

even better coincidence of the observed values and those calculated from (9.82) is present in the comparison of the averaged values. It has already been mentioned that the theoretical formula (9.82) is known to be invalid for inversion conditions. Observations confirm this conclusion. Figure 9.9 gives the results of the nighttime observations on April 5 and 6, 1955, with filtered and unfiltered instruments. At that time the sounding data found an inversion up t o 500 m, with a temperature difference inside this layer of AT = 5.5’. The calculation used (9.82), (9.86), and (9.87). The constants a and b in the last formula were taken as a = 2.5 x and b = 0.003. Figure 9.9 shows that only the “inversion” formula (9.87) gives results in good agreement with the observations. This means that the influence of the inversion stratification on the angular distribution of radiant intensity is quite notable.

Cloudy Sky. Let us start our consideration with the problem of the theoretical calculation of the relative effective radiation intensity in the finite case of solid cloud. Assuming that the radiation of the lower cloud surface can be identified with the blackbody radiation at the temperature of the lower cloud surface, Th (h is the cloud elevation above ground), we obtain

9.6. Angular Distribution of Intensity of Effective Radiation

595

9.W

FIG. 9.9 Relative intensity of the effective radiation in the conditions of inversion. (1) observational results (mean over 22 sets of measurement); (2) results of calculations from Eq. (9.87); (3) calculations from Eq. (9.82); (4) calculations from Linke’s formula f8/h= COSZ 8.

for the intensity of the atmospheric monochromatic radiation at the earth’s surface : GA(O,6 ) = E,(T2.)e-kAwh

+ :J

k, sec 0 EE(T)e-kAp

dp

Integrating this expression similarly to the integration made in deriving (9.73), we find for the full flux of atmospheric emission inside the cone, with an apex angle 2 0 in the direction of the zenith: G(0,O)

= B(To)sin2 0

+

[ P F b ) - COS’ O P J p sec O ) ]dp (9.88)

Using the relation (9.74), we then have F(0, 0 )= - 6

s” -dBj- [P&) O

- COS’

0 P& sec O ) ]dp

P

By taking 0 = n / 2 , we obtain the following expression for the radiant flux under solid cloud conditions ( F l ) calculated for the hemisphere:

596

Thermal Radiation of the Atmosphere

Making use of the last two formulas, we have

(9.89)

The following expression describing the intensity distribution over the sky can be obtained in a similar way:

(9.90)

Equations (9.89) and (9.90) are accurate expressions for the distribution over the sky of the flux and intensity of effective radiation under solid cloud conditions. They are also convenient, similarly to (9.77) and (9.78), for use with radiation charts. At the same time, on the basis of (9.89) and (9.90), it is possible to derive rather simple approximate relations. It is obvious that since the stratification corrections are generally small, they will be even smaller with uninterrupted clouds. This allows us to believe that the following relation for the relative intensity of effective radiation can be expected to be fulfilled with a sufficient degree of accuracy: (9.91) Equation (9.89) can be simplified in an analogous way. It is easy to see that (9.91) will exaggerate greatly with respect to the anisqtropy of the effective radiation because, according to the mean value theorem, the

9.6. Angular Distribution of Intensity of Effective Radiation

597

function PJ corresponding to an argument R < wh should be taken outside the integral sign. But even this formula shows that with a complete cloud cover, the anisotropy of effective radiation is considerably lower than with clear skies (see (9.85) and (9.91)). This conclusion is confirmed by experimental data. Having discussed the calculation of the intensity of effective radiation with clear skies and with a fully overcast sky, we can easily solve the problem of the distribution of the effective radiation intensity in the general case of partial cloudiness. Denote by n(8) the cloud amount a t angular distance 8 from the zenith. Then, taking account of (9.85) and (9.91), we have the following general relation for the intensity of effective radiation with partial cloud cover:

(9.92) We shall further consider an example of the use of this general relation in the problem of the depencence of effective radiation on cloud amount. 2. The Downward Atmospheric Radiation. Let us now consider the regularities of the distribution of the relative downward atmospheric radiation intensity over the sky. Obviously the solution of this problem can befound from the above results for the effective radiation.

Clear Sky. Making use of (9.85), we rewrite it as

where g,, go are the intensity of atmospheric emission in the direction of the zenith angle 8 and of the zenith, respectively, and

From this relation we have for the relative intensity of atmospheric emission : (9.93) Formula (9.93) is somewhat more general than (9.84), but cannot represent the relative atmospheric emission intensity as a function of 8 and

598

Thermal Radiation of the Atmosphere

w, only. In this connection (9.84), obtained for an isothermal atmosphere, is of interest. Since it has been established that the sky distribution of the relative intensity of effective radiation in the real atmosphere is identical with that in an isothermal atmosphere, it is natural to assume that this conclusion holds good for the atmospheric emission as well. It should be noted that the application of (9.93) is made easier if, as was shown by Falkenberg [86], the atmospheric emission in the direction of the zenith can be calculated from Angstrom’s empirical equation, with suitably changed numerical coefficients. Figure 9.10 gives the averages of 18 sets of observations (open instrument) of the relative intensity of atmospheric emission (circles) and the results calculated from (9.93), solid curve, and from (9.84), dashed curve. As seen, the agreement between the observed and calculated values may be considered satisfactory enough.

t9 ,deg

FIG. 9.10 Mean relative intensity of the atmospheric downward radiation according to 18 sets of measurements with a nonfiltered instrument.

The measurements of the angular distribution of atmospheric emission have also been realized by Yamamoto and Sasamori [87] and Bennett et al. [88].

Cloudy Sky. Making use of (9.91) for a complete cloud cover, we have (9.94) where

An equation similar to (9.92) will satisfy the case of partial cloudiness.

9.7. Some Practical Applications of Data

599

9.7. Some Practical Applications of Data on the Angular Distribution of the Intensities of Effective Radiation and Atmospheric Emission 1. Influence of Cloud, Fog, and Smoke on Eflective Radiation and Atmospheric Emission. In Sec. 9.5 we gave some observational data illustrating the considerable influence of cloudiness, fog, and smoke on the effective radiation and atmospheric emission. Let us now examine the theory of this problem.

Eflectof Cloud. Cloud is the most active factor that determines the quantities considered. As was shown earlier, the cloud cover decreases the effective radiation and increases the atmospheric emission. Some empirical formulas for the dependence of effective radiation and atmospheric emission upon cloud amount were given above. Let us now consider the influence of cloud on the effective radiation, theoretically following this author’s work [82]. The atmospheric emission can be considered in a similar way. The starting point in the solution of the problem of calculating the effective radiation with cloudy skies is clearly the distribution of the intensity of effective radiation over different directions with respect to the vertical. Iffn(€J) is the intensity of effective radiation at cloud amount n in the direction constituting an angle 0 relative to the vertical, then the effective radiation flux F0,, can be determined from the familiar relation

Fo,n = 2n

sy

fn(0) sin 0 cos 0 d€J

(9.95)

The distribution of the effective radiation intensity over the cloudy sky can be found by using the results of the preceding section. For the clear sky we have fs =hvW (9.96) where the function v(0) is given by (9.82). For the fully overcast sky,

where the function ql(0) is determined by (9.91). As was shown earlier, the radiant intensity distribution under complete cloud conditions approaches the isotropic. Thus, errors appear small even in the case that v1(€J)= 1. The lower the cloud cover, the more accurate is this approximation.

600

Thermal Radiation of the Atmosphere

For the derivation of the general formula defining the effective radiation intensity f,(e), we must allow for the fact that the cloud distribution over the sky is not uniform in the horizontal direction; a projective increase in cloudiness takes place. If no is the degree of cloudiness in the zenith, then in the general case we have for the degree of cloudiness n in the sky section at an angular distance 8 from the zenith, = n(no,

e)

(9.98)

Vaisala [89], processing observational data on cumulus clouds, obtained the following expression for the distribution of clouds over the sky:

n(no, e) = no

+ ng=yi

- no)sinyi - cOs2x2e)

(9.99)

where x is the ratio of the vertical to the horizontal dimensions of a cloud (relative thickness). Taking into account the above results we can find the general expression for the effective radiation intensity distribution over the partially cloudy sky, which is

f,(Q= [1 - 0

0

7

QlfoP,(e>

+ n(no

9

e)f,,lP@)

(9.100)

Substituting this expression into (9.95), we obtain the general equation for the effective radiation :

+ 2n&

Js’”

n(no, t9)Vl(e) sin 0 cos 0 de

(9.101)

It follows from (9.101) that the main quantities defining the effective radiation F0,, are fo ,fo,l, and n(no, 0). At the same time it is evident that the quantities fo and fo,l are not independent. They are connected by the following obvious relation :

f o -.&,I

=W(Th)PJ(W)

+ gog - g0,fZl

(9.102)

Where 6 is the emissivity (absorptivity) of the underlying surface, and Th is the absolute temperature at the lower cloud boundary z = h, go,h and g0,H are the intensities of the emission from the atmospheric layers situated between the earth’s surface (z = 0) and the altitudes z = h and z = H , respectively ( H is the “height” of the atmosphere, which in this case can be identified with the height of the tropopause).

601

9.7. Some Practical Applications of Data

The first two terms in square brackets of (9.102) determine the intensity of the atmospheric emission with a fully cloudy sky at a height z = h ; the third term gives the emission intensity for a cloudless atmosphere. The difference in effective radiation for the two extreme cases (clear and fully cloudy sky) is evidently determined by the difference in the amounts of the atmospheric emission absorbed by the underlying surface. This fact accounts for the choice of the coefficient 6 defining the absorptivity of the underlying surface as a common factor in (9.102). is Taking into consideration that g o , H = g o , & g h , H P J ( w h ) , where the emission from the atmospheric layer (h, H ) above the cloud, we can replace (9.102) by

+

f 0 -fO,l

= s[E(Th)

- gh,HIPJ(Wh)

(9.103)

Note here that the factor E(Th)- g h , H in this formula represents the intensity of the effective radiation at the upper cloud surface. Let us consider two extreme cases that can be expressed in a simpler way than in (9.103). High Clouds. In this case

gh,H

f 0 -fO.l

< E(Th), and consequently = GE(Th)PJ(wh)

(9.104)

In physical terms, the difference fo - f0,, is determined by the portion of the cloud emission E(Th) absorbed by the underlying surface.

Low Clouds. In this finite case, assuming that To = Th, we have N go,B, and since d [ E ( T h ) - g O , H ] = fo, we obtain

gh,H

The values foandf,,, contained in (9.101) define the dependence of the effective radiation Fo,n on atmospheric stratification characterized by the vertical distribution of temperature and absolute humidity. The function n(no, 0) describes the cloudiness and represents the influence of the amount and sky distribution of the cloud on the effective radiation. Let us now particularly analyze (9.101). Begin with some special cases. Assume that (1) n(no, 0) = no = n ; that is, the distribution of clouds over the sky is uniform. (2) p(0) = pl(0) = 1; that is, the effective radiation is isotropic.

602

Thermal Radiation of the Atmosphere

In this case, instead of (9.101) we have

where Fa = nfo and F,,l = nfo,l are the effective radiation fluxes. It is seen from (9.106) that, for the uniform distribution of clouds over the sky and isotropic effective radiation, the dependence of the effective radiation on cloud amount is linear. This means that the relation

FO,%= F'(1 - cn)

(9.107)

common in calculational practice, is valid. Here the coefficient c = (l/Fa) (Fa- F0,J = (l/fa) ( f a - f 0 , 1 ) . Using (9.103) through (105), we obtain the following expression for the coefficient c and its extreme values with high (c,) and low (cz) clouds: (9.108) (9.109)

It is evident from (9.108) that the coefficient depends not only on the height of cloud (as is generally assumed) but also on the stratification of the atmosphere. At a given type of stratification the variation of c in dependence upon cloud height is determined by the product of two factors: E(Th)- g h and PJ(wh), of which the latter is a decreasing function of height and the variation of the former with height depends on the vertical distribution of temperature and absolute humidity. Calculations show that with a decrease of temperature with height, the atmospheric emission g h , H decreases at a higher rate than E(Th). This accounts for the slow increase of E(Th)- g h , w with height. The decrease of PJ(wh)is, however, more rapid than the increase of E(Th)- g h , H , for which reason the coefficient c decreases at an increasing cloud altitude. The decrease of c becomes more rapid at a certain altitude for which gh,H < E(Th). In this case (9.109) applies, with both factors in the numerator of the expression for c decreasing with height. The qualitative conclusion concerning the nonuniformity of the decrease of the coefficient with height is in agreement with the vertical variation of this coefficient found from observational data by means of the formula

9.7. Some Practical Applications of Data

603

c = (l/Fo)(Fo - Fo,l).It can be seen from Table 9.11 that the coefficients c for middle clouds are less different from the corresponding coefficients for low clouds than for high clouds. However, the coefficients c obtained

from observation for the upper cloud level are small also because the emission of the lower cloud surface in this case is considerably less than the blackbody emission. These qualitative conclusions are confirmed by the detailed calculations of the effect of atmospheric stratification upon the coefficients made by Kirillova [90]. Let us now examine the anisotropy of the effective radiation, still assuming that the distribution of cloud over the sky is uniform. It is easy to see that in this case (9.107) remains valid, the only difference being that the effective radiation with clear skies and with fully cloudy skies is determined by the following formulas :

Fo = 27th

Infz v(0) sin 0 cos 0 d0 0

Fo,l = 2nf0,1

Iniz vl(0) sin 0 cos 0 d0

Thus a uniform cloud distribution over the sky is the necessary condition for a linear dependence of the effective radiation upon cloud amount. It is known, however, that the latter does not actually take place, owing to the projective increase of cloud amount in the horizontal direction. Let us now turn to the examination of (9.101) in its most general form, with allowance for the anisotropy of the effective radiation and the nonuniformity of the sky cloud distribution. For this case,

+

Fo,n = FO - n h ~ l ( n o ) nfo,iyz(no) where Fo = 2nf0

yl(no) = 2

yz(no)= 2

0

s:‘“

(9.1 11)

1”” v(0) sin 0 cos 0 d0 0

n(no,0)y(€J)sin 0 cos 0 d0 n(no, O)vl(0) sin 0 cos 0 d0

Taking into account (9.99) for n(no, 0), we replace (9.111) with the expression

+

Fo,n= (1 - ~ O FnoFo,l O - nfoy,(no)

+ nfo,ly4(no)

(9.112)

604

Thermal Radiation of the Atmosphere

It is evident from (9.113) that in the general case the effective radiation is a nonlinear function of the cloud amount in the zenith. As was mentioned earlier, Chumakova [91] obtained a relationship similar to (9.1 13) on the basis of observations over the effective radiation at Karadag (see (9.36)). In order to compare (9.113) and (9.36), it is necessary to allow for the fact that the total cloud amount ii and the cloud amount in the zenith are connected by the following relation :

A

=

J,"'" n ( n o , e) sin e de

(9.1 14)

or, using (9.99), we can write

The additional nonlinear term in the right-hand side of (9.115) acts as a correction for the effect of the projective increase in cloud amount in the horizontal direction. Using (9.1 13) together with (9.1 15), we see that, generally speaking, the effective radiation is a complex nonlinear function of the total cloud amount. Equation (9.36) is only a simplified empirical approximation of this dependence. Another important conclusion in connection with (9.1 13) is that even for a definite geographical point, the empirical coefficients contained in equations similar to (9.36) cannot be considered constant. Besides the height (type) of cloud, these coefficients are also dependent on atmospheric stratification. Unfortunately, at present there are no measurement data on the effective radiation available with simultaneous aerological soundings and in-

605

9.7. Some Practical Applications of Data

vestigations of the distribution of clouds over the sky. This is a serious obstruction in the numerical comparison of the above theory with observations. Certain numerical calculations of the dependence of the effective radiant flux and atmospheric emission upon cloud amount, together with the sky cloud distribution, confirm that this dependence must be nonlinear. Efect of Fog and Smoke. The theory of the influence of fog and smoke on the effective radiation has been dealt with by quite a number of Soviet investigators: Berland [92, 931, Krasikov [94, 951, Shifrin [96, 971, and Malkevich [98]. The theory of the considered problem reduces to the solution of the transfer equations for a medium containing suspended particles of smoke or water droplets. Since in the given case the dimensions of the particles are comparable to the wavelengths of thermal radiation, it becomes necessary to allow for the processes of absorption, scattering, and emission of radiation. This seriously complicates the general solution of the problem. In practice all the above investigations employed various approximate methods. The problem can be considerably simplified if the suitable approximate equations are used instead of the accurate integrodifferential transfer equations. For example, Berland [92] showed that if the scattering and diffusivity of thermal radiation are ignored, it is possible to derive the following formula for the reduction of the effective radiation caused by a smoke screen:

where Fois the effective radiation of an area free of smoke, k is the absorption coefficient of smoke, v is the wind speed, and M is the expenditure of the smoking substance per unit time and unit length of the smoke screen (the source of smoke is assumed to be linear). The calculation of the reduction in effective radiation by means of this formula yields values close to the observed values dF, which were given in Sec. 9.5. A similar equation in valid in the case of fog. According to Shifrin [96], we may write for this case

dFo = Fo[l - e-(a+R)]

(9.1 17)

where a and R are the coefficients of absorption and reflection calculated for the whole fog thickness. At a small optical mass (low a R values) (9.1 17) will be replaced by

+

d F ~ -(a

-- -

F O

+ R)

(9.118)

606

Thermal Radiation of the Atmosphere

The coefficients a and R calculated by Shifrin show that for all fogs with the particle size a < 1 4 p , the absorption and reflection coefficients are independent of the size of the droplet and that R is by about ten times smaller than a. This means that the radiative properties of fog are practically independent of its structure and are determined mainly by the absorptivity and emissivity of the droplets. Introducing mass coefficients of absorption, a, = a / M , and of reflection, R, = R/M (where M is the mass of droplets in a fog column of unit section), we can calculate their numerical values as a, = 550 cm2/g, R, = 55 cm2/g, and a, R,,= 605 cm2/g. Let I denote the thickness of fog and q its water content. Expressing the former quantity in meters and the latter in g/cm2, in place of (9.1 18) we now obtain the approximate relation

+

A FO = -6ql= 6~ -

(9.119)

r 0

where w is the amount of precipitated water in the fog, expressed in g/cm2. The formula (9.119) is valid only for the case of monodispersional fog and small (less than unity) optical mass. Analyzing the effect of the polydispersional property of fog Shifrin and Bogdanova [97] showed that in this case the numerical factor in (9.119) was 6.4. Hence it follows, as has already been mentioned, that the influence of the size distribution is insignificant. In the case of large (above unity) optical mass when multiple scattering becomes important, the formula for the reduction of effective radiation is (9.1 20)

where 1

p = - (1 V

+Y -d m ) a,

= a* -

a'

1

E

= - (1

a

= a*

V

+ v + d+)

+ a'

in which a*, a', and a, are the volume coefficients of attenuation, back scattering (reflection), and absorption, respectively, and H is the fog thickness. The observations of Bogdanova fully confirm the validity of (9.1 19) (with a correction for polydispersion) and (9.120).

607

9.7. Some Practical Applications of Data

2. Variation of Effective Radiation with Height under Forest Cover. It is evident that the above considered methods for calculating effective radiation and atmospheric emission cannot be applied in the case when it is necessary to determine these quantities under the forest cover. The solution of this problem is of much practical importance. It is also important to investigaje the regularities of the variation in effective radiation under the trees. Let us consider this problem, following the results obtained by this author [99]. In order to approach the theoretical solution of this problem we must first schematize the phenomenon in question. Let us assume that the variation of effective radiation with height in a forest can be simplified as follows. Introduce the concept of the “effective” solid angle of visibility of the sky from the radiating surface at ground level, and represent this solid angle as a cone with apex angle 2 0 . Thus we assume that at the point of observation, the space free of forest can be represented as a cylinder of radius R and height H (Fig. 9.11). Here H is the average height of trees and R is a certain “effective” radius of the cylinder. Let us now find by elementary geometry the relation between the apex angles of the cone of solid angle of the open sky at the height z and at ground level. We have R tan 0 = H

tan 6

=

~

R H-z

tan 8

=

~

H-z

tan

o

Thus sec 0

=

Substituting the last expression into (9.81), we obtain a relationship between the effective radiation at the height F, under the forest cover and on the area free of trees (or above the forest), Fo:

It should be noted that in solving this problem we neglect the temperature difference between the emitting black surface of the instrument and the trunks and branches of surrounding trees. Also ignored is the temperature difference between the ground and the level of the forest cover. These as-

608

Thermal Radiation of the Atmosphere

FIG. 9.11

The derivation of the formula for the variation of the effective radiation with height under the forest cover.

sumptions have a fair basis, since the main factor influencing the variation of effective radiation with height under the forest cover is the vertical change of the sky solid angle. It is perfectly evident that for our purpose the absorption function suffices to be represented even in a very approximate way. Let us assume, therefore, that PF(w,) = 0.25e-k.wm. In this case,

Fz _ -1FO

1 H2

tan2 0

4- ( H - z ) ~ x {exp[ -kw,

-I(

l)]}

(9.122)

In order to calculate the ratio FJF, from this formula, it is necessary to know the value 0.If our purpose is a purely theoretical solution, the value 0 should be determined from the data of forest estimation (height, density, and closeness of the tree tops, forest density, etc.). Our approach, however, will be semiempirical. Let the relation between the effective radiation at ground level outside and inside the forest be known. Then the value 0 can be found from (9.122). Now compare with experimental results the vertical variation of Fz/Fo thus calculated. Let us use Kuzmin’s measurement data [IOO] on the variation of effective radiation with height in a forest at a clear sky.

9.7. Some Practical Applications of Data

609

Kuzmin made two sets of measurements of the ratio Fz/Fofor two different forest areas. In one case measurements were taken in a mature forest without young stand and undergrowth, the average height of the trees being 20 m and the total density 0.5 (remember that forest density is the ratio of the sum of the tree cross sections in a given area to that in a standard forest). In the other case there were young trees and undergrowth, with the average height of the nature trees being about 20 m and the total density 0.79. The average height of the young stand was not estimated. It follows from this general description of both areas that (9.122) can be directly applied when H = 20 m, after the value 0 has been found in the above way. In the second case it appears to be necessary to allow for the effect of the low stand, since the forest has two layers. We assumed that the average height of the first layer was 10 m with the “effective” height taken equal to 5 m. (The influence of young trees and undergrowth is clearly seen not to extend to the uppermost surface, which is evident in Fig. 9.12). The calculation of the value Fz/Fowas therefore performed as follows. Up to the height z = 5 m, this value was calculated at H = 10 m. At z > 5 m, the effect of plants of the first layer was ignored and Fz/Fo was calculated at H = 15 m (the second layer beginning at z = 5 m). In both cases the mass of atmospheric water vapor was taken to be w, = 1.5 g/cm2, which corresponds to average conditions at moderate latitudes (calculations show that the accurate value w, is of no importance here).

FIG. 9.12 Variation of the effective radiation with height under the forest cover.

The following values of FJF, were obtained in the first and the second cases, respectively: 0 = 27’30’ and 0 = 23’. Figure 9.12 displays Kuzmin’s curves of the vertical variation of Fz/Fofor both cases, where the solid curve is for the mature forest without undergrowth and the interrupted curve is for the mature forest with young stand and undergrowth. Crosses

610

Thermal Radiation of the Atmosphere

and circles mark the corresponding values calculated theoretically. The agreement of the calculated and measured values is quite satisfactory. Even in the case of two layers the departure of the calculated values from observations is small. Thus we may conclude that the proposed semiempirica1 scheme of the vertical variation of effective radiation under the cover of trees closely imitates the reality. Let us now briefly examine the determination of 0,which is the apex angle of the conic effective solid angle for the sky visibility at the ground level. This author made an attempt to find the value 0 for a mature forest without undergrowth, using the estimation of the density, number of trees per unit area, mean cross section, and height of trees. The value 0 was easily obtained with an accuracy of several degrees, that is, to 15 to 20 percent. Such accuracy, however, is not sufficient for this case. On the other hand, it is doubtful whether higher accuracy can be reached with such calculations even if they are made more complicated. This is evident from the rough character of the usual forest estimation and from the difficulty of the “geometrical” representation of the actual forest. Hence it appears that the complete theoretical solution of the considered problem is hopeless. A semiempirical solution has been suggested above, assuming a definite relation between the effective radiation at ground level outside and inside the forest. This method of empirical determination of 0 from pyrgeometric measurement data at the earth’s surface is not, however, the only one possible. It is very likely that the empirical determination of the dependence of 0 on the data of forest estimation may be expedient. Comparison of the obtained values shows, for example, that in the considered cases, tan2 0 l / p (where p is the forest density) is actually correct to a high degree. The proportionality coefficient determined from these two examples turned out to be 0.14, that is tan2 0 = 0.14/p. The lack of adequate experimental data does permit testing this dependence or deriving another empirical relationship. The method itself, however, promises to be fruitful, and by realizing its possibilities one may reach further theoretical generalizations. The above results concern the variation of effective radiation in a forest. It is possible to believe, however, that analogous laws hold for other types of vegetation as well.

-

3. Eflective Radiation of Slopes. The determination of the effective radiation of nonhorizontal surfaces is of great practical importance, particularly for agricultural applications. Recently this problem has been satisfactorily solved by the author and Podolskaya [loll.

61 1

9.7. Some Practical Applications of Data

Let us first derive the theoretical formulas for the effective radiation of slopes. Let the following assumptions be made (to be dispensed with further on) : (1) The slope and the horizontal surface in front of it are perfectly black. (2) The temperatures of the slope and the adjacent horizontal surface are equal. The first of these assumptions makes it possible to neglect the effect of the multiple reflection of longwave radiation between the sloping and the adjacent horizontal surfaces; the second allows neglect of the radiative heat exchange between these two surfaces. Both assumptions will be given detailed treatment later. Let us introduce the following notation (Fig. 9.13): fh,v is the effective radiation intensity in a direction determined by the spherical coordinates h (angular height relative to horizontal plane) and y (azimuth): Q is the angle of inclination of the slope relative to the horizon. P

> 0 a=90°

FIG. 9.13 To the derivation of the formula for the effective radiation of slopes.

According to the general relation connecting the flux and intensity of radiation, for the flux F, of effective radiation at the slope we have

Fs =

1'" Ji;:, dy

f h , v cos i cos h dh

(9.123)

612

Thermal Radiation of the Atmosphere

where i is the angle between the normal to the slope, 2, and an arbitrary -+ direction 1 . Taking account of the symmetry in relation to the plane normal to the slope (the effective radiation intensity of a clear as well as a cloudy sky is independent of azimuth) it is possible to replace (9.123) by F,

=

2

In l;:,",fh,q dy

cos i cos h dh

(9.124)

We have for cos i, cos i = COS$,

+ -

2)

A

A

= cos(nx)

+ cos(ny) cos(2y) + cos(n2) cos(Z2) A

COS(ZX)

A

A

A

If the axis OX is chosen according to Fig. 9.13, then A

A

cos(nx)

=

sin a cos y

A

cos(2x) = cos h

A

cos(2y) = 0

cos(nz)

A

= cos

cos(2z) = sin h

a

Hence we obtain cos i = sin a cos y cos h

+ cos a sin h

(9.125)

Let us now determine the function h ( y ) . It is evident that h ( y ) = 0 at 0 5 y 5 n/2. In the interval n/2 5 y I n, the function h ( y ) can be found from the below relations, following from the right-angled spherical triangle ABC (Fig. 9.13):

that is, cos a

= cos

cos B

= sin

A cos b

cos A

= cos

a sin B

= sin

a cos y'

h sin B. Using these relations we have cos a sin 8 ,

cos h

= --

h(w') .,

= arc

cos a

41- sin2 a cos2 y'

Thus cos

cos a

4I - sin2 a cos2 y

9.7. Some Practical Applications of Data

or, since y

=

(n/2)

613

+ y', we have finally

h ( y ) = arc cos

cos a

4 1 - sin2 a sin2 y

(9.126)

Taking into account (9.125) and (9.126), in place of (9.124) we write F,

=2

fh,,[sin a cos y cos h

dy

+ cos a sin h ] cos h dh

(9.127)

where 0

at O _ ( y _ ( n / 2 cos a

at n/2 L y L n

1 - sin2a sin2 y Equation (9.127) gives the familiar finite relations for the effective radiation at horizontal and vertical surfaces. (1) Horizontal Plane (a = 0, h ( y ) = 0 at 0 I y L n). Relations for the horizontal plane are

dy

1:

fh,tpsin h cos h dh

(9.128)

When the effective radiation intensity is isotropic, fh,p =f = constant, whence FH = nJ (2) Vertical Plane ( a = n/2, h ( y ) = 0 at 0 i y 5 n/2, and h(y) = 7c/2 at 4 2 5 y _( y. For this we get Fv

=2

1""

cos y d y

fh,, cos2 h dh

(9.129)

0

When the intensity of the effective radiation is isotropic, fh,, = f = constant whence F, = (1/2)nJ Equation (9.127) shows that the general formula for calculating the effective radiation of slopes is complicated. Let us therefore try to derive some simplified approximate relations following from (9.127). It was shown in Sec. 9.6 that under clear or fully cloudy conditions, the effective radiation intensity fh,v as a function of direction depends only on the angular height h of a given sky section relative to the horizontal plane. Therefore effective radiation intensity close to the horizontal is insignificant. Thus, if the angle of inclination of the slope is not too large (not exceeding 25 to 30°), it can be approximated that h ( y ) = a at n/2 I y L n.

614

Thermal Radiation of the Atmosphere

Considering the above discussion we obtain from (9.127): F,

=2

I" Y'' dy

&(sin a cos y cos h

a

+ cos a sin h) cos h dh

(9.130)

where

Using (9.128), the last formula can be transformed as

At small a values the second term in this formula is considerably smaller than the first, from which we approximate for slopes of small inclination:

F,

2:

FH cos a

(9.132)

This relation shows that for low slopes, the effect of radiative diffusivity may be ignored. The connection between the fluxes on the horizontal and slanting surfaces is defined by the same relation as in the case of a parallel beam (the cosine law). Let us now compare the theoretical calculations with the observations. Measurements of the effective radiation at slopes were carried out on clear nights of June-July, 1952, at the Karadag actinometric observatory (Crimea). In measuring the effective radiation of the blackened receiving surface, the, Yanishevsky pyrgeometer (effective pyranometer) was used. The instrument was mounted on a theodolite stand, which enabled its free orientation with respect to the horizontal plane. Each set of measurements included readings of the pyrgeometer, whose receiving surface angle of inclination was varyied from 0' to 90' and back at intervals of 15'. It is a known fact that pyrgeometer readings are greatly affected by wind. With the instrument inclined and the wind strong, this effect may get out of control. The measurements were therefore made in light wind not exceeding 2 to 3 m/sec. Under such conditions wind influence on the readings of the pyrgeometer was comparatively small. However, the mean relative error of individual measurements should be estimated at about 10 to 12 percent. Since the random errors of particular measurements were large enough, only the averaged results of all the observations on clear nights could be compared with the theoretical calculations. Figure 9.14 shows the curve (solid) representing the ratio F,/F, in

9.7. Some Practical Applications of Data

61 5

Q,deg

FIG. 9.14 The effective radiation of slopes. (1) averaged results of 28 sets of observation, (2) cos a.

dependence on the angle a, calculated from (9.127). The calculation was made by graphical integration. Equation (9.82) was used to compute the effective radiation intensity. The total atmospheric water vapor content was determined from the empirical equation obtained by Sivkov [lo21 for Karadag : W,

= 0.38

(9.133)

where e is the water vapor pressure at the earth's surface in millimeters of mercury. Since in this case the mean water vapor pressure was 10.7 mm, w, = 1.8 g/cm2. The circles of Fig. 9.14 indicate the averaged results of 28 individual sets of observation. The dashed curve represents cos a. The agreement between the ratio F,/F, values, calculated from (9.127) and measured, is seen to be quite satisfactory. According to Fig. 9.14 the approximate formula (9.132) makes possible sufficiently accurate calculation of the effective radiation at slopes from measurement data on the effective radiation at the horizontal surface with angles of inclination a not exceeding 30'. But even at slightly larger angles a, the errors are not too important. It appears that with fully overcast skies, when the effective radiation intensity is much more isotropic than with a clear sky, (9.132) must be valid at even greater a values. The above formulas refer to the case where the sloping surface and the adjacent horizontal surface are at equal temperature and perfectly black. Let us now consider how these assumptions can be dispensed with. It is known that the albedo of natural underlying surfaces for longwave radiation is very small, averaging about 0.05 to 0.10. The effect of multiple reflection between the slope and the adjacent horizontal surface can there-

616

Thermal Kadiation of the Atmosphere

fore obviously be ignored. The absorptivity of natural surfaces being less than unity, the form of the above equations will not be changed. For instance, in (9.132) the values F, and FH are directly proportional to 6. This equation is therefore valid not only for 6 = 1 but also for any value of 6. The same applies to the rest of the above equations. The effect of radiative heat exchange between the slope and the adjacent horizontal surface may be easily taken account of. If the temperatures of these surfaces are T, and Th, respectively, then the intensity of the isotropic thermal radiation of these surface is S(a/n)T,“ and 6(a/n)Th4.Here 6 is the relative emissivity (absorptivity) of the slope and the horizontal surface assumed to be the same, a is the radiation constant. Ignore the comparatively small absorption of longwave radiation along the path between these two surfaces and also the emission of the intermediate air layer (it should be noted that the absorbed radiation is fairly compensated by the emission of this layer). Then the value of the radiative heat exchange between the sloping and horizontal surfaces F, will be determined by the following relation :

F,

=2

0

dy‘

Io

h(v’)

where h ( y r )= arc cos

60 n

-(T:

-

Th4) sin h cos h dh

cos a

4 1 - sin2 a cos2 y‘

(9.134)

at 0 i y L n/2

Integrating in the last equation we find

F,

=

a da(T,d - TH4)sin2 2

(9.136)

A similar equation was derived by Eisenstadt [42]. For a vertical slope (a = n/2) we have from this equation:

F,

=

4 b(T: - T H ~ )

(9.137)

Using (9.127) and (9.136) we transform the general equation for the effective radiation at slopes as

F,

=2

I” Ji::) dy

fh,,(sin a cos y cos h

a + Sa(T,d - TH4)sin22

+ cos a sin h) cos h dh (9.138)

617

9.8. Distribution of Energy in the Spectrum of Effective Radiation

or, for low slopes,

F,

= FH

cos a

+ &(T:

-

a

TH4)sin22

(9.139 )

The correction term in the last two equations may become essential mainly in daytime when the temperature difference between the slope and the horizontal surface is notable.

9.8. Distribution of Energy in the Spectrum of Effective Radiation and Downward Atmospheric Radiationt At the beginning of this chapter we described the methods for calculating the integral fluxes of effective radiation and of downward atmospheric radiation. Let us now consider the results of some calculations of the spectral distribution of effective radiation and downward atmospheric radiation performed by means of similar methods. Figures 9.15 and 9.16 give the results of calculations by Kondratyev e l al. [103, 1041 of the spectral distribution of these quantities for clear skies (solid curves). Figure 9.15 confirms the above-mentioned conclusion

A* P

FIG. 9.15 Spectral distribution of the effective radiation with clear skies.

that the transfer of thermal radiation in the atmosphere is effected in the tegion of maximal transparency. It is evident from Figs. 9.16 and 9.17 rhat only in the spectral regions adjacent to the transparency region does + S . V . Ashcheulov and D . B. Styro are coauthors of this section.

618

Thermal Radiation of the Atmosphere

30 -

5

.-... 25 -

7

9

15-

Y

x

10-

5-

0'

5

10

15

D

LP

FIG. 9.16 Spectral distribution of the atmospheric downward radiation with clear skies in the intervals 5-15 p . (1) the intensities of the atmospheric downward radiation measured in zenith at 0200 hr, Nov. 16, 1963, Rostov-on-Don; (2) the same calculated for zenith; (3) measured at the angle '8 above the horizon, 0300 hr, Nov. 16,63, Rostov-on-Don; (4) the same calculated; (5) curve of the radiant intensity of a blackbody at the temperature of the surface air 6.5OC.

the downward atmospheric radiation differ from the blackbody radiation at the temperature of the surface air layer (dashed curve). The interrupted curves represent the intensities of the downward atmospheric radiation at different angles to the horizon, measured for the corresponding meteorological situations. The coincidence between the calculated and the measured curves may be found satisfactory. The slight departure of the curves in the transparency window results from the neglect of the contribution into the radiation made by methane, nitrous dioxide, aerosols, and other components. Outside the transparency window the calculated and the measured spectral distributions show good coincidence and differ little from the blackbody radiation at the surface air temperature. Figures 9.15, 9.16, and 9.17 illustrate the fact that the thermal radiation of the atmospheric ozone does not greatly affect the downward atmospheric and effective radiations (less than 5 percent).

9.8. Distribution of Energy in the Spectrum of Effective Radiation

619

I

51 0 1

c

15

20

25

A. P

FIG. 9.17 Spectral distribution of the atmospheric downward radiation with clear skies in the interval 14-25 p. (1) intensities measured in zenith at 2100 hr, Nov. 16, 1963, Rostov-on-Don; (2) the same calculated for zenith; (3) curve of the radiant intensity of the black body at the temperature of the surface air layer 2.OoC.

Figure 9,18 gives the calculated thicknesses of atmospheric layers that generate 99 percent of the downward atmospheric radiation in various spectral regions at ground level [103]. The curves of Figs. 9.18 and 9.19 are the reverse of those of Figs. 9.16 and 9.17. The observed coincidence between these curves (except the band of ozone radiation centered at 9.65 p) is quite natural, since Figs. 9.16 and 9.17 and Figs. 9.18 and 9.19 are plotted from the same characteristics of the absorption of thermal radiation in the atmosphere. As seen from Figs. 9.18 and 9.19, 99 percent of the full flux of the downward atmospheric radiation comes from a layer about 4 km thick inside and about 1 km thick outside the transparency window. Hence it follows that in calculating the downward atmospheric radiation for clear skies, it is practically insignificant what temperature and humidity distributions are in the layers above 4 km for the transparency window and above 1 km for other spectral divisions. As has been already mentioned, this fact is one of the reasons that empirical formulas containing only the temperature and absolute air humidity near the underlying surface can give satisfactory results in the calculation of mean values of the downward atmospheric radiation.

:

620

Thermal Radiation of the Atmosphere

I 50

20 10

5

I

"

00

A+ 10

0 00 .0I2 5

15

FIG. 9.18 Thickness of the atmospheric layer responsible for 99 percent of the downward radiation in the interval 5-15 p. (1) layers accounting for 99 percent of the radiation from zenith 0200 h , Nov. 16, 1963, Rostov-on-Don; (2) the same at the angle '8 above the horizon at 0300 hr, Nov. 16, 1963, Rostov-on-Don.

It should be stressed that these conclusions are not valid for the calculation of the effective radiation. Since the effective radiation is a small dif-

O Ool

15

o

20

2

25

2

A+

FIG. 9.19 Thickness of the atmospheric layer determining 99 percent of the downward radiation in the interval 14-25 from zenith at 2100 h, Nov. 16, 1963, Rostov-on-Don.

ference between two large quantities (see Figs. 9.20 and 9.21 borrowed from [lOS]) Uo and Go, it is clear that even a slight error in the determination of Go may lead to a considerable mistake in calculating the effective radiation. The physical meaning of this is that, as the major portion of the

9.8. Distribution of Energy in the Spectrum of Effective Radiation

Spectral intensities of the downcoming radiation from zenith at different levels in the atmosphere at 2000 hr, Nov. 14, 1963, Rostov-on-Don. (1) at the level 1,000 mb (100 m); (2) 875 mb (1,200 m); (3) 725 mb (2,800 m); (4) 425 mb (6,750 m); (5) 125 mb (15,000 m); (6) 37.5 mb (22,500 m). FIG. 9.20

621

o\

N N

FIG. 9.21 Spectral intensities of the up going radiation from zenith at diffkrent levels in the atmosphere at 2000 hr, Nov. 14, 1963, Rostov-on-Don. (1) at the level 1,000 rnb (100 rn); (2) 575 rnb (4,600 rn); (3) 275 mb (10,000 rn); (4) 75 rnb (18,000 rn); (5) 17.5 rnb (28,000 rn).

9.8. Distribution of Energy in the Spectrum of Effective Radiation

623

effective radiation comes from the region of maximal transparency, it is necessary to take into account the character of the vertical distribution of temperature and humidity in such atmospheric layers, whose thickness is determined by the ordinates of the curve in Fig. 9.18, that fall on the transparency region. Thus, in calculating the effective radiation it is essential to consider the atmospheric stratification up to great altitudes. The results of the calculations of the spectral distribution of downward atmospheric radiation with fully overcast skies, made by Kondratyev et al. [I061 are given in Fig. 9.22. The calculations correspond to an actual meteorological situation (July 23, 1965, 13.00 Kaunas in Lithuania) and also to other cases when it was assumed that the lower cloud boundary was at 3, 6, and 9 km. The calculations were based on the absorption function of water vapor for the transparency window as found at Leningrad University [I071 and for other spectral regions according to the data of Wark et al. [108]. The curve 5 in Fig. 9.22 characterizes the spectral distribution of a blackbody at t = 19OC.

I

5

10

15

20

25

30

*

A+ FIG. 9.22 Spectral distribution of the atmospheric downward radiation with furry overcast skies. (1) cloud height 750 m; (2) 3 km; (3) 6 km; (4) 9 km; ( 5 ) curve of the blackbody radiation at the temperature of the surface air of 19OC.

Figure 9.22 is spectacular in illustrating the effect of complete clouds of different levels on the spectral distribution and magnitude of the flux of downward atmospheric radiative heat transfer. Besides the above-considered works the spectral investigation of the

624

Thermal Radiation of the Atmosphere

thermal atmospheric radiation is treated in many other studies [109-1301, etc. Of particular interest is Bolle’s [128, 1291 research in the fine structure of the infrared emission spectrum of the atmosphere. 9.9. Emission Spectroscopy as a Means of Investigating the Structure and Composition of the Atmospheret

What has been said above clearly indicates that the thermal atmospheric radiation is considerably dependent on the stratification of the atmosphere. This makes it possible to approach the inverse problems of the determination of atmospheric stratification, basing their solutions on measurement data about its thermal radiation. One of such problems to be considered later on as an example presents an attempt to find the vertical distribution of air and ground temperatures from satellite measurement data on the outgoing thermal radiation. In the general form this problem was discussed by Marchuk [131]. Physically the possibility of determining the atmospheric thermal stratification is based on a strong dependence of the outgoing radiation upon atmospheric temperature and also on the marked selectivity of the infrared absorption spectrum of the atmosphere. Owing to the selectivity, the “mean free path” of the thermal radiation is considerably dependent upon frequency (wavelength). This means that in different spectral regions the radiation reaching the satellite-mounted recorder originates in different atmospheric layers. In the region where the atmospheric absorption is very low, the instrument records the thermal radiation of the earth’s surface. In the region of weak absorption the received radiation is a mixture of the transformed radiation of the earth’s surface and of the atmospheric thermal radiation. In the area of strong absorption the recorder detects the radiation of the outer atmospheric layers. The thickness of the emitting layer is much dependent on the intensity of absorption in the given spectral region : the more intensive the absorption, the thinner the emitting layer. This is illustrated by Fig. 9.23, taken from the work by this author and Timofeyev [ 1321. Figure 9.23 presents the dependence of the contribution into the upward radiation recorded at the height z = 20 km upon the radiating atmospheric layers situated from 0 to 20 km. It is clearly seen for frequencies close to the center of the line with strong absorption that the radiating layer is thin and adjoining the level at which the recording takes place. For frequencies farther from the center of the t Y. M. Timofeyev is coauthor of this section.

625

9.9. Emission Spectroscopy

line, the absorption is small, the extension of the radiating layer increases, and the contribution from more distant atmospheric layers becomes important. A similar picture is observed in the case of the outgoing radiation recorded by a satellite. For a certain spectral interval the magnitude of the outgoing radiation depends on the distribution of the absorbing com-

I

0

10

1

I

30

FIG. 9.23 Dependence of the contribution to the upgoing radiation at the altitude z = 20 knz on the elevation of the layer ( y = 1000). v - v,,: (1) 0.2862 cm-l; (2) 0.7155 cm-l; (3) 1.431 cm-l; (4) 2.862 cm-l.

ponents (of which the most important are water vapor, ozone, and carbon dioxide) and of atmospheric temperature. In the spectral regions where the radiation is due to water vapor or ozone whose concentrations vary in time and space, it is impossible to distinguish between the effect of the vertical distribution of temperature and the corresponding component on the outgoing radiation without knowledge of one of the mentioned distributions. However, the matter is easier with the radiation of carbon dioxide. It is known that the relative volume concentration of carbon dioxide in the atmosphere may be considered constant and averaging 0.03 percent. Thus in the spectral region of carbon dioxide the outgoing radiation depends

626

Thermal Radiation of the Atmosphere

mainly on the temperature stratification of the atmosphere. Taking into account what has been said about the considerable dependence of the position and thickness of the radiating layers upon the intensity of absorption, it is clear that a set of measurements of CO, radiation in different spectral regions can yield information on the thermal state of the atmosphere. This is one possibility of solving the problem of determination of the atmospheric temperature stratification. Another is the measurement of the outgoing radiation from a satellite in the same spectral interval at different angles. This way is connected with the strong dependence of the position and extension of the radiating layers on the direction of sighting. Making use of the results presented in the current chapter, it can be easily shown that the intensity of the monochromatic thermal radiation of frequency v coming in the direction of the zenith angle 0 through the upper atmospheric boundary ( p = 0) is determined by the relation (9.139) where Ev[T(p)]is the intensity of the monochromatic radiation of a blackbody at temperature T of the level p (the Planck function); p o is the pressure at ground level; ,u = cos 0 ;and P,(p, p ) is the atmospheric transmission function. The first term in the right-hand part of (9.139) represents the atmospheric thermal radiation, while the second term characterizes the radiation of the earth’s surface reaching the upper atmospheric boundary. Note here that (9.139) is obtained for the case when the conditions of local thermodynamic equilibrium are fulfilled and the emissivity of the earth’s surface in the examined spectral region is assumed to be unity. In order to determine the atmospheric temperature (thermal sounding) (9.139) is most often reduced to the integral Fredholm equation of the first type, according to any of the following methods: (1) The dependence of the outgoing radiation upon sighting angle for a fixed frequency is considered. (2) The frequency dependence of the outgoing radiation for a vertical direction is considered. In this case in order to derive the integral Fredholm equation of the first type (and not a more complex one), it is necessary either to ignore the frequency dependence of the Planck function or to give its approximate representation.

It is known that the integral Fredholm equations of the first type are not correct in the classical sense. This, in particular, leads to difficulties in the numerical solution. Besides, not all the mentioned integral equations have

9.9. Emission Spectroscopy

627

a solution and a unique solution. In our case there is no doubt of the possibility of solution, since if the instrument measures the outgoing thermal radiation it is clear that a certain temperature distribution provoking this radiation exists. Let us discuss in short the uniqueness of the solution of the problem of thermal sounding. In mathematics there are various methods to prove the uniqueness of a solution. However, in using these methods it is necessary to give form to all the functions in (9.139). Since such a procedure is related to certain simplifying assumptions, it is natural that the demonstration of the uniqueness of the solution will be valid on when these assumptions are taken into account. It may turn out, for example, that such demonstration cannot be applied to the real atmosphere. For this reason it is preferable to prove the uniqueness of our solution that is based on the general form of (9.139). Let us consider the integral equation derived in the case of measurements of the frequency dependence of the outgoing radiation (,u = 1): (9.140) Here the dependence E(v) has been ignored in view of the measurements being conducted in a small interval dv and also the term E(T,,)P(v, po). In (9.140), however, we did not take into consideration an important peculiarity of the generation of the outgoing radiation at different frequencies, namely, the notable dependence of the position of the radiating layers upon frequency. From Fig. 9.24 it is clear that taking account of the limited sensitivity of the instrument allows us to rewrite (9.140) as (9.141) that is, at least one of the limits of the integration depends on frequency (and both in the general case). Equation (9.141) can be easily reduced to the common Volterra equation of the first type and then of the second type, which has a unique solution. Similarly, it can be demonstrated that the uniqueness of the solution of the problem of thermal sounding also applies to that for angular scanning. Thus, for an instrument with infinite resolving power, the problem has a simple solution. In the real conditions, however, the spectral resolution of the instrument is finite, and when an instrument with a very low resolving power is used such as that for measuring the integral radiation of the entire 15-p carbon dioxide band, the solution, of course, is not unique.

628

Thermal Radiation of the Atmosphere

4900

T T

FIG. 9.24 Reconstruction of the vertical temperature distribution over land (Bismark) by the method of optimal approximation.

The simplest way to solve the problem of thermal sounding might be the determination of such intervals dv in the CO, band for which the calculated effective temperatures would correspond to the true temperature of the atmosphere under approximately the same atmospheric conditions, independent of thermal stratification. Such measurements would then yield a set of temperatures for some atmospheric levels, while the intermediate values might be obtained by means of interpolation. The results of the calculations performed by Kondratyev and Yakushevskaya [133] for comparatively wide spectral intervals and for three atmospheric stratifications (Table 9.15) show that the levels at which the true and the effective temperatures coincide are strongly dependent on stratification. This means that in the given case it is impossible to state a simple correspondence between the outgoing radiation in some spectral region and the air temperature at a certain altitude. The calculations of Kondratyev and Yakushevskaya relate to those relatively wide spectral intervals in which the great variability of the absorption coefficient leads to the “washing-out” of the radiating layer. It is there-

629

9.9. Emission Spectroscopy

TABLE 9.15 The Effective Temperatures of the Earth-Atmosphere System. After Kondratyev and Yakushevskaya [133]

Spectral Region, E l

Stratification I11 t&-, C

Stratification I1

Stratification I

P . mb

t h ,C

P, mb

t h ,C

P. mb

1.54-1.67

16.9

960

-17.0

1000

-30.0

900

2.08-2.15

15.5

885

-18.0

765

-31.1

870

4.00463

1.6

700

-27.5

650

-38.2

690

800

-19.0

755

-31.2

865

13-14

-15.5

485

-36.0

555

-47.5

520

14-15

-47.7

200

-54.2

415

-61.2

310

15-1 6

-37.2

205

-53.0

435

-60.0

325

16-17

-20.8

440

-36.1

550

-46.2

540

17-18

-20.0

445

-30.8

615

-38.0

700

12-13

11.2

fore of interest to know what results can be obtained in calculating the outgoing radiation for more narrow regions. Such calculations were realized by Houghton and Shaw [134] for intervals of the 15-p carbon dioxide band of 1- and 0.1-cm-l width for four temperature stratifications. The calculations showed that the heights at which the true and the effective temperatures coincided were considerably less varying. The same authors, however, stated that this can be explained by a great similarity between the examined temperature distributions. We now consider some methods for solving the integral equations of the problem of thermal sounding. King [135-1381 who was the first to express the idea of using the outgoing radiation of the atmosphere to determine its thermal state, suggests engaging the angular dependence of the outgoing radiation. The intensity of the outgoing monochromatic radiation can be written as I,,(O,p ) =

lcoE,,(t)e-(T’p) 1 dt P

(9.142)

where z is the optical thickness of the atmosphere, reckoned from its upper boundary.

630

Thermal Radiation of the Atmosphere

Let us examine one of the first variants of the solution of (9.142) which was based on the application of the approximate method of Volterra as proposed by King. Let the Planck function be written as a series: n

E,.(z) =

C. aiSi

(9.143)

i=1

where

si = {

1 in the interval ziPl < z < zi 0 in the remaining atmosphere

Here the ai are numerical coefficients, and the summing up over the index i is equivalent to the division of the atmosphere by n levels. Substituting (9.143) in (9.142), we have (9.144) where (9.145)

- -

Select n values of ,u equal to ,u = ,uj ( j= 1, 2 . n). Designating Ii = Z(0, pi),write (9.144) in the form of a totality of n equations for the determination of n coefficients a i : (9.146) Let us now extend the derived relations to the case of nonmonochromatic radiation within the limits of the frequency band dv. In this case, instead of (9.142) we have (see Sec. 9.1) (9.147) where P is the transmission function characterizing the portion of radiation transmitted by the given absorbing layer (this quantity can also be determined as a probability of the photon emitted in the band dv at the optical z, leaving the boundary of the field unaffected by the absorption). The substitution of (9.143) in (9.147) gives (9.148)

9.9. Emission Spectroscopy

63 1

Similarly, n

ailij

Ij=

j = 1, 2 - v . n

(9.149)

i=l

where

lij = P(zi-l,uj)

- P(ti/%j)

(9.150)

Solving the system of (9.149) with respect to the coefficients a i , it is possible to use (9.143) for the calculation of the Planck function E,(T). Making use of the familiar expression for this function, we can easily calculate the dependence of temperature upon optical thickness T(t).The optical thickness in its turn can be expressed as a function of the height z or pressure p , which enables determination of T ( z ) or T ( p ) . It is obvious that in the given case the transmission function as well as the vertical distribution of the concentration of the absorbing and radiating matter must be known. King illustrated the proposed method of calculation with an example based on the use of the results of surface observations over the emission of the earth‘s atmosphere at various zenith angles in the region of the 9.6-,u ozone band. The calculations made for the modeled “gray” horizontally homogeneous plane atmosphere showed that with a sufficient number of measurements of the angular distribution of the intensity of the downward atmospheric radiation, the calculated vertical temperature distribution is in qualitative coincidence with the observed. In later works [138] King revealed the possible formal application of the operational methods to the solution of the problem considered, and of the Laplace transformation in particular. King also worked out methods for determining the radiant fluxes and rates of heating of the atmosphere, the distribution of the radiation absorbents, the absorption band shapes, and the pressure at ground level or at the upper cloud boundary. We shall now consider the improved technique of defining the atmospheric temperature distribution as given by King [138]. Formerly, according to the earlier mentioned method of King, the atmosphere was arbitrarily divided into isothermal layers. It appeared, then, that the solution thus obtained was strongly dependent on the approach to such division because the choice of arbitrary isothermal levels would force the solution into very severe limits. King proposed forego such division in favor of an optimal one in the process of solving the problem. Integrating by parts in (9.142), we easily obtain (9.151)

632

Thermal Radiation of the Atmosphere

Let us present the function sought for in the form (9.152)

tj)

where d(z - ti)is the Dirac delta function. In dividing (9.152), the layer limits are not yet fixed. From (9.151) and (9.152) it is possible to derive a system of linear equations for the set of angles i (i = 1, 2 . n):

-.

I,@, pi)- E,,(O)= 2 (LIE), e-W@l)

(9.153)

i

Substituting x j

= exp(-tj),

we come to a system of nonlinear equations:

Z,,(O, p ) - E,.(O)=

xi ( o E ) j

(9.154)

Xl’@l

With a subsequent selection of directions, 1

-i = 0, 1 , 2

Pi

- - - 2n - 1

(9.155)

we obtain a series of equations of the increasing power ai=xujx:

i=0,1,2..-2n-

1

(9.156)

i

where ai= Z(i) - E,(O)

uj = ( d E ) j

(9.157)

Using the known algorithm of the solution of the system of nonlinear equations, it is possible to determine not only but also the optimal division of the atmosphere into layers. The numerical example treated by King showed the advantage of the new method, which is also evident when it is applied with the quadratic formulas. The Gauss quadratic formula is more precise than the NewtonCoutes, which includes the predetermined division of the integration interval. It is natural that the most vulnerable point of the method of thermal sounding based on measurement of the outgoing radiation is the assumption of the horizontal optical homogeneity of the atmosphere. If we take into consideration that, in order to derive the information required for the solution of the problem, we need measurements in an extensive range of angles, that is, measurements covering large atmospheric regions horizontally,

9.9. Emission Spectroscopy

633

it becomes clear that the presence of clouds and also of horizontal temperature gradients practically prohibit the application of the above method to the determination of the temperature in the troposphere. The stratosphere is surely a more favorable region from the point of view of the possibilities that are provided by the measurement of the angular distribution of the outgoing radiation. The idea of the thermal sounding of the atmosphere by means of measuring the frequency dependence of the outgoing radiation was first expressed by Kaplan [139-1411, who suggested using these measurements of the radiation in the spectral intervals of 5-cm-1 width in the carbon dioxide emission band. After determining the thermal stratification from such measurement data it becomes possible to use the results of similar measurements in the region of the absorption of radiation by water vapor for the determination of the vertical water vapor concentration. Later Kaplan [141] calculated an example of practical application of the proposed method. The intensities of the upward thermal radiation at a level of 50 mb were calculated for a great number of the vertical temperature distributions given for seven isobaric surfaces: 1000 mb (earth’s surface), 700, 400, 300, 200, 100, and 50 mb. Since the vertical temperature distribution in the given case is characterized by temperature values at seven levels, it suffices to calculate the radiant intensity for seven frequencies. With a table of radiant intensities Iv,o calculated for different stratifications at hand, and knowing the measured outgoing radiation for seven frequencies I,, it is possible to select the temperature distribution from the table in such a way as to secure the minimal value of X,,(I,, In order to find the “best” temperature distribution, Kaplan proposed to use the formalism of the theory of perturbations. After certain analyses Kaplan reached a conclusion that in a number of cases the coincidence of the true temperature distribution and that calculated from the outgoing radiation (assumed to be measured) was quite satisfactory. In other cases, however, even these idealized calculations could not give a satisfactory agreement. Kaplan was the first to examine the effect of measurement errors on the solution of inverse problems. As was already mentioned above, the numerical solution of the integral Fredholm equation of the first type presents great difficulties. It is therefore particularly important to investigate whether the actual primary data contain an unknown random error. Kaplan came to a conclusion that the noise of radio installations can interfere with the process of determining the atmospheric thermal distribution, and at times can prevent it entirely. Wark [I421 considered a simplified method of thermal sounding based

634

Thermal Radiation of the Atmosphere

on measurement of the outgoing radiation in three spectral intervals. Such measurements allow obtaining either mean temperatures over the three layers or the temperature and its mean gradients in two layers. A twolayer model was also suggested (see [143]). Another method to be applied in thermal sounding was suggested by Yamamoto [144]. If the considered region of the spectrum of the 15-p CO, band is divided into m intervals, the intensity of the outgoing radiation in each of them can be presented as dp

i = 1,2

... m

(9.158)

Since in the considered region there is very strong absorption, we may assume p,, = 00. Besides, it is possible to assume that

where E J p ) is the Planck function for a certain wave number (taken v = 680 cm-l) and ai are constant and dependent upon the considered spectral interval i. Instead of (9.158) we now have

1, E ( p ) ?@-?- dp 00

aP

i = 1,2,

- - .m

(9.160)

where

7$ =Zi

(9.161)

Qi

Yamamoto suggested that (9.160) be solved by giving form to E ( p ) as a certain analytical representation, for example, as a finite series. Yamamoto investigated a case where the initial experimental data are the results of measurements of the outgoing radiation in four spectral intervals of 5-cm-1 width in the region of the 15-p CO, band: 665-670, 675-680, 686-691, and 692-697 cm-l. The first of these intervals contains the Q branch of the band with the most intensive absorption. The above assumptions demand that the function E ( p ) be presented in the form of a series with four terms. The use of the power series and also of the expansion into the Legendre and Chebyshev polynomials showed that in all the three cases the accuracy of the analytical presentation of the function E ( p ) corresponding to various real vertical temperature profiles is roughly the same. This, however, does not eliminate the problem of determining the system of functions whose participation in the expansion might give the best approximation of E ( p ) with a given number of the terms in the series.

635

9.9. Emission Spectroscopy

Let us consider the solution of the problem with the use of the Legendre polynomials. Introduce, instead of the pressure p, a new constant:

x

= 0.6970~'" -

1.5050

(9.162)

It should be noted that the interval of pressure variations embraces values practically from 0.2 to 600 mb. The transmission of the atmosphere over the level 0.2 mb is 0.99, even for the intervals of strong absorption 665 to 670 cm-l. The atmospheric thickness containing the levels under 600 mb completely absorbs the radiation in the interval of the least intensive absorption. The variation in pressure from 0.2 to 600 mb corresponds to the variation of the constant x from - 1 to + l . Thus, instead of (9.160) we have

3'i = -

I

1

-1

E ( x ) apio dx

ax

i

=

1,2,3,4

(9.163)

Let us now use the following representation: (9.164)

where P,"(x) is the Legendre polynomial of the nth order, and cin are constants that can be determined from the known values of the transmission function Pi on the basis of the use of the orthogonality of the Legendre polynomials according to

-1

i = 1,2, 3 , 4 n

Pi(x)PAo(x)dx

= 0,

1

..

m

(9.165)

Using the Gauss quadrature formula, we find m

1

I

-1

Here x1

Pi(x)P,"(x) dx

=

2 j=1

aiPi(xj)Pko(xj)

- .. x m are the zeros of the Legendre polynomials, 1

The values xi and a j can be taken from tables.

(9.166)

and (9.167)

636

Thermal Radiation of the Atmosphere

Let us further write (9.168) Substituting (9.164) and (9.168) into (9.163), we obtain the following system of equations for the determination of the coefficients u i : 3-i =

-2{c~,u,'

+ gcilulr + &ci,u,r +

+Ci3U3'}

(9.169)

The constant cin values are tabulated in Table 9.16. TABLE 9.16 Constants c,

Wave Number Interval

CiO

Ci1

ciz

Ci3

-0.08274

665670 (i = 1)

-0.4950

0.28109

0.62328

675-680 (i

2)

-0.5000

0.00786

0.91 893

0.26016

686-691 (i = 3)

-0.5000

-0.14694

0.78455

0.53207

692-697 (i = 4)

-0.5000

-0.32042

0.55958

0.64852

=

As seen, the Yamamoto method of determining the vertical temperature profile from measurement data on the outgoing radiation in four spectral intervals is very simple and convenient. It should be expected, however, that the use of only four spectral intervals does not allow obtaining finer detail of the vertical temperature distribution. Even the position of the tropopause cannot be fixed (note here that the above analytical representation suffices to describe the temperature variation solely within the range 1 to 400 mb). It is obvious that the detailed temperature profile can be obtained only under the condition that the division of the spectral region containing the 15-p CO, band be more elaborate, which demand meets certain difficulties. They are, first, that the use of intervals narrower than 5 cm-l leads to technical difficulties of measuring very small radiant fluxes. Second, with an increasing number of linear equations (due to more numerous measurements), inaccuracies begin to appear in the calculation, which tend to compound themselves so that the solution gets worse instead of getting better and especially so when interpretating real measurements containing the unknown random error. The first difficulty can be

9.9. Emission Spectroscopy

637

overcome by using the new methods of infrared spectroscopy or the idea expressed by Houghton [145, 1461. Since the 15-p CO, band has a fairly regular structure, it is natural to use the interferometric method, which enables summation of the energy of small spectral intervals of various lines. According to Houghton, the use of the Fabri-Perrault interferometer (with the half-width of the transmission band about 0.1 cm-' and supplied with an interferential filter isolating the spectral interval about 21 cm-I wide) allows receiving enough energy to be recorded. Note that, for example, the interval 669.7 to 693.7 cm-l contains 15 lines of the band 00'0 to 01'0. In this case the total width of the spectral interval of the recorded radiation is 1.5 cm-'. The receiver whose area is 1 mm2 and the sighting angle 5' will be irradiated by the energy amounting to 2.10-1° W, provided the temperature of the radiating layer of carbon dioxide is 255'K. In order to measure such a flux of radiation, it is necessary to have a receiver with the threshold sensitivity of the order of 10-l1 W, which is quite possible. Smith, in discussing the idea of Houghton, showed that the interferometer with germanium-filled interstices provides a still greater light power. In such a case it suffices to have a receiving area of 9 mm2, with the threshold sensitivity 10-lo W, which approximately corresponds to the standard optical-acoustical receivers. The application of the interferometric method opens up prospects of a considerable increase in the spectral resolving power and therefore of a higher accuracy of the thermal sounding. The second difficulty connected with the Fredholm equation can be overcome in two ways: (1) Malkevich and Tatarsky [147, 1481 suggested using the statistical methods in thermal sounding. As has been already mentioned, with this problem it is important to find the optimal approximation of the function T ( p ) . Such approximation is provided by systems of statistically orthogonal functions whose exact theory was worked out by Obukhov [149]. This theory is applicable to the finding of T ( p ) . If T ( p ) is presented in the form (9.170) where the T ' ( p ) are random departures from the normal T ( p ) (that is, T' T),which with the corresponding aerological data available is known for any point of the globe and any time interval, then the integral equation for determining T ' ( p ) is easily obtained. As was shown by Obukhov [149], the optimal approximation of the random function (for example, T ' ( p ) ) is realized through a system of orthogonal functions that are the proper

<

638

Thermal Radiation of the Atmosphere

functions of the correlation function, which in the given caset is B, . Thus, expanding T ’ ( p ) into a series, (9.171) we obtain a system of algebraic equations for the determination of t k . The use of optimal systems does not eliminate the unstability of solution related with the inaccuracy of the inverse problem; however, since these systems provide the best approximation with a small number of the serial terms (9.171), the obtained solution is sufficiently accurate. Figure 9.24 gives an example of the reconstruction of the vertical temperature distribution according to the method of optimal approximation. Evident are the advantages of this method compared with that of Yamamoto. Malkevich and Tatarsky [147] also considered the effect of measurement errors on the determination of T’(p).In order to lessen this effect the authors suggested starting with the initial data for more spectral intervals than required by the number of the basic vectors considered. The method of optimal approximation was further developed by Kozlov [150]. Kozlov analyzed the measurement possibilities of the method of thermal sounding, taking into account the statistical properties of random errors. He treated the case of independent normal distribution of errors with zero mean values and equal dispersions. The system of linear equations obtained in this case (by the method of the least squares) is not well grounded, particularly because of the unstability of the inverse problem. The direct solution of such a system is meaningless, since at least some of the unknown coefficients determined would introduce error, leading to the obtained temperature profile’s being far different from the actual. Thus, in a certain sense, the errors of the determination of the quantitizs sought with the help of the system considered (or of the measurement method considered, to put it more exactly) exceed the possible variation limits of these quantities. In order to isolate the stable part of the solution of this system, it is necessary to compare the range of variation of the measured quantities with their measurement errors in an actual experiment. For this purpose a very general method is used, based on the application of the quadratic informational metrics, which to a certain degree is in accord with the measurement method. Kozlov then further developed the optimal method of processing the experimental data; this may be taken as In setting BT at a discrete number of levels, this constitutes a basic system of vectors that are proper vectors of the correlation matrix BT(Px,).

9.9. Emission Spectroscopy

639

the method of stabilization of the inverse problem. The peculiarity of this method consists in the fact that the solution is sought for in the expansion over a system of equations naturally agreeing both with the characteristics of the actual atmospheric temperature distributions and with the properties of the measurement method itself, including the character of the dependence of the directly measured spectral radiant fluxes upon the temperature distribution sought for, and the statistics of the measurement errors. In order to reach the stability of the mentioned expansion, only such terms are kept whose coefficients are determined with error not exceeding their real variation range. Kozlov worked out a method of evaluating the effectivness of this determination of the vertical temperature profile. The quantitative effectivness characteristic is taken as the number of the temperature distributions that are distinguishable according to the given measurement method. Analysis of the numerical material, borrowed from Malkevich and Tatarsky’s work [147], showed that with a given selection of spectral intervals to measure the outgoing radiation fluxes and with a fixed accuracy of measurement of these fluxes (error dispersion of 0.16’K), it is impossible to separate from measurement data more than a hundred of all temperature profile varieties. Generally speaking, the obtained characteristic of the effectivness of this method is not high. For instance, the direct measurement of temperature at ten atmospheric levels with about 1 O acuracy secures the determination of approximately 1500 temperature profiles. A better effectivness can be reached by changing the selected spectral intervals as well as by increasing the accuracy of measurement of radiant fluxes. From the above discussion it is clear that the method of optimal approximation requires knowledge of mean vertical temperature distributions and also of the correlation functions, which can be obtained by processing direct measurement data on a great number of temperature distributions. Ideally, this information must be available for any season and location. Only the first steps have been taken in this direction and much preparatory work is needed before this method can be applied. However, Malkevich [147] points out that to this effect “foreign” correlation functions may be used, but these must be specified in further studies.

2. Method to Avoid Dijiculties of Solving Integral Fredholm Equations. This method for solving Fredholm equations of the first type is based on recent works [151, 1521 devoted to approximate solutions of the integral equations. Let us consider this method, analyzing the results of Philipps’s work

640

Thermal Radiation of the Atmosphere

[153]. Here the problem of thermal sounding reduces to the solution of the following equation : (9.172) where Z(x) and K(x, y ) are given functions, E ( X ) is an arbitrary function characterizing the random measurement error, and f(y) is the function sought. It is assumed that

When solving (9.172) we obtain a family of solutions F. The main problem now is to choose the true solution fs. For this purpose additional information about the desired function is needed. One may reckon that the true solutionf, is in some sense the smoothest function. Out of various smoothness conditions Philipps suggests the following:

In order to solve (9.172) and (9.173) numerically, let us consider a matrix approximation of this problem. Equation (9.172) will be changed for n i=O

wiKijfi = g j

+

j

E~

= 0, 1

---n

(9.174) (9.175)

where wi symbolizes weights of the quadratic formula. Introduce the following matrix designations :

A

= {wiKji}

d =ki>

A-l = { a i j ) --t

&

(9.176)

= {&j}

Equation (9.174) can be rewritten as (9.177) The use of the condition (9.173) leads to the “true” solution’s being determined from the relation

2 = ( A + yB)-’Z!

(9.178)

where B is the matrix whose terms are conventionally expressed through

64 1

9.9. Emission Spectroscopy

the elements aii of the inverse matrix A - l , and y is the selected smoothing parameter. After considering the application of this method to a number of integral equations Philipps came to the conclusion that it will operate if the unknown function is relatively smooth. Twomey [154] examined other conditions of smoothness referred to the function sought and also somewhat improved the method of Philipps. On the basis of these methods for solving the integral equations Wark [ 1551 reconsidered the problem of atmospheric thermal sounding. Wark points out three stages of the solution of (9.139). The first is the introduction under the integral sign of the term corresponding to the earth’s radiation. This is easily done if we assume that the emissivity of the earth’s surface equals unity. Then the atmosphere can be considered infinitely deep and isothermal below the level of the earth’s surface. Another important step is the selection of the independent variable, which is connected with the necessity of obtaining an approximately equal amount of information on the temperature profile over the whole investigated thickness of the atmosphere. Since the radiation is measured in different spectral intervals, the dependence of the Planck function upon frequency must be taken account of. Wark suggests that this be done as follows: B ( I i , T)= U i B ( B , ,

T)+ pi

(9.179)

where I, is a fixed frequency near the center of the spectral region considered, and the iji are the centers of the ith spectral intervals. Using the calculated radiant intensities for six spectral intervals of 5 cm-l width in the 15-,u CO, band (Table 9.17) as the initial data, Wark found the atmospheric TABLE 9.17 The Calculated Values of Radiation for Six Spectral Intervals of 5-cm-’ Width in the 15-p CO, Band for the Standard Atmosphere. After Wark [I551

Interval

Central Frequency, cm-I

Radiation, erg/cmz sec ster cm-I

669.0 677.5 691 .O 697.0 703 .O 709.0

56.900 46.249 44.206 46.094 53.518 64.244

642

Thermal Radiation of the Atmosphere

temperature distribution by means of a method similar to those of Philipps [153] and Twomey [154]. In the methods considered above for solving the problem of thermal sounding, there is no account taken of a number of complicating circumstances that may interfere with the practical solution. In particular, when measuring the radiant fluxes in the 15-,u C 0 2 band, it is necessary to take account of the fact that this band overlaps the absorption bands of water vapor and ozone. If we assume that the transmission functions will be fulfilled in the region of the 15-,u band according to the following relation:

(which must be carefully checked), we have the general formula for the radiant intensity in this spectral region :

(9.180) where P c o , , P H s o ,and Po, are the transmission functions of carbon dioxide, water vapor, and ozone, respectively. The exact solution of the problem of thermal sounding on the basis of (9.140) is extremely complicated, if at all possible. Therefore it was suggested, assuming that the contributions into the radiation due to water vapor and ozone are small, that the second and third terms in the right-hand side of (9.180) be evaluated for the conditions of the standard atmosphere. Introducing similar corrections to the measured intensity of the outgoing radiation, it is possible to find the first term in the right-hand side of (9.180) and to use these data in order to solve the problem of thermal sounding in the first approximation. The second and subsequent approximations, if necessary, can be obtained by specifying the contributions to the radiation of water vapor and ozone on the basis of data about the atmospheric stratification for the previous approximation, and also by specifying the water vapor and ozone content with the help of analogous measurements in the respective absorption bands. When treating the problem of thermal sounding we also neglected the fact that the transmission functions are, generally speaking, dependent on temperature. Taking account of the temperature dependence of the trans-

9.9. Emission Spectroscopy

643

mission functions should obviously be made also by means of successive approximations. Besides, it is necessary to consider the temperature dependence of the transmission functions when selecting the spectral intervals for the thermal sounding. The optimal selection of spectral intervals is a complex task. It is determined first of all by the optical density. Also important are the homogeneity of the optical density along the entire interval overlapping the absorption bands of water vapor and ozone, temperature dependence of the intensities of the spectral lines responsible for the transmission of the spectral interval considered, and the possibilities of experimental instrument error. So, in selecting the spectral intervals, the “microstructure” of each interval must be considered; that is, there must be full account of the available theoretical and experimental data on the position, intensities, and half-widths of the spectral lines of various gases, and also account of the dependence of line parameters upon temperature and pressure. The analysis of the possible errors in the solution of the problem of thermal sounding must not overlook the fact that the constancy of the volume concentration of CO, assumed in the above methods may not hold true in the atmosphere. Certain difficulties are also presented by the finding of the transmission function of narrow spectral intervals for the real atmosphere and by the distortion of instrument readings. The planning of satellite experiments, and in particular the thermal sounding, was given detailed treatment in Marchuk’s work [131]. Analyzing the general transfer equation and the conjugate equation, Marchuk studied the so-called conjugate functions, which he referred to as the functions of the value of information. With the help of these functions it is possible to solve in the general statement the problem of planning the outgoing radiation measurements, particularly in application to the thermal sounding. At the present time in the United States there are models of instruments for thermal sounding. One of the more recent models [156, 1571 is a spectrometer patterned after the Fasti-Ebert scheme, having a 541-1. grating. The instrument has six channels to measure radiation in the 15-,u CO, band. The seventh channel measures radiation in the transparency window at 11.1 p. The balloon tests conducted in September, 1964 [157], were successful and demonstrated the possibility of obtaining a temperature profile approaching the real. It is stressed that the radiation measurements should be carried out with a high degree of accuracy, since random errors of 3 to 5 percent may completely exclude the possibility of a solution. One of the essential shortcomings of the method of thermal sounding based on the measurements of the outgoing radiation in the 15-p CO,

644

Thermal Radiation of the Atmosphere

band is that it cannot supply satellite data on the temperature in the undercloud layer. Since a great and very important part of the atmosphere is often obliterated by clouds, it is necessary to have methods for determining the temperature distribution under the cloud cover. In this connection it is of interest to use the radiation in the microwave range [158, 1591. In particular, the oxygen line near the 5-mm wavelength can be used (as known, oxygen, as is carbon dioxide, is uniformly mixed in the atmosphere). The measurements in this spectral region have at least two advantages:

(I) The microwave radiation can pass through the majority of clouds. (2) In the microwave region it is possible to obtain a much higher spectral resolution. In particular, the emission line shape can be observed, which simplifies the interpretation of the measurement results. However, further investigations are necessary for successful application of the radiation in this spectral interval. Let us now dwell on the methods for determining the temperature of the underlying surface. For this purpose it is necessary to use the spectral interval without atmospheric absorption. Then the satellite measurements of the radiation in this interval will allow finding the temperature of the underlying surface, provided the emissivity of the earth’s surface is known. However, it is practically impossible to find such a spectral interval in view of the absence of intervals completely free of absorption. Wark et al. [I601 proposed using the transparency window 8 to 12 p, excluding the atmospheric “background” from the measured outgoing radiation (that is, excluding the absorption and emission by water vapor and ozone). The importance of atmospheric background becomes evident in the analysis of the measurements of the outgoing radiation in the region 8 to 12 p, made from the TIROS meteorological satellites. The measured temperatures of the underlying surface are often much lower than in reality, which is accounted for by the effect of the atmospheric background. Thus, in order to find the exact underlying surface temperature, it is necessary to determine the contribution of the atmosphere to the outgoing radiation, that is, to solve the problem of the atmospheric radiative transfer, which is impossible without knowledge of the temperature distributions and the absorbents. Wark et al. [I601 proposed to take account of the absorption and radiation of the atmosphere on the basis of the calculated radiation for numerous atmospheric models. However, since this radiation is strongly dependent on concrete conditions, the use of mean values may lead to considerable errors in the determination of the underlying surface temperature. Another

9.9. Emission Spectroscopy

645

means of taking account of the atmospheric background was suggested by Malkevich [161]. Let us consider the so-called transmission function of the atmosphere : (9.181) which is also dependent on the variance of T ( p ) with time and on the distribution of the absorbents. In order to take into account these variations Malkevich suggests the use of data on the vertical structure of the temperature fields T ( p ) and the specific humidity q ( p ) (that is, the mean vertical profiles T ( p ) and q ( p ) ) , and also the correlation functions (or matrices in the measurement of T and q at discrete levels):

where T’ and q’ stand for the departures from the means, and the averaging is performed over all the realized vertical profiles. By using the mean profiles for the area above which the radiation was measured, it is easy to find the transmission function for these mean profiles, and consequently the first approximation of the underlying surface temperature Tl‘(po)and its departures from the mean Tl’(po)= Tl(po)- F(po). With the help of the method of optimal extrapolation it is then possible to specify the transmission function and to obtain the subsequent approximation of the underlying surface temperature. Boldyrev [i 62, 1631 calculated the transmission functions for the interval 8 to 12 ,u in the Northern Hemisphere. He found the latitudinal variation of the transmission function, its dependence upon season, the earth’s surface temperature, and the angle of observation. Boldyrev [162] noted at the atmospheric transmission functions show a marked variation in time and space. In this connection, when calculating the underlying surface temperature from satellite measurement data, it is necessary to use charts or tables of the transmission function, compiled for a given territory and season. Kunde [164] studied this problem for the 4-,u window. Summing up what has been said about the methods for determining the vertical temperature profile in the atmosphere and the underlying surface temperature, it should be stressed that the most urgent requirement for problem solution is the practical testing of the worked-out theory and the

646

Thermal Radiation of the Atmosphere

techniques of interpretating experimental results on the basis of balloon, rocket, and satellite data. Doubtless, the theoretical evaluation of the role of the restrictions introduced in the above schematic solutions is likewise important. The problem of thermal sounding, and in particular the determination of the underlying surface temperature, can be solved empirically by comparison of the fields of the outgoing radiation and of the temperature. Examples of such empirical analysis will be given in the next chapter. REFERENCES 1. Kuznetzov, E. S. (1940). On the setting up of the balance of radiative energy for absorbing and scattering media. Bull. Acad. Sci. USSR, Ser. Geograph. Geophys. No. 6. 2. Kuznetzov, E. S. (1941). Radiative heat exchange in moving liquids. Bull. Acad. Sci. USR, Ser. Geograph. Geophys. No. 1. 2a. Kostianoy, G. N. (1965). Effect of the reflectivity of the underlying surface on longwave radiant fluxes in a free atmosphere. Proc. Acad. Sci. USSR, Ser. Atmospheric Phys. Ocean 1, No. 9. 3. Lebedinsky, A. I. (1939). Radiative equilibrium in the earth’s atmosphere. Proc. Leningrad Univ., Ser. Math. 3, No. 31. 4. Kondratyev, K. Ya. (1949). Some problems concerning radiative heat exchange. Proc. Leningrad Univ., Ser. Phys. No. 7. 5. Kondratyev, K. Ya. (1950). “Transfer of Long-Wave Radiation in the Atmosphere.” Gostekhizdat, Moscow-Leningrad. 6. Miigge, R., and Moller, F. (1932). Zur Berechnung von Strahlungsstromen und Temperaturanderungen in Atmospharen von beliebigen Aufbau. Z. Geophys. 8, Nos. 1 and 2. 7. Moller, F. (1943). “Das Strahlungsdiagramm. Reichsamt fur Wetterdienst.” Springer, Berlin. 7a. Moller, F. (1944). Grundlagen eines Diagramms zur Berechnung langwelligen Strahlungsstrom. Meteorol. Z. 61, No. 2. 8. Dmitriyev, A. A. (1940). On the net radiation of the atmosphere. Meteorol. Hydrol. No. 11. 9. Dmitriyev, A. A., and Narovlansky, G. I. (1944). The calculation of long-wave radiant fluxes coming from a limited solid angle and from a semiinfinite atmosphere. Proc. Acad. Sci. USSR, Ser. Geograph. Geophys. No. 6. 10. Elsasser, W. M. (1942). Heat transfer by infrared radiation in the atmosphere. Harward Meteorol. Studies No. 6. 11. Elsasser, W. M., with Culberston, M. F. (1960). Atmospheric radiation tables. Meteorol. Monographs 4, No. 23. 12. Yamamoto, G. (1952). On a radiation chart. Sci. Rept. Tohoku Univ. Fifth Ser. 4, No. 1. 13. Robinson, G . D. (1947). Notes on the measurement and estimation of atmospheric radiation. Quart. J. Roy. Meteorol. SOC.73, Nos. 315 and 316.

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14. Shekhter, F. N. (1950). Calculation of thermal radiant fluxes in the atmopshere. Trans. Main Geophys. Obs. No. 22 (84). 15. Kondratyev, K. Ya. (1956). “Radiative Heat Exchange in the Atmosphere.” Gidrometeoizdat, Leningrad. 16. Kondratyev, K. Ya., and Nijlisk, H. J. (1960). Comparison of radiation charts. Geofs. Pura Appl. 46. 17. Shekhter, F. N. (1964). On radiation charts. Trans. Main Geophys. Obs. No. 150. 18. Niilisk, H. Y. (1962). On the determination of the intensity of long-wave radiation in the atmosphere. invest. Atmospheric Phys. (Tartu) No. 3. 19. Bolle, H.-J. (1964). Atmospheric emission in single water vapor lines and its calculation by means of radiation diagrams. Mem. Soc. Roy. Sci. Liege 9. 20. Takahashi, K., Katayama, A., and Asakura, T. (1960). A numerical experiment of the atmospheric radiation. J. Meteorol. SOC.Japan, Ser. i Z 38, No. 4 . 21. Kagan, R. L. (1965). Calculation of thermal radiant fluxes in a cloudless atmosphere. Trans. Main Geophys. Obs. No. 174. 22. Davis, P. A. (1965). The application of infrared flux models to atmospheric data. J. Appl. Meteorol. 4, No. 2. 23. Kondratyev, K. Ya., and Niilisk, H. Y. (1961). The new radiation chart. Geofs. Pura Appl. 49, Part 11. 24. Niilisk, H. Y. (1961). A new radiation chart. Proc. Esthonian Acad. Sci.,Ser. Phys. Math. No. 4. 25. Yamamoto, G., Tanaka, M., and Kamitani, K. (1966). Radiative transfer in water clouds in the 10-micron window region. J. Atmospheric Sci. 23, No. 3. 26. Shifrin, K. S. (1954). To the theory of the radiative properties of clouds. Proc. Acad. Sci. USSR 94, No. 4. 27. Gayevsky, V. L. (1955). Measurement of radiative fluxes in clouds. Trans. Main Gophys. Obs. No. 46 (108). 28. Marshunova, M. S. (1959). Calculation of net long-wave radiation with cloudy Arctic skies. Trans. Arctic Inst. 226. 29. Kuhn, P. M. (1963). Measured effective long-wave emissivity of clouds. Monthly Weather Rev. 91, Nos. 10-12. 30. Kuhn, P. M., and Suomi, V. E. (1965). Airborne radiometer measurements of effects of particulates on terrestrial flux. J. Appl. Meteorol. 4, No. 2. 31. Gates, D. M., and Shaw, C. C. (1960). Infrared transmission of clouds. J. Opt. SOC.Am. No. 9. 32. MacDonald, J. E. (1960). Absorption of atmospheric radiation by water films and water clouds. J. Meteorol. 17, No. 2. 33. Deirmendjian, D. (1964). Scattering and polarization properties of water clouds and hazes in the visible and infrared. Appl. Opt. 3, No. 2. 34. McDonald, R. K. and Delterne, R. W. (1963). Cirrus infrared reflection measurements. J. Opt. Soc. Am. No. 7. 35. Bauer, E. (1964). The scattering of infrared radiation from clouds. Appl. Opt. No. 2. 36. Chumakova, M. S. (1947). On the nocturnal variation of effective radiation in Karadag. Trans. Main Geophys. O h . No. 5 (67). 37. Pivovarova, Z. I. (1947). The net radiation of the effective surface. Trans. NZU Hydrometserv., Ser. i No. 39. 38. Czepa, 0. (1950). Uber die Nachtliche Abkulung der Erdoberflache und der bodennahen Luftschicht. Z. Meteorol. 4, No. 12.

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39. Yaroslavtsev, I. N. (1946). Nocturnal effective radiation in Tashkent. Proc. Acad. Sci. USSR, Ser. Geograph. Geophys. No. 5. 40. Beletzky, F.A. (1947). Data on radiation in Odessa. Trans. Local Obs. Hydrometserv. No. 1. 41. Galperin, B. M.,and Yegorenkova, G. S. (1962). On the daily variation of atmospheric radiation. Sci. Rept. Znst. Geol. Geograph. Lithuanian Acad. Sci. 13. 42. Eisenstadt, B. A., and Zuyev, M. V. (1952). Some features of the thermal balance of a sand desert. Trans. Main Geophys. 06s. No. 6 (7). 43. Berland, M.E. (1954). Diurnal variations of temperature, turbulent exchange and net radiation. Trans. Main Geophys. Obs. No. 48 (110). 44. Kondratyev, K. Ya., and Holm, T. S. (1959). Some results of measurements of thermal atmospheric radiation in daytime. Mefeorol. Hydro/. No. 11. 45. Gayevsky, V. L. (1959). Certain peculiarities of the radiation regime of central arctic. Trans. Arctic Znst. 226. 46. Paulsen, H.S., and Torheim, K. A. (1964). Atmospheric radiation in Bergen, December 1957-June 1959. Arbok Univ. Bergen, Mat.-Nut. Ser. No. ll. 47. Kirillova, T. V., and Koocherov, N. V. (1935). A comparison of measurements of radiant fluxes made with different instruments. Trans. Main Geophys. Obs. No. 39 (101). 48. Evfimov, N.G. (1938). Effective radiation totals for Slutsk. Meteorol. Hydrol. NO. 8. 49. Evfimov, N. G. (1951). Totals of effective radiation for some places in USSR and other quantities characterizing effective radiation with a cloudy sky. Trans. Main Geophys. Obs. No. 26 (88). 50. Sauberer, F. (1951). Registrierung der Nachtlichen Ausstrahlung. Arch. Meteorol., Geophys. Bioklimatol. B2, No. 4. 51. Chumakova, M. S. (1947). A contribution to the method of approximate calculation. Meteorol. Hydrol. No. 4. 52. Barashkova, E.P. (1960). Dependence of atmospheric long-wave radiation on meteorological elements. Trans. Main Geophys. Obs. No. 100. 53. Bolz, H.M. (1949). Die Abhangigkeit der infraroten Gegenstrahlung von der Bewolkung. Z. Meteorol. No. 7. 54. Kreitz, E. (1954). Registrierungen der langwelligen Gegenstrahlung in Frankfurt. Geofis. Pura Appl. No. 28. 55. Sauberer, F. (1954). Zur Abschatzung der Gegenstrahlung in den Ostalpen. Wetter Leben 6, Nos. 3 and 4. 56. Budyko, M. I., Berland, T. G., and Zubenok, L. I. (1954). The net radiation of the earth’s surface. Proc. Acad. Sci. USSR, Ser. Geograph. No. 3. 57. Berland, M. E.,and Krasikov, P. N. (1948). Experiments on a smoke screen as protection against early and late frosts. Trans. Main Geophys. Obs. No. 12 (74). 58. Berland, M. E., and Berland, T. G. (1952). Measurement of the effective radiation of the earth with varying cloud amounts. Proc. Acad. Sci. USSR, Ser. Ceophys. No. 1 . 58a. Bolz, H.M.,and Fritz, H. (1950). Tabellen und Diagramme zur berechnung der Gegenstrahlung und Ausstrahlung. Z. Meteorol. 4, No. 10. 59. Yamamoto, G. (1950). On nocturnal radiation. Sci. Rept. Tohoku Univ., Fifrh Ser. 2, No. 1. 60. Henschel, K. (1956). Teoretische Betrachtungen uber empirische Formeln der Gegenstrahlung. Z. Meteorol. 10, No. 10.

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61. Tavartkiladze, K. A. (1963). A formula for calculating effective radiation. Proc. Acad. Sci. Georgian SSR 32, No. 2. 62. Monteith, J. L. (1961). An empirical method for estimating long-wave radiation exchanges in the British Isles. Quart. J. Roy. Meteorol. Soc. 87, No. 372. 63. Swinbank, W. C. (1963). Long-wave radiation from clear skies. Quart. J. Roy. Meteorol. Soc. 87, No. 372. 64. Mani, A., and Chacko, 0. (1963). Studies of nocturnal radiation at Poona and Delhi. Indian f. Meteorol. Geophys. 14, No. 2. 65. De Coster, M., and Schuepp, W. (1957). Mesures de rayonnement effectif a Leopoldville. Bull. Acad. Roy. Soc. Colon. 3, No. 3. 66. Knepple, R. (1959). Zum Klima der atmospharischen Warmestrahlung. Ber. Deut. Wetferdiensfes7 , No. 51. 67. Dyachenko, L. N. (1960). Comparison of some methods for determining long-wave radiation. Trans. Main Geophys. Obs. No. 100. 68. Goisa, N. I. (1963). To the dependence of effective radiation on the temperature and humidity regime in the surface air layer. Trans. Ukra. Hydromefeorol. Znst. No. 35. 69. Kovaleva, E. D. (1951). The calculation of effective radiation at the earth’s surface and of atmospheric downward radiation. Trans. Main Geophys. Obs. No. 27 (89). 70. Kovaleva, E. D. (1952). Allowance for the distribution of water vapor in measuring effective radiation and atmospheric downward radiation. Trans. Main Geophys. Obs. No. 37 (99). 71. Mani, A., Chacko, O., and Iyer, N. V. (1965). Studies of terrestrial radiation fluxes at the ground in India. Indian f. Meteorol. Geophys. 16, No. 3. 72. Efimova, N. A. (1961). To the technique of calculating monthly values of effective radiation. Meteorol. HydroI. No. 10. 73. Galperin, B. M. (1949). The net radiation at the Lower Volga during the warm season. Trans. Main Geophys. Obs. No. 18 (80). 74. Kirillova, T. V. (1952). On the dependence of atmospheric downward radiation on cloud amount. Trans. Main Geophys. Obs. No. 37. 75. Kirillova, T. V. (1955). To the measurement and calculations of effective radiation. Trans. Main Geophys. Obs. No. 53. 75a. Kondratyev, K. Ya., and Yelovskikh, M. P. (1955). The distribution of effective and atmospheric downward radiation over the sky. Proc. Acad. Sci. USSR, Ser. Geophys. No. 5 . 76. Braslavsky, A. N., and Vikulina, Z. A. (1954). “Evaporation Rates from Surfaces of Reservoirs.” Gidrometeoizdat, Leningrad. 77. Schmidt, K.-H. (1951). Priifung der Strahlungsrechendiagramma von Moller und Elsasser durch Gegenstrahlungsmessungen bei gleichzeitig durchgefiihrten Radiosondenaufstiegen. 2.Meteorol. 5 , No. 11. 78. Kondratyev, K. Ya. (1949). On the calculation of effective radiation at the earth’s surface. Sci. Bull. Leningrad Univ. No. 24. 79. Kondratyev, K. Ya., and Yelovskih, M. P. (1953). On the radiation topography of the sky. Sci. Bull. Leningrad Univ. No. 31. 80. Kondratyev, K. Ya. (1950). Calculation of effective radiation at the earth’s surface. Sci. Bull. Leningrad Univ. No. 25. 81. Kondratyev, K. Ya. (1952). The distribution of effective radiation with zenith angle at a fully overcast sky. Sci. Bull. Leningrad Univ. No. 30.

650

Thermal Radiation of the Atmosphere

82. Kondratyev, K. Ya. (1953). The dependence of effective radiation on cloud amount. Proc. Leningrad Univ. No. 6. 83. Lohnquist, 0. (1951). On the effective radiation to various parts of a cloudless sky. Tellus 3, No. 3. 84. Lohnquist, 0. (1931). Calculation of effective radiation considering its diffuse nature. Arch. Geophys. 1, No. 15. 85. Kondratyev, K. Ya., and Vasilyeva, M. A. (1953). An approximate method for the calculation of effective radiation in inversions. Proc. Leningrad Univ. No. 5. 86. Falkenberg, G . (1954). Die Konstanten der hgstromschen Formel zur Berechnung der infraroten Eigenstrahlung der Atmosphare aus dem Zenit. 2. Meteorol. 8, Nos. 7 and 8. 87. Yamamoto, G . , and Sasamori, T. (1954). Measurement of atmospheric radiation. Sci. Rept. Tohoku Univ., Fifth Ser. 6, No. 1 . 88. Bennett, H. E., Bennett, J. M., and Nagel, M. R. (1960). Distribution of infrared radiance over a clear sky. J. Opt. SOC.Am. 50, No. 2. 89. Vaisala, V. (1929). Uber die Verteilung der Bewolkung auf dem Himmelsgewolbe. SOC. Sci. Fennica, Commentationes Phys.-Math. 4. 90. Kirillova, T. V. (1951). A comparison of various methods of determining effective radiation at the earths’ surface. Trans. Main Geophys. Obs. No. 27 (89). 91. Chumakova, M. S. (1949). A comparison of the thermoelectric and the condensation method of determining totals of nocturnal effective radiation. Trans. Main Geophys. Obs. No. 14 (76). 92. Berland, M. E. (1948). The theoretical foundations of the protection of plants against early and late frosts by means of smoke screens. Trans. Main Geophys. Obs. No. 12 (74). 93. Berland, M. E. (1952). On the scattering of long-wave radiation by aerosols and the selection of effective smokes for protection against early and late frosts. Trans. Main Geophys. Obs. No. 29 (91). 94. Krasikov, P. N. (1948). Protection against the early and late frosts by means of smoke and fog. Trans. Main Geophys. Obs. No. 12 (74). 95. Krasikov, P. N. (1952). The attenuation of long-wave radiation by smokes. Trans. Main Geophys. Obs. No. 29 (91). 96. Shifrin, K. S. (1951). The effect of fog on net radiation. Trans. Main Geophys. Obs. No. 27 (89). 97. Shifrin, K. S., and Bogdanova, N. P. (1955). The effect of mist on the net radiation. Trans. Main Geophys. Obs. No. 46 (108). 98. Malkevich, M. S. (1954). On the protection of plants against early and late frosts by means of smoke screens. Trans. Geophys. Inst. Acad. Sci. USSR No. 23 (150). 99. Kondratyev, K. Ya. (1950). Effective radiation at various heights in a forest. Sci. Bull. Leningrad Univ. No. 26. 100. Kuzmin, P. P. (1949). The net radiation in a forest during the melting of snow. Trans. Hydrol. Znst. No. 16 (70). 101. Kondratyev, K. Ya., and Podolskaya, E. L. (1953). The effective radiation of slopes. Proc. Acad. Sci. USSR, Ser. Geophys. No. 4. 102. Sivkov, S. I. (1949). On the determination of the water content of the atmosphere. Trans. Main Geophys. Obs. No. 14 (76). 102a. Bolz, H. (1948). Uber die Wirkung der Temperaturstrahlung des atmospharischen Ozons am Erdboden. Z. Meteorol. Nos. 7 and 8.

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103. Kondratyev, K. Ya., Ashcheulov, S. V., and Styro, D. B. (1965). Comparison of measured and calculated spectra of atmospheric thermal emission. Proc. Leningrad Univ., Ser. Phys. Chem. No. 16. 104. Kondratyev, K. Ya., Ashcheulov, S. V., and Styro, D. B. (1965). The spectrum of atmospheric thermal emission in the wavelength range 25-35 p. Probl. Atmospheric Phys., Leningrad Univ. Publ. No. 4. 105. Kondratyev, K. Ya., Styro, D. B., and Zhvalev, V. F. (1966). Radiative flux divergence in the spectral region 4 4 0 p at different atmospheric levels. Proc. Acad. Sci. USSR, Ser. Phys. Atrn.ospheric Ocean No. 1. 106. Kondratyev, K. Ya., Ashcheulov, S.V., and Styro, D. B. (1967). Analysis of spectra of atmospheric downward radiation. Probl. Atmsopheric Phys., Leningrad Univ. Publ. No. 5. 107. Kondratyev, K. Ya., Badinov, I. Y., Ashcheulov, S. V., and Andreyev, S. D. (1965). Some results of ground investigations of the infrared absorption spectrum and of atmospheric thermal emission. Proc. Acad. Sci. USSR, Ser. Phys. Atmosphere Ocean 1, No. 4. 108. Wark, D. Q., Yamamoto, G., and Lienesch, J. H. (1962). Methods of estimating infrared flux and surface temperature from meteorological satellites. J. Atmospheric Sci. 19, No. 5. 109. Sloan, R., Shaw, J. H., and Williams, D. (1955). Infrared emission spectrum of the atmosphere. J. Opt. SOC.Am. 45, No. 6. 110. Bastin, J. A., Gear, A. E., Jones, G. O., Smith, H. J. T., and Wright, P. J. (1964). Spectroscopy of extreme infrared wavelengths. 111. Astrophysical and atmospheric measurements. Proc. Roy. SOC.A278, No. 1375. 111. Bell, E. E., and Eisner, L. (1956). Infrared radiation from the White Sands at the White Sands National Monument, New Mexico. J. Opt. SOC.Am. 46, No. 4. 112. Bell, E. E., Eisner, L., Young, J., and Oetjen, R. (1960). Spectral radiance of sky and terrain at wavelengths between 1 and 20 microns. 11. Sky measurements. J. Opt. SOC.Am. 50, No. 12. 113. Bennett, H. E., and Bennett, J. M. (1962). Predicting the distribution of infrared radiation from the clear sky. J. Opt. SOC.Am. 52, No. 11. 114. Burch, D. E., and Shaw, J. H. (1957). Infrared emission spectra of the atmosphere between 14.5 and 22.5 microns. J. Opt. SOC.Am. 47, No. 3. 115. Bignell, K., Saiedy, F., and Sheppard, P. A. (1963). On the atmospheric continuum. J. Opt. SOC.Am. 53, No. 4. 116. Westphal, J. (1964). New observations of atmospheric emission and absorption in the 8-14 micron region. Mem. SOC.Roy. Sci. Liege [ 5 ] 9. 117. Williamson, E. J. (1964). Balloon measurements of emission from the 6.3 micron water vapor band. Mem. SOC.Roy. Sci. Liege [ 5 ] 9. 118. Vigroux, E., and Dehaix, A. (1963). Resultats d'observations de l'ozone atmospherique par la methode infrarouge. Ann. Geophys. 19, No. 1. 119. Goody, R. M. (1957). A simple grating spectrometer for skyemission studies. Quart. J. Roy, Meteorol. SOC. 83, No. 358. 120. Murcray, W. B. (1963). Infrared radiation from the atmosphere over Arctic Ocean. Science 141, No. 3583. 121. Sloan, R., Shaw, J. H., and Williams, D. (1956). Thermal radiation from the atmosphere. J. Opt. SOC.Am. 46, No. 7.

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122. Bolle, H.-J. (1958). Spektrale Messungen der Infraroten atmospharischen Gegenstrahlung. Ber. Deut. Wetterdienstes No. 51. 123. Bolle, H.-J. (1950). Ein einfacher Prismenspektrograph zur absoluten Messung infraroter atmospharischer Strahlung (4 bis 15 micron). Z. Angew. Phys. 12, No. 3. 124. Bolle, H.-J., and Quenzel, H. (1962). Messung der Atmoshparischen Gegenstrahlung und Wasserdampfabsorption auf der Hochalpinen Forschungstation Jungfraujoch. Arch. Meteorol. Geophys. Bioklimatol. B l l , No. 4. 125. Bolle, H.-J. (1962). Untersuchungen der atmospharischen Infrarotstrahlung (7.522 micron) am Golf von Neapol. I. Die Messapparatur. Geofis. Pura Appl. 53, No. 3. 126. Bolle, H.-J. (1963). The 15-26 Microm Sky Emission Spectrum at Jungsfraujoch (3570 m). Appl. Opt. 2, No. 6. 127. Bolle, H.-J., Quenzel, H., and Zdunkowski, W. (1963). Vergleiche zwischen berechneter und auf dem Jungfraujoch gemessener infraroter Himmelstrahlung (7.526 Micron). Geofis. Meteorol. (Genoa) 11, 281-294. 128. Bolle, H.-J. (1964). Atmospheric emission in single water vapor lines and its calculation by means of radiation diagrams. Mem. SOC.Roy. Sci. Liege [ 5 ] 9. 129. Bolle, H.-J. (1965). The influence of atmospheric absorption and emission on infrared detection range. Infrared Phys. 5, No. 3. 130. Gray, L. D., and McClatchey, R. A. (1965). Calculations of atmospheric radiation from 4.2 to 5 micron. Appl. Opt. 4, No. 12. 131. Marchuk, G. I. (1964). Equations for evaluation of meteorological satellites information and the setting up of inverse problems. Space Invest. 2, No. 3. 132. Kondratyev, K. Ya., and Timofeyev, Y. M. (1964). On the fine structure of thespectrum of thermal atmospheric emission. Space Invest. 2, No. 4. 133. Kondratyev, K. Ya., and Yakushevskaya, K. E. (1964). Angular distribution of the thermal radiation of the earth-atmosphere system in different spectral regions. Trans. Main Geophys. Obs. No. 166. 134. Houghton, J. T., and Shaw, J. H. (1964). The deduction of stratospheric temperature from satellite observations of emission by the 15 micron CO, band. Mem. SOC. Roy. Sci. Liege [ 5 ] 9. 135. King, J. J. F. (1961). Deduction of vertical thermal structure of a planetary atmosphere from a satellite. Planetary Space Sci. 7 . 136. King, J. J. F. (1956). The earth satellite vehicle as a stratospheric temperature probe. Proc. 7th Intern. Astron. Congr., Rome, 1956. 137. King, J. J. F. (1956). “The Radiative Heat Transfer of Planet Earth. Scientific Uses of the Earth Satellites.” Univ. of Michigan Press, Ann Arbor, Michigan. 138. King, J. J. F. (1965). “Meteorological Inferences from Radiance Measurements,” Final Rept. Contract Cwb-10883. G.C.A. Corp. 139. Kaplan, L. D. (1959). Inference of atmospheric structure from remote radiation measurements. J. Opt. SOC.Am. No. 10. 140. Kaplan, L. D. (1960). Practicability of stratospheric sounding from satellites. I.U.G.G. 12th Gen. Assembly, Helsinki, 1959. 141. Kaplan, L.D. (1960). The spectroscope as a tool for atmospheric sounding by satellites. FaN Instr. Zutom. Conj, 1960. Instr. SOC.Am., New York. 142. Wark, D. Q., and Popham, R. W. (1961). On indirect temperature soundings of the stratosphere from satellites. J. Geophys. Res. 66, No. 1.

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143. Merrit, E. S., and Chang, D. (1966). Satellite sensing of the lower stratospheric temperature structure to support SST operations. Conf. Aerospace Meteorol., Los Angeles, 1966. 144. Yamamoto, G. (1961). Numerical method for estimating the stratospheric temperature distribution from satellite measurements in the CO, band. J. Meteorol. 18, No. 5. 145. Houghton, J. T. (1961). The meteorological significance of remote measurements of infrared emission from atmospheric carbon dioxide. Quart. J. Roy. Meteorol. Soc. 87, No. 371. 146. Houghton, J. T. (1964). Stratospheric temperature measurements from satellites. J. Brit. Interplanet. Soc. 19, No. 9. 147. Malkevich, M. S., and Tatarsky, V. I. (1965). Determination of the vertical atmospheric temperature from outgoing radiation in the CO, absorption band. Space Invest. 3, No. 3. 148. Malkevich, M. S. (1965). On the relationship between the characteristics of the vertical structure of the long-wave radiation field and the temperature and humidity fields. Proc. Acad. Sci. USSR, Ser. Phys. Atmosphere Ocean No. 10. 149. Obukhov, A. M. (1960). On the statistically orthogonal expansions of empirical functions. Proc. Acad. Sci. USSR, Ser. Geophys. No. 3. 150. Kozlov, V. P. (1966). Reconstruction of the vertical temperature profile from the outgoing radiation spectrum. Proc. Acad. Sci. USSR, Ser. Phys. Atmosphere Ocean 11, No. 2. 151. Tikhonov, A. N. (1963). Solution of noncorrect problems and regularization method. Rept. Acad. Sci. USSR 151, No. 3. 152. Lavrentyev, M. M. (1959). On the integral equations of the first kind. Rept. Acad. Sci. USSR 127, No. 1. 153. Philipps, D. L. (1962). A technique for the numerical solution of certain integral equations of the first kind. J. Assoc. Computing Mach. No. 9. 154. Twomey, S. (1963). On the numerical solution of Fredholm integral equations of the first kind by the inversion of the linear system produced by quadrature. J. Assoc. Computing Machine 10. 155. Wark, D. Q. (1964). Theory of infrared and microwave sounding. Radiation Symp., Leningrad, 1964. 156. Dreyfus, M. G., and Hilleary, D. T. (1962). Satellite infrared spectrometer. Design and Development. Aerospace Eng. 21, No. 2. 157. Hilleary, D. T., Wark, D. Q., and James, G. (1965). An empirical determination of the atmospheric temperature profile by indirect means. Astron. Aeronaut. 205. 158. Gurvich, A. S. and Yegorov, S. T. (1966). Determination of the ocean surface temperature from its thermal radioemission. Proc. Acad. Sci. USSR, Ser. Phys. Atmosphere Ocean 2, No. 3. 159. Croom, D. L. (1966). The possible detection of atmospheric water vapor from a satellite by observations of the 13.5 and 1.64 mm H,O lines. J. Atmospheric Terrest. Phys. 28, No. 3. 160. Wark, D. Q., Yamamoto, G., and Lienesch, J. H. (1962). Methods of estimating infrared flux and surface temperature from meteorological satellite. J. Atmospheric Sci. 19, No. 5. 161. Malkevich, M. S. (1964). Some problems of the interpretation of satellite radiation measurements. Space Invest. 2, No. 2.

654

Thermal Radiation of the Atmosphere

162. Boldyrev, V. G. (1965). Calculation of atmospheric transmission functions in the spectral interval 8-12 micron for the Northern Hemisphere. Proc. Acad. Sci. USSR,

Phys. Atmosphere Ocean No. 7 . 163. Boldyrev, V. G. (1965). Use of outgoing radiation measurements for calculating the earth’s surface temperature and upper cloud boundary. Trans. Moscow Meteorol. Center No. 8. 164. Kunde, V. G. (1965). Theoretical relationship between equivalent blackbody temperatures and surface temperatures measured by the NIMBUS high resolution infrared radiometer. NASA, Spec. Publ. 89.

10 NET RADIATION

The net radiation is defined as the difference between the radiant energy absorbed and that emitted by the underlying surface, by the atmosphere, or by the system earth-atmopshere. The net radiation R of the ground surface consists of the direct and diffuse radiation and also of the atmospheric emission (downward atmospheric radiation) absorbed and retained by the underlying surface after heat losses due to the thermal emission of the underlying surface. This can be expressed as the following equation for the net radiation of the underlying surface :

R

=

Q(1 - A )

+ 6Go - Usurf

(10.1)

where Q are the fluxes (or totals) of incoming direct solar and diffuse radiation, A is the albedo of the underlying surface, Go and Usurfare the fluxes (or totals) of atmospheric emission and of the thermal emission of the underlying surface, and 6 is the absorptivity of the underlying surface. Since Usurf- 6G, = F, is the effective radiation of the underlying surface, (10.1) can be written as

R

=

Q(1 - A ) - Fo

(10.1 a)

It should be borne in mind that the concept of the net radiation is to a certain degree abstract. In actual practice we deal with the net radiation of an active layer whose thickness varies within a very wide range. In some cases (smooth underlying surfaces devoid of vegetation) it is very small; in others (plantations or water basins) it reaches high values of the order of meters and tens of meters. The net radiation of the atmosphere, Ra, consists of the direct solar and 655

656

Net Radiation

diffuse radiation q' absorbed by the atmosphere and also of the absorbed thermal radiation UgWfof the underlying surface. The emitted radiation consists of the heat losses due to the atmospheric thermal emission toward the earth's surface and spaceward. The first outgoing component is the atmospheric emission Go and the second is the radiation of the atmosphere U, to space. Thus, the net radiation of the atmosphere is expressed as

If P is the transmissivity of the atmosphere for thermal radiation then the thermal radiation of the underlying surface absorbed by the atmosphere can be presented as Usurf= (1 - P)Uo, where Uo is the upward flux of thermal radiation at ground level. The quantity Uo - Go = Fo , as has already been pointed out, is the effective radiation of the underlying surface, and PU,, U, = F, is the outgoing radiation to space from both ground and atmosphere. In view of this consideration the equation for the net radiation of the atmosphere can have yet another 'form:

+

(10.2)

In the net radiation R, of the system earth-atmosphere the incoming portion consists of the direct solar and diffuse radiation absorbed by the underlying surface and the atmosphere, while the expenditure or loss is determined by the outgoing radiation. The equation for the net radiation of the system earth-atmosphere therefore has the form

(10.3) This equation can also be rewritten: (10.4)

where Q, is the flux of direct solar radiation outside the atmosphere, A , is the albedo of the planet Earth. Let us now turn to the fundamental regularities of the net radiation for the ground surface, the atmosphere, and the system earth-atmosphere in view of their application for meteorological purposes. The importance of radiation data for dynamical meteorology has been discussed in Van Mieghem's work [l].

10.1. Observed Regularities in Variation of Net Radiation

657

10.1. Observed Regularities in Variation of Net Radiation of the Underlying Surface The study of the net radiation of the underlying surface is of extreme importance, as the net radiation is the main determinant of the climate. The magnitude of the net radiation of the underlying surface greatly affects the distribution of temperature in the soil and the adjacent air layers, which accounts for the particular role of the net radiation in the calculation of evaporation and snow melting, and also in a number of problems concerned with weather forecasting, such as the forecasting of late and early frosts and of fog. The study of the net radiation is also important to synoptic meteorology; for example, the lack of data on the net radiation makes it impossible to solve the problem of the formation and transformation of air masses. Finally, the investigation of the net radiation is connected with the effect of radiation on plant and animal life. All this accounts for the great interest in the study of the net radiation of the underlying surface shown by many investigators [2-271. It should, however, be noted that the number of stations engaged in prolonged observations of all the quantities involved in the net radiation or with direct measurements of the total net radiation is very small. This is due to the fact that until quite recently there have been no sufficiently reliable instruments for the direct measurement of the net radiation. Although in recent years satisfactorily constructed balance meters have been developed (such as the Yanishevsky and the Funk net radiometers and others), nevertheless the problem of reliable measurements of the net radiation still lacks solution. It is evident that the main features of the net radiation are determined by the factors that most influence the net radiation components. These include the duration of sunshine, the conditions of cloudiness and atmospheric transparency, the stratification of the atmosphere, and the character and state of the underlying surface. The net radiation of the underlying surface will be positive if the heat gain exceeds its loss, and negative in the reverse case. In the diurnal variation of the net radiation the former is usually observed in daytime and the latter at night. In the annual range for latitudes between 40' N and 4OoS, monthly values of the net radiation are positive for land and sea. In higher latitudes the monthly values become negative in winter months.

1. Diurnal Variation. Let us first characterize the diurnal variation of the net radiation. This problem has been treated in a great number of investi-

658

Net Radiation

gations. Observations show that, as a rule, the maximal positive values of the net radiation occur around noon, and the highest negative values at night. The night variation of the net radiation (the nighttime range of effective radiation) is small in comparison with its variation during the day. The curve of the diurnal cycle of the net radiation is usually asymmetrical in relation to noon; the afternoon values are somewhat lower, for the afternoon effective radiation exceeds the morning values, a feature that is clearly marked in southern and especially so in desert areas. The mentioned regularities in the diurnal variation of the net radiation can be seen in Fig. 10.1, which shows the daily cycle of the net radiation and its components according to Eisenstadt and Zuyev [2] for Tashkent (the measurement employed the Yanishevsky pyranometer and net radiometer). The values of direct solar, diffuse, and the net radiation were averaged from observations on clear summer days (August, 1969) in a sand desert. The other components of the net radiation were obtained by calculation. As seen from Fig. 10.1, in these conditions the leading component of the daytime net radiation is the direct solar radiation. Accordingly, the maximum in the net radiation is observed almost exactly at noon, between 11 and 12 h,

HOUR

FIG. 10.1 Diurnal variation of net radiation components (means of clear days). (1) Net radiation; (2) direct solar radiation; (3) downward atmospheric radiation absorbed by the ground surface; (4) diffuse radiation; (5) reflected shortwave radiation; (6) effective radiation; (7) radiation of the underlying surface.

659

10.1. Observed Regularities in Variation of Net Radiation

and is equal to 0.68 cal/cm2 min. The minimum of -0.15 cal/cm2 min occurs soon after sunset. The zero point in this cycle was observed between 06 and 07 h and between 17 and 18 h. Figure 10.1 shows that these points do not coincide with the moments of sunrise and sunset. In the morning the negative net radiation is observed 40 to 60 min after rise, while the transition from positive to negative values in the evening outruns sunset by about 1.5 h. This can be explained by the fact that the morning influx of heat due to the absorption of direct solar and diffuse radiation can compensate for the loss of heat due to effective radiation only some time after sunrise. In the evening hours the effective radiation predominates over the incoming part of the net radiation before sunset. Observations show that the transition from the positive net radiation to the negative, and vice versa, usually occurs at solar heights of about 5 to 15'. Table 10.1 gives the mean time of the zero point in the net radiation

Month

1

40

Latitude, deg 50

I

60 tl

April May June July August September October

5-7 5-7 4-6 5-7 5-7 5-7 6-8

16-18 17-19 17-19 17-1 9 17-19 16-18 15-1 7

5-7 4-6 4-6 4-6 5-7 5-7 6-8

16-18 17-19 18-20 17-1 9 16-18 16-18 15-17

5-7 4-6 4-6 4-6 5-7 6-8 7-9

tz

1618 17-19 18-20 17-19 17-19 15-17 . -14-16

for morning (tl) and evening ( t z ) on the fifteenth day of every month in dependence on latitude. The data were compiled by Sapozhnikova [3] and later corrected in view of more recent observations. It should be remembered that Table 10.1 does not include the case of snow cover, where the positive net radiation is observed at much higher solar altitudes (about 10 to 25'). This is caused by the great snow albedo affecting the net radiation to lesser values. The peculiarities of the dependence of the net radiation upon solar height

660

Net Radiation

at a clear sky can be illustrated by the data of Table 10.2, borrowed from [Chapter 7, Ref. 37bl. Table 10.1 clearly shows a later transition of the net radiation through the zero point and lower net radiation values in the presence of snow. In winter months in the north, and partly in the intermediate latitudes, the net radiation remains negative throughout 24 h. TABLE 10.2 Average Dependence of the Net Radiation (cal/cnz2min) upon Solar Height with Clear Skies. After Kirillova (Chapter 7 [37b])

State of Ground

Albedo,

Snowless

Snow

%

Solar Height, deg 0

5

15-25

-0.07

-0.04

50-80

-0.05

-0.04

10

15

20

25

30

35

0.03 0.12 0.21 0.32 0.41 0.48 -0.01

0.05 0.10 0.17 0.23 0.29

Cloudiness also greatly affects the transition of the net radiation through the zero point. With overcast skies the setting of the negative net radiation is retarded, since in these conditions the effective radiation as a cooling component of the net radiation greatly decreases. Sapozhnikova [3] found a high correlation between the moment of the zero net radiation and the time of the setting and disappearance of the night inversion in the lower air layer of 1.5- to 2-m thickness. Table 10.1 therefore also characterizes the time of appearance (f2) and disappearance ( t l ) of the night inversion in the above air layer. It should be noted that a number of investigations have found a parallelism between the diurnal variation of the net radiation and the vertical temperature gradients near the underlying surface. Sapozhnikova showed a correlation between the net radiation and the temperature difference at 20 to 150 cm above ground. The observations of Eisenstadt and Zuyev [2] give evidence of a parallelism between the net radiation variation and the temperature difference at the level 0 to 20 cm. Eisenstadt [4] also found a close correlation between the net radiation and the temperature of the soil surface. Doubtless, the cause of all these correlations is the close relationship between the temperature of the underlying surface and the adjacent vertical temperature gradients, and the effective radiation as the cooling component of the net radiation.

10.1. Observed Regularities in Variation of Net Radiation

66 1

The smooth diurnal range in the net radiation presented in Fig. 10.1 is observed, naturally, only with clear or fully cloudy skies (in the latter case, of course, the amplitude of the daily variation is notably smaller than with clear skies). In the case of partial cloudiness the daily variation of the net radiation becomes very irregular. This is illustrated by the curves of Fig. 10.2, plotted by Sapozhnikova [3]. The observations made at Koltushy (Leningrad region) on a clear and on a dull day show that the cloudy day gives a smaller range of the diurnal variation in the net radiation, with the variation itself being also more complicated.

Hour

FIG. 10.2 Diurnal variation of net radiation. (1) June average for Tashkent; (2) at Koltushy (Leningrad region) in a clear July day; (3) at Koltushy in a cloudy July day.

Along with the strong effect of the solar altitude on the direct solar and diffuse radiation, and with the albedo of the underlying surface, the cloud amount is the main factor determining the variability of the net radiation. In daytime the appearance and increase of cloudiness leads to reduction in both global radiation and effective radiation (it should be noted that the global radiation only shows a decrease with increasing cloudiness when averaged; isolated cases can show a reverse picture). At night the variation of cloudiness can affect only the effective radiation. Thus, on the average, the daytime positive values of net radiation with a cloudy sky decrease and the nighttime values also decrease in absolute estimation. However, with the partial cloud and the unobliterated sun when the global radiation is at its highest and the effective radiation is smaller than with a clear sky, the maximal positive values of the net radiation are observed. For instance, Chikirova [ 5 ] found during the observations at

662

Net Radiation

Dolgoprudnaya station (Moscow region) in August, 1947, that the net radiation with partial cloud was equal to 1.07 cal/cm2 min. To characterize the peculiarities in the daily variation of the net radiation with clear and with dull skies, Table 10.3 gives Mukhenberg’s [6] results of observations by means of the Yanishevsky net radiometer at Koltushy (Leningrad region). TABLE 10.3 Seasonal Changes in the Diurnal Variation of Net Radiation (cal/cmzmin). After Mukhenberg [6]

-

Clear

Cloudy

Hour

0 4 8

12 16 20 0

Summer Autumn -0.063 -0.022 0.392 0.632 0.312 -0.042 -0.070

Winter

-0.092 -0.088 -0.070 -0.102 0.045 0.077 0.390 0.028 0.120 -0.064 -0.080 -0.082 -0.079 -0.070

Spring

Summer Autumn

Winter

-0.076 -0.068 0.123 0.355 0.154 -0.048 -0.060

-0.018 -0.018 -0.021 -0.013 0.098 0.053 0.247 0.085 0.175 -0.006 -0.004 -0.012 -0.019 -0.003

-0.034 -0.029 -0.016 -0.024 -0.007 0.054 0.019 0.107 -0.008 0.068 -0.007 0.004 -0.005 -0.019

Spring

It is evident from Table 10.3 that cloudiness in summer leads to a considerable decrease in the net radiation, while on the contrary, cloudiness in winter helps to reduce the negative daily net radiation. The investigations of E. P. Barashkova et al. [Chapter 7, Ref. 37a] show that the averaged dependence of the net radiation upon solar altitude can be described by the following empirical equation :

R

= a(h, -

(10.5)

b)

where a, b are constant and h, is the height of the sun in degrees. For the constants a and b, the dependence on the albedo of the underlying surface is shown below: Albedo, %

a

b

10-20 20-30

0.013 0.012 0.006 0.007 0.004

10.0 9.8 7.4 7.4 8.5

5 W O 60-70 70-80

663

10.1. Observed Regularities in Variation of Net Radiation

With constant solar height (40’) and albedo, the variation of the net radiation in dependence upon cloud amount is characterized by the following data: Cloud amount, in tenths Net radiation, cal/cm* min

3

4

5

6

7

8

0.46 0.45 0.43 0.42 0.40 0.38

We see that with the increase in albedo from 10 to 80 percent, the net radiation shows an almost triple decrease. The increase in cloud amount from 3 to 8 reduces the net radiation by only 0.08 cal/cm2 min, that is, by about 20 percent. Thus we see that the net radiation is more “sensitive” to variations in albedo than to variations in cloud amount. The investigations of Barashkova et al. found a close correlation between the net radiation of a grassy surface and the magnitude of the absorbed shortwave radiation: R,$, - 0.06 (10.6) R= 1.20 At R,, > 0.06 cal/cm2 min this formula enables calculation of the net radiation from the absorbed radiation with an error of about f 10 percent. 2. Annual Variation. In order to characterize the regularities of the annual variation of the net radiation, let us consider Fig. 10.3, which presents isopleths of the net radiation for four points in different climatic zones (Yakutsk, Omsk, Vladivostok, Tashkent). These data of [Chapter 7, Ref. 37a] are also complementary to the daily variation of the net radiation. From examination of Fig. 10.3 it is evident that the maximal net radiation values are usually observed in June to July and are 0.45 to 0.55 cal/cm2 rnin in the north (62 to 64ON lat.) and 0.6 to 0.7 cal/cm2rnin in the south (40’s lat.). The minima fall during December and January, when the obtained values are 0.02 cal/cm2 rnin in the north (Yakutsk) and 0.2 cal/cm2 rnin in the south (Tashkent). In summer the nighttime values of the net radiation of the considered area vary but little and are -0.06 and -0.07 cal/cm2 min. In winter their variation is more pronounced, but the absolute values are very small (from -0.02 to -0.05 cal/cm2 min). The exception is the monsoon area (Vladivostok), where the highest net radiation occurs in April to May, and a small secondary maximum is observed in September.

3. The Effect of Wetting. Observations show that the net radiation of underlying surfaces greatly changes after their wetting by rain or irrigation. Table 10.4 gives the results of measurements of the net radiation over an

664

Net Radiation

-0.05

-0.05

-0.05

FIG. 10.3 Isopleths of net radiation (cal/cm2min). (a) Yakutsk; (b) Omsk; (c) Vladivostok; (d) Tashkent.

irrigated cotton field and over a semidesert in July, 1952 (by means of the Yanishevsky pyrgeometer and balance meter). These results were averaged by Eisenstadt et al. [7] from eight sets of observations on clear days at the state farm “Pakhta Aral” (Uzbek S.S.R.). It is evident from Table 10.4 that at night the net radiation of an irrigated field and a semidesert are approximately equal. In daytime the net radiation of the irrigated field considerably exceeds that of the semidesert. In the hours around midday the difference between these quantities is 0.30 cal/ cmz min. It appears that this difference is determined almost entirely by the inequality of their outgoing components: Owing to the higher surface temperature, the effective radiation of the semidesert exceeds by 0.25 cal/cm2

10.1. Observed Regularities in Variation of Net Radiation

665

TABLE 10.4 Daily Variation of the Net Radiation of an Irrigation Field and a Semidesert (cal/cm2rnin). After Eisenstadt et al. [7]

Area

I

Time, h 0-1

4-5

6-7

8-9

1 6 1 1 12-13 14-15 16-17

Irrigated cottonfield -0.07

-0.06

0.18 0.62 0.89 0.96 0.87 0.46

Semidesert -0.08

-0.06

0.13 0.47 0.66 0.66 0.56 0.30

18-19

0.03 -0.05

20-21

-0.07 -0.10

rnin that of the irrigated field. The remaining 0.05 cal/cm2 min characterizes the difference between the fluxes of global radiation reflected by the two surfaces (the semidesert whose albedo is higher than that of the irrigated cottonfield reflects 0.05 cal/cm2 min more radiation). Thus the reduction of the albedo and temperature of the underlying surface due to irrigation provokes a notable increase in its net radiation. This fact was first stated by Skvortzov [8]. The earlier reversal of sign of the net radiation in the morning and the later in the evening are also due to the reduction in the temperature of the irrigated field and the consequent decrease in its effective radiation, compared with that of the semidesert. Many investigations (see [9-111) have been carried out with the purpose of detailed study of the effect of irrigation on the net radiation. Their results all confirm the notable increase in net radiation due to irrigation. It can be said that in a moderate climate the increase in net radiation due to irrigation averages 20 percent, in steppe and wooded steppe it is 40 percent, and in the semideserts of central Asia 60 percent. It should be noted, however, that the effect of irrigation is strongly dependent on the development of vegetation. Since the change of net radiation by irrigation is caused by a change of albedo and the decrease in effective radiation due to the reduction in the temperature of the underlying surface, Eisenstadt et al. [7] proposed the following formula for the calculation in the net radiation A R (cal/cm2 min):

dR

=

(Q

+ q ) ( A - A') +

B(to

- to')

(10.7)

where Q + q is the global radiation flux, A and A' are the albedos of the ground before and after irrigation, to and to' are the temperatures of the

666

Net Radiation

ground before and after irrigation, and 4, is a coefficient equal to 0.008. Using (10.7), it is easy to explain why A R / R increases southward. This is obviously due to the fact that irrigation in the south causes a stronger reduction of the ground temperature than in the north. Kirillova [12] showed that (10.7) can also be used for the calculation of the difference in net radiation between water and land at two close points. Such calculations as well as observations revealed that in summer the net radiation of water basins, R , , exceeds that of land, RL,while in winter the reverse is the case. The observations of Kirillova at Lake Sevan (Armenia) give the following results: July RwlRL 1.39

Aug. 1.36

Sept. 1.04

Oct. 0.93

This can be accounted for by the fact that the water albedo notably increases in fall as the solar height reduces, and also by the reversal in sign of the temperature difference water-land (water becomes warmer than land, which causes the effective radiation for water to be greater than for land). 4. Effect of Forest. Also important is the variation of net radiation in forests. Under the trees both income (global radiation) and expenditure (effective radiation) in the net radiation will obviously decrease. Observations show that in summer the reduction of global radiation under the trees in daytime by far exceeds the decrease of effective radiation, due to which the net radiation in a forest is much less than in an open place. This conclusion follows, for example, from the observations of Golubova [28, 291 in forest barriers of the Kamennaya Steppe (Ukraine) and in the Saratov region (the Volga area). Table 10.5 displays Golubova’s results [29], obtained by means of the Yanishevsky pyrgeometer for two points of a forest barrier in the Kamennaya Steppe in June-July, 1951. As seen, at small solar heights (in the morning or evening) the net radiation in the forest is, in individual cases, only a few percent of the open field value. At night the net radiation in these cases was practically zero. It is natural that the net radiation under the trees must be strongly dependent on their quality and composition and on the development of foliage. For example, according to Golubova [29], the net radiation inside the forest barrier without foliage on a clear April day was 53 to 88 percent of the open field value. Comparison of these figures with Table 10.5 shows the great influence of leaves on the forest net radiation. In Rauner’s [30] estimation the ratio of the net radiation values over the forest ( R ) and inside ( R 2 , R3) shows strong change during 24 h and

667

10.1. Observed Regularities in Variation of Net Radiation

TABLE 10.5 Net Radiation in Forests. After Golubova [29] Net Radiation

% in Open Field

cal/cm2min

Hours

Point 1

Point 2

Point 1

08-09

0.02

10-1 1

0.06

4 8

12-13 14-15 1617

0.09 0.03 0.02

18-19

0.01

0.21 0.43 0.04 0.02

Point 2

10

24

4 5

61 10

14

15

is considerably dependent on the phase of vegetation (see Table 10.6 where R2 and R, denote the net radiation values under the coniferous trees with insignificant undergrowth of stand density 0.9, and in a decidious forest with thin young trees and a lower growth of stand density 0.7). It can be seen from Table 10.6 that the “clarification” of the forest after TABLE 10.6 Change in Net Radiation under the Trees Compared with That over the Forest. After Rauner [30]

Period

Full vegetation (July)

Time, h

&side -

Rover

R2 R

4

6

8

10

12

14

16

18

20

22

0.00 0.09 0.05 0.06 0.05 0.06 0.08 0.07 0.00 0.00 0.00 0.17 0.10 0.07 0.16 0.07 0.08 0.11 0.00 0.00

After vegetation (October)

R2 R

0.33 0 . 3 3 0.00 0.03 0.04 0.10 0.67 0.67

-

0.10 0.15

0.29

-

0.20 0.16 0.33

-

0.40 0.33 0.33

668

Net Radiation

the final period of vegetation is most apparent in the morning and evening hours. A different relationship between the net radiation of a forest and of an open area is observed in winter with snow cover. According to Kuzmin's [Chapter 9, Ref. 1001 observations in the Valdai Hills (W. Russia), the daily net radiation in winter snow-covered forests can be positive, while the open snow field has a negative value of net radiation.

5. Net Radiation Totals. The above data on the diurnal variation of the net radiation show that in the warm half-year, the daytime positive value exceeds the negative value during the night, which leads to the daily net radiation being positive. In winter the opposite picture is observed : the daily net radiation is negative. According to the observations of Pivovarova [Chapter 9, Ref. 371 near Tashkent, the mean diurnal net radiation was 136 cal/cm2 at the end of August and in the first half of September. The fluctuations of the net radiation at times were quite notable. For example, the highest observed daily total was 205 cal/cm2. These are the desert data. At the same area the mean net radiation of a cottonfield was 454 cal/cm2 in July, while for the semidesert it equaled 270 cal/cm2. The reduction in the net radiation of the desert is due to the higher albedo of sand. The results of observations near Leningrad [31] give the daily totals of net radiation in summer of 200 cal/cm2.Table 10.7 (from Barashkova et al. Chapter 7 [37a]) summarizes the observed monthly and annual totals of net radiation. In the annual variation of net radiation the monthly totals usually follow the variation in global radiation. However, in areas with stable snow cover the variation of the surface albedo (especially during spring) is also important. In the north of the territory considered, the negative net radiation is observed during five to six months. To the south of 46' N lat. on the European U.S.S.R. territory and of 43' N lat. on the Asiatic territory, the net radiation is positive throughout the year. During the whole year the totals of net radiation increase southward. Figure 10.4 is a map of the geographical distribution of the annual net radiation totals over the U.S.S.R. territory, plotted by Barashkova et al. [Chapter 7, Ref. 37al. The annual net radiation is seen to be positive everywhere and varies from 20 kcal/cm2 yr in the north to 60 kcal/cm2 yr in the south. This geographical distribution is mainly zonal. However, there is a slight tendency to an increase of net radiation in the west and an opposite tendency in the east of the examined territory.

TABLE

10.7

Meon Latitudinal Totals of Net Radiation (kcallcd). After Barashkova et al. (Chapter 7 [37a])

Latitude, deg

Jan.

Feb.

Mar.

Apr.

May

June

July

Aug.

Sept.

Oct.

Nov.

Dec.

Year CI

0 CI

38

1.5

2.7

40

0.8

2.0

3.6

5.8

8.3

8.8

8.5

7.6

5.3

42

0.4

1.3

3.4

5.6

8.2

8.8

8.5

7.5

5.2

44

0.0

0.8

3.0

5.5

8.1

8.8

8.5

7.3

5.0

2.3

0.5

-0.2

49.6

0.3

-0.4

47.4

0.0

3.8

6.1

8.4

8.8

8.5

7.8

5.4

3.5

1.7

0.8

59.0

3.0

1.3

0.5

55.5

2.6

0.8

0.1

52.4

46

-0.2

0.5

2.7

5.4

8.0

8.8

8.5

7.1

4.7

2.0

48

-0.4

0.2

2.3

5.3

7.9

8.8

8.5

6.9

4.5

1.8

-0.5

45.3

50

-0.5

0.0

2.0

5.2

7.8

8.8

8.4

6.7

4.2

1.5

-0.2

-0.7

43.2

52

-0.5

1.6

5.1

7.6

8.7

8.3

6.4

3.8

1.1

-0.4

-0.7

40.8

54

-0.6

-0.3

1.2

4.7

7.5

8.5

8.2

6.1

3.4

0.8

-0.5

-0.8

38.2

56

-0.6

-0.4

0.7

4.3

7.4

8.4

8.0

5.8

3.0

0.5

-0.6

-0.8

35.7

58

-0.7

-0.5

0.2

3.7

7.2

8.3

7.9

5.6

2.7

0.3

-0.6

-0.8

33.3

60

-0.8

-0.6

-0.2

3.2

6.9

8.2

7.8

5.4

2.4

0.1

-0.7

-0.8

31 .O

62

-0.8

-0.6

-0.4

2.2

6.5

8.2

7.8

5.3

2.1

-0.1

-0.7

-0.9

28.6

64

-0.7

-0.6

-0.4

1.3

6.0

8.2

7.7

5.1

1.9

-0.2

-0.8

-1

66

-0.7

-0.6

-0.4

0.6

5.5

8.2

7.7

4.9

1.6

-0.4

-0.9

-1.0

68

-0.7

-0.6

-0.4

0.1

5.0

8.3

7.6

4.5

1.3

-0.7

-0.7

-1

-0.2

.o .o

26.5 24.5 22.7

FIG. 10.4 Geographical distribution of annual totals of net radiation on the USSR territory (kcal/cm2min).

10.1. Observed Regularities in Variation of Net Radiation

67 1

Of much interest are the results of actinometric measurements in the Arctic and Antarctic, obtained during and after the International Geophysical Year. It appeared, for instance, that even for the major part of the Arctic (except its central area) the mean annual net radiation was positive. According to Chernigovsky’s [32] data, averaged from ten years of observations at six stations, the annual net radiation in the central Arctic was close to zero (0.5 kcal/cm2 yr). In Marshunova’s calculations [33, 341 the maximal annual totals of net radiation occur at Cape Schmidt (1 1.6 kcal/cm2 yr) and at the Chukotskoye Sea (14.4 kcal/cm2 yr). The lowest totals in the central part of the Arctic basin vary from -2.5 kcal/cm2 yr in its western wing to -3.5 kcal/cm2 yr in the east. Interesting data on the net radiation in the Arctic conditions were obtained by Vowinckel and Orwig [35] who contend that there are three main types of the radiation regime of the underlying surface in the Arctic: (1) the Norwegian Sea regime, (2) the continental type, and (3) the regime of pack. The first type is characteristic of the open ocean surface from the Arctic Circle northward. During the year here is observed reciprocal compensation of high positive net radiation values in summer and high negative in winter (the cause for the latter is the relatively high ocean surface temperature; it is of interest that the effect of cold or warm currents is not essential, the primary factor being only the cloud effect). The main feature of this type is a high radiative heat loss of the open ocean surface. The continental type (Siberian continent and western Canada) is entirely different. In winter the radiative heat loss from the surface is small and the downward atmospheric radiation becomes less pronounced owing to scarce cloudiness. As a result the annual net radiation total for the continent is greater than for the ocean. The radiation regime of pack is typical for those ocean areas that are only temporarily free of ice. The contribution of the surface emission to the net radiation here is intermediate compared with that of land and open water. The slow change of the net radiation in spring, due to the high surface albedo, is typical. Contrary to the continental conditions the maximum in net radiation occurs here after the solar culmination. The analysis of observational data performed by Rusin [36] shows that the Antarctic radiation regime is very peculiar. Its annual net radiation is everywhere (except surfaces free of ice and snow) negative. For example, the annual total of net radiation is - 12.2 kcal/cm2 yr at Komsomolskaya station, and from -3.5 to -9.1 kcal/cm2 yr (in different years) at Mirny. With surfaces free of ice and snow the annual net radiation reaches high positive values (over 75 kcal/cm2 yr at Oasis). Thus, on the main part of

672

Net Radiation

the Antarctic territory, the high values of the incoming solar radiation mentioned in Chapter 8 are not “realized” because of the high ground albedo and the long period of radiative cooling during the polar night. In the annual range the monthly net radiation totals are positive for only three to four months. 10.2. Results of Calculations of Net Radiation at the Underlying Surface The relatively small number of observations of the net radiation and its components is not sufficient to give information on the net radiation totals and their temporal and spatial variability. In this connection, theoretical calculations of the net radiation as a whole and of its components are widely used. The methods for such calculations of totals of direct solar, diffuse and global radiation, and also of the effective radiation, have been considered in Chapters 6 , 7, 8, and 9. Calculations show that, as a rule, we deal with a simple annual variation of the net radiation with a summer maximum and a minimum in winter. This is evident, for example, from Table 10.8, compiled by Mukhenberg [ 6 ] from the results of observations and calculations of the net radiation for the Leningrad region. This table shows that the increase of net radiation TABLE 10.8 Annual Variation of Monthly Totals of the Net Radiation and Its Components in the Leningrad Region, kcallcm2. After Mukhenberg [6]

Month

Global Radiation

Radiation Absorbed

Effective Radiation

Net Radiation

January February March April May June July August September October November December

0.4 1.5 4.8 8.1 11.9 13.6 12.8 8.9 5.5 2.0 0.6 0.2

0.2 0.7 4.6 6.8 10.0 11.4 10.6 7.3 4.4 1.6 0.4 0.1

0.7 1 .o 3.4 2.5 3.1 3.5 3.2 2.4 2.1 1.5 0.9 0.7

-0.5 -0.3 1.2 4.3 6.9 7.9 7.4 4.9 2.3 0.1 -0.5 -0.6

Year

70.2

58.1

25.0

33.1

10.2. Results of Calculations of Net Radiation at the Underlying Surface

673

from winter to summer is provoked by the more rapid increase of the absorbed global radiation than of the effective radiation. Although the net radiation in the Leningrad region is negative for six months, for the year as a whole it is positive on account of the higher positive totals in summer. Figure 10.5 shows the annual variation of monthly totals of the net radiation and its components for a number of sites in different climatic zones in Efimova’s estimation [37]. It is evident that the annual range of net CAPE CHELYUSKIN

DUDINKA

BATUMI

SVERDLOVSK

PARAMARIBO

-1-2----3---4

FIG. 10.5 Annual variation of monthly totals of the net radiation and its components in diferent climatic zones. (1) net I.adiation; (2) global radiation; (3) reflected radiation; (4) effective radiation.

674

Net Radiation

radiation usually has a maximum in June to July and a winter minimum. With increasing latitude the duration of the period of positive net radiation values becomes shorter. It is not always, however, that the net radiation has a simple annual cycle. In some cases the special features of the local climatic regime cause the “abnormal” annual variation of net radiation. For example, at Bombay (India) there is a “double” annual range of net radiation, with two maxima in May and October. The minima are observed in December and August. The appearance of the summer minimum is brought about by the increase of monsoon cloudiness at this time of the year, which provokes a considerable decrease in the incoming global radiation (in the preceding section we dealt with a similar case concerning Vladivostok). Interesting results are revealed in the comparison of the values of net radiation and global radiation. At the coast of the Arctic Ocean (Cape Chelyuskin) the annual net radiation of 6.7 kcal/cm2 yr is only about 10 percent of the yearly income of global radiation (63 kcal/cm2 yr). In tundra (Dudinka on the Yenisey near the Kara Sea) the portion of net radiation (17 kcal/cm2 yr) increases to 23 percent (the global radiation being 74 kcal/cm2 yr). An even greater increase of the relative net radiation is observed in the forest zone (Sverdlovsk) to 35 percent, that is, 30 and 86 kcal/cm2 yr. In humid subtropics (Batum) this ratio is 50 percent (64 and 129 kcal/cm2 yr), and in the zone of humid tropical forests (Paramaribo of South America) it is at its highest, about 60 percent (94 and 161 kcal/cm2 yr). The main factor bringing about the change of the relationship between the net and the global radiation is the albedo of the underlying surface (at Cape Chelyuskin of 61 percent and at Paramaribo 18 percent). The relation between the net and the global radiation serves as a spectacular index of the potential energetical possibilities of a given geographical area. For example, irrigation of desert can increase its net radiation by 60 to 70 percent. A still greater effect can’follow the melting of ice or snow in the Arctic. The above data are characteristic of the annual cycle of net radiation at different points of the continent. Table 10.9 gives the results of calculating the net radiation for the Atlantic Ocean and the White Sea, performed by Budyko et al. [38] and by Shiskho [39]. It is natural that the general form of the annual variation of net radiation at sea is the same as that of the inland (maximum in June to July, minimum in January to December). Kirillova [40] has given detailed study to the net radiation of small water basins. Having described in short the peculiarities of the annual variation of

TABLE 10.9 Annual Variation of Monthly Totals of the Net Radiation at the White Sea and the Atlantic Ocean (kcal/cm2mo). After Budyko et al. [38]

Jan.

Feb.

Mar.

Apr.

May

June

July

Aug.

Sept.

Oct.

Nov.

Dec.

Yearly Total

White Sea

-3.2

-2.2

-0.8

1.6

5.6

5.9

5.7

3.2

1.2

-1.3

-2.7

-3.3

10.7

Atlantic Ocean (40°N, m o w )

-0.5

6.6

9.2

9.7

10.2

9.1

6.9

.0.8

60.4

Site

1.6

4.6

3.0

0.8

676

Net Radiation

net radiation at the ground, let us now turn to the examination of its geographical variability. One of the main features of the geographical distribution of net radiation totals is their relatively low latitudinal range in summer and winter and considerable differences during the transitional seasons. This is evident in Table 10.10, borrowed from Alisov et al. [41]. If, for example, the maximal difference between the winter totals of the net radiation is 1.7 kcal/cm2, the corresponding difference will be 16.3 TABLE 10.10 Seasonal and Annual Totals of Net Radiation for Diferent Points in the USSR (kcallcnP). After Alisov et a/. [41]

Place Ust Tzilma Syktyvkar Moscow Kiev Rostov-on-Don Guryev Range of variation

Latitude

Winter

Spring

Summer

Fall

Year

65'27' 61'40'

-7.0 -7.4

55'45' 50'24' 47'15' 36'01'

-6.1 -6 . 9 -5.7 -6.8

0.8 3.6 8.1 11.2 14.5 17.1

23.0 25.5 21.3 23.7 25.7 24.5

-1.7 -1.7 1.2 2.4 5.6 3.2

15.2 20.0 24.5 30.4 40.1 38.0

. ..

1.7

16.3

4.4

7.3

24.9

kcal/cm2 in spring. It is important that in winter or summer there is no regular variation of net radiation in dependence on latitude, whereas during the transitional seasons, particularly in spring, the net radiation clearly increases southward. The annual net radiation totals are therefore regularly increasing with decreasing latitude. The data of Table 10.10 relate to a relatively limited territory. In the case of vaster areas the winter totals of net radiation show a strong dependence on latitude. According to Sauberer and Dirmhirn [13, 141, in December the diurnal totals of the net radiation of oceans in the Northern Hemisphere vary from -I20 kal/cm2 at 60" to 260-300 kal/cm2 near the equator. Table 10.11 gives the results of calculations of the mean latitudinal distribution of annual net radiation totals for land, oceans, and the entire surface of the globe, obtained by Budyko et al. [38]. The data of Table 10.1I characterize the average distribution of annual net radiation totals over the latitudes. It is evident, however, that in actual fact these totals are also dependent on longitude as well. Let us therefore

10.2. Results of Calculations of Net Radiation at the Underlying Surface

677

turn to consideration of the geographical distribution of annual net radiation totals over the entire global territory. In recent years Soviet investigators were the first to carry out the calculations of the geographical distribution of the net radiation for vast territories and for the entire globe. Budyko et al. [Chapter 7, Ref. 34; 15, 241 have plotted maps of the geographical distribution of monthly and annual net radiation means for the earth as a whole. Figure 10.6 is a map compiled by Efimova [37] and describes the distribution of annual means. One’s attention is drawn by the “jumping” character of the variation of the net radiation in the transition from land to sea, which finds expression in the discontinuity of the isolines at the coastal area. This is due to the sharp change of the ground albedo, as the lesser ocean albedos usually lead to larger net radiation compared with continents. TABLE 10.11 Mean Latitudinal Distribution of Annual Net Radiation Totals over Land, Ocean, and the Earth as a Whole (kcal/cm2yr). After Budyko et al. (Chapter 7 [38]) Net Radiation

Latitude, ON

60-50 5040 40-30 30-20 20-10 10-0

Earth as a whole

1

Latitude,

Ocean

Land

Globe

34 54 78 100 110 107

23 38 56 64 74 79

28 46 69 86 101 101

46

68

77

OS

0-10 10-20 20-30 30-40 40-50 50-60

I 1

Net Radiation Ocean

Land

Globe

107 107 94 73 53 31

75 69 62 55 39 26

99 99

87 71 53 61

It is evident from Fig. 10.6 and Table 10.11 that the annual net radiation is positive over all the globe, varying from values close to zero in the central Arctic and 10 kcal/cm2 yr near the boundary of permanent ice t o 80 to 95 kcal/cm2 yr in tropical latitudes. This does not mean, however, that the annual totals cannot be negative. As has already been mentioned in the preceding section, negative totals may be expected in regions with stable or long-time ice or snow cover, that is, in certain Arctic or Antarctic areas. The distribution of the net radiation over the colder and warmer parts of the globe is roughly zonal (the underlying surface being relatively ho-

FIG. 10.6 Geographical distribution of annual totals of net radiation over the earth's surface (kcaZ/cm2).

10.2. Results of Calculations of Net Radiation at the Underlying Surface

679

mogeneous through large areas in both summer and in winter). I n d r y continental regions (Sahara, deserts of central Asia, etc.) a considerable decrease of the net radiation is observed on account of the high albedo of desert and the great amount of the outgoing radiative heat due to the high desert surface temperatures. Reduced values of net radiation also occur in the monsoon areas and are caused by the increased cloudiness during the warm season. The highest values of the net radiation (over 140 kcal/cm2 yr) are observed in two areas of the Indian Ocean: to the east of the Arabian Peninsula and to the northwest of Australia. On land, the maximal net radiation of about 80 to 95 kcal/cm2 yr is observed in the regions with little cloud but sufficient humidity, such as savanna and evergreen tropical forests. Analysis of the maps of the geographical distribution of net radiation leads to the following conclusions: In January the net radiation is negative north of 45 to 47’N and positive over the rest of the earth (excluding Antarctic). The negative net radiation on land is smaller in absolute value (not more than 1 kcal/cm2 mo) than over the ocean (up to 4 kcal/cm2 mo) because the ocean surface has a higher temperature and loses more heat in radiation. To the south of these latitudes the positive net radiation increases as far as the equator, where it reaches about 8 to 12 kcal/cm2 mo. From the equator southward the net radiation at sea and inland varies little and is respectively 8 to 12 and 6 to 8 kcal/cm2 mo. In January and July the moderate latitudes of the Northern Hemisphere are characterized by a notable homogeneity of the net radiation fields, which accounts for the local formation of continental air masses. In the tropics and at the equator the distribution of net radiation is, on the contrary, rather “motley,” which is due first of all to the inhomogeneity of the geographical distribution of cloudiness. The zero isoline of the net radiation in July passes at 45 to 47’s. According to Sauberer and Dirmhirn [14], the maximal deviations of the geographical distribution of the net radiation of oceans from the zonal norm take place in June. For example, in the middle of the Pacific Ocean there is a sharp minimum of net radiation (240 cal/cm2 day), and in the region where the Gulf Stream originates there is a clear maximum of about 400 cal/cm2 day. The main features of the geographical distribution of the net radiation at sea in summer are determined by cloudiness. The position of the zero line of the net radiation in March is as follows: It passes through Eurasia from northwest to sutheast, through regions of southern Scandinavia, Lithuania, White Russia, northern Ukraine, Saratov region and northern Kazakhstan, and in East Asia passes roughly

680

Net Radiation

along the 48' N circle. In North America it passes through the lower current of the St. Lawrence river and in the west runs up northwards as far as 55'N. 10.3. The Net Radiation of Slopes The results discussed in the preceding sections reveal the regularities of the net radiation of horizontal surfaces. However, agricultural and other important applications demand information on the net radiation of slanting surfaces. In this connection the problem of the laws governing the net radiation of slopes becomes quite vital. Unfortunately, this problem has not been adequately investigated, although much work has been done with respect to the quantity of incoming direct solar and diffuse radiation to slopes and the effective radiation at slopes. The main results concerning incoming shortwave radiation were described in Chapters 5 and 8. The effective radiation on slopes was discussed in Sec. 9.7. Let us now examine some data on the net radiation of the slopes of a sand dune, obtained by Eisenstadt [2] from theoretical calculations based on the following considerations. Let us write the equation for the net radiation R,,of a slope:

where Ssl, DS1and riz are the fluxes of the direct solar diffuse and reflected radiation on the sloping surface, r is the flux of the shortwave radiation reflected from the slope, GSzis the flux of the atmospheric emission on the slope, G,,szis the flux of the emission reflected by the adjacent horizontal surface slopeward, UiUrfis the flux of the thermal radiation of the horizontal surface on the slope, and Us,is the thermal flux emitted by the slope itself. The quantity S,, in (10.8) can be calculated from the familiar equation S,,

= S,[cos

h, sin a cos(A - a )

+ sin h, cos a ]

(10.9)

where S , is the flux of solar radiation incident on the perpendicular to the rays surface, h, is the height of the sun, a is the angle of the slope, A is the azimuth of the sun, and a is the azimuth of the slope. In the considered case it is assumed that for the strewing sand, a = 135', a = 33', and for the windward side, A = 315', a = 16'. If we assume that the diffuse radiation and the radiation reflected by the adjacent horizontal surface and also the thermal radiation of the atmosphere

10.3. The Net Radiation of Slopes

68 1

and of the horizontal surface are isotropic, and that the temperature and optical properties of the slope and the horizontal surface are equal, the following equations for the net radiation components at the slope may easily be obtained (see Chapter 8): a 2 a riZ = r sin2 2 a G,I = Go C O S ~ 2

Dsl

= D COS' -

a G,,sl = (1 - 6 ) sin22 a Uiuri= Usurfsin22

where D, r, G o , Usurf are the fluxes of the diffuse radiation, reflected radiation, downward atmospheric emission, and self-emission for a horizontal surface. All these quantities can be directly measured. Also obvious is the validity of the formulas

where A is the albedo and Tszis the temperature of the slope. Making use of these formulas, Eisenstadt calculated the components of the net radiation of slopes and the net radiation as a whole from data of actinometric observations over a horizontal part on the top of a sand dune carried out on August 22, 1949. The results are summarized in Table 10.12. They should, of course, be treated as very approximate in view of the assumptions adopted. However, even such rough calculations disclose a number of interesting features of the net radiation of slopes. Examination of Table 10.12 shows that the greatest difference between the components of the net radiation for the slopes and the top occurs in the case of the direct solar radiation and the thermal radiation of the underlying surface (see quantities S,, and Us,). The differences of the other components are very small. This means that the peculiarities of the net radiation of slopes compared with a horizontal surface are determined first of all by the differences in the incoming direct solar radiation and the thermal emission of the sloping and horizontal surfaces. It is quite ob-

682

Net Radiation

vious that the difference in thermal emission is due to the difference in surface temperatures. The data of Table 10.12 make it also possible to analyze the special features of the diurnal variation of the net radiation of slopes. This variation is roughly symmetrical with respect to noon on top of the dune, and is considerably asymmetrical for both slopes. The maximum in the net radiation on the leeward side facing southeast takes place before noon (at about lO.OO), while on the windward slope it is observed after noon (at about 13.00). It is easily understandable that the earlier maximum on the leeward slope is caused by a great increase of the temperature of this surface directed to the sun and by the consequent increase in radiative heat loss. Since the steepness of the windward and of the leeward slopes is different, it is also possible to use Table 10.12 for evaluation of the effect of steepness on the peculiarities of the net radiation of slopes. For example, it can be seen that the net radiation of the steeper leeward slope is greater from that of the dune top than in the case of the windward slope. Chizhevskaya [42] measured the net radiation at the north and south slope of the gradient 12 to 17' at Voyeikovo near Leningrad. According to her observations, in spring and fall on clear days the net radiation of the south slope is by 15 percent larger than of the north one. In summer this difference reduces to 5 to 7 percent. Eisenstadt [43, 441 carried out similar measurements at the north, south, east, and west slopes of the Kumbel Pass (the Turkestan mountain range) with angles of 33, 31, 33, and 23O, respectively, and also at the horizontal surface of the ridge of the pass. The revealed differences in the net radiation of the oriented slopes were mainly due to the differences in their thermal emission. The net radiation at the south slope reached 1.01 cal/cm2 min, and at the north slope it was 0.36 cal/cm2 min, on the average. The asymmetry of the daily range of the net radiation at the east and west slopes was quite significant; for example, at 8 h, Rsl = 0.76 cal/cm2 min for the east slope and Rsl = 0 at the west slope. The daytime total of the net radiation of the south slope was by three times and the total during 24 h by 7.5 times greater than for the north slope. The daytime total for the west slope was somewhat larger than for the east, as the latter has a higher temperature during the day and consequently a more intensive radiative cooling. In order to investigate the regularities of the variation of net radiation, Kondratyev and Fedorova [Chapter 8, Refs. 5.9-551 carried out daytime measurements of the net radiation of differently oriented blackened surfaces by means of a ventilated Yanishevsky pyrgeometer. As the result,

TABLE

10.12

Net Radiation of Sloping Sand Dunes and of Horizontal Surfaces on the Top of a Sand Hill (callcm2min). After Eisenstadt and Zuyev [2]

Net Radiation and Its Components

Time, h ~~

6

7

O.OO0 0.125 0.090 0.330 0.645 0.365 0.050 0.085 0.050 0.085 0.045 0.080 0.000 0.000 0.005 0.010 0.010 0.050 0.035 0.100 0.100 0.175 0.390 0.395 0.400 0.405 0.370 0.370 0.000 0.000 0.005 0.005 0.010 0.010 0.040 0.045 0.540 0.585 0.545 0.595 0.560 0.640 -0.100 -0.020 -0.040 0.125 0.170 0.340

~~

8

9

10

11

12

13

14

15

0.355 0.600 0.940 0.110 0.110 0.100 0.005 0.015 0.015 0.170 0.250 0.470 0.480

0.585 0.850 1.165 0.120 0.120 0.110 0,005 0.020 0.170 0.220

0.785 1.045 1.305 0.125 0.125 0.115 0.005 0.025 0.220 0.280 0.345 0.565 0.575 0.530 0.000 0.005 0.015 0.060 0.740 0.800 0.925 0.535 0.665 0.770

0.955 1.210 1.350 0.135 0.140 0.130 0.005 0.025 0.265 0.325 0.360 0.535 0.545 0.500 0.000 0.005 0.015 0.060 0.790 0.820 0.995 0.590 0.750 0.715

1.050 1.230 1.275 0.130 0.135 0.125 0.005 0.025 0.285 0.325 0.340 0.560 0.570 0.525 0.000 0.005 0.015 0.060 0.840 0.845 1.025 0.635 0.765 0.650

0.090 1.185 1.090 0.140 0.145 0.135 0.005 0.025 0.295 0.320 0.300 0.600 0.510 0.560 0.000 0.005 0.015 0.060 0.860 0.840 1.005 0.695 0.780 0.570

0.990 1.ooo 0.780 0.160 0.165 0.150 0.005 0.025 0.275 0.780 0.340 0.560 0.570 0.525 0.000 0.005 0.015 0.055 0.845 0.780 0.980 0.610 0.675 0.330

0.840 0.775 0.455 0.175 0.180 0.165 0.005 0.020 0.245 0.230 0.155 0.500 0.510 0.470 0.000 0.005 0.015

0.440 0.000 0.005 0.010 0.050 0.635 0.670 0.730 0.200 0.350 0.570

0.310 0.515 0.525 0.485 0.000 0.005 0.015 0.055 0.685 0.735 0.830 0.385 0.540 0.700

0.055

0.805 0.770 0.870 0.485 0.465 0.145

16

17

18

0.690 0.470 0.080 0.560 0.310 0.030 0.170 0.000 0.000 0.140 0.095 0.065 0.145 0.095 0.065 0.135 0.085 0.060 0.000 0.000 0.000 0.015 0.010 0.000 0.200 0.135 0.035 0.170 0.100 0.025 0.075 0.025 0.015 0.475 0.455 0.465 0.485 0.465 0.475 0.445 0.430 0.435 0.000 0.000 0.000 0.005 0.005 0.005 0.015 0.010 0.010 0.055 0.050 0.050 0.760 0.720 0.685 0.750 0.710 0.660 0.640 0.695 0.758 0.395 0.175 -0.100 0.270 0.070 -0.115 -0.035 -0.140 -0.105

Note. Subscript 1 denotes the windward slope, subscript 2 the leeward slope, while the quantities without index refer to the top of the dune.

684

Net Radiation

curves were plotted to represent the dependence of the ratio RslIRh of the net radiation for the slope and for a horizontal surface upon the angle of inclination and the orientation of the slope with various solar heights. Some of these curves are given in Figs. 10.7 and 10.8. As seen from these figures, with increasing height of the sun above the horizon, the dependence of the ratio Rsl/Rhupon surface azimuth becomes less important.

pw

200

a

FIG. 10.7 Dependence of the relative net radiation of the slope Rs,/Rh upon inclination and orientation. July 20, 1956, h, = 27O, yo = 268O, 16.53 h, clear, south-easterly wind of v = 1 mlsec.

For slopes facing the sun, the maximum of Rsl/Rhis observed at a of the order of 90' - h, , being most pronounced at low altitudes of the sun.

40

20 0

N In IU

qn LU

In JU

nn

-ru

cn JV

cn

w

-m

iu

on

ou

J

90

I

a

FIG. 10.8 Dependence of the relative net radiation of the dope Rsl/Rhupon inclination and orientation. June 20, 1956, h, = 66O,yo = 199O, 12.32 h, south-easterly wind, v = I mlsec, clear.

10.3. The Net Radiation of Slopes

685

The curves of Rsl/Rhfor slopes directed opposite to the sun have a minimum, and with certain a the net radiation of the slope is negative when the solar height is not too big. With increasing angle of inclination the net radiation passes through zero at such values of the angle when the direct solar radiation does not hit the sloping surface (a 2 A,). Steep slopes (a> 50') directed opposite to the sun again show positive values, apparently on account of the increasing flux of the reflected radiation to the slope and the reduction in the effective radiation. For slopes with azimuths 90' and 270' relative to the sun, a monotonic decrease of Rsl/Rh with increasing angle of inclination is observed (east and west slopes in Fig. 10.8). It is of interest that with low and moderate solar heights, slopes facing the sun have a far greater net radiation value than do horizontal surfaces. At high elevations of the sun (Fig. 10.8), when the angle of incidence of solar radiation is very large, the net radiation of any oriented surface is either the same or less (in the majority of cases) than that of the horizontal surface. Figure 10.9 gives the results of measurements of the net radiation with continuous cirrus clouds. It is eident from comparison of Figs. 10.7 and 10.9 that the azimuthal dependence of the net radiation in this case tells

I40

a

FIG. 10.9 Dependence of the relative net radiation of the slope RsllRh upon its angle of inclination and orientation. June 16, 1956, h, = 28O, yo = 269O, 16.49 h, cirrus cloud of force 10, south-easterly wind of v = I mlsec.

in a weaker degree than with a clear sky. It is also interesting that in Fig. 10.9 no zone of negative net radiation values appears to be connected with the increase of incoming diffuse radiation and decrease of effective radiation due to the effect of cloud.

686

Net Radiation

The above data are only roughly illustrative with respect to certain regularities of the net radiation of slopes. It should be stressed that there is an extreme need for experimental investigations of the net radiation of slopes and for the perfection of methods for calculating the components of this quantity.

10.4. Net Radiation and Its Components in a Free Atmosphere The boundary atmospheric layer, which is so effective in transforming the integral solar and longwave radiation fluxes, has a thickness of the order of 30 to 50 km. The regularities of the variation of radiant fluxes in the free atmosphere can therefore be investigated by means of instruments mounted on aircrafts, balloons, and rockets. Such investigations provide data on the vertical profiles of the net radiation and its components, which is particularly important in view of the interpretation of satellite measurement data on the outgoing radiation. A special feature of aircraft and balloon investigations is the facility they give in obtaining the information to be used with the purpose of checking the reliability of satellite measurements. Experiments in radiant fluxes in the free atmosphere by means of aircraft and balloons were begun as early as 30 years ago, but only during the past 10 to 15 years has it become possible to conduct them at a sufficiently high level of investigation. In this section we shall first consider the available measurement methods and their accuracy, and then discuss the results of the present investigations. 1. Peculiarities of the Use of the Standard Actinometric Instruments. The standard actinometric instruments mounted on aircraft and balloons were used for measuring radiant fluxes in the free atmosphere in a great number of research works [45-841. It is natural that the use of the standard instruments in such unusual conditions required special methods of measurement and processing, and also careful study of their characteristics with changing parameters of the medium. The most complicated turned out to be the problem of the use of pyranometers and balance meters for measuring global and net radiation. Important investigations of the methods applied in aircraft pyranometric measurements were carried out by Kastrov [51, 531 who had worked out a theory of pyranometers to be installed on aircraft. Omitting the standard corrections (angular, spectral, temperature dependence, etc. ; see

10.4. Net Radiation and Its Components in a Free Atmosphere

687

[Chapter 1, Ref. I]), let us consider only the special features of aircraft or balloon pyranometric measurements. The most important factor controlling the processing of the measurement data is in this case the nonhorizontality of the receiving surfaces. For the lower pyranometer, which measures the upward shortwave radiation flux, the departure of the receiving surface from the horizontal position is of no importance, since the angular distribution of the upward radiation may in the first approximation be considered isotropic. The readings of the upper pyranometer, however, whose receiving surface is exposed to the direct solar radiation as the main component of the global radiation with clear skies, are very sensitive to displacement from the horizontal. Let us assess, following Kastrov [54], how the nonhorizontality of the upper pyranometer affects the angle of incidence of solar radiation and how it changes computation. Let the axis 0.2(Fig. 10.10) correspond to the vertical direction and OZ’ be normal to the pyranometer receiving surface. The position of the sun in the sky is determined by the point M ; a is the azimuth of the plane OZD with respect to the solar vertical (the other designations are evident from the drawing or will be interpreted later on).

FIG. 10.10 The eflect of the nonhorizontality of the pyranometer’s receiving surface.

In the departure from the normal of the receiving surface by angle E relative to the vertical O Z , the solar radiation flux incident on the receiving surface will change by a quantity equal to S (cos (@‘ - cos lo). Here S is (the flux of solar radiation on the surface perpendicular to the rays)

68 8

Net Radiation

the normally incident flux of solar radiation, To is the solar zenith angle, and Cot is the angle of incidence of solar radiation on the receiving surface, defined by the relation

+ sin Co sin

cos lo’= cos Co cos E

E

cos a

When averaging over azimuth the mean error of measurement of the solar radiation flux on the horizontal surface will equal

=

If the angle

E

- S cos lo(l - cos

E)

=

E - 2 s sin2 cos Co 2

(10.10)

is small, we have approximately AS’

= -&2 s c o s 50

2

(10.11)

Hence it is evident that the relative measurement error S’ is equal to 2 / 2 ; that is, it is independent of the solar zenith angle to. The obtained result is apparently valid also for the diffuse radiation propagating in a given direction, and therefore (10.1 1) can be easily generalized for evaluating the measurement error with the shortwave net radiation d(F+ - F f ) : E2

d(F4 - F J . )= - (FL - F f ) 2

(10.12)

where F+ and F f are the downward and the upward fluxes of shortwave radiation, respectively. Differentiating (10.12) along the vertical coordinate 2 and introducing the notation q = (d/dz)(FJ.- F f ) for the radiative heat influx, we obtain the following expression for the evaluation of the error in the determination of 9: &2

(10.13)

dq= - - q 2

The above formulas show that at insignificant disturbances of the horizontality of the receiving surface, the measurement error is not important. For example, even at E = 6’ the value (4)~~0.005; that is, the relative error is 0.5 percent. It is essential, however, that the above equations assume azimuthal averaging of the pyranometric readings. In the estimation of random errors from single records their values will be much greater, even at small angles E .

-

10.4. Net Radiation and Its Components in a Free Atmosphere

Let us assume, as before, that

E

= 6’

cos CO’ - cos 5‘0 cos C@

and C0

-

= 60’.

689

In this case

0.2

that is, the relative error will increase up to 20 percent. If To = 70°, the relative error is about 30 percent. At large solar zenith distances, therefore, and without the azimuthal averaging of measurement data (relative to the sun), the errors due to the nonhorizontality of the upper pyranometer receiving surface become very significant. That is why the practice of aircraft pyranometric measurements includes repeated taking of the pyranometric readings for the same “area” from two opposite flight directions; “the left-hand side sun” and “the right-hand side sun.” An even more complicated situation takes place when interpreting the results of measurements from balloons because this method cannot be realized there. In this case, however, the problem is alleviated by the balloon’s rotation around its vertical axis as it ascends. Lopukhin [56] compiled a table for the introduction of corrections for the nonhorizontality of the receiving surface in dependence upon solar height. When the horizontality of the pyranometer receiving surface is disturbed, its sighting field changes: instead of the sky section ABB‘C, the inclined upper pyranometer (Fig. 10.10) views the section AD‘DC positioned below the horizontal plane. The change of the sighting field of the lower instrument is similar. This naturally leads to distortion of the readings. Analyzing the measurement errors in this case, Kastrov [54] came to the conclusion that they are not essential. Aircraft or balloons always give rise to certain “perturbations” in the radiation field considered. In the case of aircraft pyranometric measurements it is more important to take account of the “underlighting” of the pyranometer due to the reflection of radiation from the aircraft surface. In Kastrov’s estimation [54] the errors appearing in this case are of secondary importance. With balloons, one must consider the shading of a portion of the sky the by casing. However, if the suspender, at the end of which the pyranometers are fixed, is long enough, the obliterating effect of the casing may be ignored. The usual purpose of aircraft or balloon measurements of the net radiation is to obtain data on its vertical profile and the components and also to determine the radiative flux divergence. It is natural that in order to obtain such data there are needed, strictly speaking, simultaneous actinometric measurements at different atmospheric levels above the given

690

Net Radiation

underlying surface. The net radiation B ( z ) is determined from the formula B(z) = F J ( z ) - F q z )

(10.14)

The radiative flux divergence q is further calculated from the relation q z - dB dz

(10.15)

derived on assumption of the horizontal optical homogeneity of the atmosphere and underlying surface. In reality, if the condition of the horizontal optical homogeneity is upset, the simultaneous measurements of radiant fluxes at different levels are practically impossible. This fact makes it necessary to solve the following two problems. The first problem is related to the need for evaluating the errors caused by the effect of the horizontal optical nonhomogeneity. Rough estimates of this kind were provided by Kastrov [51]. Suppose that the radiant fluxes are measured over a circular area with the albedo A’, which is surrounded by a surface whose albedo is A . Let the medium between the underlying surface and the level considered be vacuum. Designating 0 as the angle at which the radius r of the area is seen from the given level z, obtain the following relationship between the upward and downward radiant fluxes : F ~ ( z= ) F + ( z ) Asin28

+ F+(z)A’cos28

(10.16)

Further, it can easily be seen that the difference in net radiation at the levels z1 and z2 due to the “parasitic” influence of the horizontal nonhomogeneity of the underlying surface will equal d(z,, z2) = F + ( A - A ’ )

(

+ z22 - + z12 ) ‘12

r2

(10.17)

r2

The value d(z,, z 2 ) is the error sought. Since the radiation transformation by the intermediate layer can only smooth the optical contrasts of the underlying surface, the formula (10.17) gives the upper limit of the error considered. Making use of (10.17), Kastrov evaluated the errors appearing in the measurement of the shortwave net radiation for the case where the shiny snow cover of the Rybinskoye reservoir (near Moscow) with the albedo A = 0.82 was surrounded by a surface with almost melted snow (A’ = 0.27). These calculations showed that the effect of the horizontal optical nonhomogeneity of the underlying surface should be particularly

10.4. Net Radiation and Its Components in a Free Atmosphere

69 1

considered in processing measurement data at elevations above 1 km. In the case where the albedo of the investigated area is lower than that of the environment (reservoir in summer) the effect of the horizontal nonhomogeneity turned out to be insignificant. In the real conditions the horizontal optical nonhomogeneity of the underlying surface and atmosphere can be extremely variegated, which makes difficult the accounting of it in processing measurement data. It should be concluded therefore that the most reliable way to solve the problem of the vertical profile of the net radiation and radiative heat inflow consists in measurements over a homogeneous surface with a clear sky. When applying balloons the horizontal homogeneity must be checked by means of photographing the area considered. In the absence of the horizontal homogeneity the measured net radiation values have a narrow local meaning, and the determination of the radiative heat inflow from (10.15) is related to more or less marked errors whose quantitative evaluation is very difficult. The second problem to be solved when plotting the vertical net radiation profile from aircraft or balloon measurement data consists in the reduction of these data to a given moment of time. In the case of measuring the shortwave radiation fluxes, this problem is solved by using approximate empirical formulas that express the dependence of these fluxes upon solar height. For example, Faraponova and Kastrov [64] used the following formula:

Fl

=

So sin h,

=a-bfi

(10.18)

where So is the solar constant for the given day, h, is the solar height, m is the atmospheric mass in the direction to the sun, and a and b are constant, with b > 0 in the case of the downward and b < 0 in the case of the upward flux. The reduction of the results of measurements of the net radiation or longwave radiation fluxes to a given time presents a problem for which a method of solution is not clearly established. In the case of daytime net radiation measurements it appears to be possible to use empirical formulas similar to (10.18). For the longwave radiation fluxes, variability within 1.5 to 2 h is usually considered insignificant. The correct solution of the problem on the basis of theoretical calculations demands knowledge of the transformation of the vertical temperature and humidity profiles over relatively short time intervals (of the order of several hours), which cannot as yet be obtained.

692

Net Radiation

An important factor to be considered in processing the standard actinometric readings (of pyranometers and net radiometers used in aircraft or balloons) is their dependence on temperature and pressure. The calibration of the instruments in the thermobarochamber, imitating the real conditions of the vertical variation of temperature and pressure, should be taken as the simplest and most reliable. It is obvious that such an artificial “ascent” of the instrument can be accepted only as an approximate standard of atmospheric stratification. This is the kind of calibration that was used at the Chair of Atmospheric Physics of Leningrad University [67-721 in the preparation for balloon measurements. 2. Actinometric Radiosondes. It is obvious that the practical use of data on the net radiation and radiative flux divergence in the free atmosphere is possible only when a sufficient volume of the observational material is available. In this connection many recent attempts have been made to construct simple and light net radiometers that might be launched with the standard radiosondes. Such systems, consisting of standard radiosonde and radiometer, are called actinometric radiosondes. Since the problem of daytime net radiation measurements is very difficult and cannot be solved with the help of radiosonde-borne net radiometers, the latter instruments are used at present for nighttime soundings only. The instrument most widely used with actinometric radiosondes is the so-called economical radiometer of Suomi and Kuhn [84a-89], which will be later indentified as the Suomi net radiometer. This instrument has a multilayer system, schematically presented in Fig. 10.11. Here the surfaces

FIG. 10.11 The scheme of the Suomi net radiometer.

labeled P are polyethylene membranes of 12.7-,u thickness. The inside of the radiometer contains mailar membranes, M , of 6.4-p tnickness. The surfaces of the inner mylar membranes are faced with aluminum. The outer mylar membranes are receiving surfaces and are blackened from the outside and coated with aluminum inside. The polyethylene membranes

10.4. Net Radiation and Its Components in a Free Atmosphere

693

P serve as windshields. The instrument is not in vacuum, and the inside pressure therefore always equals the atmospheric pressure of the corresponding level. The curve T depicts the temperature profile of the net radiometer wall characteristic found for conditions of measurement in the middle troposphere. The casing Z is good thermal insulation. Figure 10.12 presents the outside appearance of the actinometric radiosonde. The net radiometer is Seen to be of triangular form. Note here that the board contacting the net radiometer is coated with aluminum. The

FIG. 10.12 Actinometric radiosonde.

+

vertical dimension of the net radiometer ( D 2 4 = 5.6 cm (the membranes P and A4 are at 0.7-cm distance) and the side of the triangle equals 30 cm. Direct measurement is made of the temperatures Tt and Tb of the receiving surfaces, for which temperature sensors are used. The readings of these sensors are transmitted to the ground in the same way as the readings of the standard air temperature sensor of the radiosonde system. The polyethylene light filters pass from 80 to 90 percent of the integral longwave radiation. The remainder (10 to 20 percent) is dispersed as 90 percent reflected and 10 percent absorbed by the polyethylene film. The latter means that the absorption of radiation can be practically ignored. The elementary theory of the net radiometer [87], based on approximate heat balance equations of the receiving surfaces, leads to the following

694

Net Radiation

expression for the net radiation (effective radiation) :

where u is the Stefan-Boltzmann constant, k = l/a(l - ar) (a = 0.85 is the absorptivity of the receiving surfaces, r = 0.16 is the reflectivity of the polyethylene membranes), ci = (I/D)ki(Tb- T t ) is the heat flux due to the molecular thermal conductivity of air (calculations show that the effect of convection can in this case be ignored), il = 5.10-3 cal/cm2 deg is the thermal capacity of the receiving surfaces, and En is the residual term characterizing the influence of secondary errors. Knowing the constant values of the given instrument, the net radiation sought can be easily determined from the measured Tb and Tt by means of (10.19). Since the receiving surfaces in the Suomi net radiometer are separated by thermal air insulation, it is natural that the temperature difference Tb - Tt is quite high, varying from several degrees near the earth’s surface to scores of degrees in the stratosphere. When deriving (10.19) a number of error sources were not taken into account, such as the radiation of the polyethylene filters and of the radiosonde’s casing, the leaking of air from the inside of the ascending net radiometer, the precipitation of white frost or moisture on the outer surfaces and inside the net radiometer, and the adiabatic air cooling inside the net radiometer in ascension. Calculations show, however, that the first two terms of (10.19) are roughly equal and by far exceed the other terms in the right-hand side of (10.19), whose value is not more than 10 percent of the main terms. It should be noted that almost all secondary factors tend to decrease the temperature difference Tb - Tt , which leads to a small systematic underestimation of the measured net radiation value. The ground testing and comparison of the Suomi net radiometer with other instruments [89] showed that this model may be considered quite reliable for nighttime measurements. It should be noted that the multilayer structure of the net radiometer greatly complicates the interpretation of the obtained results, owing primarily to two causes: First, as seen from (10.19), in processing the measurement data it is necessary to know the value ci characterizing the thermal flux from one receiving surface to another due to the molecular thermal conductivity. In (10.19) this flux is taken account of within the “one-dimensional” theory of thermal conductivity. It is obvious, however, that

10.4. Net Radiation and Its Components in a Free Atmosphere

695

the “side” heat fluxes in the horizontal direction can also be effective. Nor can the influence of the convection inside the net radiometer always be ignored, although its quantitative consideration is extremely difficult. Second, it is clear that the many layers of the net radiometer considerably increase its inertia, which is undesirable. Both unfavorable factors are practically eliminated in the net radiometer of the “disc” type, supplied with wind protection in the form of hemispherical polyethylene covers. It is known that with a disc net radiometer (for example the Yanishevsky type [Chapter 1, Ref. 21) adequately calibrated, there is direct proportionality between the net radiation and the temperature difference of the receiving surfaces, while the time constant can be made sufficiently low. From this standpoint the use of the disc net radiometer is preferable. All these considerations however, have little practical importance. The simultaneous balloon flight [90] of the Suomi and the disc types showed that the results reparted by not more than 2 percent. The ground comparisons of the Suomi net radiometer with the ventilated disc model by the same constructor showed that the ratio of their respective readings was 1.01029, with the mean quadratic deviation of 0.02017. The evaluation of the random errors in the net radiation measurements caused by the errors of the determination of the Suomi, type receiving surface temperature (assumed to equal 0.2OC) led to 0.0027 cal/cm2 min for the upper troposphere and stratosphere, and 0.0041 cal/cm2 min for the lower troposphere. This corresponds to the errors in the determination of the temperature variation in the layers of 50-mb thickness, equal to 0.45 and 0.68 deg/day, respectively. For 100-mb thicknesses these errors decrease by twice. Thus, as.can be judged from these data, the accuracy of measurements by means of the Suomi net radiometer is fairly satisfactory. A number of investigations offered modifications of the Suomi net radiometer. For example, Gayevsky [45] described a similar construction, repeating the pyrgeometer type (the “one-side” net radiometer). Kostianoy [76-781 used the principle of the “multilayer” net radiometer in constructing an actinometric radiosonde. Fritschen and Van Wijk [91, 921 constructed a miniature net radiometer whose receiving surface is a standard thermopile closed on both sides with coupled mica filters of 5-p thickness. The instrument has a cylindrical form of 2.54-cm diameter and about 0.5-cm height. The preference of mica to polyethylene as a filter was dictated by the desire to use the wind protection with steadfast water-repelling properties. However, mica has a strong absorption band in the interval 8.8 to 10.3 ,LA

696

Net Radiation

(where the filter transmission is zero) and a very low transmission in the interval 10.3 to 15 p. In this connection Fritschen [91] concluded that the filter should best be made of the polymer membrane Saran wrap (chemical composition not given). Pohl and Muller [93-961 measured the effective radiation with a miniature heated thermistor net radiometer, substituting it for the temperature sensor in the standard American radiosonde. The variation in the resistance of the thermistors caused by the variation in the effective radiation is used for the frequency modulation of the transmitted radio signal. This allows recording of the vertical variation of the effective radiation during the ascension of the radiosonde. Figure 10.13 presents the scheme of the Pohl net radiometer. As seen, it consists of two disc net radiometers whose receiving surface temperature is measured with the help of thermistors 1, 2, 3, 4. The receiving surface DOWNWARD RADIANT FLUX

U?WARO RADIANT FLUX

FIG. 10.13 The scheme of the Pohl net radiometer.

diameter is 3.9 cm. The upper receiving plate of one net radiometer and the lower of the other have electric heating, which generates H amount of heat in these plates. In order to secure an equal heat exchange between the receiving surfaces and the air, the net radiometer is rotated around its vertical axis in ascension. The fundamental theory of this instrument (see

10.4. Net Radiation and Its Components in a Free Atmosphere

697

[97]) gives the following expression for the net radiation (effective):

(10.20) I

where T I ,T2, T, , T4 are the temperatures of the receiving surfaces, and N is the corrective term for the selectivity of the receiving surfaces, the difference in ventilation of the upper and lower plates, and for the effect of their displacement from horizontal. The value N ’ does not exceed 10 percent of F. Since the directly measured values are the receiving surface temperatures and the strength of current in the heating coils, the accuracy of measurement is much dependent on the reliability of the determination of these quantities. With the accuracy of the temperature difference data of 0.2O and the error in the determination of the strength of current not over 2 percent, the total error in the effective radiation measurement is about 15 percent. An interesting model of net radiometer for the actinometric radiosonde was realized by Aagard [98, 991. His instrument, called by the author “a double radiometer,” consists of two discs (detectors A and B ) of 4-cm diameter and 0.5-mm thickness made by tightly coiling constantan wire insulated with epoxyde resin. The upward receiving surfaces are blackened, while the lower are aluminized and coated with thin quartz protection. The lower receiving surface of one disc is screened by an aluminum membrane positioned at 2-mm distance. The detector A operates in the pyrgeometric regime and serves for measuring the downward radiant fluxes. The deviations of the unscreened detector B are determined by the total of the upward and downward fiuxes. Its lower surface is striped black to keep it cooling even in the case when the radiative heat inflow is positive. In winter the entire receiving surface is blackened. Both detectors are self-compensating : The equality in temperature between the discs and the air is reached by passing the current through the constantan wire (the process of temperature equalization is automatic). Knowing the amount of the joule heat emitted in the discs and the air temperature, it is possible to determine the downward and upward radiant fluxes and their difference (the net radiation). The rough theory of the instrument expresses the net (effective) radiation as

1

+%A* (%+1-)

dT dil

(10.21)

698

Net Radiation

Here H A , HB are the generated heat in the detectors A and B due to heating, a, and a, are the emissivities of the upper receiving surfaces and of the lower surface of the detector By A , is the area of the receiving surfaces; and 1, T are the thermal capacity and temperature of the detectors. According to Aagard [98], the accuracy of measurement of the net radiation during the ascension is 20 percent, while during the horizontal drift the errors decrease to 3 percent. Businger and Kuhn [ 1001 performed simultaneous measurements of the atmospheric thermal radiation by means of the follownig four radiation detectors, launched on an automatic balloon up to the height of 25 km during the night of July 30, 1958, Madison, Wisconsin: (1) a Suomi net radiometer; (2) a disc radiometer with blackened upper and lower receiving surfaces (the resistance thermometer inside the disc measures the mean temperature of the upper and lower surfaces, characterizing the total of the radiant fluxes absorbed by both surfaces); (3) a black sphere; (4)a silver sphere blackened from the outside. In the latter two cases the measured value is the temperature of the black sphere. All the mentioned receiving surfaces are protected from wind by polyethylene film. If the intensity of the atmospheric thermal radiation as a function of the angle between the direction of the beam and the vertical Z(0) is changed for the effective temperature T@), determined from the relation (a/n)T,"(0) = I@), where (T is the Stefan-Boltzmann constant, then it is possible to express all the directly measured temperature values (of the black sphere, disc, or detectors of the net radiometer) in terms of Te(0). The results of measurements are therefore presented in the form of the curves of the vertical temperature distribution. If T,,Tb are the temperatures of the upper and lower surfaces of the net radiometer and Tdis the disc's temperature, then Td4= (1/2)(T: Tb4). The agreement of the measured values Tt, Tb, and Tdshowed that they fully satisfy this relation, and consequently the readings of the net radiometer and of the disc radiometer are in agreement. The temperatures of the black and of the silver spheres turned out to be unequal: in the lower zone of the sounded layer the black sphere is warmer, and in the upper, much colder than the black. The authors explain it by the effect of the convection inside the black sphere on the readings of the resistance thermometer inside the sphere. The net radiometer readings determine the vertical variation of the upward, downward, and effective fluxes of thermal radiation. These data are used to calculate the radiation temperature variations at different heights. The latter are compared with the temperature differences of the black sphere

+

10.4. Net Radiation and Its Components in a Free Atmosphere

699

and the air and also of the silver sphere and the air at the corresponding heights. There is found a similarity between the curves of the vertical distribution of the radiation temperature variations (radiative heat inflow) and the temperature difference of the black sphere and the air.

3. Special Instruments for Balloon and Aircraft Measurements. Since the standard pyranometers and actinometers have proved to be sufficiently reliable in measuring the shortwave radiation in the free atmosphere, up to now there was no great need to construct their special aircraft or balloon modifications. It is otherwise with net radiometers. We know that the errors of ground measurements of the net radiation by means of the standard Yanishevsky net radiometers are quite large. Their use for measuring the net radiation in the free atmosphere is made particularly difficult because in processing the obtained results it is necessary to introduce a correction for wind speed, which demands knowledge of this quantity (because wind currents are inevitably present in horizontal attitudes). For aircraft and balloon measurements (up to now realized during the night only) special net radiometers were constructed whose readings are free of the wind effect. Some of them, applied with actinometric radiosondes, have already been described. We shall now speak of other models used by aircraft or free balloon investigators on board the vehicles. Shliakhov [65] used a compensational Yanishevsky net radiometer to measure the effective radiation (the net radiation at night) from free balloons. The instrument imitates the double pyrgeometer, whose upper receiving lamina is electrically heated to the temperature at which the zero balance takes place. In the first approximation the measured net radiation is simply determined by the quantity of the joule heat generated in the lamina (that is, by strength of current passing through the plate). Investigations of the compensating net radiometer showed that its readings are much dependent on the regime of ventilation. In particular, the vertical ventilation can be greatly effective. If, however, the vertical speed of ascension is low (not more than 1 m/sec), the transfer factor of the instrument varies by not more than f 7 percent. In order to eliminate the varying effect of natural ventilation on the readings of the compensating net radiometer, in a number of flights Shliakhov used artificial ventilation with speed of 4 m/sec. When measuring the longwave radiation fluxes Shliakhov [65] also used his own design of a compensating pyrgeometer with higher sensitivity, resembling the “black shining” Angstrom type (see [l, 21). Depending on the sign of the net radiation the electric current is passed through the black

700

Net Radiation

or shiny (polished constantan) stripes for the purpose of equalizing their temperature by generating the joule heat. The rough theory of the instrument shows that the measured net radiation of the black stripes is directly proportional to the emitted joule heat (to the square of the strength of current). Since the readings of the compensating pyrgeometer depend on the regime of ventilation, a forced ventilation was used to “stabilize” its effect. Investigations showed that with the artificial horizontal ventilation of 3.8 m/sec speed and the vertical speeds of ascension not above 1.5 m/sec, the latter do not affect the results of measurements. It was found experimentally that in ventilating the pyrgeometer, the strength of compensation current must be increased. This means that the temperature of the shining stripes is higher than that of the black stripes. In this connection, to eliminate the wind effect in certain models of the pyrgeometer, Shliakhov also used a method of increasing the thermal resistance between the shining stripes and the connected thermojunctions. The empirical selection of thermal insulation practically allows complete avoidance of the influence of vertical ventilation. The evaluation of errors in the measurement of longwave radiation by means of the compensating net radiometer and pyrgeometer showed that they are never in excess of a few percent. The works of Gayevsky [45-48] described aircraft measurements of longwave radiation fluxes in daytime made with a radiation detector consisting of a linear thermopile assembled in a thick brass casing and supplied with fluoric calcium or KRS-5 filters. The main features of the thermopile are [44] : sensitivity 0.7 V/W; inertia, 3 sec; diameter of the receiving surface, 10 mm; resistance, 22 Q; sighting angle (aperture) 90’ (a later model [45] had an almost hemispherical sighting angle and slightly different parameters). The field and laboratory experiments showed that in this case the measurement errors were not above 0.01 cal/cm2 min. The instruments are fixed in the cockpit against the outlets in the ceiling and floor of the cabin. Since the filter transmits some shortwave radiation, special corrections were introduced to eliminate its “parasitic” effect. For this purpose the daytime measurements employed an auxiliary glass filter. The filter completely excluded the wind effect. The numerous calibrations of the instrument after the blackbody resealed the linearity of its scale in a wide temperature range. Houghton and Brewer [101, 1021, for aircraft measurements of thermal radiation fluxes, constructed a vacuum bolometer with a KRS-5 filter. Investigations showed that its readings were strongly dependent on the filter temperature. At small elevations this source of error is not influential.

10.4. Net Radiation and Its Components in a Free Atmosphere

70 1

The total measurement error varies from a few to 10 percent. The description of their aircraft investigations of the radiant fluxes can be found in [103-1071. The first complex actinometric observations from helicopter were realized by Malevsky-Malevich et al. [108-1101. The preceding discussion shows that only the technique of measuring the shortwave radiation and thermal radiant fluxes may be considered sufficiently refined. As to the net radiation measurement, up to now there have been no daytime attempts in the free atmosphere. The matter stands almost the same with respect to the direct solar radiation. In this connection recently there were first steps made in preparation of the complex automatic instruments for daytime balloon measurements of the net radiation and its components [67-721. The primary stage of such investigations was based on the standard actinometric instruments. 4. The Set of Automatic Instruments for Balloon Measurements of Net

Radiation and Its Components: General Features. These instruments enable continuous measurement and recording of the global, direct solar, and reflected radiation of the net radiation, the full upward and downward radiant fluxes, and also of the air temperature, humidity, and pressure of the temperature of the actinometric and recording instruments and the total ozone content. For measuring the global and reflected radiation two standard Yanishevsky pyranometers are used. The direct solar radiation is measured by means of a thermoelectrical actinometer automatically sighted on the sun with the help of a photoelectric system [I 111. The full upward and downward fluxes and the net radiation are measured by means of Yanishevsky and double net radiometers supplied with special windshields. The air temperature is taken by a platinum resistance thermometer, as is the temperature of the instruments. For measuring the pressure and humidity the respective sensors of the standard radiosonde are used. The ozone content is determined according to the spectroscopic method from the ozone absorption bands, in the ultraviolet spectrum.

Construction of the Instruments. The measurement of the direct solar radiation in a free atmosphere is made possible solely by continuous observation of the sun. The main features of the photoelectric watching system are the following: There is a sun-seeker with an all-round survey and three degrees of accuracy in targeting. The accuracy of the continuous watch is

702

Net Radiation

5 angular minutes. The weight of the installation without the power supply is 8 kg, the consumed capacity 20 W, the range of the working temperatures from -70' to +40°C. There are alternating-current amplifiers with transistors. The use of the standard net radiometer for balloon measurements was made possible only after providing special wind protection. The construction of the windshield is shown in Fig. 10.14. The receiving surfaces, 4

2

FIG. 10.14

The windshield of net radiometers.

10.4. Net Radiation and Its Components in a Free Atmosphere

703

and 5 , of the net radiometer are covered with two polyethylene hemispheres, 2, resting on the wire frames, 3. The rings, 1, press the hemispheres to the base of the framework. The testing of the instrument showed that this protection excludes almost completely the wind effect on the readings, and that the system receiving surface protection satisfies the Lambert law. The upward and downward radiant fluxes are measured by a double net radiometer consisting of two net radiometers positioned in parallel. The space between them is divided into three cavities. The middle one is meant for radiative separation of the upper and lower net radiometers. The side cavities are blackened inside, while the bottom temperature is taken with a platinum resistance thermometer. The net radiometers serve as covers for the intermediate cavities. The other receiving surfaces of the net radiometers are protected by polyethylene filters. The construction of the protection is similar to that of Fig. 10.14. Recording of Measurement Data. As has already been mentioned, the measurement of the radiant fluxes is continuous throughout the flight. The zero is fixed for each sensor regularly (every 36 sec) by disconnecting its circuit. The surrounding air temperature is also measured continuously. The platinum resistance thermometer intended for this measurement contacts the bridge scheme of the self-recorder. The other parameters are measured periodically at 36-sec intervals. All the data, except air temperature, are put on record at the tape of a 13-train oscillograph. The time marks are spaced at I-min intervals. The pressure and humidity are ciphered. Also taped are the train readings taken in the check up of their standard tension and sensitivity. Location of Actinometric Instruments of the Recording and Auxiliary Devices. The instruments are fixed to a frame made of duralumin tubes, and suspended by steel ropes and a strap, which increases the distance from the balloon casing to the center of the frame up to 100 m, thus practically eliminating the shading effect of the balloon. The length of the frame is 8 m (Fig. 10.15). The pyranometers, the net radiometer, and pyrgeometer are placed at the ends of the frame, as far as possible from the container with the recording instruments. The pyranometers are situated at the ends of a short-period pendulum fixed at the gimbals. The receiving surface of one is directed downward; of the other, upward. The pendulum period is about 0.3 sec. Such swings of low amplitude are insignificant for the pyranometers, whose time constant is about 20 sec. The net radiometer and pyrgeometer are stiffly fixed to the frame before the flight.

704

Net Radiation

The watching system with an actinometer and ozonometer is fixed at the cover of the thermoinsulating container positioned at the center of the frame (Fig. 10.15).

\

/6

0

FIG. 10.15 Arrangement of instruments on the suspension frame. (1) upper pyranometer; (2) lower pyranometer; (3) cloud photorecorder; (4) ozonometer; (5) actinometer; (6) orienting device; (7) radiosonde; (8) air temperature sensor; (9) net radiometers; (10) ozonometer; (1 1) tracking system.

Outside the container are placed the temperature, pressure, and air humidity sensors, and also an instrument for checking the horizontal plane of the frame and the angular height and position of the sun relative to the longitudinal frame axis. The latter instrument was found to be necessary when examining the actinometric recordings on the tape of the oscillograph. The slowly varying (in dependence on height) values of the global and net radiation were affected by periodical disturbances, with an alternating amplitude and periods of about 10 and 72 sec. The readings of this instrument not only enable determination of the cause for such disturbances but also permit introducing corrections to the readings of the upper pyranometer and net radiometer. Inside the thermoinsulating containter there are the recording devices, the program and order block, the amplifiers of the watching system, and the power source (anode batteries, accumulators).

5. Vertical Projile of Radiant Fluxes in the Free Atmosphere. Let us now give a short review of the results of measurement of the radiant fluxes in the free atmosphere. Consider first the more numerous data on the thermal radiation.

10.4. Net Radiation and Its Components in a Free Atmosphere

705

Thermal Radiation. In 1952-1954 Shliakhov [65] conducted a long series of nighttime balloon flights for the purpose of studying the vertical profile of the net radiation during the night. These measurements were realized by means of a special thermoelectric net radiometer and a pyrgeometer constructed by Shliakhov for balloon flights (the main features of the instruments have been given above). The observations were conducted at “platforms.” During the night the balloon managed from 4 to 6 “platforms” at 1-, 2-, 4-,6-, and 8-km elevations. Along with the actinometric measurements, meteorological observations were made and the measurement of the dust content by means of an impactor. According to Shliakhov, the net radiation F = F f - I;+ increases with height and at the highest elevation (8 km) reaches 0.346 cal/cm2 min. The effective radiation of the blackened receiving surface of the pyrgeometer (F‘ = oT4 - I;+, where T is the temperature of the receiving surface of the pyrgeometer) increases up to a certain level ( 5 to 6 km) and then begins to decrease. The comparison of the observed effective radiation with the values calculated from the Shekhter chart reveals a satisfactory agreement in the case where the effective water vapor mass necessary for the calculation of the radiant fluxes with the chart is made after the formula

dpz,

eZcand p is the absolute humidity and atmospheric where f ( p ) = pressure at the level z; and p o = 1000 mb. Observations show that the introduction of the effective water vapor mass, intended to account for the dependence of the absorption upon pressure, is quite important in the conditions of a free atmosphere. It should be noted, however, that the aircraft measurements by Gayevsky [45] showed that above 1 to 2 km the observed longwave radiation fluxes appeared to be by 15 percent more, on the average, than those computed with the help and that of the Shekhter chart (the correction for pressure f ( p ) = the departure of the computed values from the observed values increases with height. This conclusion was justified by the results of the actinometric radiosonde measurements obtained by Kostianoy [76]. It is therefore possible that Shliakhov’s data can be accounted for by the compensation of systematic errors in the calculation of the upward and downward thermal fluxes. Thus the problem of the actually justified form of the correction for the pressure cannot be considered clarified. Gayevsky’s data clearly show that the cause for the increase of the effective radiation with height is a

G),

706

Net Radiation

more rapid vertical decrease of the downward long-wave radiation flux rather than of the upward. Similar results are revealed in Lopukhin’s work [57]. Given in Table 10.13 are the ratios of the downward to upward fluxes with clear skies according to Lopukhin [57]. TABLE 10.13 Ratios of Downward to Upward FIuxes Pressure, mb: 966 FJ/Ff :

965

950

900

814

800

748

697

600

505

422

0.82 0.87 0.81 0.75 0.59 0.69 0.64 0.60 0.53 0.41 0.38

Shliakhov’s values of the radiative air cooling were from a few hundredths to 0.25 deg/h, which means that the corresponding quantities calculated from the Shekhter chart are in a good agreement with the experimental data. The measurements of the dust content at different heights, conducted by Shliakhov [65], showed it to be insignificant over all cases. The effect of dust on the longwave radiation absorption could not therefore be traced. The theoretical calculations confirmed the conclusion that during the night observations, the dust effect cannot be notable. The above investigations refer to the conditions of a clear sky. It is natural that with cloudy skies the vertical profile of the thermal radiation fluxes undergoes considerable transformation. For instance, according to the aircraft measurements of Lopukhin [58], the nighttime vertical increase of the net radiation is not so clear with overcast skies as without clouds. Inside the dense cloud cover of the lower level, as a rule there is the state of radiative equilibrium; the net radiation is either very small or zero. The last conclusion was justified by the results of aircraft measurements of longwave radiation fluxes obtained by Gayevsky [47]. Since near the cloud boundaries the observed vertical gradients of the longwave radiation fluxes are large, the radiative temperature variations, especially above the clouds, are greater than with clear skies. With net radiation in the region of lower clouds decreasing, observations indicate radiative heating in this area. At the upper cloud boundary a strong radiative cooling takes place. For example, Lopukhin [58] observed the radiative heating under the lower stratus clouds equal to 0.06 deg/h; and above the cloud tops, the cooling of 0.28 deg/h. The above results of aircraft and balloon measurements of the longwave radiation fluxes relate to comparatively small heights. Certain recent investigations have been busy with measuring the thermal radiation fluxes

10.4. Net Radiation and Its Components in a Free Atmosphere

707

at different atmospheric heights, including the stratosphere. They were realized mainly by means of actinometric radiosondes. The data on the stratosphere give some experimental information about the outgoing radiation. Kuhn et al. [112], for example, obtained the following thermal radiation flux values at 20 km on a clear summer night: F f = 0.36 cal/cm2 min, and FC = 0.04 cal/cm2 min. On a clear winter night at 12 km, F f = 0.27 and F J = 0.08 cal/cm2 min. The authors of [112] assume that the outgoing radiation Fm = F f - FC. On this assumption the experimental data for the outgoing radiation considerably depart from the theoretical calculation of the upward thermal radiation flux at the level of the tropopause. It appears that the assumption F, FT', where FTf is the upward thermal flux at the tropopause, should be considered better substantiated than the identification of the outgoing radiation with the flux difference F f - FC. This is justified not only by theoretical calculations but also by experiments. Actually, according to Kuhn et al. [112], in all cases F f F J . This means that the upward flux is only slightly transformed by the stratosphere and the difference F, - F t is clearly less than F J . In Kagan's estimation [113] the outgoing longwave radiation may be identified with the upward flux at the level of 100 mb. As given in [112], with a clear sky the observed effective radiation increases with height up to the stratosphere. It is natural though that the vertical gradient of the effective radiation should decreases as the height increases, especially in winter. For example, the actinometric radiosonde launched on February 17, 1958, showed that already in the layer 200 to 300 mb there is a radiative equilibrium (the vertical gradient of the effective radiation is zero). Kuhn and Suomi [114] published the results of 15 simultaneous launchings of actinometric radiosondes made at different points on the territory of the central and western United States on July 29, 1959. Thus the first experimental data characterizing the geographical variability of the outgoing radiation were obtained. At a later date similar results were derived by Kostianoy and Pakhomova [77] and by Kostianoy [115, 1161. The authors of [114] used two vertical cross sections to characterize the spatial distribution of the effective radiation flux F f - F J in the layer from the earth's surface to about 30 km elevation. Figure 10.16 gives a vertical cross section of the effective radiation field (expressed in hundredths of cal/cm2 min) for 0600 h of July, 1959 from Las Vegas to the International Falls (Minnesota). Analysis of the vertical sections shows that the effective radiation distribution near the ceiling of sounding is similar to the field of radiation in the

-

>

708

Net Radiation

km

mb

FIG. 10.16 Vertical cross section of the net radiation field (IOF cal/cma min) between Las Vegas and International Falls (Minnesota), July 29, 1959, 0600 h.

troposphere in that it is determined first of all by the meteorological regime of the troposphere. For example, the minimal effective radiation is observed in the frontal region. The radiation field is found to be greatly variable at 25 to 30 km elevation, depending on the type of air masses. With tropical sea air the mean value of the effective radiation at 25 km is 0.35 cal/cm2 min. With conditions of a weak front, the measurements reveal a variation in

10.4. Net Radiation and Its Components in a Free Atmosphere

709

the effective radiation of 22 percent in the passage from one air mass to another. The maximal heat influx due to the longwave radiation occurs in the lower half of the troposphere. For the layer, for example, situated between the earth’s surface and the 400-mb level, the mean difference in effective radiation at the borderline is 0.20 cal/cm2 min, whereas the corresponding value for the layer from 400 to 15 mb is only 0.07 cal/cm2 min. Thus, about three-quarters of the total heat influx takes place in the tropospheric layer below 7 km. Figure 10.17 illustrates the data characterizing the vertical section of the radiative flux divergence (in cal/cm2min for a 100-mb layer) and clearly

SOUNDING POINTS

FIG. 10.17

cal/cm2min, for a Vertical cross section of the radiative divergence 100-mb layer) under the same conditions as in Fig. 10.16.

710

Net Radiation

shows the complexity of the spatial structure of the radiative heat inflow field. Here the regions of radiative cooling are marked C and those of heating are W . Staley and Kuhn [90] carried out two launchings of actinometric sondes in the area of intensive baroclynic zones in the middle troposphere. One of the ascents (April 5, 1958, southwestern United States) revealed an anomalous vertical profile of the effective radiation and the upward longwave radiant flux. Beginning from the upper boundary of the temperature inversion in the baroclynic zone to the ceiling of sounding (near 125 mb), the upward flux increases with height in spite of the monotonic decrease of the air temperature. Analysis of the possible causes for this abnormality shows that the most probable of them is the transfer of the radiosonde from the area obliterated by clouds to the cloudless atmosphere with a warm underlying surface. This justifies the importance of taking into account the horizontal nonhomogeneity of the atmosphere and underlying surface in the interpretation of the measurement data on the radiant fluxes in the free atmosphere. This also makes it possible to expect, in particular, that the results of measurements of the radiant fluxes from satellites and from balloons will be different. The calculation of the vertical profiles of the radiative temperature variations in both cases yielded quantitatively similar results. The obtained vertical dependences of the radiative temperature variations were sharply nonmonotonic with a transition from radiative cooling to heating in the baroclynic zones. The radiative cooling above and below these zones very from 0 to 4-5 deg/day. The radiative heating in the baroclynic zones is slightly less, but with a maximum of 4 deg/day. The curves of the vertical distribution of the radiative temperature variations at maximal height enables us to suppose that radiative heating occurs in the tropopause. The qualitative comparison of the experimental data with the results of theoretical calculations reveals a satisfactory agreement (except that the measurements do not show the radiative cooling at the upper inversion layer boundary predicted by theory). This may be explained by the low “resolving power” of the instruments, which does not allow detection of the radiative temperature variations in relatively thin atmospheric layers. According to 50 actinometric radiosoundings (with a Pohl net radiometer) performed by Ronicke [I171 at San Salvador (Central America) in different seasons, the mean vaules of the upward longwave radiation flux in the lower stratosphere is 0.35 cal/cm2 min in the dry season, 0.32 cal/cm2 min in the transitional period, and 0.29 cal/cm2 min during the rainy season. In all cases the observed effective radiation increased up to

10.4. Net Radiation and Its Components in a Free Atmosphere

71 1

about the tropopause and then slightly decreased with height in the lower stratosphere. According to the measurements of Fenn and Weickmann [97] in the Thule area (Greenland), made on February 14, 1959, the effective radiant flux at about 30-km height was approximately 0.16 cal/cm2 min. The vertical profile of the effective radiation is characterized by its increase with height from 0.05 cal/cm2 rnin at ground level to 0.20 cal/cm2 rnin within 15 to 20 km. Above 20 km the effective radiation was observed to decrease with height. Comparison of the results of these measurements with the above data for the intermediate latitudes shows that the Arctic values of the effective radiation are much smaller. It is obviously explained, in the main, by the low temperature of the underlying surface during the polar night. The actinometric radiosounding carried out by Muller [93], mostly in the cloudless conditions, showed, that on May 4, 1959 (18.21 to 20.20 GMT), the net longwave radiation increased from 0.075 cal/cm2 rnin at ground level to 0.315 cal/cm2 rnin at the height of 10.2 km (this level does not coincide with the minimal temperature level in the tropopause), then decreased to 0.195 cal/cm2 min at 13.75 km, and finally increased to 0.210 cal/cm2 rnin near 24 km. Comparison with the calculations from the Moller chart shows that the agreement in the troposphere is good but that the calculations do not reveal the inversion in the variation of the net longwave radiation in the stratosphere. The mean (over 46 launchings) vertical profile of the net radiation clearly demonstrates the above-mentioned inversion in the net radiation variation in the stratosphere. Analysis of the averaged profiles of the net longwave radiation relating to different circulation types showed that the minimal values of the outgoing radiation are observed with the zonal circulation ; the maximal, with the meridional. Mantis [118] studied the regularities in the variation of the outgoing radiation from measurement data on the temperature T R of a black sphere. The sounding with the black-sphere method showed that above 60 mb, the temperature TR had no vertical variation (the decrease with height of the downward longwave radiation flux in the stratosphere appears to be compensated by the increase of the upward flux). For the determination of the outgoing radiation it was therefore possible to use the following empirical formula F, = 2.12aTR4(50 mb) (10.22) where a is the Stefan-Boltzmann constant, and T R (50 mb) is the temperature of the black sphere at 50-mb level. Using this formula for the calcula-

712

Net Radiation

tion of F,, Mantis found out a strong decrease of the outgoing radiation connected with the increase in the upper cloud amount. Since the cirrus cloud has little effect on the incoming shortwave radiation, the net radiation of the system earth-atmosphere should therefore be expected to increase with the upper level clouds. With clear skies the outgoing radiation showed little variation, although the total water vapor content in a vertical column of atmosphere was greatly varying. The temperature dependence of the outgoing radiation was likewise weakly expressed. Three nighttime soundings at different places were used to plot charts of the isolines of the outgoing radiation over the eastern United States. These charts clearly show that cloudiness is the main factor that determines the variability of radiation. The latitudinal dependence of the outgoing radiation is characterized by a decrease northward. The obtained values of the outgoing radiation are in satisfactory agreement with the results of theoretical calculations. The averaging of all observational data gave a value of 0.345 cal/cm2 min. The mean effective radiation of the earth’s surface is 0.064 cal/cm2 min. Thus the heat loss due to the radiative cooling of the entire atmospheric thickness is 0.281 cal/cm2 min. The accuracy of these determinations is about 5 percent. Shortwave Radiation. Although the measurements of shortwave radiation fluxes in the free atmosphere are fairly numerous, all relate to relatively low heights. Since 1946-1947 the Central Aerological Observatory and then the Geophysical Observatory of Tashkent as well as some other local geophysical observatories (in Kiev, Minsk, Odessa) have been undertaking investigations of various elements of the net radiation in the free atmosphere, of the shortwave radiant fluxes above all. A great contribution to the development and direction of these studies was made by an outstanding Soviet specialists in actinometry and atmospheric optics, V. G. Kastrov, whose work dealt with the methods for measurements and with a number of important actinometric problems concerning the free atmosphere. The above research employed free balloons, automatic stratostats, and aircraft. The first stage of the investigations was devoted mainly to the vertical profiles of shortwave (global and reflected) radiation. Observations were usually conducted only during clear or almost clear days. The purpose of this research was to study the factors of the shortwave radiation attenuation in the atmosphere and to obtain information on the radiant heat inflow due to shortwave radiation at different atmospheric levels.

10.4. Net Radiation and Its Components in a Free Atmosphere

713

The applied instruments were standard actinometers used with surface measurements (pyranometers, actinometers, pyrheliometers). In the measurements from free balloons the pyranometers were fixed on gimbals and dropped from a cord at 50 to 70 m below the balloon cabin. The balloon ceiling was usually 7 to 9 km. The heights reached by aircraft varied from 3 to 6 km. In all cases along with the actinometric measurements recorded were the temperature, pressure, and humidity of the air. In 1953-1954 Piatovskaya [60, 61 ] conducted aircraft measurements of shortwave radiation fluxes up to 3 km, over various homogeneous surfaces with clear skies and with clouds of force 2, not more (Leningrad region). In the observation of the global radiation a standard Yanishevksy pyranometer was used. To measure the reflected radiation a special high sensitivity pyranometer was applied. The readings were taken at 200,500,1000,1500,2000,2500, and 3000 m. The technique of processing the readings was usually in combination with Kastrov’s method, which takes into account the peculiarities of aircraft actinometric measurements. According to [60], the global radiation almost always increases with height. Its decrease occurred only in the cases where the sun was obscured by cirrus clouds invisible to the eye. The gradient of the increase was in all cases decreasing with height, especially above 2 km. For example, in the layer 0 to 1 km it averaged 0.076 cal/cm2 min km, while for 2 to 3 km it decreased to 0.034 cal/cm2 min km. The dependence of the global radiation fluxes upon the altitude of the sun is in all cases nonlinear, The attenuation of the global radiation in the 0- to 3-km layer has an annual mean of 0.18 cal/cm2 min. The measurements of the reflected radiation (upward flux) were made above Lake Ladoga (Leningrad region) over all seasons and above a homogeneous area with crop planting (in winter a smooth snow field). The reflected radiation almost always increased with height, and its vertical profile depended upon the underlying surface type. As did global radiation, the reflected radiation showed a notable vertical increase only in the layer 1.5 to 2 km thick, above which it either rapidly slowed down or completely stopped. The vertical gradient of the reflected radiation was greatly variable. The maximal gradients were observed in the layer 1 to 2 km. For example, averaged over a year, the vertical gradient in this layer is 0.018 cal/cm2 min km, while in the underlying layer 0 to 1 km, it is 0.009 kal/cm2 min km and 0.008 kal/cm2 min km for the 2- to 3-km level. Calculations of the net shortwave radiation as a difference between the global and reflected radiation give, as a rule, an increase of the net radiation with height. The net radiation difference between two levels was used to

714

Net Radiation

calculate the shortwave radiation absorption by l-km layers. The absorbed radiation greatly varies in dependence upon season and decreases with height. For a year the absorbed radiation in the entire 0- to 3-km layer was, on the average, 0.135 cal/cm2 min. Calculations of the shortwave radiation absorption by water vapor from the Moller and Kastrov formulas showed that it decreased with height, and had a summer maximum and a winter minimum, averaging 0.042 cal/cm2 min for a year in the 0- to 3-km layer. The ratio of the measured absorbed radiation to that calculated by taking account of the absorption by water vapor only decreases with height. In the 0- to 3-km layer, its mean shows little seasonal variation, and the above ratio is, on the average, 3.2 for a year. The absorbed radiation values were used to calculate the radiative heating. Individual values of the radiative heating vary within a wide range from hundredths to 0.3 deg/h. Averaged over a year, the radiative heating is 0.1 deg/h in the 0- to 3-km layer. When calculated, taking into account the absorption by water vapor only, it was 0.03 deg/h. Interesting aircraft measurements of the shortwave radiation fluxes up to relatively high elevations (13 km) were carried out by Roach [119]. The data of these observations were processed to obtain the albedo and radiative heat inflow due to the absorption of shortwave radiation. Similarly to the above mentioned Soviet results, Roach found a considerable residual absorption of the shortwave radiation caused by aerosols. In atmospheric layers strongly polluted with aerosols, the radiative air heating can exceed 10 deg/day. Brewer and Wilson [120] have carried out measurements of solar ultraviolet radiation in a free atmosphere.

The Net Radiation and Its Components. Above has been described a set of automatic balloon instruments intended for daytime measurements of the net radiation and its components. Let us now consider some results of the balloon measurements, as given in [67-771. The daytime sounding with actionometric balloons up to 25 to 32 km was started in 1961. The data on the radiant flux profiles, obtained in 19611962 are summarized in [55, 681. We shall treat the results of 1963-1964 [71]. Knowledge of the radiant flux profiles makes it possible to study the energy transformation in the atmosphere, to find the seasonal peculiarities in the variation of the radiant fluxes, and to estimate the outgoing radiation. The gradients of the radiant fluxes enable calculation of the radiative flux divergence over each atmospheric component in the thermal regime. One of the soundings with a set of actinometric instruments was perfor-

715

10.4. Net Radiation and Its Components in a Free Atmosphere

med on October 23, 1964. The ascension took 2 h 30 min. The variation in the solar height during this period was 4' (from 26'21' to 22'36'). The area was exposed to the influence of a high-pressure ridge expanding over the eastern European territory of the U.S.S.R. in the zone of the washed out polar front that separated the continental temperature air from the continental tropical air. The branch of the Arctic front was near the polar. For this reason at some elevation there was observed a zone of close pressure isolines. It represented the stream flow passing from Tselinograd (North Kazakh, S.S.R.) to the southern Urals-Kuibyshev (on theVo1ga)Syktyvkar (on the N. Dvina). The launching was carried out in the outer anticyclonic area of the stream flow. Radiant Fluxes. Direct Solar Radiation S (Fig. 10.18). After the instruments were removed from the cloud at 1.9-km altitude, the actionometer was automatically targeted on the sun. The value S at this altitude was 1.22 cal/cm2 min. In the layer from 2 to 5 km, the solar radiation increased by a quantity AS = 0.35 cal/cm2 min. The increase of S then slowed down, but continued up to a height of 22 km, where S = 1.94 cal/cm2 min. Above 22 km, the flux of direct solar radiation was practically constant.

The Reflected Radiation Rk (Fig. 10.18). The ground value of the reflected radiation flux was 0.035 cal/cm2 min. It was the same up to the level of the continuous cloudiness (1 km).

Q

------ --___-__

Q-R

-- _-__

6 18 20 22 24 26 28 H, km

FIG. 10.18

30

-R

B

32

Vertical profiles of net radiation and its components measured in the ascent on October 23, 1964.

716

Net Radiation

In the cloud layer, Rk increased with a gradient 0.086 cal/cm2 rnin km in the lower part, with 0.187 cal/cm2 rnin km in the middle, and with 0.16 cal/cm2 rnin km in the upper layer. The maximal value during the given flight was 0.5 cal/cm2 min. With increasing balloon elevation above the cloud the reflected radiation decreased of 0.3 cal/cm2 rnin at the heights over 24 km (h, = 23'). The fall of 0.05 cal/cm2 rnin at the very end of the ascension related to the discontinuity in the cloud underlying the balloon.

The Net Radiation B (Fig. 10.18). Two types of net radiometer with a polyethylene wind protection were used for measuring B with the usual Yanishevsky and a double net radiometers. The vertical profiles of the net radiation obtained by means of both instruments fully coincide, but the range of fluctuations of B as shown by the double net radiometer was probably due to the use of gimbals. The systematic deviation between the respective net radiation readings was 15 to 20 percent. Those of the usual net radiometer were taken to be the principal readings because it was calibrated in the wind channel, whereas the double net radiometer was calibrated with respect to the sun at the horizontal position of its receiving surface. The ground value of the net radiation was 0.125 cal/cm2 rnin and did not vary in the space below the cloud nor even up to the middle cloud level. At about 7 km there was a small minimum of 0.1 cal/cm2 min, while above this altitude the observed B gradually increased to 0.24 cal/cm2 rnin at 28.5 km, which can be explained mainly by the changing properties of the underliyng surface. The Albedo A (Fig. 10.19, curve 3). In the layer under the cloud the albedo varied from 18 to 20 percent. The " peak " albedo inside the cloud reached 85 percent, after which it showed a monotonic decrease. At 20 km its value was already 42 percent, remaining at that up to 27 km and then decreasing slowly (with a decrease in R k ) down to 35 percent at the level of 31.5 km. The decrease of the albedo over 27 km resulted from the change in the nature of the underlying surface (cloud conditions). The Global Radiation Q (Fig. 10.18). The global radiation value at ground level equaled 0.2 cal/cm2 min. In the layer below the cloud and in the lower cloud, Q was practically constant. It increased to 0.66 cal/cm2 min. closer to the upper cloud boundary and was equal to 0.65 cal/cm2 rnin above the cloud top. The variation of the global radiation flux in the layer from 2 to 6 km was by about 0.12 cal/cm2 min. From 8 to 20 km it increased only by 0.03 cal/cm2 min, which is explained by the increase of S and the vertical decrease of the diffuse radiation. Above 26 km, Q was slowly increasing.

10.4. Net Radiation and Its Components in a Free Atmosphere

717

The value of the global radiation at the level of the sounding ceiling (31.5 km) was fully determined by S' = S sin h, , that is, the contribution of the diffuse radiation was insignificant. A 100

90

80 70 60

$. 50 a 40

O'

2

4

6 8 10 I2 14 16 18 20 22 24 26 28 30 32H, km

FIG. 10.19 Vertical albedo profiles. (1,2) for clear weather; (3) for cloudiness.

Net Shortwave Radiation Q - Rk (Fig. 10.18). This flux was calculated from the profiles Q and R k and was characteristic of the shortwave radiation transfer. The values Q - Rk within the cloud and below fluctuate near 0.15 cal/cm2 min. Their minimum in the middle cloud was 0.08 cal/cm2 min. In the upper layer Q - Rk reached 0.18 cal/cm2 min, while closer to the upper cloud boundary it decreased to 0.14 cal/cm2 min. Above the cloud the variation of the net shortwave radiation is caused by the increase of Q (up to 10 km) and decrease of Rk (within 10 to 20 km). Analysis of the results of a number of soundings reveals some general regularities in the transformation of the radiant fluxes and the fields of the meteorological elements during the summer and fall periods. Let us consider the profiles of the net radiation and its main components typical of these seasons, obtained in the soundings of 1963-1964.

The Net Radiation with a summer homogeneous underlying surface is transformed chiefly in the troposphere (Fig. 10.20). A state close to radiative equilibrium was observed in the morning hours at the solar height h, = = 25 to 30' (curves 1,2). Here the dashes indicate a probable variation of the net radiation for a homogeneous underlying surface. In fall the profile of the net radiation is likewise little variable (curve 3). With solar heights

718

Net Radiation

25 to 30°, the change in the net radiation usually occurred only within the lower 5-km layer. The exceptions are the data corresponding to the curve 3. The increase of the net radiation in the 8- to 21-km layer from 0.1 to 0.24

0

2

FIG. 10.20 (1,2) summer, h,

4

6

8

-

10 12 14 16 18 20 22 24 26 28 30 32 H,km

Vertical profiles of net radiation for summer and autumn. (3) autumn, h, = 26'; (4) summer, h, = 58'.

= 26-30';

cal/cm2 min was related to the decrease of the global radiation (see Fig. 10.18). It should be noted that the profiles 1 to 3 are reduced to the fixed solar altitudes. The profile 3, for example, is reduced to 26'32'. The minimal net radiation was obtained at low altitudes of the sun and with snow cover in November ( B = 0.07 cal/cm2 min). The Direct Solar Radiation (Fig. 10.21) undergoes changes up to about 15 to 22 km. Its vertical profile depends on the content and vertical distribution of the attenuating components : water vapor and aerosol particles.

0.8 0.6

/ *

Figure 10.21 gives four profiles of the direct solar radiation plotted from the data for the summer (curves 1, 2, 3) and fall (curve 4) periods. The air masses whose attenuating properties are characterized by the considered profiles are essentially different with respect to the content of water vapor and aerosol particles. Curve 1 was plotted with a mean atmospheric mois-

10.4. Net Radiation and Its Components in a Free Atmosphere

719

ture content equal to 2.4 " cm " of precipitated water. Profile 2 refers to the air mass of the type " warm sea," with the moisture content of 4.7 cm of precipitated water. The profiles 3 and 4 correspond to 3.5 and 1.78 cm, respectively. The considerable turbidity of the troposphere was an almost constant peculiarity of the zone of sounding. The exception was the case where there was an air mass of high transparency (profile 1) above the layer of the continuous lower cloud. The aerosol layers in the lower atmosphere are usually situated at 1 to 2 and 3 to 5 km, which is indicated by the change in the inclination of the direct solar radiation profiles of these obtained at different solar altitudes in July (1963-1964). It should be noted that differences between individual profiles disappeared above 21 km. The difference between the fluxes of solar radiation near the ceiling of sounding for the summer and fall measurements is determined by the variation in the sun-earth distance. The solar radiation profiles obtained at the heights of the sun 55' to 60' were characterized by variation of S only within the troposphere (curves 1,2). A notable increase of the direct solar radiation at h, = 25' to 30' continued up to 21 km. Above that height the increase of S continued by about 0.007 cal/cm2 min km. The measured S at the peak point of sounding was 1.87 cal/cm2 min in summer and 1.97 cal/cm2 min in fall. Reducing these values to the mean sun-earth distance, we have 1.93 and 1.94 cal/cm2 min, respectively. The given S profiles were obtained with the use of new corrections for the effect of surrounding temperature on the readings of the actinometer. Laboratory investigations revealed that the temperature correction is very important and that the correction for the joint effect of temperature and pressure cannot be determined with sufficient accuracy. The value of the mean temperature correction to the sensitivity of the actinometer is 0.0875 percent per degree.

Global Radiation Q (Fig. 10.22). The global radiation fluxes obtained at the altitudes of the sun close to 30' (curves 2,3,4) undergo transformation practically only in the lower 10-km layer. In the 10- to 15-km layer the decrease in the diffuse radiation is compensated by an increase of the direct solar radiation, and in this layer Q remains constant. Above 15 km the global radiation slowly increases with a decreasing attenuating air mass. The contribution of the diffuse radiation above 16 km can be ignored. The vertical global radiation profiles 2, 3, 4 are reduced to the fixed solar altitudes. Profile 2, for example, is reduced to h, = 30°, the other two to h, = 26'10'. The difference of the global radiation fluxes in the stratosphere results from the inequality of the solar altitudes during the measurements.

720

Net Radiation

Profiles 1 and 4 were obtained with complete cloud at the heights 2 to 3 and 1 to 2 km, respectively.

L

,

.

0

2

4

8

6

10 12

14

16

18 20 22 24 26 28 3032-

Y km

FIG. 10.22 Vertical profiles of global radiation for summer and fall. (1) summer, h, = 60°; (2, 3) summer, h, = 26-30'; (4) fall, h, = 2 6 O .

Reflected Radiation Rk (Fig. 10.23). The reflected radiation fluxes with clear skies depend more on solar altitude than on cloud. For example, the close profiles 2, 4, and 5 were obtained at a clear sky, but the profile 2

0

2

4

6

8

10 12

14

16 18 20 22 2 4 26 28 30 32 H, km

FIG. 10.23 Vertical profiles of reflected shortwave radiation. (1) summer, h, = 5 8 O , Sc force 10; (2) summer, h, = 5 5 O , clear; (3) summer, h, = 53O, cloud of three layers; (4) summer, h, = 30°, clear; (5) summer, h, = 2S0, clear; (6) autumn, h, = 2 6 O , Sc force 10.

corresponds to h, = 56' to 60°, while the profiles 4, 5 correspond to h, = 30' to 35'. The profiles 1, 3, and 6, relating to the conditions of a cloudy sky, are notably different because the profiles 1 and 3 correspond to greater solar heights than does profile 6. With an increasing height the value Rk in the sounding zone for a clear sky increased by 20 to 25 percent, and

10.4. Net Radiation and Its Components in a Free Atmosphere

72 1

for an overcast sky increased by 30 to 40 percent above the cloud level. Near the upper cloud boundary, Rk reached 0.9 cal/cm2 min. This value appears to be the highest possible for the temperate latitudes. At ground level, Rk varied from 0.04 to 0.2 cal/cm2 min. Profile 3 characterizes the variation of Rkwith three level clouds. The decrease of Rk above 10 km resulted from the variation in the underlying clouds. The same factor influenced the variation of Rk near the ceiling of sounding in the case of curves 5 and 6. The Albedo (Fig. 10.19). The variation of albedo with height is represented by only three profiles, all obtained at h, = 25' to 35'. Profile 1 is typical of the summer season, although its values are slightly higher than the usual summer means. As to the strong variability of the albedo represented by curve 3, this, as has already been mentioned, is caused by the cloud effect. Radiative Flux Divergence. The set of measurements compiled for net radiation and its components made it possible to obtain detailed profiles of the shortwave and longwave radiant fluxes. The vertical profiles of the total shortwave and longwave net radiation were used to calculate the radiative flux divergence and also the radiation temperature variations for 50-mb layers.t At this point certain details of the profiles of the radiant fluxes related to the effect of the horizontal nonhomogeneity of the underlying surface were corrected. Figure 10.24 displays the vertical profiles of the

t

1

408

V

H. km

FIG. 10.24 Radiation temperature variations for 50-mb layers from measurements of the integral net radiation. (1,2) summer; (3) autumn. t Kostianoy has discussed possibilities of direct measurements of the radiative flux divergence in his work [121].

722

Net Radiation

radiation temperature variations calculated from the total net radiation. Profiles 1 and 2 determine (ar/at),&d for the morning hours (h, = 25' to 35') on a clear July day. Profile 3 refers to the noon (h, = 25') of October 23, 1964. As is evident, the profiles of the radiation temperature variations in the considered cases are very different, although there are some common features expressed in the transition from the zones of radiative heating to the zones of cooling. Since the vertical net radiation profile varies only slightly, the values (aT/at),&dare small (except profile 3, obtained with an anomalous variation of the net radiation) and fluctuate near zero (radiative equilibrium). In the 3- to E-km layer there occurred insignificant cooling of about 0.02 deg/h, while in the 8- to 28-km layer a tendency to heating was observed. Figures 10.25 and 10.26 present the vertical profiles of the

t

3

0.30

0.221 0.20

! !I II

0.lOC

'Id

-0.02-0.06 -0.10

-0.14-018 -0.22-026 -

-0.30-

/

I\

0.l8I 0 161

/

IA*

I

I-----/-;b

0

I

,?

4

6

8

10 12

14

16

18

20 22 24 26 28 30 32

H , km

I

FIG. 10.25 Radiative heating for 50-mb layers due to absorption of shortwave radiation. (1,2) summer; (3) autumn.

radiation temperature variations, calculated from the shortwave and longwave net radiation. Here the sharp variations of (aT/at),,, for profile 3 relating to a cloud layer should be noted. It should also be noted that the great variations of (ar/at),,d calculated from Q - Rk correspond to a notable variation of the opposite sign for (aT/at),&d,calculated from Blw. Thus, in the majority of cases, the marked radiative cooling due to the

10.4. Net Radiation and Its Components in a Free Atmosphere

723

longwave radiation relate to the atmospheric layers that strongly absorb shortwave radiation.

c 0 24 0 20 0 16 0 12

0 08 0.l -34

0 02

-0.02 -006 -010 -0.14

-0.18

-0.22

FIG. 10.26 Radiation temperature variation for 50-mb layers due to longwave radiation. (1) summer; (2) autumn.

As has already been mentioned, the total net radiation in the sounding zone varied little, especially at low solar altitudes. Therefore the effect of the nonhomogeneity of the underlying surface (for instance, the underlying cloud variation) would lead to a notable exaggeration or underestimation of the radiation temperature variations. Such distortions appear to have taken place for the case described by profile 3. For the profiles (b’T/b’t)rad calculated from Q - R, negative values were at times obtained. For example, during the measurements inside a cloud layer there were obtained considerable values of -0.26 deg/h (the cloud layer is dashed). Such results can be explained by two causes: (1) by the horizontal nonhomogeneity of the shortwave radiation field inside the cloud; (2) by the precipitation of moisture on the protecting covers of the instruments during the passage through clouds. Rocket Measurements. Recent years have seen the first steps in devoloping rocket methods for investigation of outgoing radiation. Kasatkin [1221 described the construction of a high-altitude optical

724

Net Radiation

station designed for complex investigations of the radiative field at great elevations, in a wide spectral region. The station was envisaged as comprising a great number of instruments, such as telephotometers, teleradiometers, telespectrometers, spectroanalyzers, and net radiometers. Liventzov et al. [I231 made a rocket radiometer for measuring the integral longwave outgoing radiation. The radiation detector of this instrument, based on the principle of differentiation (the outgoing radiation is measured as the difference in the radiation of the earth and space), is a bismuth bolometer. Table 10.14 gives the results of the measurements carried out by Liventzov et al. [123] in 1958-1961. For comparison (which should be considered only conditional) are also given certain theoretical calculations. The outgoing radiation values are expressed in watts per square meter and also through the effective temperature TOK. Although comparison of the results of measurements and calculations is conditional (the latter are of climatological nature), it may still be noted that the measured values are in all cases less than those calculated. Since 1958 Markov et al. [124, 1251 have been carrying out systematic investigations of the outgoing infrared radiation by means of rocket- and balloon-borne instruments. The angular distribution of the outgoing radiation intensity was measured from rockets in the 0.8- to 40-p spectral region at 100 to 400 km, and simultaneously in the atmospheric thickness up to 30 km from geophysical balloons (about 50 measurements in all). The angular resolving power of the instruments was 2 x rad. The authors of [I241 have made the following conclusions of their research: (1) The observed agreement between the results of measurements and calculations is satisfactory. (2) The contribution of the upper atmospheric layers to the outgoing radiation is considerably greater than it was supposed (the height of the radiating thickness in the atmosphere can reach 150 km, and the vertical distribution of contributions to the radiation takes place over the layers, in particular, for the 2.5- to 8-p spectral region over 280, 430, and 500 km). (3) The angular distribution of the outgoing radiation does not show any small-scale nonhomogeneities. (4) There is no marked daily variation in the angular distribution of the outgoing radiation (of chief importance are the meteorological conditions of the moment).

Recently [124, 126, 1271 detailed investigations of the angular and spectral distribution of the outgoing longwave radiation have been made. Using instruments mounted on geophysical rockets that were launched to 500

TABLE 10.14 Results of Rocket Measurements and Theoretical Calculations of the Integral Longwave Outgoing Radiation. Afrer Liventzov et al. [123 ] Theoretical Calculations

Experiments Conditions of Measurement

Clear

4

...

...

Complete cloud

...

...

0 . 9 ~ 1 0200 ~

...

...

2.1 x102 248

1 . 4 ~ 1 0224 ~

...

.. .

1 . 8 ~ 1 0238 ~

470

200

200

20

16

15

Date

8/27/58

7110159

6/15/60

2/15/61

Moscow time

0.8, 0.6

0.4, 12

0.5, 42

Near noon

Mean ground temperature, O C

Kondratyev Filipovich

Simpson

Bauer, Philipps

1 . 9 ~ 1 0242 ~

1 . 8 ~ 1 0238 ~

2.2x1O2 250

1.2x1O2 216

Moderate cloud

Elevation, km

3

2

1

100

-2

726

Net Radiation

km on November 18, 1962, and June 18, 1963, Markov et al. [124, 126, 1281 realized simultaneous measurements of the spectral composition (wavelength interval 4 to 38 p ) and angular distribution (range of angles f90’ relative to the nadir) of the outgoing radiation. The measurements were made during both upward and downward flight by means of an impulse infrared rocket spectrometer. The spectral resolution of the radiation was performed with the help of modulation filters that transmitted in the region of the absorption bands of the modulating materials (quartz, lithium fluoride, fluorite, and nontransparent metallic membrane). In this case the measured value was the difference in radiation between the earth and the modulator. Since the temperature of the modulator was not fixed, in order to check its constancy and also to take account of the “parasitic” radiation of the inlets and other optical elements, the radiation of the space in the direction close to horizontal (the radiation of space is assumed to be zero) was recorded as standard. The constancy of the sensitivity of the detectingamplifying scheme was achieved by calibration to the standard incandescent lamp. The use of filters allows reaching a spectral resolving power of the order of several microns. The channels of the spectrometer embrace the following wavelength intervals : 4.5-38, 12.5-38, 4.5-8.5 p. To increase the amount of energy reaching the instrument, a slit diaphragm with the side ratios of 1:lO or 1:30 was used with the minimal resolvable angle of 2 x rad. A low inertia bolometer served as a radiation detector. All joints of the instrument with moving details were hermeticized. The spectrometer was calibrated to the black radiators with the temperatures from 77 to 350’K. The sensitivity threshold in the flying conditions was 1.5 x 10” W/cm-2, which corresponds to the resolution with respect to the effective temperature of 2.7’K. The measurement data showed that in the range of 200 to 500 km heights, the shape of the angular distribution curves are little dependent on altitude. For wide spectral regions the angular distribution was close to isotropic and almost without small-scale fluctuations. In the narrow regions the outgoing radiation intensity fluctuations were much greater (up to 50 percent). In the majority of cases the maximum in the spectral radiation distribution was observed in the 4.5- to 8.5-,u region, with the effective temperature of 270 to 280’K, whereas in the transparency window it was about 240’K (the meteorological conditions were determined by variable cloud from 1 to 2 to 7 to 8 tenths, with the lower boundary from 1 to 7 km). Comparison of the effective temperatures with the real values at various

10.4. Net Radiation and Its Components in a Free Atmosphere

727

levels revealed that in the case of the integral radiation, the effective temperature corresponded to air temperature at the 6-km level. For the region of the water vapor absorption band (4.5 to 8 . 5 ~ this ) level was 20 km. However, the high effective temperatures should in this case be attributed to the fact that the source of radiation lay in the upper atmospheric layers. In this connection the intensive infrared radiation of the atmosphere, as observed by the authors of [124, 1261, is of greatest importance. With different conditions of investigation the intensive infrared radiation of the atmospheric layers was observed at 250 to 300, 420 to 450, and about 500 km. This radiation was concentrated mainly in the wavelength range of 2.5 to 8 p and in the part of the atmosphere illuminated by the sun. The radiant flux in the sighting along the tangent, when the ray’s path through the atmosphere extended over 1000 km, reached (3 to 7) x lo2 W m-’. The radiant intensity was increasing during the maximal solar activity. Careful analysis of the conditions of instrument functioning showed that the above conclusions are not affected by any measurement errors but are quite objective. Concerning the nature of the emission it is assumed in [I261 that it was due to the excitation by the corpuscular solar radiation of certain gas molecules contained in the ionosphere. One of these gases appears to be NO (in any case with respect to 280 km). Calculations showed that with the observed radiant intensity the effected temperature must be about 2000OK. In the given case, however, it is impossible to compare the effective and the kinetic temperatures in view of the absence of the state of thermodynamic equilibrium. Lebedinsky et al. [129] carried out measurements of the spectral composition of the outgoing thermal radiation in the wavelength range of 7 to 38 p by means of a diffraction scanning spectrometer mounted on the satellite “Cosmos 45” (206-km perigee, 327-km apogee, 65’ orbital angle of inclination). The terrestrial radiation in the direction of the nadir was measured as the difference with respect to the space radiation at the satellite level in the horizontal direction. The type of cloud underlying the satellite was determined photometrically by measuring the nadir radiance in the spectral region of 0.6 to 0.8 p (the spatial resolving power of the photometer was about 30 km). The infrared spectrophotometer consisted of two monochromators with plane reflecting diffraction gratings of 24 lines/mm (of 7- to 20-p wavelength) and 12 lines/mm (14 to 38 p). In the first wavelength range the spectral resolution was increasing with an increase in wavelength from 1.4 to 1.1 p ; in the second, it varied from 2.8 to 2.1 p. The instrument angle of

728

Net Radiation

sighting field was 1'46' x 2'20', which corresponds to a ground area of 75 km2 with a mean orbital altitude of 250 km. Radiation was detected by a bolometer with a 1 x 1 mm2 surface. The instrument was calibrated to blackbody characteristics. The duration of a complete measurement cycle was 81 sec. The readings were recorded on a 35-mm tape of the sixtrain oscillograph (container with the tape was landed). The operation of the instruments produced 2880 recordings. Typical curves of outgoing radiation spectral distribution with cloudy and clear skies clearly indicated the 9.6-,u ozone band and the 15-,u carbon dioxide band. The observed correlation between photometric readings and measurement data for the 8-7 to 12-,u transparency window was markedly negative, which reflects the opposite effect of clouds on the shortwave and the longwave outgoing radiation. A similar negative correlation was observed with the outgoing radiation of 18.5-,u wavelength. The effect of clouds was found to be considerable in all the studied intervals of the spectrum. This is due to the fact that the main contribution to the outgoing radiation of the considered wavelength range is made by the lower troposphere. The temporal variability of the radiation in the region of the 9.6-,u ozone absorption band was observed to be very great, which resulted from the variations of the ozone content in the atmosphere. It is supposed that the radiation in the 14.1-,u ozone band may greatly contribute to the outgoing radiation corresponding to this wavelength. Interesting spectral measurements of the infrared outgoing radiation were made by Band and Block [129a]. Many investigations (for example, [130-1321) were devoted measurements of the ultraviolet (solar and diffuse) radiation. Important information on the outgoing radiation in the 3.4- to 4.2-,u transparency window was obtained by means of a high-resolution infrared radiometer during the operation of the Nimbus I meteorological satellite (see [133]).

10.5. Climatology of Net Radiation of the Earth The first measurements of the net radiation of the atmosphere and the system earth-atmosphere (to be considered in the next section) are quite recent. For this reason various methods of their theoretical calculation are used in the determination of these quantities, particularly when the investigation is concerned with the climatology of the net radiation. We shall consider later some results of such theoretical calculations, mainly following the works [134-1411.

10.5. Climatology of Net Radiation of the Earth

729

1. Net Radiation of the Atmosphere. Equation (10.2), which determines the atmospheric net radiation, contains three components : the effective radiation F,, the outgoing radiation F,, and the solar radiation absorbed by the atmosphere, 4'. Calculations and observations show that the latter value. is much smaller than the other components. The atmospheric net radiation is therefore determined mainly by the thermal radiative influx I;, - F,. It is easy to see that, on the average, F,, < Fm , and consequently the atmospheric net radiation is negative. This is due to the fact that the atmosphere absorbs only the earth's thermal radiation (and considerably less so the solar radiation), while its emission is directed toward the earth's surface and toward space. Kondratyev and Dyachenko [1351 calculated the longwave net radiation of the atmosphere as Fa = F,, - F,, which made possible the plotting of charts to show its geographical distribution (the given longwave radiation values are absolute throughout the section). For these charts the data of 260 points (165 continental and 95 maritime) equally spaced over the earth's surface were used. The polar latitudes (over 80" N and 70" S ) and high altitudes were not considered because of the lack of a necessary amount of data. The total area of the investigated surface is 460.1 million km2 out of the earth's 510 million km2. The isolines of the chart of the distribution of the annual longwave net radiation totals are plotted at 20 kcal/cm2. The monthly isolines have a 2 kcal/cm2 spacing. The work [135] gives a map of the distribution of the annual totals of the atmospheric longwave net radiation (Fig. 10.27) and 12 monthly charts. It follows from considerations of the charts that the field of the atmospheric longwave net radiation is farly homogeneous, with a small range of the monthly and even the annual totals variation. The considered values show a monotonic increase from the poles equatorward in all the charts. Analysis of the chart of the annual longwave net radiation totals (Fig. 10.27) reveals that these totals vary from less than 100 kcal/cm2 yr in the polar latitudes to 160 kcal/cm2 yr at the equator. The isolines are mostly in the latitudinal direction. At the sea-continent border, discontinuities of the isolines result from the horizontal nonhomogeneity of the temperature field. A certain break of zonality in the variation of the isolines is observed over cold and warm ocean currents. With the cold currents the longwave net radiation of the atmosphere Fa decreases. This occurs, for example, over the cold Peruvian current or above the currents caused by the westerlies near the southwestern coasts of Australia. The warm currents are connected

730 Net Radiation

10.5. Climatology of Net Radiation of the Earth

731

with an increase in the longwave net radiation totals. Such an increase is observed over the warm Gulf Stream, over a branch of the northern Pacific ocean current, and over the warm currents in the Indian ocean. These data reveal a possibility of satellite detection of ocean currents, since the fields of the outgoing radiation and radiative flux divergence in the atmosphere indicate their presence. The great ocean area of the Southern hemisphere and the position of the thermal equator north of the geographical borderline causes the mean heat inflow to the atmosphere of the Southern hemisphere to be larger than that of the Northern hemisphere. The maximal longwave net radiation of 140 to 160 kcal/cm2 yr takes place over the warm equatorial currents with a frequent recurrence of clouds. The absolute maximum in the longwave radiative heat influx to the atmosphere was observed over the Pacific Ocean and equaled 163 kcal/cm2 yr. Comparison of the geographical distribution charts for the longwave net radiation of the atmosphere, F,, with analogous charts for the outgoing radiation F, and for the effective radiation of the underlying surface, Fo ,shows that the distribution of the longwave net radiation over the ocean is largely determined by the influence of the outgoing radiation. On continents the longwave radiation is determined by the effective radiation of the earth’s surface. In this connection it should be noted that the minimal F, values are related to deserts, where the observed effective radiation of the surface, Fo ,is at its highest (due to the high temperatures, low atmospheric moisture content, and insignificant cloud). For example, in the Northern hemisphere a notable decrease of the annual totals, F, , was observed in the deserts of North Africa and Arabia, where F, was less than 120 kcal/cm2 yr. In the Southern hemisphere the longwave net radiation was observed to decrease considerably over the Great Sandy Desert of Australia and over the deserts of South Africa. The absolute minimum was noted over the Kalahari desert (South Africa), with F, less than 100 kcal/cm2 yr. The mean atmospheric longwave net radiation for the globe is 131.5 kcal/cm2 yr (Table 10.15). The mean continental value is 119.0 kcal/cm2 yr, and the maritime is 136.0 kcal/cm2 yr. These values were calculated with account taken of the fact that the land and the sea areas are 29 and 71 percent of the globe, respectively. The authors of [I351 also obtained the annual totals of the atmospheric longwave net radiation for different latitudinal zones (Table 10.15). In their determination the sea-land proportionality for each latitudinal belt was taken into account. The mean annual total for a zone was calculated

732

Net Radiation

from the formula

where S is the total zonal area, S, and S, are the respective area of land and sea, and (Fa)L and (Fa)sare the atmospheric longwave net radiation over land and sea, respectively. The analysis of Table 10.15 shows that the longwave radiative heat inflow for the whole atmospheric thickness increases from the poles equatorwards. Note here that the mean net radiation for the Southern hemisphere is slightly larger than for the Northern hemisphere. This is due to the great ocean area of the former. Let us now characterize the results of the analysis of the monthly distribution charts of the atmospheric longwave net radiation (Figs. 10.28 and 10.29 give the January and July maps). The monthly maps of the longwave net radiation distribution are to some degree similar to the annual charts. In winter the Northern hemisphere has a clear minimum of F, in the area of the Siberian anticyclone, while the Gulf Stream somewhat increases the heat inflow to the atmosphere of the North Atlantic region. The tropical maxima slightly displace from season to season. In summer they are observed near the tropics; in winter, closer to the equator. From the consideration of these maps it is possible to state that the maximal absolute values of the monthly totals Fa occur over the equatorial ocean. In July the maximum of the heat inflow displaces north of the equator and is more than 12 kcal/cm2 mo(in the Pacificocean more than 13 kcal/cm2 mo). A reverse picture is observed in January. The F, maxima displace from the equator southward and exceed 12 kcal/cm2 mo (13 kcal/cm2 mo in the Pacific Ocean). Further analysis of the monthly charts shows that the maxima in the longwave radiative heat inflow to the atmosphere in this case are also connected with the maximal outgoing radiation (F, N 16 kcal/cm2 mo) observed over ocean. This is evident from the comparison of the above charts and also from a graph of the dependence of the atmospheric longwave net radiation on the outgoing radiation plotted for oceans (Fig. 10.30). The relationship is farly close and linear. The correlation coefficient r = 0.94. The effective radiation of the surface appears to have little influence on the radiative heat inflow to the atmosphere over the ocean. This is explained by the fact that the maritime Fa values are little variable and relatively small (3 to 4 kcal/cm2 mo) as compared with F,.

TABLE 10.15 Mean Latitudinal Values Fa (kcal/cm2yr). Afrer Kondratyev and Dyachenko [135]

Latitude, deg

90-80 80-70 70-60 60-50 5040 40-30 30-20 20-10 10-0 0-10 10-20 20-30 3040 40-50 50-60 60-70 70-80 80-90 From 80' N lat. to 70" S lat.

Land Area,

Sea Area,

Total Zonal Area, mln (lo6) km2

%

%

10 29 72 57 52 43 38 26 23 24 22 23 12 3 1 10 78

90 71 28 43 48 57 62 74 77 76 78 77 88 97 99 90 22

3.9 11.6 18.9 25.6 31.5 36.4 40.2 42.8 44.1 44.1 42.8 40.2 36.4 31.5 25.6 18.9 11.6 3.9

71

460.1

119.0

Mean

Mean

Fa

Fo

Mean Zonal

over Land, kcal/cm2 yr

over Sca, kcal/cm2 yr

kcal/cm2 yr

Fa

C Fa Zonal, kcal/cm2 yr

L

0 01

e3

111 115 114 112 110 119 126 132 131 131 113 111 114 118

118 121 120 129 139 150 150 151 147 135 127 125 120

136.0

131.5

116 117 116 121 131 144 146 146 144 130 125 124 120

21.9 x 1OI8 30.0 x 1OI8 36.6 x los8 44.1 x 10'8 52.7 x 1Ol8 61.7 x 1OI8 64.5 x 10'8 64.5 x los8 61.7 X 1Ol8 52.3 x lor8 45.6 x 39.1 x 1Ol8 30.8 x 1Ol8

5

B s z z

s

w

4

w w

FIG. 10.28 Geographical distribution of monthly totals of the atmospheric longwave budget, January.

FIG. 10.29 Geographical distribution of monthly totals of the atmospheric longwave budget, July.

736

Net Radiation

..

*

.... . ..... ... .......

.. . . : : :

.. ... ...:....... .... .. .

. ...

*

i

FIG. 10.30 Dependence between the longwave budget and the outgoing radiation.

The continental Fa values are strongly affected by the effective radiation of the earth's surface, whose influence becomes less only with a complete low cloud cover. In this case the temperatures of the earth's surface and the lower cloud boundary are equalized, which leads to almost zero values of the effective radiation. The relationship between the atmospheric longwave net radiation, Fa, and the continental outgoing radiation F, is not simple. Let us consider the graphs of Fig. 10.31 for the purpose of analyzing the annual variation in the atmospheric longwave net radiation. The presented

Fig. 10.31

737

I............ I I U P r n I x x I

-

MONTH (d 1

7

I

mnnrrIxxI month

(e 1

t

13

7 1 . . . . . . . - - . . -

r m s ! m l x s I

&

MONTH (9)

FIG. 10.31 Annual variation of the longwave net radiation in different landscape and climatic zones.

(a) tundra [(l) Amderma, Siberian Arctic; (2) Cape Barrow]; (b) forest of the temperate latitudes [(l) Sverdlovsk, (2) Forth Nelson]; (c) wooded steppe and forest of the temperate latitudes [(l) Rostov-on-Don, (2) El Passo]; (d) arid zones of the subtropical latitudes and the Mediterranean area [(l) Tashkent, (2) Rome]; (e) region of subequatorial monsoons [(l) Fort Larni, (2) Calcutta]; (f) tropical deserts [(I) Vadi Halfa, (2) Alice Springs]; (g) equatorial climate of the tropical forests [(l) Sao Gabriel, (2) Batavia, (3) Singapore].

738

Net Radiation

curves depict the annual variation of Fa with different climatological and local features. To secure direct comparison of Fa and F, throughout the year, the selected points were coincident or close climatologically to those used in the work of Yefimova and Strokina [37]. It is evident from the consideration of Fig. 10.31 that the annual variation of the atmospheric longwave net radiation in tundra has a May and September maxima of 10.5 kcal/cm2 mo at Amderma (Siberian Arctic) and 9.5 kcal/cm2 mo at Point Barrow (N. Alaska). These maxima are due to a considerable cloud amount and the resulting low F, values. During the warm period the observed minimum is fairly deep (9.0 to 9.2 kcal/ cm2 mo) and is due to the maximal F,. Figure 10.31(b) presents the annual variation of the longwave net radiation for the wooded area of the temperature latitudes. The curves are rather smooth, with a summer maximum of 10.2 to 10.5 and a winter minimum of 8.7 to 9.0 kcal/cm2 mo. In the zone of wooded steppe, and the steppe presented in Fig. 10.31(c), the quantity considered has a deep spring minimum (at Rostov-on-Don, 9.1 kcal/cm2 mo; at El Paso, 8.0 kcal/cm2 mo). This period is characterized by large effective radiation values. In early summer (June and July) a maximum Fa is observed (9.7 to 10.2 kcal/cm2 mo). The atmospheric longwave net radiation decreases notably closer toward fall. Figure 10.31(d) gives the curves of the annual Fa variation for the dry areas of the temperate latitudes (Tashkent) and for the Mediterranean (Rome). The annual variation of Fa is here quite great, with a range of about 2 kcal/cm2 mo. The minimum Fa occurs in summer and is due to the high values of the effective radiation of the earth’s surface, resulting from hot and cloudless weather. The great monthly F, totals in the dry temperate zones are caused by the high temperatures and the presence of cloud. A similar increase of Fa in the Mediterranean follows the considarable cloud amount and increased humidity during winter. In the dry temperate zone (Tashkent) the maximal Fa is observed in spring, with low F, and high F, values. In summer there is a minimum of Fa due to the maximal F,, though in this period the outgoing radiation F, exceeds 16 kcal/cm2 mo. Returning to the analysis of the charts of the longwave net radiation geographical distribution, we note that in deserts (Sahara, Kalahari, Arabian) the January Fa values are fairly large (6 to 8 kcal/cm2 mo), which can be accounted for by the high temperatures and little cloud. The F, values here are small, with minima less than 10 kcal/cm2 mo (less than 8 kcal/cm2 mo in Australia).

739

10.5. Climatology of Net Radiation of the Earth

In the Northern hemisphere, in America as well as in Eurasia, the longwave net radiation of the atmosphere in January is little variable, with a range from 9 to 10 kcal/cm2 mo, lowering to about 8 kcal/cm2 mo only in the area of the Siberian anticyclone. This Fa minimum is the result of the great cooling of the underlying surface, which causes the outgoing radiation to fall to 10 kcal/cm2 mo. In July in Eurasia there is a washed-out field of the atmospheric longwave net radiation. In the high latitudes (behind the Artic Circle), Fais of the order of 9 kcal/cm2 mo. The rest of the continent has Fa less than 10 kcal/cm2 mo. In North America at the same latitudes, Fa has the same values. The exception is the Arctic regions, where Fa is about 8 to 9 kcal/cm2 mo. In the zone of deserts and semideserts the minimal F, of less than 10 kcal/ cm2 mo are observed in July, which is caused by the high effective radiation. The areas of the equatorial monsoons are presented in Fig. 10.31(e). We see a considerable range of the annual F, variation. The maxima of Fa are observed in fall (11.8 to 12.6 kcal/cm2) and the minima in winter (9.2 to 9.6 kcal/cm2 mo). A small-range annual variation of Fa takes place in the tropical deserts (see Fig. 10.31(f)). In African deserts Fa is 9.0 to 10.5 kcal/cm2, while in Australian deserts, it is somewhat lower (8.5 to 9.0 kcal/cm2 mo). Figure 10.31 (g) displays the curves of the annual variation of the quantity considered for areas with the climate of the equatorial tropical woods, where the cloud amount is significant and humidity high. The range of Fa here is smooth throughout the year, with large values of the order of 11.0 kcal/cm2 mo (San Gabriel, California). On the islands of the same climatic zone (Batavia and Singapore) the variation of F, has a large amplitude of the order of 1.0 to 1.5 kcal/cm2 mo. Calculations of the shortwave (incoming) component of the atmospheric net radiation are rather few, which results in limited information for the total net radiation of the atmosphere. This subject has been given most detailed treatment by Vinnikov [142]. According to Budyko (137), the latitudinal distribution of the annual atmospheric net radiation totals in the Northern hemisphere is characterized by the following values: Latitude, deg:

0-10

R, , kcal/cm2yr :

-56

10-20 20-30

-54

-50

30-40 40-50 50-60 60-90 0-90

-59

-69

-76

-73

-60

As seen, the atmospheric net radiation slightly decreases in absolute value within the latitudinal belt from the equator to about 25' and then increases

740

Net Radiation

again up to about 60' latitude. To the north of 60' the absolute value is again observed to decrease. The mean annual net radiation in the Northern Hemisphere is -60 kcal/cm2 yr. Its components are Fa = 50, F, = 145, q' = 35 kcal/cm2 yr. Later, these components were calculated for the Southern hemisphere. The calculation of the atmospheric net radiation components for the earth as a whole gives the following values: F, = 150, Fa = 43, q' = 39, R, = 68 kcal/cm2 yr. In Budyko's estimation the negative pet radiation of the atmosphere is by about three-quarters compensated by the heat gain due to condensation, and by one-quarter by that due to the turbulent heat loss of the underlying surface. Moller [143] studied the regularities in the annual variation of the atmospheric net radiation in various climatic zones to find that the maximal absolute value always occurred in November and was equal to 250 to 300 cal/cm2 day. The minima were observed from May to July and varied from 140 to 190 cal/cm2 day. Table 10.16, compiled by Moller, compares the results of the calculation of the annual means of the atmospheric net radiation and its components for various latitudinal zones performed by different authors. TABLE 10.16 Comparison of the Calculations of the Aimospheric Nei Radiaiion and Its Components (cal/cm2day). After Moller [143] Calculation

30-50'N

40-60'N

60-90'N

90 217 127

82 233 151

81 227 146

121 359 238

99 333 234

59 287 228

96 305 209

83 280 197

65 230 165

92 300 208

79 29 1 212

55 262 207

1. Baur and Philipps (1934) q'

Fo

- Fa

I Ra I 2. Houghton (1954)

d Fo - Fm I Ra I

3. London (1957) q'

Fo - Fa

IRa I 4. Moller (1959)

d Fo - Fa

I Ra I

741

10.5. Climatology of Net Radiation of the Earth

This comparison shows that the data of the recent calculations are in fairly good agreement, whereas those obtained earlier are either underestimated (Baur and Philipps) or exaggerated (Houghton). Vinnikov’s calculations of the components of the net and thermal atmospheric radiation [144-1461 are, as was already mentioned, the most detailed. Table 10.17, borrowed from Vinnikov [146], gives the mean latitudinal distribution of the annual totals of the thermal balance components for various latitudinal belts. Here P denotes the turbulent heat loss of the underlying surface toward the atmosphere, L, is the heat income due to condensation, C is the advective heat transfer caused by the horizontal movement in the atmosphere. We see that the data of this table reveal a low latitudinal variability of the atmospheric net radiation. TABLE 10.17 Components of the Atmospheric Thermal Balance (kcal/cm2yr). After Vinnikov [I461

I Ra I

P

Lr

C

9 13 17 23 24 15 9 8 11 15

11 9 8

29 45 46 45 44 72 119 94 71 53 57 63 61

-32

40-50 50-60

70 60 60 69 82 83 76 74 16 74 71 64 57

Earth as a whole

72

13

59

0

Latitude

70-60’ N 60-50 50-40

40-30 30-20 20-10 10-0

O-loo s

10-20 20-30 30-40

-2 3 -1 - 14 4 52 28 12 -6 -3 8 12

Berland [138] plotted the annual and monthly (for the four seasons) maps of the geographical distribution of the atmospheric net radiation for the Northern hemisphere. Analysis of these maps likewise reveals a comparatively low spatial variation of the net radiation. The field of the net radiation is particularly “washed out” in summer in view of the small latitudinal variations of the absorbed solar and outgoing radiation (the net radiation is about 4 to 6 kcal/cm2 mo). In December the variation of the net radiation is from - 10 kcal/cm2 mo in the high latitudes and to 4 kcal/cm2 mo in the

742

Net Radiation

low latitudes. The annual net radiation also has a notable geographical variation. Similar results are obtained by Vinnikov [146] for the entire globe. The annual totals on the global scale vary approximately from -40 to - 100 kcal/ cm2yr. The smallest (in absolute value) totals are observed in the intermediate and high latitudes in summer, where the atmosphere absorbs the largest amount of solar radiation.

2. The Outgoing Radiation. The investigations of the outgoing radiation have a long history (see [l, 21). We shall consider here only the more complete and recent results of Vinnikov’s study [ 1441. Vinnikov proposed an approximate method for calculating the outgoing longwave radiation, and plotted monthly and annual charts of the planetary distribution of the outgoing radiation. Figure 10.32 gives one of his charts, the global distribution of the annual totals of the outgoing radiation. Analyzing these monthly and annual maps we see that the field of the outgoing radiation is fairly homogeneous. The outgoing radiation varies within a relatively small range of monthly and annual total. In all maps, F, is seen to increase from the poles toward the tropics. This is explained first by the increasing mean temperature of the troposphere, and second by the decrease in cloud in the high-pressure belts. At the equator there is a minimum of the outgoing radiation, due to the increasing cloud amount. The monthly 10, totals in January vary from 10 kcal/cm2 mo in the high latitudes to 16 kcal/cm2 mo in the tropics, while in July this variation is from 12 to 18 kcal/cm2 mo, respectively. The annual totals vary from less than 140 kcal/cm2 in the polar latitudes to over 200 kcal/cm2yr in North Africa and Arabia. The maxima in the outgoing radiation are observed over the low-latitude deserts with high air temperatures and insignificant moisture content and cloud. At places where the cloud amount and air humidity are high (equator) the outgoing radiation has considerably smaller values, but there is no continuance of small F., The variation of F, is dependent on atmospheric moisture and cloudiness. The zonal distribution of the outgoing radiation is seen to be broken by the effect of cold and warm ocean currents. With powerful currents the isolines are strongly fractured. The warm ocean currents cause an increase, whereas the cold currents effect a decrease of both monthly and annual totals of the outgoing radiation. This is evident from the effect of the warm Gulf Stream and the cold Peruvian current. Also effective with respect to the outgoing radiation field distribution are

10.5. Climatology of Net Radiation of the Earth

744

Net Radiation

the displacement of the thermal equator north of the geographical and the great ocean area of the Southern Hemisphere. As a result the thermal radiation of the Southern Hemisphere is less, on the average, than that of the Northern Hemisphere.

3. The Net Radiation of the Earth-Atmosphere System. As seen from (10.3), the net incoming radiation of this system is determined by the direct solar and diffuse radiation absorbed by the atmosphere and of the losses from the outgoing radiation. Calculations show that the quantity considered can be both positive and negative. In its annual range net radiation of the system in the intermediate latitudes is positive in summer and negative throughout the remainder of the year. Table 10.18 gives Vinnikov's calculations [1461, which are characteristic of the latitudinal and seasonal variability of the net radiation. (One can also consult Shneerov's work [147].) TABLE 10.18 Mean Latitudinal Distribution of the Net Radiation of the Earth-Atmosphere System (kcaI/cm2).After Vinnikov [146J

Latitude

January

July

Year

70-60' N 60-50

-10 -8.7 -6.8 -4.7 -2.6 -0.5 1.5 3.4 4.9 6.1 6.8 6.7

3.9 4.4 4.9 4.8 4 3.2 2.4 1

-49 -30 -12 4 14 23 29

50-40

40-30 30-20 20-10 10-0 &loo 10-20 20-30 3 0 40-50 50-60

s

-1 -3 -5.3 -7.3

-

31 28 20 9 -8 -29

It is evident from Table 10.18 that the transition from positive to negative values (northward) takes place near 40'. An interesting conclusion reached from consideration of the above data is that the net radiation of the Southern Hemisphere exceeds that of the Northern Hemisphere. Since there are no marked variations in the thermal regime of the earth

L

0

TABLE 10.19

p

Mean Annual Latitudinal Distribution of the Net Radiation and Its Components of the Earth-Atmosphere System (cal/cmamin). After London 11391

P 8

N Latitudes, deg

8-

Net Radiation Component 0-10

10-20

20-30

30-40

40-50

50-60

60-70

70-80

80-90

Mean

% 2

a

P Absorbed solar radiation

0.403

0.409

0.387

0.341

0.276

0.224

0.169

0.122

0.160

0.324

Outgoing radiation

0.347

0.354

0.353

0.327

0.306

0.287

0.270

0.253

0.245

0.324

Net radiation

0.056

0.055

0.034

0.014 -0.030

0.139

O.OO0

-0.063

-0.101

-0.131

P

5.

8

746

Net Radiation

as a whole, it follows that the mean annual net radiation (identical with the thermal balance) of the earth-atmosphere system must be zero. The same conclusion is reached by London [I391 in particular, whose data are given in Table 10.19. At the same time Table 10.19 gives an idea of the relationship among the components of the net radiation of the earth-atmosphere system. Berland’s data on the mean latitudinal range of the monthly net radiation and its componets for the same system are presented in Table 10.20. Comparison with Vinnikov’s data (Table 10.18) reveals a satisfactory agreement between these results. TABLE 10.20 Mean Latitudinal Variation of the Monthly Net Radiation and Its Components for the Earth-Atmosphere System (kcaI/cm2).After Berland [138]

December

June N Lat., deg

70 60 50

40 30 20 10 0

15.4 17.2 18.3 19 19.1 18.5 17.2 15.5

12.9 13.4 13.6 14.1 14.6 14.2 13.3 12.7

2.5 3.8 4.7 4.9 4.5 4.3 3.9 2.8

0.0 0.7 2.6 5.7 9.6 12.8 15.5 16.9

11.4 11.7 12.3 13.1 14.3 14.8 14.1 13.1

-11.4 -11 - 9.7 - 7.4 - 4.7 - 2 1.4 3.8

London [I391 has also calculated the seasonal and annual net radiation and its components of the system averaged for the entire Northern hemisphere (Table 10.21). The data of Table 10.21 make it possible to analyze the relation between the net radiation components. As seen, the main contribution to the income of the net radiation is made by the absorption of shortwave radiation by the earth’s surface. A significantly smaller portion of the radiation is absorbed by the atmosphere and even less by clouds. Accordingly, the greatest losses of the shortwave radiation are due to its scattering spaceward by the atmosphere and clouds. All absorbed (reflected) radiation components have an annual variation. For example, a radiant flux scattered by the atmosphere and clouds is maximal in summer, which is caused by the maximum of isolation outside the earth and an increasing cloud amount during this period. As to the reflection of radion by the earth’s

747

10.5. Climatology of Net Radiation of the Earth

TABLE 10.21 Seasonal Distribution of the Mean Net Radiation and Its Components for the Northern Hemisphere in the Average Cloud Conditions (cal/cmzmin). After London [139]

Net Radiation Components

Winter

Spring

0.348

0.580

Summer Autumn

Year

I. Incoming shortwave radiation Insolation at the upper boundary of the atmosphere

0.645

0.424

0.500

Radiation absorption in the atmosphere: (a) by ozone (b) by water vapor and dust (c) by clouds

0.011 0.044

0.016

0.019

0.010

0.014

0.067

0.092

0.057

0.065

0.005

0.010

0.011

0.007

0.008

Total absorption

0.060

0.093

0.122

0.074

0.087

0.023 0.078

0.037 0.141

0.015

0.029

0.048 0.162 0.024

0.028 0.103 0.018

0.034 0.121 0.021

0.116

0.207

0.234

0.149

0.176

0.085

0.142

0.129

0.045 0.043

0.091 0.050

0.090 0.070

0.091 0.064 0.048

0.112 0.072 0.053

0.173

0.283

0.289

0.203

0.237

0.530

0.564

0.614

0.581

0.572

0.439 0.091

0.473 0.091

0.523 0.091

0.494 0.087

0.482 0.090

Reflection and scattering of radiation spaceward: (a) by atmosphere (b) by clouds (c) by earth’s surface Total reflection Absorption of radiation by the earth’s surface : (a) of direct solar (b) of transmitted by clouds (c) of diffuse Total absorption by earths’ surface 11. Longwave radiation

Effective radiation of earth’s surface: (a) thermal terrestrial radiation (b) downward atmospheric radiation (c) effective radiation

748

Net Radiation

TABLE 10.21 (continued) Net Radiation Components

Winter

Spring Summer Autumn

Year

Thermal radiation of troposphere: (a) thermal radiation absorbed by troposphere (b) tropospheric thermal radiation

0.501

0.535

0.588

0.555

0.545

0.716

0.749

0.817

0.778

0.765

Longwave net radiation of troposphere

0.215

0.214

0.229

0.223

0.220

0.029 0.277 0.011

0.029 0.276 0.016

0.026 0.294 0.019

0.026 0.283 0.010

0.027 0.283 0.014

0.317

0.321

0.339

0.319

0.324

Thermal radiation spaceward: (a) of earth’s surface (in transparency windows) (b) of troposphere (c) of stratosphere Total outgoing radiation

surface, it has a spring maximum. Although the albedo is maximal in winter, the lowest insolation value at this time causes the reflected radiation to reach maximum only in the spring, with greater insolation values and the snow cover remaining on the major part of the hemisphere. The heat losses of the earth-atmosphere system are determined first of all by the thermal radiation of the troposphere, which makes the largest contribution to the outgoing radiation. The radiation of the earth’s surface (in the transparency windows) is less than 10 percent of the outgoing radiation, while the stratospheric contribution is still less, not exceeding 3 to 6 percent. The seasonal variability of the outgoing radiation and its components is seen to be very low. In the Northern Hemisphere the mean outgoing radiation is 0.324 cal/cm2 min, which corresponds to the effective temperature of -22’. The seasonal values of the outgoing radiation depart from the mean by not more than 2 to 5 percent. The low variability of the radiation of the earth’s surface is explained by the mutually compensating effects of its temperature increase from winter to summer, on the one hand, and by the increasing cloud amount and decreasing atmospheric transparency due to the increase of the total water vapor content, on the other. It is important to note, however, that the conclusion about the low variability of the out-

10.5. Climatology of Net Radiation of the Earth

749

going radiation holds good only with respect to the outgoing radiation values averaged over the whole Northern Hemisphere. As is evident from Table 10.21, the net radiation of the earth-atmosphere system for the Northern Hemisphere is positive in spring and summer, and negative in autumn and winter. The maximum summer value is 0.072 cal/ cm2 min, while the winter minimum is 0.084 cal/cm2 min. Vinnikov [ 1461 plotted monthly (for all months) and annual charts of the geographical distribution of the net radiation of the considered system. The most characteristic feature of this geographical distribution is its closeness to the zonal, which gives evidence of the leading role of the astronomical factors in the radiation regime. The zonality is seen to be broken in the desert areas. Also evident in the net radiation field is the nonhomogeneity of the underlying surface at the ocean-land division, which is indicated by the discontinuity of isolines. The net radiation of the earth-atmosphere system is positive throughout the year only in a narrow equatorial zone of f 10’ latitudes. At all other places the net radiation changes sign twice a year. For about three months (in summer) the net radiation is positive over the entire hemisphere (souhern and northern). The zone of negative values appears at the poles in summer and then gradually expands southwards, occupying the territory south of 30’ for five months. In spring the zero isoline begins to recede toward the north. The maximal positive values reach 40 kcal/cm2 yr, while the negative become 60 kcai/cm2 yr. Making use of the charts given in the “Atlas of the heat balance of the globe” [141], it is possible to determine all components of the thermal and water balances for the land, ocean, and the earth as a whole. Calculations show that the earth as a planet absorbs 168 kcal/cm2 yearly. Of this, 112 kcal/cm2 yr, or two-thirds, is absorbed at the earth’s surface and 56 kcal/ cm2yr, or one-third, in the atmosphere. The earth‘s surface loses yearly, on the average, 40 kcal/cm2, owing to the long-wave effective radiation, which results in its mean net radiation being 72 kcal/cm2 yr. Of this, 59 kcal/cm2 yr is spent in evaporation and 13 kcal/cm2 yr is given to the atmosphere in the turbulent heat loss. It should be noted that the heat expenditure in evaporation is 82 percent of the net radiation for the whole earth, this expenditure constitutes 90 percent for the ocean and almost 50 percent for land. In accordance with the amount of heat lost in evaporation the mean annual total of evaporation from the earth’s surfaces appears to equal 100 cm. This is, at the same time, the annual total of precipitation at the earth’s surface. The numerical values of the main components of the thermal and water

750

Net Radiation

balances of the earth’s surface obtained in the recent investigations somewhat exceed the formerly found values. In particular, the majority of the earlier studies would give the annual precipitation and evaporation for the whole earth equal to 80 to 90 cm. Their increase to 100 cm/yr is explained mainly by the greater accuracy in determining the normal precipitation on oceans, where it was not sufficient in the former years. As is known, the solar radiation is the chief climate-forming factor. It is of interest, however, that even now the amount of energy used by man is comparable with the net radiation value (see Budyko’s work [140]. According to latest data, the mean net radiation of the total continental surface is 49 kcal/cm2 yr, while the energy consumed by mankind is about 0.02 kcal/cm2 yr, with almost 1 kcal/cm2 for large areas in individual countries. Assuming that the annual increase of the energy generation be 10 percent we may expect that the total amount of energy due to man’s efforts will exceed the net radiation in less than a hundred years. In such conditions the role of the main climate-forming factor will pass to the energy generated by mankind. It is natural to expect, then, considerable changes in the regularities of the radiation regime and climate. The above-considered calculations performed by different authors show certain discrepancies, which stresses the necessity of experimental investigations of the quantity considered. An important step in this direction has been made by using data on the outgoing radiation obtained from meteorological satellites. Let us now discuss these data. 10.6. Investigations of the Earth’s Net Radiation by Means of Satellites The instruments used with the TIROS and NIMBUS meteorological satellites to measure the outgoing radiation in various spectral regions and the first results of these measurements are described by this author (see [148]). This author’s other work [149] is devoted to the practical application of such data. In the present section we shall reduce the subject to consideration of the recent satellite data on the componentes of the net radiation of the earth-atmosphere system (the significance of degradation errors in satellite radiometers must be mentioned, as for example, in [150, 1511). In accordance with the theoretical calculations discussed in the preceding paragraph the satellite data show that the outgoing longwave radiation has an equatorial minimum, almost symmetrical maxima in the subtropics, and a monotonic decrease polewards. Such results were obtained, for example, by Winston and Rao [152, 1531 (five-channel radiometer of Tiros 11; see Fig. 10.33), House [I541 (hemispheric sensors of the outgoing radiation in Tiros

10.6. Investigations of the Earth’s Net Radiation by Means of Satellites

751

550 I

Y.

500 “E

2 450 8 c

._ 5 5 400

7

.z 0

350

c

a 0

300 50

40

30

10 0 Latitude

20

S

10

20 30

40

50 N

FIG. 10.33 Mean latitudinal distribution of the outgoing Iongwave radiation measured from TIROS II over 26 days in comparison with theoretical calculations, (1) data of Houghton (annual means; (2) Retien (winter); (3) London (winter); (4) TIROS I1 (November-January); (5) Simpson (November-December); (6) Baur and Philipps (January).

IV; see Fig. 10.34),Bandeen et al. [ I S ] , and others. An interesting research in this field belongs to Hanel and Stroud [156].

.‘6

e

z

0.35

3

P

a 0.300 . 2 5 ~ 1I 1 I I 9060 30

0s

(II) 510 (I) so1 I

I

I

0

I

I

I

30

I

I

I I L

6090

ON

FIG. 10.34 Latitudinal variation of the outgoing longwave radiation measured from TIROS I V (semispherical sensors). (I) Feb. 8-Apr. 10, 1962; (11) Apr. 11-June 10, 1962.

The main factors of the mentioned peculiarities of the latitudinal distribution of the outgoing radiation are the earth’s cloud and temperature. For example, the outgoing radiation maxima in the subtropics are caused by the high temperature of the earth’s surface and the relatively small cloud amount (subtropical high-pressure zones). The equatorial minimum is con-

752

Net Radiation

nected with the area of the intertropical convergence zone. The results of measurements and calculations are seen to depart in a rather marked way (Figs. 10.33 and 10.34). Rao [157] investigated the Tiros I1 and Tiros I11 data on the outgoing thermal radiation relating to the belt 55' S to 55' N lat. (see Fig. 10.35).

LAT

FIG. 10.35 Latitudinal variation of the outgoing longwave radiation in summer and autumn. (1) from data of TIROS I1 (over 26 days, Nov.-Dec. 1960 and Jan. 1961); (2) data of TIROS I11 (9 days of July 1961).

As seen, in the intermediate latitudes of the Northern hemisphere the outgoing radiation increases from winter to summer, which can be caused by both a decrease in cloud and increase in the temperatue of the underlying surface. The inverse varibility of the outgoing radiation is observed in the southern intermediate latitudes. The two decreases of the outgoing radiation near the equator as observed by Tiros I1 can be explained by two local zones of greater cloudiness. The most detailed investigation of the variability of the net radiation components of the earth-atmosphere system has been made by Bandeen et al. [155] on the basis of 14 months of continuous Tiros VII operation. The method for processing observational data used in [155] is characterized in [149]. Table 10.22 gives seasonal totals of the net radiation components for the earth as a whole. Figures 10.36 and 10.37 present the curves of the latitudinal variation in the mean annual outgoing longwave radiation and albedo compared with London's calculations [ 1391. The seasonal variation of the outgoing long-wave radiation totals is seen to vary little (Table 10.22). The seasonal range of the latitudinal profile of the outgoing radiation as presented in [I551 shows that the tropical mini-

10.6. Investigations of the Earth's Net Radiation by Means of Satellites

753

TABLE 10.22 Seasonal and Annual Totals of the Net Radiation Components for the Earth. After Bandeen et al. 11551

Monthly Period

June-July Sept.-Nov. Dec.-Feb. Mar.-May

Outgoing Longwave Radiation 10l6 cal/min 1725.2 1725.2 1712.2 1746.5

Solar Radiation loz6cal/min

Albedo of Earth,

Reflected

Incident

%

699.0 922.2 859.7 796.0

2457.8 2574.8 2609.8 2544.2

28.4 35.8 32.9 31.3

mum occurs in the Northern hemisphere during most of the year, except from December to February. This causes the minimum in Fig. 10.36 to take place at 5' N lat., thus presenting asymmetry between the hemispheres.

FIG. 10.36 Latitudinal variation of annual means of the outgoing radiation (the scale of the axis of abscissas is proportional to the area of the latitudinal belts). (1) data of TIROS VII (region 8-12 p, June 1963-May 1964); (2) calculations of J. London.

The Northern hemisphere also contains the absolute maximum of outgoing radiation during as long a period (except from June to August), which asymmetry is clearly evident from Fig. 10.36. As a rule, the outgoing radiation in the northern latitudes outside the tropics is greater than in the southern, but the equatorial zones reveal an inverse relationship, which balances the heat losses in radiation by the two hemispheres.

754

Net Radiation

FIG. 10.37 Latitudinal variation of annual means of the planetary albedo (the scale of the axis of abscissae proportional to the income of solar radiation outside the atmosphere and corresponding latitudinal belts). (1) data of TIROS VII (spectral region 0.55-0.75 p , June 1963-May 1964); (2) calculations of J. London.

The latitudinal variation of albedo is almost directly opposite to that of the outgoing radiation and is caused by the cloud effect (see Fig. 10.37 and also Fig. 10.38, borrowed from [154]). The dashed curves of Fig. 10.38

-1

70

011 I 9060

I

1

I

I

I

0 OS

I

I

I

30

1

1

1

1

6090

ON

FIG. 10.38 Latitudinal variation of albedo measuredfrom TIROS IV (sensing halfspheres). (I) Feb. 8-Apr. 10, 1962; (11) Apr. 11-June 10, 1962, [averaged area 48ON48OS; albedo: (I) 28.9 percent, (11) 37.2 percent; reflected radiation: (I) 598 cal/cma day, (11) 500 cal/crna day).

10.6. Investigations of the Earth's Net Radiation by Means of Satellites

755

characterize a possible measurement errors influence. The maximal albedo near the equator takes place at about 5' N lat. (Fig. 10.37)' coinciding with the minimum of the outgoing long-wave radiation. The minimal albedo is, however, observed at 16' N lat., that is, is somewhat displaced south of the outgoing radiation maximum. The albedo of the Northern Hemisphere exceeds that of the southern within the latitudinal belt from the equator to 12' and is smaller at higher latitudes. Analysis of the annual albedo variation reveals that its minima occur approximately in July, and maxima in October. This finds reflection in the variability of the seasonal means as well (Table 10.22). The average latitudinal distribution of the net radiation of the earth-atmosphere system shows asymmetry in relation to the equator, which results in the transition from positive values in the low latitudes to negative in the high, taking place in different latitudinal belts of the Northern and Southern Hemispheres. This is clearly seen in Fig. 10.39 [154]. 0.20

1

0.15 .E

0.10

; 2

0.05

0

H

0

3

-0.05

I-

-0.10

2 W

2

-0.1 5 -0.20

90 60

30 OS

0 LATITUDE

30

60 90

ON

FIG. 10.39 Latitudinal variation of net radiation measured from TIROS IV (sensing half-spheres). (1) data of Simpson (1928) for March-May; (2) London (autumn and spring of 1957); (3) Tiros IV (March-May, 1962).

Astling and Horn [ 1581 used the Tiros I1 measurement data for 27 days (November 26, 1960 to January 6, 1961) to plot the mean latitudinal distributions of the outgoing longwave radiation for the entire global surface and separately for continent and ocean. The measurements were concerned with the zone 50' S to 50' N lat., except a part of central Asia and northern South America, including the adjacent regions of the Pacific and Atlantic

756

Net Radiation

Oceans. The initial data for each 24 h were averages (over at least 10 values) with respect to squares of 2.5' equatorial side and then used in plotting charts of geographical distribution of the outgoing radiation for every day of the 27 days. All data relate to the angles with respect to nadir not less than 56'. The averaged meridional profiles of the outgoing radiation were calculated by reading the charted values (there were 7269 points) and the mean latitudinal magnitudes were calculated for zones of 5' width. The obtained meridional profile represents, as in the above-mentioned works, an equatorial minimum, maxima in the warm and relatively cloudless subtropical zones and decreasing radiation with a further increase in latitude. The absolute outgoing radiation values appeared to be less (especially in the zone from 5' N to 10' S Iat.) than the earlier data of the Explorer VII satellite and actinometric radiosondes. Such deviation should evidently be explained by the inaccurate consideration of the effect of the edgeward darkening of the planet's disk in processing the Tiros I1 data (the outgoing radiation measured at large angles relative to the nadir appears to be underestimated due to the effect of darkening as compared with the corresponding "undersatellite" values). This conclusion is confirmed by the fact that comparison of the outgoing radiation values, averaged over all the measurements with the values selected for the nadir angles less than 26', revealed the former's underestimation. Since, however, even the "undersatellite" values obtained with Tiros I1 are less than the Explorer VII data, it should be believed that one of the causes for this deviation is the different calibration of the respective instruments. According to data for nadir angles less than 26', the outgoing radiation in the subtropical minimal zone (15-25' N, 10-35" S lat.) is about 480 cal/cm2 day. Astling and Horn [158] found a notable difference between the meridional outgoing radiation profiles relating to continent and ocean. With continents there is a sharp minimum of the outgoing radiation within 5' N to 15' S lat., while with ocean this minimum is weaker and displaces to the interval 5' to 10" N lat. It appears that the continental displacement of the minimum southward is due to the effect of clouds related to the intertropical convergence zone. Another important difference is observed in the subtropics and relates to the influence of the underlying surface temperature in these relatively cloudless areas. Over ocean (low variability of temperature) the outgoing radiation in both hemispheres in the subtropics is about 500 cal/cm2 day, while the corresponding continental values are 540 cal/cm2 day (the summer hemisphere with high temperatures) and 475 cal/cm2 day (the winter hemisphere with low temperatures). The data of [158] on the variation of the outgoing radiation relative to

10.6. Investigations of the Earth’s Net Radiation by Means of Satellites

757

its mean show that this variation is most marked over the continents. Recently there have been attempts made at investigation of the diurnal range of the outgoing longwave radiation. Astling and Horn, for example, in processing the Tiros I1 data for 27 selected days (Nov. 26, 1960 to Jan. 6, 1961) found a notable daily variation over both continents and oceans. In the case of continents, the daytime values exceeded the diurnal means by 0.04 cal/ cm2 min, while the nighttime values were systematically lower by 0.02 cal/cm2 min. With oceans, the daily variation appeared to be less marked though still notable. The statement of daily variation in the outgoing radiation over the ocean whose surface temperature is stable enough was quite unexpected and required careful analysis of the measurement errors. So far the effect of errors has not been taken sufficiently into account, which does not allow acceptance of the above conclusion on the daily variation of the outgoing radiation for ocean as being reliable. It has been noted above that the main factors determining the outgoing longwave radiation are clouds and atmospheric stratification. The recent calculations and measurements reveal a considerable effect of the aerosol layers on the outgoing radiation (see [159]). The available experimental data are, however, contradictory. For example, Gupta [160] analyzed the daily totals (averaged over 5 days) of the outgoing longwave radiation for various Indian stations as given by the Tiros IV meteorological satellite in April, May, and June of 1962. The obtained values were then compared with the calculations of the outgoing radiation with a clear sky from the averaged aerological sounding data derived by means of the Elsasser chart. The deviation between the measured and calculated values was not more than 10 percent. Such good agreement gives evidence, in particular, of the insignificant effect of aerosol dust particles on the longwave radiation transfer. This, however, does not change the conclusion about the notable influence of aerosol on the radiative heat inflow, reached on the basis of measurements and calculations of the radiative temperature variations due to longwave radiation. The attempt to use the Tiros I1 data for evaluating the outgoing radiation and albedo means for the earth as a whole with calculated high-latitudes Values led to the planetary outgoing radiation of 0.31 1 cal/cm2min and albedo of 0.38 [158]. It should be noted that the albedo is usually assumed to equal 0.34, while according to the Explorer VII data, it is 0.33. This discrepancy should be attributed to the limited (with respect to the time interval) use of data and also to measurement errors. In processing the Tiros I11 data, Wexler [25] evaluated albedo at from 4 to 10 percent (with clear skies) to 30 percent with continuous cloud (in the

758

Net Radiation

Sahara region the albedo reaches 20 percent in clear weather). The albedo attained 54 percent for channel 5 and 47 percent for channel 3 only during the first five orbits. The mean value for this period was 20 percent. Considering that the mean cloud albedo as measured from aircraft is 50 percent with individual values of 80 percent, it becomes clear that the satellite data should be regarded as underestimated. Conover [I611 worked out a method for determining the albedo of the earth-atmosphere system based on the use of television data (cloud cover and terrestrial photographs) obtained from meteorological satellites. This method was applied in the processing of satellite pictures and the simultaneous photographs made from a U-2 aircraft at 20 km elevation. The spectral sensitivity of the satellite TV camera and of the aerophotographing instrument was approximately the same. The TV system (including video recording) was calibrated by successive irradiation of the screen from three tungsten incandescent lamps (the use of several sources was caused by the necessity to model a wide range of radiances). The determination of the absolute cloud radiance (in energy units) from the signals recorded at the ground-receiving video installation was carried out by comparing with the amplitude of signals recorded in the irradiation of the TV camera during the calibration before the fight. As shown in [161], in the photometric film processing it is necessary to consider the following factors : (1) picture-to-picture variation in exposition; (2) effect of temperature variations on the TV functioning; (3) nonhomogeneous distribution of blackening density even with homogeneous lighting; and (4) non homogeneity of the film. When calculating the albedo from cloud radiances it was assumed that the radiation reflection by clouds is isotropic; the atmosphere overlying the cloud was taken into account solely by computing the contribution of the Rayleigh scattering for the nadir angle 22’ (the angular dependence of multiply scattered light ignored) and of the absorption by ozone, whose total content was 0.28 “cm”. The values of cloud and of terrestrial radiance in satellite and aircraft estimation agree satisfactorily. The averaged satellite albedo values vary from 7 percent (the Pacific, cloudless) to 92 percent (dense and extensive cumulonimbus clouds). The snow albedo was found to decrease from 70 to 51 percent four days after snowfall. For the White Sands Desert (New Mexico) the albedo was recorded at 68 percent with solar altitude of 78’ and nadir angles from 17 to 28’. Comparison with albedo data known from literature shows that similar satellite and aircraft measurements yield exaggerated values. Nordberg et al. [I621 analyzed the results of the effective temperature and

10.6. Investigations of the Earth's Net Radiation by Means of Satellites

759

albedo determination from the Tiros 111measurement data on the outgoing radiation by making use of simultaneous wide-angle photographs of cloud distribution. The three typical situations considered in [162] relate to the Atlantic Ocean and North Africa (two representing a cloudless atmosphere) and also to the eastern United States (cloudy sky). All data were obtained for local noon. It is of essence that in all the three case the TV cameras and wide-angle radiometer were almost exactly oriented with respect to the nadir. The nadir sighting angle for the five-channel radiometer varied from 0 to 45'. The results obtained by the authors of El621 are discussed in [148, 1491. Rasool and Prabhakara [I631 were the first to realize the climatological processing of the Tiros data for 1962-63; their purpose was to derive information on the average planetary distribution of the net radiation components for the esrth-atmosphere system in the zone 60' S to 60' N lat. For more accuracy with respect to the five-channel radiometer data, only those readings were used which related to the solar zenith angles of less than 60' and nadir sighting angles less than 45'. The Tiros IV data from February to June of 1962 averaged for 5' lat 5' long squares were used to plot the planetary albedo chart. In this plot only such points were considered that corresponded to not less than a hundred individual albedo values (most often 500). The albedo calculation was made from the channel 3 readings (spectral region 0.2 to 5 p ) on the assumption of isotropy of the solar radiation reflection by the earth. In processing the results, the temporal variation of the radiometer's sensitivity was taken into account. The most characteristic features of the geographical albedo distribution may be summarized as follows: The ocean albedo, especially in subtropics, is about 20 percent. The continental albedo varies from 30 to 40 percent. On the average, the albedo is 26 and 34 percent, respectively. The mean albedo of the earth-atmosphere system within the investigated zone is 31 percent. The albedo isolines follow the oceanic contours. At the coastal line, large albedo gradients were observed. In the Southern Hemisphere the longitudinal variation of albedo is weak, except the subtropical zone with three minima over oceans. During the whole period of measurements the albedo for the Sahara and Arabian Deserts is about 45 percent and comparable with that of Central Africa and South America where the intensive cloud was observed (these data are justified by the Tiros I11 and Tiros IV results). Such a high albedo value leads to believe that the earlier estimations of the absorbed solar radiation for desert areas were exaggerated. Since the outgoing longwave radiation

760

Net Radiation

in this case is quite great, it turns out that in the Sahara conditions, the net radiation of the earth-atmosphere system approaches zero, which contradicts the familiar calculations that present a positive value. Figure 10.40 gives the isopleths of absorbed radiation q', plotted by Rasool and Prabhakara [123a, 1631 from the Tiros IV and Tiros VII data and considered as annual means. It is evident that the q' values in the temperate and subtropical northern latitudes in June and July are unusually high. Also notable is the zone of maximal absorbed radiation in the southern subtropical latitudes in summer. Both peculiarities in the absorbed radiation variation are due mainly to the effect of the low ocean albedo.

FIG. 10.40 Isopleths of daily totals of absorbed solar radiation (cal/cm2day).

The data of Fig. 10.41 characterize the latitudinal and annual variation of the outgoing longwave radiation. Hence it is clear that the latitudinal range of considered means is very weak. A still weaker range is observed with the annual variation. Nevertheless these data, as well as the above considered results, confirm the presence of two weak subtropical maxima of the outgoing radiation along the equator, observed in August, September, and October (in the same months in spite of the opposite seasonal phases). The equatorial minimum indicates the intertropical convergence zone with more cloud.

10.6. Investigations of the Earth’s Net Radiation by Means of Satellites

761

FIG. 10.41 Zsopleths of daily totals of the outgoing longwave radiation (callcine day).

In Fig. 10.42 are given isopleths of the net radiation for the earth-atmosphere system. Since the outgoing longwave radiation field is very “diluted,” these isopleths are similar to those of the absorbed solar radiation of Fig. 10.40. It is evident from Fig. 10.42 that the maximal net radiation zones occupy the belts 20 to 40’ in both hemispheres in summer. Within 20’ N to 15’s lat. the net radiation is positive throughout the year. In Rasool and Prabhakara’s estimation [163] the net radiation of the earth as a whole is practically zero. In the Northern hemisphere this quantity is -77 x 10l8 and +81 x 10ls cal/day to the north and south of the 50’ parallel, respectively. In the Southern Hemisphere the analogous values are -92.5 x 10l8 and +94 x 10ls cal/day. Vowinckel and Orvig [35] used calculations of the solar and outgoing longwave radiation absorbed by the atmosphere and also the known values of the net radiation and its components for the underlying surface, to obtain the net radiation of the atmosphere and the earth-atmosphere system. Extremal evaluation of the effect of errors in the determination of the cloudless atmosphere moisture content on the calculation of the absorbed shortwave radiation showed that they were not in excess of 4 percent. The absorption by stratus clouds was taken into account with the help of corrections‘

762

Net Radiation

FIG. 10.42 Zsopleths of daily totals of the net radiation for the system earth's surface atmosphere (cal/cm8day).

determined from the familiar literature data. With respect to cirrus clouds, it was assumed that they would not affect the absorption. The outgoing longwave radiation was calculated from the Elsasser chart, considering water vapor only and assuming identity of the outgoing radiation and of the upward longwave radiation flux at the 300-mb level. The altitudes of the lower, middle, and upper clouds were taken respectively as 1.2, 4.0, and 5.5. km. Consideration of the calculations of the annual variation in the atmospheric net radiation and its components for various points with clear skies and in the real cloud conditions showed that its main difference from the net radiation of the underlying surface consists in a smaller amplitude due to the low value of the shortwave component as compared with the stable longwave. In all cases the atmospheric net radiation is negative, being lowest in late summer and autumn and highest in spring and early summer. All components are maximal in summer. The most characteristic feature is an exceptional stability of the outgoing longwave radiation during the year. As a rule, the negative net radiation of the atmosphere increases with cloud, since the radiation toward the earth's surface increases with the appearance of clouds and is not compensated by an increase in the absorped shortwave

10.6. Investigations of the Earth’s Net Radiation by Means of Satellites

763

radiation (an inverse situation may be observed only in the conditions of a very dry atmosphere). An unusual Arctic feature, compared with the temperate latitudes, is the increase in the outgoing radiation due to inversions with the appearance of clouds, and consequently the increasing radiative cooling of the entire atmospheric thickness. The amount of atmospheric net radiation consists mainly of radiation from the underlying surface. The zonal and meridional sections of the net radiation for the earthatmosphere system repeat the main features of the corresponding sections for the underlying surface. Monthly charts of its geographical distribution reveal a very low latitudinal variability in winter. The differences are greatest for land and oceans. In midwinter the maximum of net radiation occurs in the Norwegian Sea and moves toward the zone of pack ice closer to spring. The maximal cooling is observed over the pole only in late summer. In spring there is a notable increase of the meridional net radiation gradients (under the influence of the albedo nonhomogeneity). The typical winter contrast between ocean and continent disappears in summer. In midsummer and early autumn the regional and latitudinal variation of the net radiation strongly smooths out. In August and September the net radiation distribution presents a rather simple picture: its negative values decrease from the pole with increasing latitude. Along with the climatological characteristics of the net radiation and its components for the earth-atmosphere system, it is of interest to know their variability. Such information obtained in processing the Tiros I11 and Tiros IV data with its four orbits (middle July, 1961) and by using theoretical data is given by Davis [164] and Kennedy [165]. Table 10.23 summarizes the atmospheric net radiation values of the total radiative heat influx to the whole thickness of the atmosphere and its components (the longwave radiative heat influx and the absorbed solar radiation) and also the outgoing longwave radiation and the reflected solar radiation. It is evident from this table that these quantities have relatively small variations. The exception is the reflected solar radiation (outgoing shortwave radiation) whose greater variation is due to the effect of cloud nonhomogeneity . The outgoing longwave radiation was found to have a more “motley” field than given in [I641 by Gergen [166], who used actinometric radiosoundings at 25 United States locations during May 26 to 30, 1959. In his estimation the outgoing longwave radiation varied from 15.6 to 26.6 mW/ cm2, that is, almost by twice, which was mainly due to the cloud cover nonhomogeneity. An interesting example of the spatial variability of the outgoing radiation in the transparency window 8 to 12 p, from the Tiros

TABLE 10.23 Mean Values and Standard Deviations of the Components of the Radiative Heat Inflow for the Whole Atmospheric Thickness, of the Outgoing Longwave Radiations, and of the Reflected Solar Radiation. After Davis 11641 ~

~~~

~~

Orbit 3

Orbit 4

Measurement Mean

Std' Deviation

Mean

Std' Deviation

Orbit 29 Mean

Std' Deviation

Orbit 44 Mean

Std. Deviation

z

s w L Total radiative heat inflow, cal/cm2 min

0.1347

0.0277

0.1408

0.0228

0.1394

0.0262

0.1454

0.0233

Longwave radiative heat inflow, cal/cm2 min

0.2436

0.0258

0.2551

0.0219

0.2578

0.0231

0.2535

0.0327

Absorbed solar radiation, cal/cmz min

0.1090

0.0155

0.1143

0.0187

0.1183

0.0128

0.1084

0.0192

Outgoing longwave radiation, cal/cm2 min

0.3580

0.0513

0.3633

0.0481

0.3359

0.0360

0.3846

0.0494

Reflected solar radiation (channel 3, W/mZ)

94.4

46.7

29.6

62.2

E B'

10.7. Statistical Features of Net Radiation of the Earth-Atmosphere System 765

I11 data on July 16, 1961, is considered by Allison et al. [167]. According to their data, the effective temperature (measured in the transparency window) varies from 225 to 300’k and more in the belt 55’ N to 5 5 ’ s lat. The isolated character of the available data on the variation of the net radiation components in the earth-atmosphere system underlines the necessity of further investigations in this direction.

10.7. Statistical Features of the Net Radiation of the Earth-Atmosphere System At present the available satellite data on the outgoing radiation are so numerous as to make any complete “individual” analysis impossible, not even with high-speed electronic computers. Certain statistical methods of analysis are therefore needed to represent and generalize the material of observations that iiust be processed. This problem is also essentially interesting from the viewpoint of using outgoing radiation data as a field of random values. Statistical methods for analyzing satellite meteorological information are in their infancy (see [167a, 168-1721). We shall consider here some results obtained by Borisenkov et al. [167a, 1681, who in analyzing the outgoing radiation fields used the statistical theory of turbulence. The need to know the structure of the thermal terrestrial radiation field is essential because the strict solution of certain problems of the methods for observation and objective analysis of the radiation field is possible only if the structural characteristics of the field is known. Such knowledge, for example, can help to pinpoint the expedient frequency of satellite measurements of the outgoing radiation. It can also be used in objective analysis of radiation fields, in their comparison with the fields of other meteorological elements (temperature, for example), and so on. Even the first results of the outgoing radiation, as obtained by satellite measurements characterize the radiative field structure to a certain degree. In the work [167a] are considered the statistics of the outgoing radiation fields as obtained by the Tiros I1 meteorological satellite. The authors made use of the information summarized in the Tiros I1 radiation data catalog [173]. Tiros 11, launched on November 23, 1960, was supplied with two TV cameras for recording cloud, ice, and other ground objects, and was designed for measuring the earth’s outgoing radiation in different spectral regions. Two wide-angle sensors measured the integral fluxes of long- and short-

766

Net Radiation

wave radiation within the 50' sighting angle. For spectral measurements, a five-channel radiometer with a sighting angle of 5' and supplied with filters to isolate certain spectral regions was used. The five channels corresponded to the following intervals:

No. 1 to NO. 2 to NO. 3 to NO. 4 to No. 5 to

6-6.15 p : 8-12 PFC: 0.2-6 p : 8-30 p : 0.55-0.75 p :

water vapor absorption band atmospheric transparency window region of the reflected solar radiation integral thermal radiation region of the camera's spectral sensitivity

The above-mentioned catalog gives the results of radiation measurements by each channel for 52 orbits in the interpolation with a grid step of 40 miles. Besides, the channel 2 data are plotted on polar stereographic charts (scale 1 :50,000,000) with a grid step of 200 miles. Later, the catalog was followed by additional limitations to use of the given material [1741. It appeared that the geographical correspondence of the data had to be corrected in view of the errors in the determination of the scanning angle as a function of time. Besides, there were errors in the deciphering of the transmitted signals, due to radio noises. The distortion of signals was especially strong in channels 3 and 5. The supplement thus recommended that channel 5 data be ignored and those of channel 3 be used with great care. Only the results for channels 1,2, and 4 were therefore considered in [167a]. However, errors for these channels, apart from error in geographical location, were still great. According to [174], the mean error in the determination of the outgoing radiation for the 6 to 6.5 p region was f0.06 W/m2, which corresponds to the effective temperature error of f2' (at T = 240'K). With the channel 2 region, it was f 1.6 W/m2 (f2O at T = 270OK). With channel 4, it was f 1.7 W/m2 (31 2' at T = 260'K). The finite errors in the determination of radiation values by channels 1, 2, and 4 were f0.18 W/m2 (f 6' at T = 240'K), f4 W/m2 (f 5' at T = 270°K), and f5.6 W/m2 (f6' at T = 260'K), respectively. The errors in the water vapor absorption band (channel 1) may be still greater, owing to certain effects, which results in their finite value of & 0.2425 W/m2 (f8' at T = 240OK). In spite of quite a few shortcomings of the observational data given in the catalog, they are still valuable for some preliminary conclusions about the radiative field structure. The data were processed according to the following logic. Let us select rectangular surfaces with small latitudinal extension along the regular grids

10.7. Statistical Features of Net Radiation of the Earth-Atmosphere System 767

of the observational charts. Let the number of columns m and lines n that determine the selected grids be fixed for each surface. Denoting the radiation value in the grid r through F(r), we have the following mean value for a field of m bars and n lines : (10.23) In the case for determination of the spatial structural function of the element f, we can use the formula

bf(l)

1 sn

= -

z 8n

i=l

[f@i

+4 -

(10.24)

wheref(ri) = F(ri) - F and s = m/2 is rounded off with respect to the lesser side. The autocorrelational moment mf(/) and the normalized autocorrelational moment (autocorrelation factor) kf(l)for the element f were determined from the formulas :

The value I of the shift between the grid with the correlated elements varied from h to sh (h is the grid step; in this case h z 40 miles). Since, as has already been mentioned, the radiation measurements included considerable errors, we should try to evaluate the reliability of the obtained structural and correlation functions, taking into account these errors. Let f(ri) contain a random error E . In this case, f(ri

+ 1)

=f‘(ri

f(ri) = f ’ ( r i )

+ 1) + &(Ti+ I> +

&(Ti)

(10.27a) (10.27b)

where f ‘ is the true f value. If we assume that the measurement errors are in no way connected with the true ,f’ values and at various points of the field are statistically independent of each other, the following equation is easily obtained from (10.24) and (10.27): (10.28) bf(l) = bf’(Z) 2 2

+

768

Net Radiation

where 1 sn

=-

1 sn

z [&(ri>l2 8n

=-

8n

[&(Ti i=l

+ l)I2

Substituting (10.27) in to (10.26) and considering the above assumptions, we have in the numerator:

1

nn

+ 2, where

The denominc’ior of (10.26) is

In this case

[ I

at

/=o

where kf‘ is the true value of the autocorrelation factor and is the true dispersion. Since in the given case there is the following relationship between the calculated and the true dispersion, 02

=

(d)2

+

&2

the expression for kf can be rewritten as

kf(0

=

{

(1

-$)ki(1)

1

at I = O

at I f 0 (10.29)

It is easy to see from (10.29) that the observational errors cause the true autocorrelation moment always to exceed the kf obtained from the data with a random error E. It also follows from (10.29) that with the error E~ small compared with the dispersion u2, the true autocorrelation factor is little different from that calculated. For example, for the pressure at sea level, E is of the order of

10.7. Statistical Features of Net Radiation of the Earth-Atmosphere System 769

0.1 mb and cr 21 10 mb. It is obvious that such an error in the determination of the autocorrelation factor is negligible. In the investigation of the correlation moments of radiative fields, however, the error cannot be ignored, since the ratio &/cr is 0.4 to 0.5 at best, while for individual fields E and cr are comparable. The derived formulas enable certain correction of the structural and autocorrelation functions. Note here that at a given &, the autocorrelation factors obtained for orbits with higher (T values are more trustworthy, since in this case the calculated kf(l) are closer to the true values. The calculated structural characteristics of radiative fields for channels 1, 2, and 4 are given below, but analysis of them should take into consideration the preceding remarks. The structural and correlation functions were calculated for 50 fields of channel l , 56 of channel 2, and 62 of channel 4. The number of cases here exceeds the number of orbits, since the radiation fields for several orbits were divided into two regions. Figures 10.43 through 10.45 give the mean, maximal and minimal structural functions, and autocorrelation factors for the mentioned fields. All calculations were made with a Ural2 electronic digital computer. Q060L

a-0.040

a020

-

xil 0

(b)

P

-04 -02

FIG. 10.43 Structural functions ( a ) and autocorrelation factors (b) of the radiation field. Channel I. (1) maximum; (2) mean; (3) minimum.

770

Net Radiation

-0.2 -0.4

-0.6 b

FIG. 10.44 Structural functions ( a ) and autocorrelation factors (b) of the radiation field. Channel 2. Numbered as in Fig. 10.43. a

100

'n

20 2

4

6

8

10

12

Ii

I6

IbL

I .ol-

' r

-0.4

-

FIG. 10.45 Structural functions ( a ) and autocorrelation factors (b) of the radiation field. Channel 4. Numbered as in Fig. 10.43.

10.7. Statistical Features of Net Radiation of the Earth-Atmosphere System 771

To obtain the structural and correlation functions, the values of all functions for each Z were plotted on the graphs, the means were calculated, and the average b(1) and k(Z) lines were drawn. The maxima and minima with b(Z) and k(Z) were simply drawn by a smooth curve around the field of points obtained, and isolated large departures were ignored. In order to check the function isotropy, analogous calculations were made, with selections taken from the radiation values in the grids perpendicular to the initial direction. The first calculations were concerned with the linear grids but now they included the vertical columns. The linear values are plotted in solid lines and the vertical in dashes. Since the columns contain less grids than lines, the latter calculations ended with smaller 1. It is evident from the figures that, on the average, isotropy is best fulfilled for the integral radiation and is worse, for the radiation in the water vapor absorption band. It will be shown further that for individually selected radiation fields, there may be no isotropy, even in the 8- to 30-p region. We see that the variation of the structural functions first shows a rapid increase and then a practical stability with certain 1. Let us designate the distance at which b, reaches “saturation” by d. On the average, for the radiation fields of each channel, d w 16 h. The radiant flux in the 6- to 6.5-p region is practically determined by the atmospheric water vapor radiation, with a maximum at about the 400-mb area level. The structural functions of the radiant flux in this region will thus characterize the spatial structure of the water vapor distribution and its temperature. The Tiros I1 data on the channel 1 range, as was already metioned, contain far greater relative errors than on the radiation of channels 2 and 4. For this reason the channel 1 information should be considered very rough. It is possible that the considerable dispersion of points for the field k,(Z) and relatively large deviations from the conditions of isotropy may be partly explained by the low accuracy of these data. The obtained nonstandard values of the structural function for each Z are naturally less in the given case than b,(Z) and b,(l) because the absolute value of radiation for the 6- to 6.5-p region is small compared with the radiation in the 8- to 12 and 8 to 30 p regions. The field of points b,(Z) is, however, more compact than with b,(Z) and b4(Z). Thus, since the ratio of the maximal structural function to the minimal (from the rounding lines) at distances close to finite is of the order of 7 to 8, it is about 20 for b,(d) and b,(d). This fact may be explained by the more uniform spatial distribution of water vapor in comparison with that of cloudiness, which strongly affects the value and nature of the structural functions b,(Z) and b4(Z). Considering the variation of the autocorrelation factor k,(l) with distance, it is

772

Net Radiation

evident that it shows a slower decrease than in the other two channels. In the initial interval (for Z < 3h), however, the decrease of k,(Z) is more rapid than that of k2(Z)and k4(I). Besides, the difference between the maximal and minimal lines of the k,(l) field is considerably larger than for channels 2 and 4. This appears to be caused by the low accuracy of the obtained values of radiation in this spectral region. From the determination of the structural function it follows that b’, 0 = 0. Therefore we have, on the basis of (10.28),

b,(O)

= 2&2

Extrapolating the mean b,(l) value (Fig. 10.43(a)) up to I = 0, we have 2 2 N 0.006 W2/m4. As seen, the average error determined by zero extrapolation equals the average error in the radiation measurement given by the supplement to the Tiros I1 catalog [174]. At present the greatest attention is drawn to the 8- to 12-,u atmospheric window (of all the five spectral regions). The radiation detected from the satellite in this interval is by 75 percent due to the radiation of the earth’s underlying surface or clouds. It can thus be used to determine their temperature. Since the cloud surface temperature is usually much lower than that of the underlying surface, the radiation field in the 8- to 12-,u region can be used to estimate the cloud distribution and the pressure systems involving considerable cloud amount. The structural radiation functions for this part of the spectrum will therefore be strongly dependent upon the character of cloud fields. The same dependence must take place for channel 4. Comparing the structural and correlations functions of radiation of channels 2 and 4, we see that they have a close resemblance. Especially close are the mean b,(Z) and b4(Z), although the isotropy of the average radiative field for the 8- to 12-,u window is considerably more than for the 8- to 30-,u region. Comparison of the autocorrelation factors shows that the connection between radiation over different points of channels 2 and 4 undergoes approximately the same extinction. Such agreement between the structural and correlation funtions of radiation for channels 2 and 4 enables attempting transition from b(I) and k(Z) of the one channel to those of the other. Figure 10.46 is a graph of the relation between the structural functions of these radiative fields for a selected value I = 1Oh. The relationship is linear and is represented by the equation of regression : b4(II0)= o.71b2(Il,,)

+ 4.67

(10.30)

10.7. Statistical Features of Net Radiation of the Earth-Atmosphere System 773

with a relatively high correlation coefficient of 0.89. This justifies the great closeness of the connection. 80 I-

70 60

50

-*

40

-

30 20 10

0

I

10

I

I

l

20 30 40

I

I

50

60 70 80

1

1

b2

FIG. 10.46 Relationship between structural functions of the radiation fields (Channels 2 and 4, I = 10 h).

Since the radiative fields can, on the average, be considered isotropic, the following simple relationship is established between the structural function and the autocorrelation factor on the basis of (10.24)and (10.25),provided the condition for homogeneity is fulfilled : bf(l) = 20; [l - kf(Z)]

(10.31)

It follows from the expression (10.31)that closer relations between the investigated characteristics lead to a decrease of the structural function, while their separation makes it increase. In particular, if such distances L are selected for which k f ( L ) = 0, then b f ( L )= 2aP.To check the fulfillment of this relation, the data of Figs. 10.43 to 10.45 were used to determine L, for which kf = 0, and the values bf(L) = 2a2 were taken from the mean curve. The f-value dispersion was directly determined for each field and later averaged. The obtained E~ values are given in Table 10.24. It is evident from Table 10.24 that the coincidence of the obtained c2 is quite satisfactory, which justifies the validity of the statement about the isotropy and homogeneity of the mean radiative field in the above spectral intervals. Let us further consider some applications of the derived structural characteristics. Since in plotting charts the observational data are interpolated in

774

Net Radiation TABLE 10.24 Dispersion of the Radiative Field. After published data [174]

Spectral Region, p

az (from Figs. 10.43-10.44)

6-6.5 8-12 8-30

0.0118 20.0 17.5

a2 Averaged with Respect to Isolated Fields 0.0126 21.90 16.85

the grids of the regular net, it is natural to inquire about the optimal frequency of radiation measurements made from the satellite, which would have an interpolation error not exceeding the given one. The scheme for analyzing the meteorological fields in general and radiative fields in particular (adopted in the United States) did not until recently consider the structure of the analyzed fields. The interpolated value of the element f at every regular grid interesection was determined as a mean weight among several values ; the weight factor taken to be inversely proportional (without sufficient justification) to the square distance between the grid intersection and the initial point of interpolation. Such was the logic of radiation data processing as presented in the catalog [173]. If the structural and correlation functions had been taken into account, this interpolation would have been more accurate. According to Drozdov and Shepelevsky, the mean square error in the linear interpolation can be determined from the formula given in Gandin's work [175]:

Substituting the expression (10.3 1) into this formula, we have &I2 - - 6 = 2-

a '

r

I

[l - kf(f - r ) ]

+2 (10.33)

At the ends of the segment I, the error 6 is defined by the error kf.If the interpolation is performed with respect to the middle of the segment f (r = (l/2)), then (10.34)

10.7. Statistical Features of Net Radiation of the Earth-Atmosphere System

775

If we know the radiative field structure, it is obviously easy to state the finite distai-cu at which 6 will become larger than the given error. Gandin [I751 proposed a formula for interpolation which considered the minimal error condition. The interpolated fo value is thus sought in the form of a linear combination: n

fo =

c Pifi

(i = 1 , 2

- . a

n)

(10.35)

i=l

in which the weights Piare so selected that they have the minimal mean square error for interpolation with (10.35): n

E~~

=

[fo

-

x Pifil2

= min

(10.36)

i=1

In the transition from thefvalue to the autocorrelation factors, equations for determining the coefficients Pi by way of the structural characteristics of the field f are derived. The interpolation error here is defined from the formula n

6=1-

x koiPi

(10.37)

i=1

where the koi are the autocorrelation factors between the element foand the knownfi element at points i. In the case of interpolation over two points with respect to the middle of the 1 segment,

6

=

I -2Pkf(f)

(10.38)

where

The error calculated with (10.38)is slightly less, all other conditions being equal, than the error of the linear interpolation. The admissible measurement frequency in the second case therefore appears to be somewhat lower than in the first. However, for small distances, both formulas yield close results. For example, to secure an error in the integral radiation interpolation not exceeding the mean of 1.7 W/m2, the distances between the measurement points must not be larger than 60 to 70 miles. The same evolution is obtained with (10.34).It is evident that the radiation measurement errors as well as the interpolation error will affect the autocorrelation factor. More accurate estimates of the admissible I can therefore be obtained by

776

Net Radiation

taking into account the errors in the determination of k,. Depending on the position of basic points, the interpolation error can be evaluated prior to the derivation of the weight factors Pii by making use of the normalized correlation functions k,(Z). Let us present the following analogy with (10.35):

where the subscript 1 indicates the number of points at which the interpolated f value is calculated. Let us have the following system of normal equations to find the weight factors Pij by the least squares method:

(10.40)

---

where u is the dispersion with el = u2 = = on for homogeneous fields, and kii are the correlation funtions (conjugate correlation factors) dependent on the reciprocal position of the basic points only. The mean quadratic error of such a representation will be defined by the formula (10.41) where k12

kl?l

1 ... k2n ............... ...............

k21

D=

k13

kn1

k23

kna

kn3

*.'

(10.42)

1

is the determinant composed of conjugate correlation factors and D is the corresponding minor. The correlation functions can be used with (10.41) foi, the evaluation of the interpolation error and the expedient measurement frequency without preliminary knowledge of the weight factors P,. The values of the correlation function should first be corrected as noted above.

10.7. Statistical Features of Net Radiation of the Earth-Atmosphere System 777

The spatial radiation structure must be known for the calculation of radiation means in a certain satellite-orbit interval and also for the evaluation of the accuracy of such calculations. Lichtman and Kagan [I761 derived formuls for the determination of the accuracy in defining the mean for the infinitely great distance on the basis of measurement data in its final section Z : E~~ =

bAm,

+ 22 lorb,(r) dr 1

2

1

1:'

b,(r) dr

(10.43)

In addition, if a limited number of measurements n is performed in this n), according to the work cited, there will section at points ri (i = 1, 2 appear an additional error due to the limitation of the number of measurements :

If there are several parallel itineraries spaced at d or greater distance it is possible to consider that the measurements at one route are independent of those at others, and that their total length will be equal to the summed itineraries. If the distance between the routes is less than d, the second and subsequent routes must obviously be summarized with the weights proportional to b(r)/b(d),where in the given case r is the distance between the successive itineraries. Making use of the above structural functions and taking account of the admissible errors in the determination of the mean, it is easy to find with (10.43) and (10.44) the expedient dimensions of the area with respect to which the averaging should be performed. Presenting, for example, the structural function of the radiation in the 8- to 30-p region in the form b4(r) b4(d)(1 - [exp(- 0.0758r4'3)]}

-

and substituting it in (10.43), we find that for a rectangular area of 20h length and 1% width, the error in the determination of the mean radiation is approximately equal to a single measurement error. The above examples made use of the mean structural characteristics of radiative fields. However, those of individual fields can also be useful in certain investigations. When analyzing the structural functions of radiative fields, their variability in time and from region to region was noted. The authors of [I681 made an attempt at qualitative analysis of the dependence

778

Net Radiation

of the variability of b4(Z)upon the state of the atmospheric thermobaric field. It was assumed that, on the average, the type and character of cloud and humidi-y distribution would be reflected by the nature of the thermobaric field. To perform this analysis the areas employed in the calculation of the structural radiation functions were plotted on charts of absolute topography of the 500-mb isobaric surface. The baric features were then compared with the structural radiation function of the given region. Figures 10.4'1 and

FIG. 10.47 Absolute topography chart of a 500-mb isobaric surface, Dee. 13, 1960.

10.48 give an exemplary baric field at the level of 500 mb (December 13, 1960) and the structural radiation functions for three regions of this field. It appears that the greater the field's horizontal homogeneity, the smaller the value b,(d), and the best is the fulfillment of radiation isotropy. In region 3 the thermobaric field is at its calmest, with corresponding and relatively small b4(d) 10. The fulfillment of isotropy of the radiation field is satisfactory here. In regions 2 and 1 the thermal field and baric features are not horizontally homogeneous. This possibly accounts for the great values and

-

10.7. Statistical Features of Net Radiation of the Earth-Atmosphere System

779

anisotropy of the structural characteristics of the radiative field. The anisotropy is especially marked in region 2. Apart from the physical causes it may be due to the great difference in the number of intersections along the columns and lines of the area considered. Analogous comparison of the

L

FIG. 10.48 Structural functions of the integral radiation fields. (1) 0554 hr, orbit 290; (2) 0739 hr, orbit 291 ; (3) 1240 hr, orbit 294.

structural radiation functions with thermobaric fields was made for five days. The general picture is similar to the one described above, with respect to analysis of the structural and correlation functions. Thus, even the magnitude and nature of the spatial variation of the structural (correlation) functions and the fulfillment of the isotropy conditions can, to a certain degree, be indicators of the state of the thermobaric field. Further researh in this direction will enable further verification and provide more detail to this statement. Of considerable interest is the determination of the spectral density of the most energetically important perturbations. The spectral density S(o) calculated in [168] from the formula

Irn

S(w) = k ( t ) cos W T d t n o

(10.45)

where k ( t ) is the correlation function, is presented in Fig. 10.49. The calculation was based on the Tiros I1 and Tiros I11 data and performed with the Ural 2 electronic digital computer by making use of the correlation functions of radiative fields for various spectral intervals. The spectral func-

780

(a 1

0.41

400

1200

2000

2000

3600

krn

-cn

0.3

1

3 0.2

km

km

0.41

(e 1

400

1200

2000 km

2000

3600

FIG. 10.49 Spectral density of the radiation field with using radiation data selected from intersections of the 40 miles grid step (2), of 1.25’ (2) and of 2 . F (3). (a) Channel 1 ; (b) Channel 2 ; (c) Channel 3; (d) Channel 4; (e) Channel 5.

10.7. Statistical Features of Net Radiation of the Earth-Atmosphere System 781

-.

tions were obtained for w corresponding to I = h, 2h -,with h different for each satellite. It should be noted that since the correlation function at I > 10h is not realiably determined, the values S(1) at I > 10 must not be much trusted. Figure 10.49 presents three curves characterizing the spectral density of the integral outgoing radiation field for three types of correlation function, computed with a grid step of 40 miles and 1.25' and 2.5' of the meridional arc. It follows from this figure that the most probable scales of the integral outgoing radiation perturbations at h = 40 miles are 350 and 800 km. Taking into account the limits of the correlation function fluctuations, the minimal scale of perturbations is placed at 200 to 500 km, and the maximal scale at 600 to 800 km. Perturbations on such scales are observed in the fields of other meteorological elements-of pressure and geopotential, in particular. If the radiation data are selected at a grid step h = 1.25', the calculated spectral functions (curve 2) also reveal two scales of perturbations of 400 and 700 km. Taking account of possible departures of the correlation function from the means, the scale of lesser perturbations varies within 250 to 700 km; for greater perturbations it is within 700 to 900 km. In addition, still greater perturbations are revealed whose scale is of the order of 1100 to 1700 km. It is easy to see that, in spite of the notable discrepancy between the channel 4 data with the Tiros I1 and Tiros 111, the spectral characteristics of perturbations turned out to be fairly close. The same figure gives a curve of the spectral density for the outgoing radiation field as observed at channel 4 of the Tiros I11 satellite with a grid step of 2.5'. It is obvious that in the given case the authors could not isolate the spectrum of perturbations on the 200 to 500-km scale, but then revealed in more detail perturbations on the 800-1000 km scale and even greater perturbations on the planetary scale whose dimesions were from 1600-2000 to 2500-3500 km. The radiant flux in the 6- to 6.5-p region is practically determined by the atmospheric water vapor radiation, with a maximum at 7 to 8 km. The spectral density of these radiative field (Fig. 10.49) therefore makes it possible to evaluate the scales of the energetically significant perturbations at about 400,700, and also 1100 to 1700 km (curve 2). However, curve 3 of the spectral radiation density from the Tiros I11 data at h = 2.5' has only one maximum at I 21 800 km, coinciding with a similar curve of Fig. 10.45. The two curves are then almost specular. The absence of any perturbations as shown by the Tiros I1 curve of spectral density is strange. Curve 1, however

782

Net Radiation

(Fig. 10.45), has exceptionally clear maxima at distances of the order of 350 and 800 km. Also perturbations of the radiative field appear in the atmospheric window, with a scale of about 1500 km. In calculating from the radiation data with h = 1.25', though, the spectral density has but a weak perturbation only for I 800 km. According to the curve 3, there are notable perturbations on the 1200 to 1600-km scale in this spectral interval. These perturbations are generally less marked than for the fields of the integral and water vapor radiation. It is possible that cloud fields are the important factor causing the perturbations in the atmospheric window region. The large perturbations on the plantary scale are most pronounced with radiative fields formed mainly by the whole tropospheric thickness, that is, in the spectral region isolated by channels 1 and 4. In the region of the integral shortwave radiation (Fig. 10.49) curve 2 shows two scales of perturbations, of 500 to 1000 and 1200 to 1500 km. The same order of values is revealed in the perturbation of radiative fields when channel 5 uses the selections from the intersections with a grid step 2.5' (curve 3 of Fig. 10.45). The same curve shows another maximum at I 2400 km. Two other curves, curve 3 in Fig. 10.49 and curve 2 in Fig. 10.49, have least notable perturbations. The first, curve 3, shows them only with I = 500 to 1300 and 1500 to 2100 km, while with curve 2 it is practically impossible to trace any perturbations. Since the initial measurement data on the radiation with these two channels are particularly erratic, it is natural to expect great errors in the calculated spectral density. For example, the spectral density curves for channel 3 at h = 2.5' and for channel 5 at h = 1.25', calculated from the maximal and minimal surrounding lines of the autocorrelation factors field, are almost specular. The spectral density calculated from the mean k is thus very smooth. Similar spectral analysis may appear to be very useful in comparing the radiative fields plotted from satellite data with the fields of the basic meteorological elements of pressure and temperature in particular.

-

-

REFERENCES 1. Van Mieghern, J. (1958). Radiation data needed in dynamical meteorology. Arch. Meteorol., Geophys. Bioklimatol. A10, No. 4. 2. Eisenstadt, B. A., and Zuyev, M. V. (1952). Some features of the heat budget of a sand desert. Trans. Tashkent Geophys. Obs. No. 6. 3. Sapozhnikova, S. A. (1950). "Microclimate and Local Climate." Gidrometeoizdat, Moscow. 4. Eisenstadt, B. A. (1957). Net radiation and the soil surface temperature at Tashkent. Trans. Tashkent Geophys. Obs. No. 13.

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790

Net Radiation

158. Astling, E. S., and Horn, L. H. (1964). Some geophysical variations of terrestrial radiation measured by Tiros 11. J. Atmospheric Sci. 21, No. 1. 158a. Wexler, H. (1964). Infrared and visual radiation measurements from Tiros 11. Appl. Opt. 3, No. 2. 159. Zdunkowski, W., Henderson, D., and Hales, J. (1965). The influence of haze on infrared radiation measurements detected by space vehicles. Tellus 17, No. 2. 160. Gupta, M. G. (1965). Outgoing long-wave radiation from Indian stations. Nature 207, No. 4999. 161. Conover, J. H. (1965). Cloud and terrestrial albedo determination from Tiros satellite pictures. J. Appl. Meteorol. 4, No. 3. 162. Nordberg, W., Bandeen, W. R., Conrath, B. J., Kunde, V., and Persano, J. (1961). Preliminary results of radiation measurements from the Tiros 111 meteorological satellite. J. Atmospheric Sci. 19, No. 1. 163. Rasool, S. I., and Prabhakara, C. (1965). Radiation studies from meteorological satellites. New York Univ. Geophys. Sci. Lab. Rept. No. 65. 164. Davis, P. A. (1964). Satellite radiation measurements and the atmospheric heat balance. NASA Final Rept. Contract 5-2919, Stanford Res. Inst. 165. Kennedy, J. S. (1964). Energy generation through radiative processes in the lower stratosphere. Planet, Circ. Proj. Rept. No. 11. 166. Gergen, J. L. (1965). Atmospheric energy calculations related to radiation observations. J. Atmospheric Sci. 22, No. 2. 167. Allison, L. J., Gray, T. Y., Jr., and Warnecke, G. (1966). A quasi-global presentation of Tiros I11 radiation data. NASA Goddard Space Flight Center, Washington,

D.C. 167a. Borisenkov, E. P., Doronin, Y. P., and Kondratyev, K. Ya. (1963). Structural characteristics of the radiation field of the earth as a planet. Space Invest. 1, No. 1. 168. Borisenkov, E. P., Doronin, Y. P., and Kondratyev, K. Ya. (1965). Structural characteristics of the outgoing radiation fields as measured from the Tiros I1 and Tiros 111 satellites and their interpretation. Space Invest. 3, No. 3. 169. Baryshev, V. A. (1965). The mesostructure of the integral radiation field of the earth as a planet. Proc. Acad. Sci. USSR, Ser. Phys. Atmosphere Ocean No. 8. 170. Malkevich, M. S., Monin, A. S., and Rosenberg, G. V. (1964). The spatial structure of a radiative field as a source of meteorological information. Proc. Acad. Sci. USSR, Ser. Geophys. No. 3. 171. Malkevich, M. S., Malkov, I. P., Pakhomova, L. A., Rosenberg, G. V., and Faraponova, G. P. (1964). Determination of statistical characteristics of radiative fields over clouds. Space Invest. 2, No. 2. 172. Malkevich, M. S. (1966). On the spatial structure of the atmospheric long-wave radiation field. Proc. Acad. Sci. USSR,Ser. Phys. Atmosphere Ocean, 2, NO. 4. 173. Tiros I1 Radiation Data Catalogue. I. Goddard Space Flight Center, NASA and U.S. Dept. of Commerce Weather Bureau, Washington, D.C., 1961. 174. Tiros I1 Radiation Data Users’ Manual Supplement. Goddard Space Flight Center, Washington, D.C., 1962. 175. Gandin, L. S. (1960). On the optimal interpolation and extrapolation of meteorological fields. Trans. Main Geophys. Obs. No. 114. 176. Lichtman, D. L., and Kagan, R. L. (1960). Some problems of rationalizing snow W i n g . Trans. Main Geophys. Obs. No. 108.

TEMPERATURE VARIATION IN THE ATMOSPHERE DUE TO RADIATIVE HEAT EXCHANGE

Radiative heat exchange in the atmosphere is connected with changes in air temperature, which must be taken into account when investigating the thermal regime of the atmosphere. In some cases, in the lower stratosphere, for example, radiative heat exchange is one of the main mechanisms of the variation in air temperature. We shall therefore devote this chapter to the calculation of air temperature variations caused by radiative heat exchange (certain information on this subject has already been given in the preceding chapter). It should, however, be mentioned that in most cases the processes of atmospheric heat transfer are complex, and radiative exchange is but one of their components. The treatment of radiative exchange apart from the other types of heat transfer whose reciprocal action determines the air temperature variation is arbitrary. Such consideration is nevertheless useful because it makes possible the evaluation of the radiative factors in the air temperature variation, and also leads to more efficient schemes for approaching the effect of radiative heat exchange within the general problem of the atmospheric thermal regime.

11.1. Equation for the Heat Inflow The equation of heat inflow in the atmosphere can be generally written as (11.1)

where p is the pressure,

e is air density, T is the air temperature, q is the 79 1

792

Temperature Variation in the Atmosphere

amount of heat received by unit mass of air in unit time, D is the dissipation of mechanical energy, C, is the specific heat of air at constant volume, and t is time. The heat inflow can be presented with sufficient accuracy as the sum of the following three components e4 = &1

+ + 82

(1 1.2)

-53

where E~ is the heat inflow due to turbulent conductivity, E~ is the heat inflow due to radiative exchange, and E~ is the heat inflow due to phase transformations of water in the atmosphere. Taking account only of the vertical turbulent mixing, the heat influx is expressed as - 5 1 =a - ( 1 x )dT (11.3) az

where iZ is the coefficient of turbulent thermal condictivity. The heat inflow due to the phase transformations.of water in the atmosphere is usually ignored or given a estimated roughly in view of the difficulties involved in its calculation, although in some cases this component is of much importance. However we shall not consider it here. Let us examine now the expression for the radiative heat inflow (radiative fiux divergence), following Kuznetzov [l]. The heat gain due to radiation appears to be caused by the inequality of the energies absorbed and emitted by individual atmospheric layers. It is therefore necessary first to express these two quantities. Considering the results of Chapter 1 and taking into account the dependence of I,, on direction, it is easy to see that the total absorption of radiant energy by an elementary layer of mass dw can be written as

dw

Jr

k,.dv

I,& P , r ) dw

(11.4)

where the second integral is taken over all possible directions. In like manner we obtain for the total emission of the elementary layer: 47t dw

1- qlnd l

= 4nq

dw

(11.5)

0

Using (11.4) and (1 1.5) and having in mind that the main absorbing and emitting component of the atmosphere is water vapor, we find now for the

11.1. Equation for the Heat Inflow

793

radiative heat inflow per unit mass of the absorbing substance: (11.6)

where E, is the heat gain per unit volume. Let us now derive the expression for the radiative heat gain, using the accurate transfer equation (1.79). Denoting the directing cosines of the beam in the direction r by cos(r, x ) , cos(r, y), and cos(r, z), we have

Thus (1.79) can be rewritten as

Integrating both sides of the last equation with respect to r (that is, over all possible directions), we find

= - aJ

I , cos(r, x ) dw -

ax

The right-hand side of the transformed equation is now

s [sI,(P, r’)y,dP, r‘, r ) dw’l dw s [J I,@, =

=

dw’Z,(P, r’) Jy,(P, r’, r ) d o

= 432 sI,(P,

= 4n

r’)y,(P, r‘, r ) dw] dw’

J I,(P, r‘) dw’

r ) dw

Thus we have 1

- div F,

e

=

+ G,

4nyA

+

-(k~

s

02)

Z,(P, r ) dw

1

ZAP, r> dw

794

Temperature Variation in the Atmosphere

or, after simplification,

Integrating both sides of this equation with respect to all wavelengths within 0 to 00, we finally have

1

- div F = 4ny -

e

Irn ka dil J Ia(P, r ) dw

(11.7)

Comparing this relation with (1 1.6) and having in mind that here, as in e should be replaced by ew,we find

(1 1.6),

E~

= - div F

(11.8)

The result obtained allows us to reach two important conclusions. First, we are now convinced that the radiative heat influx is determined uniquely by the vector of the effective radiant flux at a given point. Since approximately, aT Qdt= E~ = - d i v F (11.9) we find that at div F > 0, radiative heat transfer causes a cooling of air temperature, while at div F < 0, its thermal effect is positive. The case div F = 0 appears to describe the state of radiative equilibrium at the given point (the heat inflow is zero). The second important conclusion from (1 1.8) is that the expression for the radiative heat influx is now devoid of the terms characterizing scattering at point P. This means that the scattering in a given elementary volume does not affect its thermal regime. However, the environmental scattering processes cause the radiation absorbed at a given point to be of the intensity Z(P, r ) , which differs from that possible without scattering. Thus scattering affects the heat influx, though it is not evident from (1 1.7). Kuznetzov has solved the problem of the vertical distribution of atmospheric temperature in the state of radiative equilibrium, to show that the effect of scattering is strongly felt in this case. Let us now consider the transformation of the above formulas in an approximate setting of the problem of radiative heat transfer in the atmosphere. Assuming that the atmosphere is in the state of local thermodynamic equilibrium, and applying the approximate radiative heat transfer equation

11.2. Methods for Calculating Radiative Heat M o w

795

(9.22), we replace (1 1.6) by the following obvious expression: (11.10) where the summation extends over all selected intervals in the infrared spectrum. Equation (1 1.9) is now transformed as (1 1.1 1) This shows that div F

=

-ew 2 ki(Gj + Uj - 2piB) i

Since with an approximate treatment of radiative heat transfer, F = U - G and the fluxes U and G vary only in the vertical direction, we obtain div F

aF

-

= --

a(U- G)

i3Z

aZ

( 11.12)

Thus the sign of the temperature changes due to radiative heat transfer is determined by the sign of the effective radiant flux. 11.2. Methods for Calculating Radiative Heat Inflow

The simplest method of practical calculation of the radiative flux divergence consists in the use of (11.12) and the approximate relation aF - A F aZ dZ

(11.13)

The finite change of the effective radiant flux for a particular A z can be calculated from any of the radiation charts (see Chapter 9). Calculations show, however, that a replacement of derivatives by finite differences can lead to considerable errors in this case. It will further be shown that the quantity dF/az is determined by the values of the derivative from the transmission function dPF/dwat different atmospheric levels. Meanwhile, according to Brooks [2], accurate values of the derivative dPF/dw,calculated for a particular level, can differ greatly from the one averaged over a layer adjacent to the given level. If, for example, w = 0.001 g/cmz, dPF/dw = 53.0, while the average values dpF/dw over different layers d w are:

796

Temperature Variation in the Atmosphere

Aw:

dp,ldw :

0.001 0.005 20.0 37.6

0.009 14.4

0.05 4.73

0.1 2.81

0.5 0.81

These very convincing results show the approximate nature of (1 1.13). Let us now seek more reliable methods of calculating the radiative heat inflow. The calculation of the radiative heat gain in the atmosphere presents a very complex problem. Therefore numerous attempts have been made to suggest simple approximate solutions of this problem. One of them is based upon the assumption of diffusion for describing the radiative heat transfer in the atmosphere. This means that the effective thermal radiation flux can be represented in the form F = -k,-

dT dZ

(11.14)

where k, is the coefficient of “radiative” diffusion. Kondratyev [3J showed, however, that the use of this assumption and of (1 1.14) is either not substantiated or inefficient. Later, Podolskaya [4] considered the replacement of the integral expressions for the integral longwave radiation flux by the differential and concluded that such a replacement might be possible, provided the transmission function and its particular derivatives have properties of the delta function. The atmospheric transmission function does not, however, satisfy this condition, whence it follows that the approximation of diffusion cannot be applied. Podolskaya then tried to solve a more general problem of deriving expressions for the longwave radiation fluxes that would not contain integrals. No conclusions could be reached, however, because of lack of information about the secondary derivatives from the transmission function. It is evident that the most reliable method of calculating the radiative heat gain is by the use of accurate formulas for this quantity. Since the corresponding equations are rather complicated, it is expedient to use radiation charts when performing numerical calculations. This was done by Bruinenberg [ 5 ] , whose work dealt with calculations of temperature changes by means of radiation charts. The theory of Bruinenberg’s presentation is relatively simple and can be briefly described as follows. Let there be two closely positioned levels A and B spaced at a distance d z corresponding to the difference in water vapor content, d w = w B - w,. For the atmospheric downward radiation at levels A and B, we have

11.2. Methods for Calculating Radiative Heat Inflow

797

The difference between these two quantities will read as G,

-

GB=

1"

a

0

d dw

(+-

w-

1,"

aGB dw

dW

(11.15)

Let us introduce the notation

(aGA4 F -ddW GB)dW

Let (AT), be the ifference in temperature between the two closely positioned levels A' and B', with a coordinate w overlying the levels A and B. Then we can write approximately

Using this relation and having in mind that dw obtain

=

e dz, we

can easily

(11.16) where (dT/dz), = 7, is the vertical temperature gradient and ew is absolute humidity. Concerning g2, Bruinenberg maintains that this integral may be approximately replaced by

tT2=

(g) Aw

(11.17)

w=wm

Making use of (1 1.6) and (1 1.7) we have

-

G . ~- G~ = AG

1 (-) d T d w WB

= AW

o

d2G

yu,-

dW ew

+ A , ~ (dG -) dw

w=woo

For the radiative temperature variation, taking account only the downard

798

Temperature Variation in the Atmosphere

radiant flux,

Or, passing to the limit at d z + 0, we find

where q = ew/e is specific humidity. In like manner the expression for (dT/dt)r in easily obtained. Here it should be remembered that in (11.18) the second term in braces is to be represented as (aU/aw),,,. It is clear, however, that this value is zero. Introducing the notation DJ. = d2G/dT/aw and DT = azU/dT/dw, and replacing the integrals by sums, we finally have for the temperature variation :

Here the first summation extends over all elementary layers situated above the level considered (the total number of layers is N ) , while the second sum concerns all underlying layers (the total number of layers being infinitely large for the earth's surface is considered as an isothermal gaseous layer of infinite thickness). Since the values of the derivatives d2G/dTaw and d2U/dTdw are identical at the given point, for the practical application of (1 1.19) it is necessary to have information about the derivatives a2G/dT dw and dG/dw. Bruinenberg plotted radiation charts that help to obtain individual values for water vapor and carbon dioxide. This enables separate calculations of the radiative temperature variations due to the absorption and emission of radiation by the above substances. As is evident, the method of Bruinenberg is not simply applied or easily handled; nor is it justified theoretically, since its theory is not adequately accurate. For this reason it has not found wide application. Brooks [2] attempted to change Bruinenberg's method both in theory and in simplification of the calculation, but without success. Later the research of Brooks was supplemented by the investigations of Zdunkowski and Johnson [5a].

799

11.2. Methods for Calculating Radiative Heat Inflow

It can be shown, however, that a very simple graphical method of calculating the radiative heat inflow is possible, the theory of which is more accurate than that of (11.19). Let us use (11.11) in the form

Substituting the expressions for radiant fluxes and assuming 6 obtain dT k (wW-W) e c p r = - e n { ~ ( w w ) Pjkj e- j

=

1, we

ci

From the determination of the transmission function we have PF(w) =

pi

eckjW

(1 1.21)

It should be noted that although the transmission function is represented by (1 1.21), its accuracy may be approximate and of varying degrees, depending on the number of the terms in the sum. For the derivative of the transmission function we find (1 1.22) Using the last two relations, the formula (11.19) can be transformed as

By performing further obvious transformations we find

dTat

- w,

dw

dB

Temperature Variation in the Almosphere

800

It should be remembered that this formula relates to the conditions of a clear atmosphere and the earth's surface as a blackbody radiator. Taking into account that

dPF(Wcu - w ) > 0 dw

d M P - w ) <0 dw

>0

- PI

@F(W

dw

we write

The last two formulas show that the expression for the radiative temperature variation is conveniently used with radiation charts. In the coordinate system ( I dPF/dwI , B), the integrals of (11.24) will be determined by the corresponding areas. Figure 11.1 presents schematically a radiation chart for calculating (C,/q) (aT/at)with the curve (dPF/dw)( B ) on the assumption of a vertical decrease of air temperature. Each area indicates the sign to be used with it. As seen from the theory of radiation charts, the main initial data for their compilation are those on the derivative of the transmission function.

1

0

I

I

I

T(wa)

T(w)

T(0)

I

FIG. 11.1 The scheme of the chart for the caIcuIation of radiative flux divergence.

801

11.2. Methods for Calculating Radiative Heat Inflow

Kondratyev and Yakushevskaya [6] have plotted a radiation chart on the basis of the derivative dPF/dw calculated from Shekhter’s data. This enables comparison of the radiative heat inflow as calculated with their chart and with that of Shekhter for thermal radiant fluxes (see Chapter 9). The values of the derivative dPF/dW rapidly decrease with the increasing w and are negligible already at w > 0.01 g/cm2. This means that the radiative heat inflow, and accordingly the radiative temperature variation, are due first of all to the effect of the radiation with a small free path relative to the considered level. The thickness of the “effective” layers for the radiative heat gain at a given level is determined by the values of water vapor mass of the order of 0.01 g/cm2. In the lower atmospheric layers this value w corresponds to air layers of about several scores of meters thick. However, it should be emphasized, that, generally speaking, the importance of radiation with that or other length of free path relative to the considered level is notably dependent on the nature of the vertical temperature distribution. It is also important to treat the “effective” layer separately with respect to the over- and underlying atmospheric layers (in relation to the given level). The chart for calculating the radiative heat inflow is presented in Fig. 11.2.

L

-

0.000l2 0.00014

-

O.OQ016

O.OOOl8 0.0002

: r

0.00022 0.00026 -

-

0.0003 0.00034I

0.0004 0.0005 a0006

:

. :

ooooa 0.001 1 0.0014I 0002 :

@ J

0’

5’

10’

15’

200

25’

30”

35”

40°

T

FIG. 11.2 Chart for calculations of radiative flux divergence.

450

802

Temperature Variation in the Atmosphere

Its use confirms the conclusion formulated at the beginning of the current section that possible errors can he made in calculating dF/dz = dF/dz. For example, the radiative temperature variation at 2-m elevation was calculated with the following atmospheric stratification :

E ~ S Z I ~

T = To - a l n z SO

The numerical values of the parameters are:

To = 290°K, a = 0.7,

h = 50 m,

cm

e0 =

y = 6 deg/km,

eu0 = 7 g/m3, /?= 4.5 x

H = 11 km I/cm

For z = 2 m, the calculation gives (aT/&)rad = 0.0896 deg/min. With Shekhter’s chart at d z = 4 m (b’T/at)rad = 0.113 deg/min. The large discrepancy is easily seen. It is evident that calculations reveal a dependence of Shekhter’s results on the value of the “step” d z . The following data illustrate this conclusion in relation to the above example: Az:

cm

(aT/af)rad:

deg/min

120

200

0.0761 0.0878

300

400

0.0993

0.1130

Hence it follows that the best coincidence with the results obtained from the chart of Fig. 11.2 is observed at LIZ = 200 cm. The value of this optimal d z must obviously be dependent on atmospheric stratification. It should be further noted that the chart of Fig. 11.2 enables detailed investigation of the vertical profile of the quantity ( a T / a f ) r a d in the atmospheric boundary layer. For the above mentioned stratification, calculations give z:

cm

(aT/af)rad:

deg/min

50

100

200

400

800

0.1931

0.1320

0.0896

0.0565

0.0388

It is evident that the charts for calculating the thermal radiation fluxes cannot be applied in similar computations of the vertical profile of ( a T / a t ) , , , Everything said above gives evidence of the fact that the chart of Fig. 11.2 for calculating the radiative temperature variation is doubtlessly advantageous in relation to the charts for the radiant fluxes. It should

.

11.2. Methods for Calculating Radiative Heat Inflow

803

only be remembered that it is intended for calculations in the boundary layer of the atmosphere. The difficulty of calculating the radiative heat inflow caused by the rapid increase of I dPF/dwI with a decrease of w has been successfully overcome by Yamamoto and Onishi [7] who used l/dp,ldw instead of I dPF/dwI as a coordinate. For plotting the chart, a polar coordinate system was used in which the radius vector was proportional to 2/dp,ldw and where the polar angle was B = aT4. However, even this nomogram turned out to be convenient only for calculating the radiative heat gain in a free atmosphere. For the atmospheric boundary layer Yamamoto and Onishi have proposed a special chart with the coordinates (radius vector) and (1/2a) ezaB (polar angle), where B is a constant. This chart is presented in Fig. 11.3.

caB.\/dPFldW

FIG. 11.3 The chart of Yamanroto and Onishi (unit area corresponding to 0.1 calJcm8min).

804

Temperature Variation in the Atmosphere

As seen, the scale of water vapor contents (the chart considers water vapor only) in this case is abbreviated, as is the chart of Fig. 11.2. The method of calculations according to this chart is principally similar to the above considered method of Kontratyev and Yakushevskaya. A new numerical method of calculating the infrared cooling rate was recently proposed by Funk [7a], Rodgers and Walshaw [8], and Zdunkowski and Johnson [9]. Integrating by parts the common expression for the radiative heat gain, it is possible to cancel out the derivatives from the transmission function at small values w, as was shown by Funk. However, the temperature derivatives remain, which means that the temperature profile must be given with a high degree of accuracy. Since in micrometeorology there are now available sufficient detailed data on the temperature stratification, the Funk method can be efficiently used for the atmospheric boundary layer. The effect of clouds and haze has been considered in several papers [lo-131. 11.3. Results of Caclulations of Radiative Flux Divergence

As was shown in Sec. 11.1 the temperature changes due to radiative heat exchange can be calculated from the following formula: (1 1.25)

It is evident that an increase of the effective flux with height causes radiative cooling of the air. If, however, dF/dz < 0, then dT/dt > 0. 1. Free Atmosphere. Theoretical calculations and observations show (see Chapter 10) that in the free atmosphere with cloudless skies the effective radiation, as a rule, increases with height. This means that radiative cooling is usually prevalent in the free atmosphere. As was mentioned above, the local variation of air temperature due to radiative heat exchange can be formulated by selective breakdown of the absorption spectrum, as

11.3. Results of Calculations of Radiative Flux Divergence

805

The last expression shows that the temperature variation contains three parts, each having a simple physical meaning. Moller [13a, 14, 14a] has noted that the first term in the right-hand side of (11.26) describes the air temperature change (cooling) due to radiation into outer space. The second term represents the heating due to the positive net radiation at a given level in relation to the atmospheric layers below, while the third term gives the cooling due to the negative net radiation at the level considered with respect to the overlying atmospheric layers. Figure 11.4 gives the results of calculations of the vertical variation in the total radiative cooling and its components, performed by Moller. The calculations were concerned with the case of a "normal" atmosphere (To = 283'K, y = 6'C/cm, H = 10 km, relative humidity constant, and equal to 70 percent in the troposphere; above the tropopause a constant mixing ratio was assumed).

DAY

FIG. 11.4 Radiative cooling at different levels in the atmosphere.

The curve a of Fig. 11.4 corresponds to the first term of (11.26). As seen, the cooling of the atmosphere due to the radiation into outer space increases from 0.4'C/day at the earth's surface to a maximal value of 0.9'C/day at a height of 7 km, and then decreases as far as the tropopause, where a new increase is observed. This vertical variation of the cooling rate due to the radiation to space can be easily explained on the basis of the formula (1 1.26). In fact, the sum in the first term of this formula shows a monotonic increase with height until a finite valued Xjpjkj is reached. This variation is overlapping with the vertical variation of specific humidity, q = ew/e,which decreases in the troposphere and is constant in the stratosphere. At this point

806

Temperature Variation in the Atmosphere

in the lower tropospheric layers (up to 7 km) the effect of the increase of the above sum prevails, while at high altitudes the considerable decrease of specific humidity is of deciding importance, which results in the reduction of the whole first term. When the tropopause is reached, the decrease of the mixing ratio ceases and the increase of the sum again becomes dominant. The curve b of Fig. 11.4 represents the second term in (11.26). We see that the heating (the heat exchange with the underlying layers) increases from 2 to 3’C/day at the earth’s surface to a maximum of 4’C/day at a height of 3 km and then decreases to 2.6’C/day at the tropopause, finally increasing with height at en even greater rate. Curve c shows the air cooling due to the negative net radiation at the level considered with respect to the overlying atmospheric layers. It decreases from 6.4’C/day at the earth’s surface to O°C/day at the tropopause. Above the tropopause the third term of (1 1.26) becomes zero, owing to the isothermal state of the stratosphere. Curve d characterizes the combined effect of the second and third factors. And finally, curve e represents the vertical variation of the total value of the radiative cooling of the atmosphere. This quantity decreases from the earth’s surface to the tropopause, where it attains positive values. In the major part of the tropsphere the radiative cooling is about 2’C/day. Similar values of the radiative cooling have been obtained in other investigations. However, the nature of the vertical distribution of radiative cooling turns out to be greatly dependent on atmospheric stratification. Many authors attribute the discontinuities in the curve e of Fig. 11.4 to the fact that radiative heat transfer is similar to the thermal conductivity that leads to the temperature differences being smoothed out. Hence it follows that in the “normal” atmosphere a sharp increase in the radiative cooling rate must take place near the earth‘s surface, while a similar sharp decrease of the cooling must occur near the tropopause, which results in the heating of the air at the tropopause. It is easily seen, however, that this logic does not always hold good. The thermal effect of the radiative heat exchange near the earth’s surface is determined not only by qualitative difference in temperature between the earth’s surface and the adjacent atmospheric layers. It is perfectly evident that in the normal atmosphere this effect is negative because the net radiation at a level close to the earth’s surface is negative with respect to the overlying layers and practically zero with respect to the underlying atmosphere and earth’s surface. But a reverse situation exists, as we have seen in Chapter 9, when we consider the presence of a surface layer with its abnormally high vertical temperature gradients. In this case, even under the conditions of a superadiabatic stratification

807

11.3. Results of Calculations of Radiative Flux Divergence

(when, as with the normal atmosphere, the earth’s surface is warmer than the adjacent atmospheric layers), these layers become heated by radiation. The sharp variation in the radiative cooling rate near the zones of temperature and humidity “discontinuities” is of extreme importance, since in this case the effect of radiative heat exchange is particularly pronounced. Moller [I41 has calculated the variations of radiative cooling near the region of sudden changes of humidity and dust content. The following data (Table 11.1), obtained in the assumption of sudden changes in relative humidity from 100 to 20 percent at the altitude 1.67 km, illustrate the results of Moller’s calculations. TABLE 11.1 Radiative Heating Rates near Discontinuities of Temperature and Humidity. After Miiller 1141

Atmospheric layer, km

z, at

04.5

OC/day -2.2

0.5-1

1-1.25

1.25-1.5

1.5-1.67

1.67-1.75

-2.5

-3.3

-4.4

-6.5

-1.5

1.75-2.0

2.0-2.5

2.5-3.0

-1.5

-1.3

-1.3

It is evident from Table 11.1 that in the region of the discontinuity in relative humidity, a considerable increase in the cooling rate is observed. This shows that the presence of humidity discontinuities brings forth the development of radiation inversions. The effect of radiative cooling and of inversions is especially marked near the surfaces of cloud. Observations justify this conclusion. Gaigerov and Kastrov [15, 161, for example, used the results of balloon soundings, which show the presence of considerable air cooling above the upper cloud boundary and in the upmost haze layer in many cases. It is most natural to suppose that this cooling is largely due to radiation. The above calculations of the radiative cooling of the atmosphere have been performed with account taken of only the water vapor as the sole absorbing component of the atmosphere. However, carbon dioxide is also an important absorbent of thermal radiation, but its effect in causing radiative temperature changes in the free atmosphere is not significant. This is easily understandable if we consider that the radiation absorption in the 15-,u CO, band is very intense, and therefore at every layer GC02N Uco,, that is, FCO,N 0 and aFCoz/& N 0. The calculations of Brooks [39] showed that the radiative cooling due to CO, is about 3 percent of the radiative cooling

808

Temperature Variation in the Atmosphere

caused by H,O. It should, however, be noted that the equality aF,,,/az 21 0 holds good only in the case where the vertical temperature lapse rate is not too large. This means that it may be quite important to consider the radiative temperature changes due to CO, in the surface atmospheric layer. It is obvious that this conclusion also applies to the conditions in the atmosphere when the vertical temperature gradients are small and the water vapor content is negligible. The above calculations of radiative cooling ignore the effect of pressure on radiative heat exchange. The calculations of Thompson [17] have shown that this factor is of secondary importance. Numerous investigations in the structure of the middle and upper tropospheie carried out during the past 15 years have revealed the presence of stable baroclinic zones, fronts and quasi-isentropical “bands,” and also a notable interaction between the troposphere and stratosphere. Staley [181 has analyzed the influence of radiation factors on the evolution of zones of low humidity in the middle troposphere on the basis of calculations of the vertical profile of radiative cooling in the vicinity of temperature inversions at different vertical distributions of specific humidity with its rapid decrease within the inversion. At a high relative humidity the radiative cooling causes the “effacement” of inversion; the minimal cooling occurs at the base of the inversion, while the maximum is near its upper boundary. In the case of a rapid vertical decrease of relative humidity in the inversion layer, the radiative cooling rate is observed to decrease over the entire inversion layer, with maxima of a few degrees per day near its base (thus in the given case the radiative cooling maintains inversion). These results have been compared with the data of ten actinometric soundings realized at Tucson (Arizona) in the early spring of 1961, 1962, and 1963. The measurements justify the conclusion that the radiative cooling facilitates the inversion; average cooling maxima of about 2 deg/day were observed at the base of the inversion and below, while minima of about 1 deg/day occurred in the upper inversion. There were no breakdowns of inversion stability due to radiative cooling (this might have been caused by the effect of measurement errors). All ten ascensions reached the stratosphere and recorded radiative heating or minimal cooling near the tropopause. This implies that the formation of the tropopause cannot be attributed to the effect of radiation factors. It may be of practical importance to compare local temperature changes due to radiative heat transfer with those due to other factors. Such comparisons have been performed by Gaigerov and Kastrov [15, 161 who used meteorological and actinometric observations with free balloons.

11.3. Results of Calculations of Radiative Flux Divergence

809

According to measurements in clear or almost clear summer weather, the radiative cooling rate during the transformation of air masses averaged 0.06'C/h. The heating rate due to absorption of solar radiation has approximately the same value. The daytime radiative temperature change in summer is therefore close to zero. During the night, however, radiative cooling is about 0.06'C/h. Comparison of temperature variations due to radiation with those due to other factors (advection, turbulent mixing, etc.) reveals that the radiation factors are undoubtedly not the main causes, but even so they must not be 0

Lot N.deg

Lat N , deg

FIG. 11.5 Meridional cross section of radiative cooling in the Northern hemisphere under average cloudiness conditions. (a) January; (b) April.

810

Temperature Variation in the Atmosphere

ignored. It should be emphasized that this conclusion was teached on the basis of observations at height from 3 to 5 km. The relative importance of different factors of the thermal air transformation is therefore not yet clear. Extensive theoretical calculations of the latitudinal distribution of radiative heating rate within the entire troposphere have been carried out by London [19, 201 and Davis [21]. Figures 11.5 and 11.6 give the results of radiative cooling for 20 to 70' N lat. and the heights from ground to the 25-mb level. The calculations were made by means of an IBM 704 electronic computer, following highly reliable methods for calculating the radiative flux divergence. C

LAT N,

FIG. 11.6

DEG

Meridional cross-section of radiative cooling in the Northern hemisphere under average cloudiness conditions. (c) July; (d) October.

11.3. Results of CalcuIations of Radiative Flax Divergence

811

Examination of these figures shows that radiative cooling is maximal in the low latitudes in summer. The inhomogeneous distribution of radiative temperature changes is mainly caused by the nonhomogeneity of the cloud cover. In particular, the cooling maximum observed in the troposphere is placed immediately above the zone of the upper cloud boundary of most frequent recurrence. In the lower stratosphere of polar latitudes there is a second pronounced cooling maximum due to the intense radiation of water vapor in the far infrared. Minimal values of radiative temperature variations are observed in the lower subtropical stratosphere (at 25' N lat. there is radiative heating in July). The meridional gradient of radiative cooling in the troposphere (cooling decreases northward) is of the opposite sign to that in the stratosphere (cooling increases northward). Very detailed and highly reliable calculations of radiative heat inflow have been made by Manabe and Moller [22] and by Manabe and Strickler [23]. The first work provides meridional sections of radiative temperature variations for the four seasons at altitudes from 0 to 30 km. Figure 11.7 presents the mean annual meridional distribution of the total radiative temperature variations.

I

i 1;

E

X

E

01

s

10

LAT N, deg

FIG. 11.7 Computed distribution of annual mean rate of temperature change ('C/day).

812

Temperature Variation in the Atmosphere

Figure 11.8 gives the mean annual vertical profiles of the total radiative heat gain and its components. (“Net” means the vertical distribution of the Scoa,and So, are the rates of temperanet rate of temperature change. SHao, ture change due to absorption of solar radiation by water vapor, carbon , LcoB,and Lo, are those due to the dioxide, and ozone respectively; LHaO longwave radiation by water vapor, carbon dioxide, and ozone, respectively). 30 -

20E

x

10

-

5

FIG. 11.8 Mean vertical distribution of various heat budget components (‘Clday).

2. The Surface Layer of the Atmosphere. The main peculiarity of the surface atmospheric layer is the presence of abnormally high vertical gradients of temperature and other meteorological elements. This means that the vertical lapse rate of the effective radiant flux, and consequently radiative temperature variations, are considerably greater here than in the free atmosphere. This can be illustrated by the observations and theoretical calculations of Robinson [24], presented in Table 11.2. In the first case (June 21, 1949) we see a large superadiabatic vertical lapse rate of temperature. The theoretical radiative heating here exceeds by far the observed heating. In the second case (June 7, 1949) of inversion stratification the calculation gives a marked radiative cooling in excess of the observed temperature variation, and finally in the third case (Nov. 26, 1948) the observed conditions approach the strate of radiative equilibrium (temperature variation is zero). Robinson also calculated the radiative heating rate for a layer between the earth’s surface and a level of 500 mb. Comparison with the observed heating

813

11.3. Results of Calculations of Radiative Flux Divergence

TABLE 11.2 Thermal Radiation Fluxes and Heating Rates over a Grass-Covered Area. After Robinson [24]

Time of Observation 6/21 149 11.00 h

6/7/49 21.30 h

11/26/48 17.45 h

Effective radiation of grass surface, cal/cm2sec, measured

5.2 x lo-'

1.6 x lo-'

1.OxlO-J

Variation of effective radiation in 0-50 cm layer, cal/cmzsec, calculated

-1.7 x lo-'

2.0 x

0.0(1 x lo4)

Radiative heating rate in 0-50 cm layer, OC/sec

1 x lo-=

-1.5

X

lo-*

0.0

Observed heating rate, O C / s e c

3 x 10-4

-3 x 10-4

-1 x 10-8

Temperature difference across 0-50 crn layer (To - T6d

20

-3.5

-4

Absolute humidity near underlying surface

10

9.3

4.8

rate (Fig. 11.9) shows that in this case the nighttime radiative temperature variation is also very notable. As seen, calculations of the radiative heating rate near the earth's surface often give higher values than those of observations. The same conclusion has been reached in [25,26]. This provides evidence of the importance of radiative heat exchange in the total heat balance. The solution of this problem should, however, be sought in the combined calculations of radiative and turbulent heat exchange rather than in the calculation of radiative heating rates alone. We shall consider this problem in Sec. 11.5. A number of investigations [25, 27-33] were concerned with attempts at experimental determination of radiative flux divergence and the subsequent comparison with calculations. In most cases the measured values of heating due to longwave radiation exceeded the calculated ones. Kondratyev et al. [9,34,35] were the first to perform detailed calculations of the spectral distribution of the longwave heating component in a free cloudless atmosphere. These calculations made it possible to analyze the relative contribution of spectral intervals to the radiative heat exchange. The work [34], for example, deals with the region of wavelengths 4 to 40 ,u at heights from 0 to 30 km. Calculations show that the most intense (almost

814

Temperature Variation in the Atmosphere

independent of altitude) cooling occurs in the transparency window 8 to 12 ,u. The cooling in the absorption bands (6-,u H,O, and 15-,u CO, bands) considerably increases with height. It is of interest that in spite of a pratically monotonic increase in the integral radiative cooling with height (in I

r

-02

0

6

12

18

24

Hour

FIG. 11.9 Mean temperature variation of the layer between the earth's surface and the 500-mb level. June 21, 1949, S. England. (1) radiation; (2) convection; (3) total.

a cloudless atmosphere), the vertical profiles of radiative-heat gain for individual spectral intervals are variegated and in some cases nonmonotonic (in particular, in the intervals with strong absorption). 11.4. Radiation Factors in the Thermal Regime of the Stratosphere and Mesosphere The radiation factors determining the thermal regime of the stratosphere and mesosphere have much in common. There is no distribution of such factors over layers. However, certain specific conditions in the stratosphere and mesosphere are doubtless contributory. Considering this and also wishing to make the presentation clearer, we shall now turn to successive examination of both layers. Regarding the higher atmosphere, the notable pe-

11.4. Radiation Factors in Thermal Regime of Stratosphere and Mesosphere 815

culiarities of the heat regime in the thermosphere and exosphere exclude the possibility of treating this problem within the limits of our presentation. 1. The Stratosphere. It has been traditional for a long time to explain the thermal regime of the stratosphere on the basis of the theory of radiative equilibrium, assuming that the stratosphere is in this state. With this assumption many important results have been obtained. Recent experimental and theoretical conclusions, however, disprove the concept of stratospheric radiative equilibrium. This is particularly so as concerns theory, since the accuracy of measurements is not yet reliable. In this connection it is interesting to mention the work of Ohring [36], devoted to calculations of the radiative budget of the stratosphere (here the stratosphere is taken to be the layer of atmosphere from the tropopause up to 55 km). Using observational data on the structure and composition of the atmosphere and on the quantitative features of the absorption of solar radiation by ozone and water vapor and of the longwave radiation also by carbon dioxide, Ohring has calculated the “shortwave,” “longwave,” and overall net radiation for the individual absorbing components in the stratosphere and for their totality. The calculations were concerned with 10’ latitude rings of the Northern Hemisphere in January, April, July, and October. As a result a fairly complete picture of latitudinal and seasonal variations in the stratospheric net radiation of the Northern Hemisphere was obtained. The calculations showed that the main contribution to the “longwave” component of the stratospheric net radiation is made by water vapor and particularly by carbon dioxide. The radiative heating rate due to the absorption and emission of longwave radiation by water vapor and carbon dioxide is negative. The mean difference of the effective radiation at the boundaries of the stratosphere is 0.016 cal/cm, min for CO, and 0.006 cal/cm2 min for H,O. The radiative cooling due to these gases is only slightly compensated by the positive net radiation of the “ozone” component of the longwave budget. According to Plass [37], the longwave radiative heating rate due to the 9.6-,u ozone band is insignificant up to heights of about 30 km and negative at higher altitudes. Thus the total longwave net radiation of the stratosphere is always negative and absolutely increasing with an increase in latitude (from near zero in low latitudes to 0.03 to 0.07 cal/cm2 min in high latitudes), having a July maximum and a January minimum at all latitudes (except temperate) in its annual range. At the temperate latitudes the absolute maxima are observed in April and the minima in October. The range of the annual variation increases with latitude. The mean annual longwave net radiation of the Northern Hemisphere

816

Temperature Variation in the Atmosphere

is 0.018 cal/cm2 min, that is, about 9 percent of the corresponding value for the troposphere. The absorption of ultraviolet solar radiation by ozone and of the infrared radiation by water vapor to a great degree compensates for the radiative cooling of the atmosphere. The absorption of solar radiation due to ozone, exceeding by about four times that due to water vapor, is of chief importance in this case. The total shortwave net radiation is at maximum and increases with latitude in the warmer half-year. During the colder period its values are minimal and decreasing as the latitude increases. The range of the latitudinal and seasonal variability of the shortwave component is less than that of the longwave in the temperate and high latitudes (north of 35 to 45’ N lat.). An inverse relationship is observed in the equatorial zone. The total net radiation of the stratosphere is therefore positive south of 34 to 35’ N lat. and negative throughout the rest of the Northern hemisphere (Fig. 11.10). Examination of Fig. 11 10 clearly reveals the absence of the state of 90

-40

\

-40-30

r l

-30

70

\

1

-20 -10 0 10

FIG. 11.10 Net radiation of the stratosphere from the tropopause to the 55-km level cal/cmamin).

radiative equilibrium in the stratosphere at all latitudes except a narrow transition zone at 35 to 45’ N lat. This obviously implies that the presence in the stratosphere of high and low latitudes of heat sinks, and sources must lead to interlatitudinal advection as an important factor in thermal redistribution. The temperature decrease observed in the stratosphere (except its lower part) poleward appears to be caused by the aboveconsidered latitudinal distribution of stratospheric net radiation.

11.4. Radiation Factors in Thermal Regime of Stratosphere and Mesosphere 817

The local net radiation of the stratosphere considerably differing from zero at almost all places, the average net radiation for the Northern hemisphere has very small values (Table 11.3). TABLE 11.3 Average Net Radiation of the Stratosphere in the Northern Hemisphere, After Ohring [36]

callcm= min.

Jan.

Apr.

July

Oct.

Year

Net radiation of the entire stratosphere

-1

-2

+5

-2

0

Net radiation of the upper stratosphere

-1

-1

+3

-4

-0.8

It is known that the annual temperature variation of the stratosphere shows a maximum in July and minimum in January. If we assume, therefore, that radiation is the only factor determining stratospheric temperature, it follows that the net radiation (with heating of the stratosphere) must be positive in April and negative in October (the cooling period). During the months of extreme temperature the net radiation must obviously be zero. As is evident from Table 11.3, the calculations do not agree with these conclusions, which may be due to two causes: (1) the presence of other determinants of the thermal regime of the stratosphere, besides radiation; (2) the low accuracy of calculations. Two main sources of errors here are inaccurate information on the structure and composition of the atmosphere and also extrapolation of laboratoty data on the infrared radiation absorption to very small pressure values. The former is particularly important, since a qualitatively correct distribution of the infrared radiative heating rate can be obtained even with erratic absorption functions, the radiative heating being determined by the temperature profile above all. The use of the tropopause as the lower boundary of the stratosphere may cause the nonradiative factors of the thermal regim (the vertical heat transfer through the tropopause and thermal advection in the inclined tropopause of the temperate latitudes) to become important. The effect of the interaction between the stratosphere and troposphere can be largely eliminated if we consider the net radiation of the stratosphere with its lower boundary a t 21-km elevation. Calculations reveal that in this case the distribution of net radiation is greatly different (Fig. 11.11). As seen, the net radiation decreases northward in the cold half-year, while during late spring and summer the latitu-

818

Temperature Variation in the Atmosphere

dinal gradient of the net radiation has the opposite sign. It is also evident that in summer the net radiation of the upper stratosphere is positive for all latitudes. These results show that with the radiative mechanism determining the thermal regime in this case, the heat transfer should be expected in the direction to the pole in winter and in the reverse direction in summer.

Season

FIG. 11.11 Net radiation of the upper stratosphere at 21-55 km

cal/cmz min).

The familiar scheme of circulation in the upper atmosphere, proposed by Kellog and Schilling [38], gives just this change of transfer. It is therefore possible to consider that radiation is the main factor in the thermal regime and dynamics. This is also justified by the fact that the notable variation of net radiation in high latitudes from April to October, revealed by Fig. 11.11, agrees with the observed maximal temperature variation in these latitudes from summer to winter. It should further be mentioned that calculations of the mean net radiation of the upper northern stratosphere lead to the same conclusions as the analogous calculations for the entire stratosphere (see Table 11.3). Since in both cases the net radiation over individual months differs from zero, it is possible to expect the presence of heat exchange between the Northern and the Southern hemispheres. Thus the main conclusion from Ohring’s work [36] is that the stratosphere in its entirety (considered in this case as an atmospheric layer between the tropopause and the level of 55 km) is not in state of radiative equilibrium. The calculations of Ohring and also of Brooks [39] show that the same holds for the local net radiation (radiative heat gain) computed for stratospheric layers of 2-km thickness. As a rule, there is radiative cooling in the lower

11.4. Radiation Factors in Thermal Regime of Stratosphere and Mesosphere 819

stratosphere and radiative heating of a few degrees per day in its upper part. Figure 11.12 illustrates the calculations of the vertical profile of radiative temperature variations for the belt 60 to 70' N lat. (April). The zone of the transition from the region of radiative cooling to that of heating greatly displaces in dependence on latitude and season. For example, in the 2

\

\

/

I 1

/

I \

42-

I 38-

I I 1

E

f34-

i

I

30-

I

/

I I I I

I

-4

O

-2

L

I

1

2 4 6 DEG/ DAY

I

I

I

8

1

0

Vertical distribution of radiative temperature variations (deglday), April, 60-70' N Lat. (1) due to radiative heat exchange; (2) due to solar radiation absorption; (3) total temperature variation.

FIG. 11.12

same belt in October the transition through zero occurs at a height of about

44 km (Fig. 11.13). However, in all cases the maximum of radiative heating takes place near the 50-km level. Relatively numerous data of other investigators confirm this conclusion. There are, however, considerable discrepancies among different results. Brooks [39], for example, calculating the radiative temperature variations due to the 15-p CO, band, obtained far smaller values of radiative cooling (not more than 2 to 2.5'C/day) than Ohring. Moreover, according to Brooks, radiative heat exchange in the 15p C-0, band at the tropopause of

820

Temperature Variation in the Atmosphere

the equatorial latitudes causes heating and not cooling. In this situation the complementary positive heating rate due to the absorption of solar radiation by ozone leads to very small totals of heat gain (radiative temperature changes, accordingly) in the region of radiative cooling of the atmosphere. 54 -

If

-

\

50

46 42

-

38

-

E

x.34

r

-

30 -

26 22 18 -

14

-

I

\ \

I \

i

\

I

I

2

I

/ /

/

/ /

/

/

/

/

I

I

I

\

I

2"

I

4'

I

6'

DEG/ DAY

FIG. 11.13

Vertical distribution of radiative temperature variations (deglduy), October, 60-70' N Lat. Notation as in Fig. 11.12.

Along with this there appears an area of radiative heating near the equatorial tropopause. The finding of a region with a marked spatial nonhomogeneity of heating rate in the 25- to 30-km layer, especially in the warmer period, is an important result of Brooks (see, for example, the April data of Fig. 11.14). A very interesting peculiarity of the stratospheric thermal regime near a temperature maximum at the 50-km altitude was stated by Craig and Ohring [40]. As is known, the concentration of ozone at this altitude can be considered equilibrated. It was found, on the other hand, that the balanced ozone concentration depends on temperature decreasing with ozone increase. Con-

11.4. Radiation Factors in Thermal Regime of Stratosphere and Mesosphere

821

sequently an increase in temperature due to absorption of solar radiation results in a decrease of ozone concentration, which in turn leads to a decrease in the absorption. The temperature dependence of ozone concentration is thus a stabilizing factor that prevents any lasting increase or decrease of

I

200

/ 250900

750

600

450

300

15'

N LAT. deg

FIG. 11.14

Vertical distribution of radiative temperature variations in the Northern hemisphere (deglday), April.

Examination of Figs. 11.12 and 1I .13 shows that the presence of a temperature maximum near the 50-km level should be attributed to the maxima radiative heat inflow at this level. As regards the vertical increase of temperature in the layer of stratosphere below 50 km, it is undoubtedly caused by the vertical increase of the absorption of ultraviolet solar radiation by ozone. The decrease of temperature above the maximum at 50 km appears to be related to the decreasing absorption of ultraviolet radiation by ozone above this level. The effect of absorption of the infrared solar radiation has been investigated by Houghton [41]. Houghton has also shown recently [42] that for the lower stratosphere (that is, the region 18 to 30 km), the total radiative flux divergence is very nearly zero. Herrman and Yarger [43] investigated some effects of multiple scattering on heating rates in the ozone layer. Recent theoretical investigations have clarified to a certain degree the problem of the causes for the daily temperature range in the stratosphere.

00

822

Temperature Variation in the Atmosphere

It appears that the main factor of this range is the daily variation in the absorption of solar radiation by ozone. According to the data of Johnson [44], for example, the daily temperature changes due to the absorption of solar radiation by ozone are maximal at about the 50-km level and are approximately 5OC, decreasing to l0C at 30 and 70 km. Similar results have also been obtained by Pressman [45]. According to Pressman, in the low and intermediate latitudes the character of the daily temperature variation at the heights in the stratosphere from 30 to 60 km is almost independent of season (inclination of the sun). Only at high latitudes (near and north of 7 5 O N lat.) is there an evident relationship between the range of the daily temperature variation and the inclination of the sun, the range decreasing with an increase in the duration of the day. Unfortunately, experimental data on the daily temperature variation in the stratosphere are scarce, which makes difficult the correction of theoretical results. The available data reveal considerable deviations between theory and experimental results.+ For example, Beyers and Miers [46] estimated the daily temperature fluctuations at the stratopause to be 15 to 2OoC, which exceeds by there to four times the theoretical calculation. Although the above results give convincing evidence of the main importance of radiation in the determination of the stratospheric thermal regime, nevertheless there still remains the problem of the relative importance of other factors, particularly of turbulent mixing. Such widely known facts as the mixing atmosphere up to high altitudes and the presence of stratospheric jet streams certainly indicate that the “radiation” theory of the stratospheric thermal regime is far from being all-inclusive. To this time, no attempt has been successful in producing a “convection-radiation’’ theory of the thermal regime in the stratosphere. The only attempt to evaluate the contribution of various factors to heat exchange in the lower stratosphere is contained in the work of Chin Wan-cheng and Greenfield [47]. These authors calculated the contribution of radiative heat gain, of thermal advection, and of the vertical convective and turbulent heat exchange, determining the temperature variation for 12 h at the level of isobaric surfaces of 200, 100, and 50 mb. It was assumed that the radiation contribution could be isolated by averaging the 12-h temperature differences over a great number of soundings at various places (it is supposed that this temperature variation is due entirely to radiation factors). The calculations showed that these temperature variations caused by radiation fluctuate from 0.15OC at 200 mb in winter to 0.35OC at 115 mb in summer. The advective temper+Some data are given, for example, in Vaisala’s work [48].

11.4. Radiation Factors in Thermal Regime of Stratosphere and Mesosphere

823

ature variation was computed from charts of pressure topography and was shown to be from 4.5' at 200 mb to 1.0'C at 50 mb in winter. In summer the effect of heat advection is somewhat less (all these results were obtained from aerological soundings over United States territory). Comparison of the above values shows that the influence of heat advection on the temperature variation in the low stratosphere is much more than that of radiation. These two factors appear to become equally influential only at the level of the 50-mb surface. This evaluation should, of course, be treated as rough and arbitrary. However, it clearly shows the importance of taking into account the dynamic factors in the theory of the stratospheric thermal regime. 2. Mesosphere. At present this is the name of an atmospheric layer between the heights 50 and 80 km. Thus, the preceding results concern not only the stratosphere as commonly understood (atmospheric layer from the tropopause to 50-km elevation) but also the mesosphere in part. We shall now proceed with the examination of these results, having in view the clarification of the theory of the thermal regime in the mesosphere as a whole. The importance of radiation in the stratospheric thermal regime, is even more so in the mesosphere. For this reason the few theoretical studies concerned with the thermal regime in the mesosphere have been mainly dedicated to calculations of radiative flux divergence and the corresponding radiative temperature changes. It follows that the main contribution to the shortwave radiative heat gain in the lower mesosphere is made by the absorption of ultraviolet solar radiation due to ozone. This factor causes the presence of a zone of maximal radiative heating in the 45- to 55-km layer. The second important constituent of the mesosphere that strongly absorbs solar radiation is molecular oxygen, which has intensive absorption bands in the vacuum ultraviolet. According to Penndorf [49], the maximal heat inflow due to the absorption of solar radiation by oxygen takes place at about 100-km elavation, with the radiative temperature variation in the layer of absorption maximum reaching 68OC/12 hr. These data agree with more recent results of Murgatroyd et al. [50-521, and also of Borisenkov and Osipov [53]. The radiative heating values obtained in [50]turned out to be much lower (of the order of 10 to lS°C/day near the 100-km level). According to the data of Shved [54, 551, the heating due to the absorption of ultraviolet solar radiation at about 100 km is of the order of 50°C/day if all the absorbed energy transforms into heat at the same level, which obviously is not so. Actually, a portion of the absorbed energy is "accumulat-

824

Temperature Variation in the Atmosphere

ed” in the dissociation of 0, by radiation as chemical energy. Atomic oxygen is transported by turbulent diffusion to the lower layers (and above 100 to 110 km by molecular diffusion) where it reintegrates as O2 and gives out the accumulated energy. Thus the correct profile of radiative heating rate due to the absorption of ultraviolet radiation in the lower thermosphere can be obtained only by taking account of the turbulent vertical transfer of atomic oxygen. The above-mentioned works of Murgatroyd, Goody, and Singleton give the fullest theoretical presentation available of the radiative heat inflow in

---------

\

E 70 .*-

40

30

00 E

-L.

September

-

7060-

40 50

30

I

March

/--._-------’-/ 0.5

0 /

0

11.4. Radiation Factors in Thermal Regime of Stratosphere and Mesosphere

825

the mesosphere (within the levels 30 to 90 km). As a result, a mean meridional profile of the radiative heating rate due to absorption of solar radiation by ozone and oxygen has been plotted for the Northern Hemisphere in all months. Comparison with the similar calculations of Pressman [45] reveals a qualitative agreement, but the quantitative results depart considerably. The latter is explained by the fact that the authors were taking account of the absorption not only by ozone but also by oxygen. Also differing are the initial vertical profiles of ozone concentration : the vertical ozone distribution according to Murgatroyd and Goody is higher than Pressman's estimate of concentrations above 40 km, and are smaller below this level. For this reason Murgatroyd and Goody obtained smaller values of heating rates (due to radiation absorption) below 40 km and higher rates above. The most important peculiarity of the data on radiation absorption is the presence of radiative heating maxima at 45 to 55 and 100 km and of a minimum (near zero) at about 80 km. This is evident from Fig. 11.15. Thus the vertical distribution of the shortwave heat influx turns out to be similar to the vertical temperature profile. It is, however, important in comparing these two profiles to know the effect of the longwave component of the radiative heat income. In the mesosphere it is mainly determined by the 15-p carbon dioxide and 9.6-p ozone absorption bands. The calculations for the 15-p C 0 2 band and standard atmospheric stratification are illustrated by Table 11.4. The 9.6-p ozone band also causes radiative cooling, but is notably smaller in magnitude (with a maximum of -3.0°C/day). The resulting radiative temperature variations obtained by taking into account the shortwave and longwave components are given in Fig. 11.16, borrowed from the work of Murgatroyd and Singleton [52]. Analogous results were derived by Davis [56]. TABLE 11.4 Radiative Temperature Variations ('Clday) due to the 15-p CO, Absorption Band.

Hcight, km

25

30

35

40

45

50

55

60

After Plass

-1.3

-2.3

-3.3

-4.5

-4.9

-4.6

-4.0

-2.3

After Murgatroyd and Goody

-1.7

-2.8

-4.1

-5.2

-6.6

-6.8

-5.0

-3.8

826

Temperature Variation in the Atmosphere

It is evident from Fig. 11.16 that the long- and shortwave components are mutually compensated to a great degree. The extensive equatorial areas of the stratosphere and mesosphere are therefore in the state close to radiative equilibrium. This implies that in these regions the vertical heat transfer due to nonradiative processes must be insignificant. Considerable deviations from the state of radiative equilibrium occur in the polar areas, which must obviously result in advective heat transportation from the summer to winter pole in the upper mesosphere.

POLE

FIG. 11.16

EQUATOR

POLE

Meridional cross section of radiative temperature variations. The shaded area represents radiative heating.

Regarding the regularities of the mean vertical temperature distribution in the mesosphere, it can only be said that their interpretation on the basis of Fig. 11.16 is not feasible, since in this case there is hardly a similarity between the vertical profiles of temperature and radiative heat gain. The data of Fig. 11.16 make even more difficult the explanation of the main features of the vertical temperature profile, as they mark neither maximal heating at about 50 km nor its minimum at the 80-km level. The last result is contradictory to the above-considered data. However, the causes for these significant discrepancies between the results of different authors are not

11.4. Radiation Factors in Thermal Regime of Stratosphere and Mesosphere 827

obvious. It appears that they may be due to the inadequately reliable (and nonidentical) initial data, for the transmission function above all, and also due to the differences in methods of calculations. Recently a whole series of investigations [56-631 was undertaken with the purpose of more accurate revision of these results. Also apparent in the upper mesosphere is the importance of those departures from the state of local thermodynamic equilibrium (and consequently the breakdown of Kirchhoffs law), which were discussed in Chapter 1. The importance of the latter factor can be seen from Fig. 11.17, which presents Shved’s [54, 551 vertical profiles of radiative temperature variations. These profiles were obtained by taking into account only the basic transformations of the main isotopic variants of the CO, molecule, which results first of all in the underestimation of cooling rates in the lower mesosphere. The same figure (right-hand side) presents the temperature distribution for which the calculation was performed.

FIG. 11.17

Vertical profiles of radiative temperature variations computed taking into account only the fundamental transition of the molecule C120~6.

It was mentioned in Chapter 1 that the local thermodynamic equilibrium is not fulfilled at high altitudes and that for this reason the deviations from Kirchhoff’s law for radiation must be taken into account. With radiative transfer in the 15-p CO, band, this factor becomes important above the 70-km level. In order to obtain radiative heat fluxes, it is necessary to know

828

Temperature Variation in the Atmosphere

the time of vibrational relaxation z, a quantity that becomes larger as the collisions of the CO, and air molecules become rarer, and which describes the energy exchange between the vibrational and translational degrees of freedom. At present there are no reliable experimental t values available. Theoretical calculations indicate a possible dispersion from 2 to 20 psec at 200°K and 1-atm pressure. In the lower thermosphere (at about 100-km elevation) at z = 20 psec, the atmospheric cooling due to the 15-p CO, band is by an order of value less than with t = 2 psec. This is easily understandable if we take into account that with increasing time of relaxation z, the importance of collisions in the population of vibrational levels of the CO, molecules becomes less and less compared with optical transitions. Thus the conditions of monochromatic radiative equilibrium are approached, when the absorbed quantum will necessarily be reemitted at the same frequency. However, in the conditions of radiative equilibrium the radiative heating is zero. The statement of the weak cooling effect of the 15-p CO, band at the mesopause sets the problem of other cooling factors. Some are given briefly as fellows : (a) Upward air movements leading to adiabatic cooling. (b) Radiative transfer in the rotational H,O band, provided there is a sufficient amount of water vapor in the upper mesosphere. (c) Nonthermal emission of hydroxyl OH (from the visible to the infrared near 4.5 p). For the total radiation by OH, calculations give a value of 3.2 erg/cm2 sec. If the OH layer is situated at the height of 80 km, this amount of the emitted energy can secure approximate cooling rates of l"C/day at these altitudes. (d) Emission in the line of atomic oxygen with the wavelengt) 63 p, if the maximal concentration of 0 takes place at the mesopause. Let us now dwell on the radiation at R = 63 p. This oxygen line is an important factor in the lower thermospheric cooling. For example, according to the measurements of atmospheric structural parameters and composition carried out by Pokhunkov [64], the cooling rates at the heights 100, 110, and 120 km are 3.3, 8.8, and 16" C/day, respectively. These results relate to an optically thin radiating layer. The accounting for an extensive layer must lower the cooling rates. Photochemical calculations show that the maximal 0 concentration should be expected at about the 100-km elevation. Observations over chemaluminescence in a cloud of nitric oxide gave maxima at the heights 103 to 107 km. Below the maximum, the degree of dissociation of 0, becomes less and the volume concentration of 0 is so insignif-

11.5. Relation between Radiative and Turbulent Heat Exchange

829

icant as to lead to very small thermal contributions of 0, and of positive sign, at the mesopause. However, the available data on atomic oxygen are too few to enable any final conclusion. Some factors (pronounced seasonal variation in the “dissociation level” of 0,, downward air movements, and turbulent transfer) point toward a possible displacement of the maximal 0 concentration below 100 km. Many peculiarities of the temperature field of the stratosphere and mesosphere and its irregular variations cannot be explained on the basis of the theory of radiation. These are, first of all, the warmer winter mesopause of high latitudes as compared with that in summer, and the usual vertical temperature range during the polar night. The unexplainable irregular variations are the abnormal warming of the polar stratosphere and mesosphere in winter and a sharp change of the type of temperature distribution in the mesosphere. The mentioned phenomena can be accounted for, and generally the correct qualitative descriptions of the thermal regime of the stratosphere and mesosphere can be obtained only in the combined consideration of radiative transfer, dynamics and the photochemical processes. So far, only first steps have been made in this direction, as, for example, in the works of Leovy 1651, Lindzen and Goody [66], and Lindzen [67]. 11.5. Relation between Radiative and Turbulent Heat Exchange in the Surface Layer of the Atmosphere

It was pointed out in the preceding section that the problem considered should be solved within the theory of the nonstationary thermal regime of the surface atmospheric layer. Let us now discuss such asolution of the problem of the relation between radiative and turbulent heat exchange in the surface atmospheric layer, using the data of Kondratyev, et al. [25]. (For more detail, see [22, 68-79].) We shall examine the following adequately consistent setting of the nonstationary problem of heat exchange. The turbulent and radiative heat transfer are assumed to be in combined action. The temperature change TT+rad during a finite time interval A t is sought. Then the temperature variation AT, due to the turbulent transfer alone is determined, as well as that due to only radiative heat exchange AT,,,. All three cases keep the same boundary and initial conditions and are concerned with the same time interval A t . The obtained values, ATT+,,,, AT,, and ATrad,are compared. It can be expected that this comparison will reveal the relative importance of each type of heat exchange. The process of temperature variations due to radiative and turbulent

830

Temperature Variation in the Atmosphere

heat exchange (without consideration of solar radiation, night) is described by the following system of equations :

(1 1.28a) (1 1.28b) (1 1.29)

Equation (1 1.27) describes the heat gain in the air, (1 1-28) are radiative transfer equations, and (1 1.29) is for the heat inflow in the soil. The notation is as follows: T is the air temperature, T* is the temperature of the soil, il is the coefficient of turbulent thermal conductivity, 8 is the zenith angle, k, is the absorption coefficient at the wavelength A, &(z) is the Planck function, k* is the coefficient of temperature conductivity of the soil, G,(z, 0 ) and UA(z,0) are the downward and upward radiative heat fluxes, respectively, F = J," (U, - GA)dil, and z is the vertical coordinate. The boundary conditions are assumed to be

dT

dT*

a* dz-

dZ = -

(1 1.30d)

Equation (11.30~)in the system is the boundary condition of temperature continuity at the level of the underlying surface, while (1 1.30d) is the condition of thermal budget at ground level; and A* is the coefficient of thermal conductivity of the soil. Let the system (11.27) through (11.29) be complemented by the initial conditions T(z, to) = f ( z ) (11.31a)

T*(z, t o ) = f * ( z )

(11.31b)

For the solution of the problem it is possible to use a modified math-

83 1

11.5. Relation between Radiative and Turbulent Heat Exchange

ematical scheme proposed by Malkevich [73]. For the vertical variation of the coefficient of turbulent transfer, the Shvetz-Yudin model was taken as

[

k(z)=k, x+= k,(x

3,

+ 1)

Y

Olz_(h (1 1.3 1c)

k,, h 5 z 5 00

where k,x is the coefficient of molecular temperature conductivity, and x is the coefficient of temperature conductivity outside the surface atmospheric layer. The numerical solution is so far obtained only for the simplest case of an atmosphere with an isothermal initial vertical temperature distribution. The results of calculations of temperature variations at 4-h intervals are presented in Table 11.5. The solution is in the first approximation. This means that the radiative heat inflow during the considered time interval TABLE 11.5 Cooling of an Isothermal Atmosphere Calculated for 4-h Intervals. After [73]

Level, crn 0 2.0 4.8 24.0 28.0 200.0 480.0 2,400.0 24,000.0

AT,, OC -4.29 -3.11 -2.82 -2.26 -2.05 -1.52 -1.22 -0.71 -0.19

Arrad+T,

OC

-3.76 -2.78 -2.54 -2.06 -1.89 -1.45 -1.20 -0.91 -0.43

remains the same as the initial flow. In the surface layer the value AT,,XC is almost independent of height and equals to 1.7SoC/4 h. Comparison of ATra,+,, AT,, and AT,,, shows that during a finite time interval, the temperature variations due to different types of heat exchange are not additive, as it is often maintained on the basis of the observed and radiative temperature changes. The resulting temperature variation ATrad+,at the lower five meter-levels appeared to be greater than AT,. This shows that interaction of both types of heat transfer can bring forth their mutual weakening. Such a conclu-

832

Temperature Variation in the Atmosphere

sion appears to be natural, as every type of heat exchange tends to decrease contrasts in temperature. This implies, in particular, that if ATrad+Tis comparable to AT, or ATrad, we cannot yet speak of the prevailing influence of turbulent or radiative transfer. One of these types, being small in magnitude, can be still greatly effective indirectly by weakening the other. The first approximation does not adequately represent the interaction between the radiative and turbulent heat exchange. It is therefore of great interest to derive a more accurate solution. Further calculations should also be extended to different parameters, time intervals, and initial conditions. To conclude what has been said above, it should be noted that usually the setting of problems using the heat inflow equation is not consistent logically, since the coefficient of turbulent mixing cannot be considered an outer parameter independent of temperature distribution and wind. The final solution of the discussed problem will apparently be given only on the basis of a closely formulated system of equations without the use of the semiempirical theory of turbulent mixing. This requires, first of all, creation of a physical theory of atmospheric turbulence. REFERENCES 1. Kumetzov, E. S. (1946). Vertical atmospheric temperature distribution with radiative equilibrium. Trans. Inst. Theoret. Geophys. l. 2. Brooks, L. F. (1950). A tabular method for the computation of temperature change by infrared radiation in the free atmosphere. J. Meteorol. 7 , No. 5. 3. Kondratyev, K. Ya. (1965). “Radiative heat exchange in the atmosphere.” Pergamon Press. 4. Podolskaya, E. L. (1954). To the problem of differential equations for the integral radiant flux. Trans. Main Geophys. Obs. No. 150. 5. Bruinenberg, A. (1946). A numerical method for the calculation of temperature changes in the free atmosphere. Koninkl. Ned. Meteorol. Inst. BiIt. Mededel. Verhandl. B125, Part 1, No. 1. 5a. Zdunkowski, W. G., and Johnson, F. G. (1965). Infrared flux divergence calculations with newly constructed radiation tables. J. AppI. Meteorol. 4, No. 3. 6. Kondratyev, K. Ya., and Yakushevskaya, K. E. (1956). A new method for calculating radiative heat divergence. Proc. Leningrad Univ., Ser. Phys. No. 9. 7. Yamamoto, G., and Onishi, G. (1953). A chart for the calculation of radiative temperature changes. Sci. Rept. Tohoku Univ., Fifrh Ser. 4, No. 3. 7a. Funk, J. P. (1961). A numerical method for the computation of the radiative flux divergence near the ground. J. Meteorol. 18, No. 3 . 8. Rodgers, C. D., and Walshaw, C. D. (1966). The computation of infrared cooling rate in planetary atmospheres. Quart. J. Roy. Meteorol. SOC.92, No. 1415.

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9. Kondratyev, K. Ya., Nijlisk, H. J., and Noorma R. J. (1968). On the spectral distribution of radiative flux divergence in the spectral region 12 to 44,44p. Proc. Acad. Sci. USSR, Ser. Phys. Atmosphere Ocean, No. 6 . 10. Kovaleva, E. D., and Shekhter, F. N. (1966). The influence of atmospheric stratification on radiative flux divergence. Trans. Main Geophys. Obs. No. 187. 11. Shekhter, F. N. (1966). Some problems of radiative heat exchange in a cloudy atmosphere. Trans. Main Geophys. Obs. No. 187. 12. Petrova, L. V., and Feugelson, E. M. (1966). Radiative heat exchange with progressing cloud. Proc. Acad. Sci. USSR, Ser. Phys. Atmosphere Ocean No. 4. 13. Zdunkowski, W., Henderson, D., and Hales, J. V. (1966). The effect of atmospheric haze on infrared radiative cooling rates. J. Atmospheric Sci. 23, No. 3. 13a. Moller, F. (1935) Die Warmequellen in die freien Atmosphare. Meteorol. Z. 52, No. 11. 14. Moller, F. (1941). Langwellige Wasserdampfstrahlung und Stratospharentemperatur. Meteorol. Z. 58, No. 10. 14a. Moller, F. (1956). The pattern of radiative heating and cooling in the troposphere and lower stratosphere. Proc. Roy. SOC.A236, No. 1205. 15. Gaigerov, S. S., and Kastrov, V. G. (1951). Investigation of the thermal transformation of drifting air from data of free balloons. Trans. Centr. Aerol. Obs. No. 6. 16. Gaigerov, S. S., and Kastrov, V. G. (1954). Some generalizations concerning thermal air transformation as observed with free balloons. Truns. Centr. Aerol. Obs. No. 13. 17. Thompson, A. H. (1953). On the pressure dependence of atmospheric cooling due to water vapor radiative exchange. Bull. Am. Meteorol. SOC.34, No. 4. 18. Staley, D. 0. (1965). Radiative cooling in the vicinity of inversions and the tropopause. Quart. J. Roy. Meteorol. SOC.91, No. 389. 19. London, J. (1952). The distribution of radiational temperature change in the northern hemisphere during March. J. Meteorol. 9, No. 2. 20. London, J. (1954). The distribution of infrared cooling in the atmosphere for winter and summer seasons. Proc. Toronto Meteorol. Con$ 1953. Roy. Meteorol. SOC., London. 21. Davis, P. A. (1961). A re-examination of the heat budget of the troposphere and lower stratosphere. Sci. Rept. No. 3, AF19(604)-6146, New York University. 22. Manabe, S., and Moller, F. (1961). On the radiation equilibrium and heat balance of the atmosphere. Monthly Weather Rev. 89, No. 12. 23. Manabe, S., and Strickler, R. F. (1964). Thremal equilibrium of the atmosphere with a convective adjustment. J. Atmospheric Sci. 21, No. 4. 24. Robinson, G. D. (1950). Two notes on the temperature changes in the troposphere due to radiation. Centr. Proc. Roy. Meteorol. SOC. 25. Kondratyev, K. Ya. Gayevskaya, G. N., and Yakushevskaya, K. E. (1962). Radiative heat flux divergence and heat regime in the lowest layer of the atmosphere. Arch. Meteorol, Geophys. Bioklimatol. B12, No. 1. 26. Yamamoto, G. (1952). Effect of radiative transfer on the vertical distribution of temperature in the troposphere. Sci. Rept. Tohoku Univ. Fijith Ser. 4, No. 2. 27. Funk, J. P. (1960). Measured radiative flux divergence near the ground at night (Plate IV). Quart. J. Roy. Meteorol. SOC. 86, No. 369. 28. Funk, J. P. (1961). Comparison of radiative divergence near the ground with blackball measurements. J. Meteorol. 18, No. 5 .

834

Temperature Variation in the Atmosphere

29. Funk, J. P. (1962). Radiative flux divergence in radiative fog. Quart. J. Roy. Meteorof. SOC. 87, No. 377. 30. Prokh, L. 2. (1964). To the sounding of the atmospheric boundary layer with double net radiometers. Trans. Ukrainian Hydrometeorol. Znst. No. 42. 31. Lieske, B. J., and Stroschen, L. A. (1965). Arctic radiative divergence measurements. Trans. Am. Geophys. Union 46, No. 1. 32. Barashkova, E. P. (1965). To the vertical distribution of long-wave radiation fluxes in the boundary air layer. Trans. Main Geophys. Obs. No. 170. 33. Elliot, W. P., and Stevens, D. W. (1966). Long-wave radiation exchange near the ground, Solar Energy 10, No. 1. 34. Kondratyev, K. Ya., Zhvalev, V. F., and Styro, D. B. (1966). Radiative flux divergence in the spectral region 4 4 0 micron at different atmospheric levels. Proc. Acad. Sci. USSR, Ser. Phys. Atmosphere Ocean, No. 1. 35. Kondratyev, K. Ya., Nijlisk, H. J., and Noorma, R. J. (1966). On the spectral distribution of radiative flux divergence in a free atmosphere. Proc. Acad. Sci. USSR, Ser. Phys. Atmosphere Ocean, No. 2. 36. Ohring, G. (1958). The radiation budget of the stratosphere. J. Meteorol. 15, No. 5 . 37. Plass, G. N. (1954). The influence of ozone on the infrared radiation influx. Bull. Am. Meteorol. SOC.35, No. 10. 38. Kellog, W. W., and Schilling, G. F. (1951). A proposed model of the circulation in the upper stratosphere. J. Meteorof. 8, No. 4. 39. Brooks, D. L. (1958). The distribution of carbon dioxide cooling in the lower stratosphere. J. Meteorol. 15, No. 2. 40. Craig, R. A,, and Ohring, G. (1958). The temperature dependence of ozone rzdiational heating rates in the vicinity of the mesopeak. J . Meteorol. 15, No. 1. 41. Houghton, J. T. (1963). The absorption of solar infrared radiation by the stratosphere. Quart. J. Roy. Meteorof. SOC.89, No. 381. 42. Houghton, J. T. (1965). Infrared emission from the stratosphere and mesosphere. Proc. Roy. SOC. A288, No. 1415. 43. Herman, B. M., and Yarger, D. N. (1966). Some effects of multiple scattering on heating rates in the ozone layer. J. Atmospheric Sci. 23, No. 3. 44. Johnson, F. (1953). Temperature variations in the upper atmosphere due to ozone. Bull. Am. Meteorol. SOC. 34, No. 3. 45. Pressman, J. (1955). Diurnal temperature variations in the middle atmosphere. Bull. Am. Meteorol. SOC. 36, No. 5 . 46. Beyers, N. J., and Miers, B. T. (1965). Diurnal temperature change in the atmosphere between 30 and 60 km over the White Sands missile range. J. Atmospheric Sci. 22, No. 3. 47. Chin Wan-cheng and Greenfield, R. S. (1959). The relative importance of different heat exchange processes in the lower stratosphere. J. Meteorol. 16, No. 3. 48. Vaisala, V. (1965). Observations concerning the daily variation of the temperature and balloon effect on the temperature measurement at high altitudes. Viiisalii News No. 28. 49. Penndorf, R. (1950). Absorption of solar energy by oxygen molecules in the E-layer. J. Meteorol. 7, No. 3. 50. Murgatroyd, R. J., and Goody, R. M. (1958). Sources and sinks of radiative energy from 30 to 90 krn. Quart. J. Roy. Meteorol. SOC.84, No. 361.

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51. Murgatroyd, R. J. (1965). Radiation sources and sinks in the stratosphere and mesosphere. Colloq. Probl. Meteorol. Stratosphere Mesosphere, Paris, 1965. 52. Murgatroyd, R. J., and Singleton, F. (1961). Possible meridional circulation in the stratosphere and mesosphere. Quart. J , Roy. Meteorol. SOC.87, No. 372. 53. Borisenkov, E. P., and Osipov, B. A. (1964). Evaluation of seasonal peculiarities of the energy budget in the upper atmosphere of the northern hemisphere. Trans. Arct. Antarct. Znst. 271. 54. Shved, G . M. (1965). Method for considering the departure from Kirchhoff‘s law in the mesosphere with radiative transfer in the 15 micron CO, band. Bull. Leningrad Univ., Ser. Phys. Chem. No. 1. 55. Shved, G. M. (1967). Radiation factors of the thermal regime of the atmospheric layer from 50 to 100 km. Probl. Atmospheric Phys., Leningrad Univ. Publ. No. 5 . 56. Davis, P. A. (1963). An analysis of the atmospheric heat budget. J. Atmospheric Sci. 20, No. 1. 57. Houghton, J. T., and Hitschfeld, W. (1961). Radiative transfer in the lower stratosphere due to the 9.6 micron band of ozone. Quart. J. Roy. Meteorol. SOC.87, No. 374. 58. Walshaw, C. D., and Rodgers, C. D. (1963). The effect of Curtis-Godson approximation on the accuracy of radiative heating-rate calculations. Quart. J. Roy. Meteorol. SOC.89, No. 379. 59. Mayot, M., and Vigroux, E. (1965). Application de l’approximation de CurtisGodson A l’ozone atmospherique. Ann. Geophys. 21, No. 1. 60. Rodgers, C. D. (1964). Radiative heating rates in the earth’s atmosphere. Zn “Le spectres infrarouge des astres. Litge.” 61. Clark, J., and Hitschfeld, W. (1964). Heating rates due to ozone computed by Curtis-Godson approximation. Quart. J. Roy. Meteorol. SOC.90, No. 383. 62. Bickert, A. (1964). Uber die Strahlungsbedingte Abkiihlung in der Stratosphare. Met. Abhandl. Inst. Meteorol. Geophys., Freien Univ. Berlin 28, No. 4. 63. Newell, R. E. (1965). A review of studies of eddy fluxes in the stratosphere and mesosphere. Dept. of Meteorology, Rept. No. 12. Massachusetts Institute of Technology. 64. Pokhunkov, A. A. (1962). Gravitational division, composition and structural parameters of a nocturnal atmosphere at the heights from 100 to 210 km. Art. Earth Satellite No. 13. 65. Leovy, C. (1964). Simple models of thermally driven mesospheric circulation. J . Atmospheric Sci. 21, No. 4. 66. Lindzen, R., and Goody, R. M. (1965). Radiative and photochemical processes in mesospheric dynamics: Part 11. Models for radiative and photochemical processes. J. Atmospheric Sci. 22, No. 4. 67. Lindzen, R. S. (1966). Radiation and photochemical processes in mesospheric dynamics. Parts 11-IV. J. Atmospheric Sci. 23, No. 3. 68. Shekhter, F. N. (1965). Solution of the problem of the boundary atmospheric layer structure considering radiative heat exchange. Trans. Main Geophys. Obs. No. 167. 69. Estoque, M. A. (1963). A numerical model of the atmospheric boundary layer. J. Geophys. Res. 68, No. 4. 70. Fleagle, R. G . (1965). The temperature distribution near a cold surface. J. Meteorol. 13, No. 2.

836

Temperature Variation in the Atmosphere

71. Feugelson, E. M. (1964). Radiative flux divergence in the atmosphere. Proc. Acad. Sci. USSR, Ser. Geophys. No. 10. 72. Moller, F. (1955). Strahlungsvorgange in Bodennahe. Z. Meteorol. 9, No. 2. 73. Malkevich, M. S. (1956). Temporal air temperature variations due to turbulent mixing and radiative heat exchange. Trans. Geophys. Znst. Acad. Sci. USSR No. 37. 74. Kluchnikova, L. A., and Shekhter, F. N. (1960). To the role of radiative and turbulent heat exchange in the forming of temperature stratification in the atmospheric boundary layer. Trans. Main Geophys. Obs. No. 94. 75. Goody, R. M. (1960). The influence of radiative transfer on the propagation of a temperature wave in a stratified diffusing atmosphere. J. Fluid Mech. 9, Part 3. 76. Goody, R. M. (1956). The influence of radiative transfer on cellular convection. J. Fluid Mech. 1, Part 4. 77. Gille, J., and Goody, R. M. (1964). Convection in a radiating gas. J. Fluid Mech. 20, Part 1. 78. Viskanta, R., and Grosh, R. J. (1962). Heat transfer by simultaneous conduction and radiation in an absorbing medium. J. Heat Transfer 84, No. 1. 79. Galin, M. B., Popov, A. K., and Rudenko, S. I. (1966). Propagation of perturbations in a baroclinic atmosphere with radiation and turbulent heat conductivity. Proc. Acad. Sci. USSR, Ser. Phys. Atmosphere Ocean No. 3. 80. Lichtman, D. L., Kazhdan, R. M., and Utkina, 2. M. (1961). Experimental determination of radiative flux divergence in the lower atmospheric layer. Trans. Main Geophys. Obs. No. 107.

Additional Bibliography

CHAPTER 1 1. Fridzon, M. B., Pakhomova, L. A., Selyukov, N. G., and Gorbatova, E. S. (1967). Spectral characteristics of the reflection of some materials used in meteorology. Meteorol. Hydrol. No. 9. 2. Vassilevsky, A. M., and Kozyrev, B. P. (1966). Determination of spectral emissivity of water and snow in the 4-15 micron region. Proc. Leningrad Electrotechn. Znst. No. 55. 3. Kozyrev, B. P., and Kropotkin, M. A. (1966). Investigation of the infrared diffuse reflection of white coatings. Proc. Leningrad Electrotechn. Znst. No. 55. 4. Buznikov, A. A,, and Kozyrev, B. P. (1966). Investigation of the coefficient of the integral emission of water. Proc. Leningrad Electrotechn. Znst. No. 55. 5 . Baker, D. J., and Brown, W. L. (1966). Presentation of spectra, Appl. Opt. 5, No. 8.

CHAPTER 2 I . Goysa, N. I. (1966). Method of actinometric measurements from the IL-14 aircraft. Trans. Ukr. Sci. Hydrometerol. Znst. No. 55. 2. Collinbourne, R. H. (1966). General principles of radiation meteorology. Zn “Light Ecology Factor.” Blackwell, Oxford. 3. Blackwell, M. J. (1966). Radiation meteorology in relation to field work. Zn “Light Ecology Factor.” Blackwell, Oxford. 4. Paulsen, H. S. (1967). Some experiences with the calibration of radiation balance meter. Arch. Meteorol. Geophys. Bioklimatol. 15, No. 1-2. 5. Schuepp, W. (1965). Der Einfluss der Zirkumsolaren Himmelstrahlung auf die Pyrheliometervergleiche. Astron.-Meteorol. Anstalt der Universitat Basel. 6. Kozyrev, B. P., and Buchenkov, V. A. (1966). Measurement of spectral sensitivity of pyranometers and balance meters. Proc. Acad. Sci. USSR,Ser. Phys. Atmosphere Ocean 2, No. 5. 7. Kyle, T. G. (1966). A crystal radiometer with f.m. output. J. Sci. Znstr. 43, No. 10. 8. Barashkova, Y. P., Lebedeva, K. D., and Yastrebova, T. K. (1966). Comparison of longwave radiation fluxes measured with various instruments. “Meteorological Investigations,” No. 15. Nauka, Moscow. 9. Sulev, M. A. (1966). On some results of the comparison of longwave radiation receivers. “Meteorological Investigations,” No. 15. Nauka, Moscow.

CHAPTER 3 1. Foitzik, L., Hebermehl, G. and Spankuch, D. (1966). Kollektiver Streuquerschnitt und kollektive spektrale Extinktion der Mie-Streuung bei logarithmischen GaussVerteilungen. Beitr. Geophysik. 75, No. 6. 2. Spankuch, D. (1966/67). Atmospharische Streufunktionen bei Grossenverteilung des Dunstes nach Jungschem Potenzgesetz. Optik 24.

837

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Additional Bibliography

3. Queney, P. (1966). Determination du pouvoir emissif d'une nappe d'eau en equilibre. Ann. Geophys. 22, No. 4. 4. Pontier, L., and Dechambenoy, C. (1966). Determination des constantes optiques de l'eau liquide entre les 40 micron. Application au calcul de son pouvoir reflecteur et de son emissivite. Ann. Geophys. 22, No. 4. 5. Ivanov, A. I. (1967). Spectrophotometric investigations of the intensity of scattered light. Trans. Astrophys. Znst. Acad. Sci. Kazakh SSR. 8. 6. Fessenkov, V. G. (1967). On the sounding of the atmosphere optical properties by means of satellites. J. Asfron. 44, No. 1. 7. Perelman, A. Ya. (1967). Calculation of the density function of the size distribution of particles on the basis of the spectral and angular characteristics of a dispersive system. Proc. Acad. Sci. USSR, Ser. Phys. Atmosphere Ocean 3, No. 4. 8. Livshitz, G. Sh., and Tashenov, B. Y. (1967). Intensity of scattered radiation at small angular distances from the Sun and the size spectrum of aerosols. Proc. Acad. Sci. USSR, Ser. Phys. Atmosphere Ocean 3, No. 6. 9. Gorchakov, G. I., and Rosenberg, G. V. (1967). Correlation in the optical characteristics of atmospheric hazes. Proc. Acad. Sci. USSR, Ser. Phys. Atmosphere Ocean 3, No. 6. 10. Rosenberg, G. V., and Gorchakov, G. I. (1967). Degree of polarization ellipticity of light scattered by atmospheric air as a tool for the investigation of the aerosol microstructure. Proc. Acad. Sci. USSR, Ser. Phys. Atmosphere Ocean 3, No. 7. 11. Fried, D. L., Mevers, G. E., and Keister, M. P., Jr. (1967). Measurements of laser beam scintillation in the atmosphere. J. Opt. Soc. Am. 57, No. 6. 12. Buck, A. L. (1967). Effects of the atmosphere on laser beam propagation. Appl. Opt. 6, No. 4. 13. Rosenberg, G. V. (1967). Atmospheric aerosol properties according to the optical investigation data. Proc. Acad. Sci. USSR, Ser. Phys. Atmosphere Ocean No. 9. 14. Kozlyaninov, M. V., and Semenchenko, I. V. (1967). On the determination of the vertical attenuation and absorption indexes on the basis of aircraft measurement of the sea spectral brightness coefficients. Proc. Acad. Sciences USSR, Ser. Phys. Atmosphere Ocean 3, No. 10. 15. Kattawar, G. W., and Plass, G. N. (1967). Electromagnetic scattering from absorbing spheres. Appl. Opt. 6, No. 8. 16. Smith, R. C., and Tyler, J. E. (1967). Optical properties of clear natural water. J. Opt. SOC.Am. 57, No. 5 . 17. Eldridge, R. G. (1967). A comparison of computed and experimental spectral transmissions through haze. AppI. Opt. 6, No. 5 . 18. Holland, A. C., and Draper, J. S. (1967). Analytical and experimental investigation of light scattering from polydispersions of Mie particles. Appl. Opt. 6, No. 3. 19. Hopfield, R. F. (1967). A comment on the scattering of coherent light. AppI. Opt. 6, No. 1. 20. Owens, J. C. (1967). Optical refractive index of air: dependence on pressure, temperature and composition. Appl. Opt. 6, No. 1. 21. Danzer, K. H., and Bullrich, K. (1966/67). Calculations of the influence of aerosol particles with certain intervals of radii on the scattering coefficient and the sky radiance. Optik 24. 22. Barteneva, 0. D., Dovgyallo, Ye. N., and Polyanova, Ye. A. (1967). Experimental investigation of the optical properties of the surface layer of the atmosphere. Trans. Main Geophys. Observatory, No. 220.

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23. Livshitz, G. Sh., Pavlov, V. Ye., and Milyutin, S. N. (1966). On light absorption by atmospheric aerosols. Trans. Astro-Phys. Znst. Acad. Sci. Kazakh SSR 7. 24. Pavlov, V. Ye. (1966). On the scattering function of the atmospheric haze. Trans. Astro-Phys. Znst. Acad. Sci. Kazakh SSR 7. 25. Shifrin, K. S., and Chayanova, E. A. (1966). Determination of the particle spectrum from the scattering function. Proc. Acad. Sci. USSR, Ser. Phys. Atmosphere Ocean No. 2. 26. Zander, R. (1966). Spectral scattering properties of ice clouds and hoarfrost. J. Geophys. Res. 71, No. 2. 27. Guermoguenova, 0. A. (1966). Influence of the electrostatic interaction upon the scattering of electromagnetic waves by atmospheric aerosols. Proc. Acad. Sci. USSR, Ser. Phys. Atmosphere Ocean No. 3. 28. Timofeyeva, V. A., and Kovetnikova, L. A. (1966). On the experimental determination of the turbid media extinction coefficient. Proc. Acad. Sci. USSR, Ser. Phys. Atmosphere Ocean No. 3. 29. Plass, G. N. (1966). Mie scattering and absorption cross sections for absorbing particles. Appl. Opt. 5 , No. 2. 30. Migeotte, M. (1965). La propagation de faisceaux lasers dans l’atmosphere terrestre. Rev. HF 6, No. 7. 31. Herman, B. M., and Yarger, D. N. (1965). The effect of absorption on a Rayleigh atmosphere. J. Atm. Sci. 22, No. 6. 32. Edwards, B. N., and Steen, R. R. (1965). Effects of atmospheric turbulence on the transmission of visible and near infrared radiation. Appl. Opt. 4. 33. Twomey, S., Jacoboitz, H., and Howell, H. B. (1966). Matrix methods for multiplescattering problems. J. Atm. Sci. 23, No. 3. 34. Gorchakov, G. I. (1966). Matrices of light scattering by ground air. Proc. Acad. Sci. USSR, Ser. Phys. Atmosphere Ocean 2, No. 6. 35. Perelman, A. Ya., and Shifrin, K. S. (1966). Calculation of the optical characteristics of the dispersive systems with the narrow distribution. Proc. Acad. Sci. USSR, Ser. Phys. Atmosphere Ocean 2, No. 6. 36. Gibson, F. W. (1966). Some applications of the laser as an atmospheric probe. Proc. ConJ Aerospace Meteorol,, Los Angeles, March 28-31, 1966. 37. Collis, R. T. H., and Ligda, M. G. H. (1966). Note on lidar observations of particulate matter in the stratosphere. J. Atrn. Sci. 23, No. 2. 38. Harris, E. D., Nugent, L. J., and Cato, G. A. (1965). Laser meteorological radar study. Electro-Opt. Systems, Inc., Pasadena, California. 39. Fioccio, G., and Grams, G. (1964). Observations of the upper atmosphere by optical radar in Alaska and Sweden during the summer 1964. Tellus 18, No. 1. 40. Northend, C. A., Honey, R. C., and Evans, W. E. (1966). Laser radar (lidar) for meteorological observations. Rev. Sci. Znstr. 37, No. 4. 41. Collis, R. T. H. (1966). Use of lidar in atmospheric resezrch. J. Spacecraft and Rockets 3, No. 4. 42. Masterson, J. E., Karney, J. L., and Hoehne, W. E. (1966). The laser as an operational meteorological tool. Bull. Am. Meteorol. SOC.47, No. 9. 43. Newkirk, G., Jr., and Kroening, J. L. (1965). Aerosols in the stratosphere: a comparison of techniques of estimating their concentration. J. Atm. Sci. 22, No. 5 .

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1963. Aerospace Corp., Tokyo. 45. Eiden, R. (1966). The elliptical polarization of light scattered by a volume of atmospheric air. Appl. Opt. 5, No. 4. 46. Blau, H. H., Jr., Espinola, R. P., and Reifenstein, E. C. (1966). Near infrared scattering by sunlit terrestrial clouds. Appl. Phys. 5, No. 4. 47. Leupolt, A. (1966). Bestimmung der Kontinuumabsorption im Spektralbereich von 0.5 bis 2.5 micron. Teil 11. Optik 23, No. 7. 48. Foitzik, L., Hebermehl, G., and Spankuch, D. (1965/66). uber die spektrale Extinktion und die spektrale Streuung von Mie-Partikeln bei Vorliegen logarithmischer Gauss-Verteilungen. Optik 23, No. 3. 49. Dietze, G., and Seidel, H. (1967). Uber die Sondierung stratospharischer und mesospharischer Staubschichted mit Laserimpulsen. 2. Meteorol. 19, No. 7/8. 50. Carrier, L. W., Cato, G. A., and von Essen, K. J. (1967). The backscattering and extinction of visible and infrared radiation by selected major cloud models. Appl. Opt. 6, No. 7. 51. Gurvich, A. S. (1968). Determination of turbulence characteristics from light propagation experiments. Proc. Acad. Sci. USSR, Ser. Phys. Atmosphere Ocean 4, No. 2. 52. Chayanova, E. A., and Shifrin, K. S. (1968). On the scattering function of the surface layer. Proc. Acad. Sci. USSR, Ser. Phys. Atmosphere Ocean 22, No. 2. 53. Knestrick, G. L., and Curcio, J. A. (1967). Atmospheric propagation of laser and nonlaser light. Appl. Opt. 6, No. 8. 54. Kronke, R. H., and Raymond, F. W. (1967). Earth radiance and reflectivities relative to the response of a silicon detector. Appl. Opt. 6, No. 12. 55. Kozyrev, B. P., and Mezenov, A. V. (1966). Radiation attenuation in the wave length band of 0.5-200 micron due to water drops suspended in the air. Proc. Leningrad Electrotechn. Znst. No. 55. 56. Zuyev, V. Ye., Pokasov, V. V., Pkhalagov, Yu. A., Sosnin, A. V., and Khmelevtsev, S. S. (1968). Transparency of the surface atmospheric layer for emission of some laser sources. Proc. Acad. Sci. USSR, Ser. Phys. Atmosphere Ocean No. 1. 57. Mikirov, A. Ye. (1967). Average size of particles at heights 70-450 km. Geomagn. Aeronomy No. 4. 58. Ivanov, A. P. (1968). New express method for investigating the optical properties of the atmosphere and ocean. Proc. Acad. Sci. USSR, Ser. Phys. Atmosphere Ocean 4, No. 2. 59. Pyldmaa, V. K., and Rosenberg, G. V. (1966). Some results of atmospheric twilight sounding and investigating its possibilities. (1966). Proc. Acad. Sci. USSR, Ser. Phys. Atmosphere Ocean 2, No. 8. 60. Shifrin, K. S., and Kolmakov, I. B. (1966). Influence of limiting the scattering function measurement interval on the small angle method accuracy. Proc. Acad. Sci. USSR, Ser. Phys. Atmosphere Ocean 2, No. 8. 61. Shifrin, K. S. (1966). Essential region of scattering angles at the measurement of particle size distribution by the method of small angles. Proc. Acad. Sci. USSR, Ser. Phys. Atmosphere Ocean 2, No. 9. 62. Heger, K. (1966). Die von der triiben Atmosphare nach aussen gestreute Strahlung 11. Ergebnisse numerischer Auswertung. Beitr. Phys. Atmosphare 39, No. 1

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63. Heger, K. (1965). Die von der triiben Atmosphsre nach aussen gestreute Strahlung I. Theoretische Grundlagen. Beitr. Phys. Atmosphiire 38, No. 314. 64. Sekera, Z. (1966). Recent developments in the theory of radiative transfer in planetary atmosphere. Rev. Geophys. 4, No. 1. 65. Livingston, P. M. (1966). Multiple scattering of light in a turbulent atmosphere. J. Opt. Soc. Am. 56, No. 12. 66. Strobehn, J. W. (1966). The feasibility of laser experiment for measuring atmospheric turbulence parameters. J. Ceophys. Res. 71, No. 24. 67. Guermoguenova, T. A., and Zege, E. P. (1967). Solution of the transfer equation taking account of the dependence of the substance absorptivity upon the radiation density. Proc. Acad. Sci. USSR, Ser. Phys. Atmosphere Ocean 3, No. 2. 68. De Bary, E. (1965). A method of evaluation of measured light intensities and degrees of polarization of a unit volume of turbid air. Geophys. Pura AppI. 62. 69. Bryant, H. C., and Cox, A. (1966). Mie theory and the glory. J. Opt. SOC.Am. 56, No. 11. 70. Twomey, S., Jacobowitz, H., and Howell, H. B. (1967). Light scattering by cloud layers. J. Atm. Sci. 24, No. 1. 71. Brinkmann, R. T., Green, A. E. S., and Barth, C. A. (1967). Atmospheric scattering of the solar flux in the middle ultraviolet. Appl. Opt. 6, No. 3. 72. Spankuch, D. (1967). Helligkeitsmessungen eines senkrechten Scheinwerferstrahls zur Bestimmung der vertikalen Triibungsschichtung der Troposphare. Teil I: Methodik und Apparatur. Teil 11: Messergebnisse. Beitr. Phys. Atmosphare 40, No. 1/2. 73. Divari, N. B. (1966). Cosmic dust cloud round the Earth. Aeron. J. 43, No. 6. 74. Miller, S. I., and Tillotson, L. C. (1966). Optical transmission research. Appl. Opt. 5, No. 10. 75. Dobbins, R. A., and Jizmagian, G. S. (1966). Particle size measurements based on use of mean scattering cross sections. J. Opt. Soe. Am. 56, No. 10. 76. Bolsenga, S. J. (1967). Near infrared radiation in northern Greenland. J. Appl. Meteorol. 6, No. 2. 77. Driving, A. Ya., Mikhailin, I. M., and Rosenberg, G. V. (1967). Polarizational functions of light scattering by the surface layer of atmospheric air. Proc. Acad. Sci. USSR, Ser. Phys. Atmosphere Ocean 3, No. 8. 78. Zuyev, V. Ye., Kabanov, M. V., and Savelyev, B. A. (1967). Limits of Buger’s law applicability in scattering media for the collimated light beams. Proc. Acad. Sci. USSR, Ser. Phys. Atmosphere Ocean 3, No. 7. 79. Kondratyev, K. Ya., and Timofeyev, Yu. M. (1967). Direct methods of the calculation of the transmission functions of atmospheric gases. Proc. Acad. Sci. USSR, Ser. Phys. Atmosphere Ocean 3, No. 2.

CHAPTER 4 1. Gray, L. D. (1967). Calculations of carbon dioxide transmission, Part I: the 9.410.4 micron bands. J. Quant. Spectrosc. and Radiat. Transfer, 7, No. 1. 2. Armstrong, B. H. (1967). Spectrum line profiles: the Voigt function. J. Quant. Spectr. Radiative Transfer 7, No. 1. 3. Wolk, M. (1967). The strength of the pressure-broadened CO, bands at 15 microns by digital integration of spectra. J. Spectr. Radiative Transfer 7 , No. 1.

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4. Yamamoto, G. (1967). Transmission due to overlapping lines. J. Quant. Spectr. Radiative Transfer 7 , No. 1. 5. Anderson, A., An-Ti, Chai, and Williams, D. (1967). Self-broadening effects in the infrared bands of gases. J. Opt. SOC.Am. 57, No. 2. 6. McCaa, D. J., and Shaw, J. H. (1967). The infrared absorption bands of ozone. The Ohio. State Univ., Rept. No. AFCRL-67-0237. 7. Eldridge, R. G. (1967). Water vapor absorption of visible and near infrared radiation. Appl. Opt. 6, No. 4. 8. Kyle, T. G. (1967). Absorption of radiation by uniformly spaced Doppler lines. Astrophys. J. 148. 9. Bolle, H.-J. (1966). Fine structure calculations of Martian CO, emission spectrum, in “Moon and Planets.” North-Holland, Amsterdam. 10. Bolle, H.-J. (1965). Stratospheric water vapor determination by absorption in single lines. Beitr. Phys. Atmosphure 38, No. 1. 11. Irvine, W. M., and Pollack, J. B. (1966). Infrared optical properties of water and ice spheres. Harvard College Obs. and Smithsonian Astrophys. Obs. 12. Quenzel, H. (1966). Ein Interferenzfilter-Actinograph zur Optischen Bestimmung der Atmospharischen Gesamtwasserdampfgehaltes. Beitr. Physik Atmosphure 39, NO. 2-4. 13. Quenzel, H. (1966). Tagesgange des atmospharischen Gesamtwasserdampfgehaltes nach Messungen mit einem Interferenzfilter-Aktinographen. Beitr. Phys. Atmosphure 39, NO. 2-4. 14. Kisseleva, M. S., and Neporent, B. S. (1965). Humidity measurement of gaseous mixtures according to the infrared absorption spectra. Opt. Spectr. (USSR) (English Transl.) 19, No. 6. 15. Kisseleva, M. S., Neporent, B. S., and Fedorova, E. 0. (1967). Infrared radiation absorption with the unresolved spectrum structure for slant paths in the atmosphere (H,O and CO, effect). Proc. Acad. Sci. USSR, Ser. Phys. Atmosphere Ocean 3, No. 6. 16. Zolotarev, B. M. (1967). Absorption absolute intensities and optical constants of light (H,O) and heavy (D,O) water in the spectral region 4000-1000cm-1. Opt. Spectr. (USSR) (English Trans!.) 23, No. 5. 17. Kondratyev, K. Ya., and Timofeyev, Yu. M. (1967). On the applicability of the approximative methods for atmospheric nonhomogeneity calculation and the calculation of the transmission function of the water vapor rotational band. Proc. Acad. Sci. USSR, Ser. Phys. Atmosphere Ocean, 3, No. 2. 18. Dave, J. V., and Furukawa, P. M. (1967). The effect of scattering and ground reflection on the solar energy absorbed by ozone in a Rayleigh atmosphere. J. Atm. Sci. 24, No. 2. 19. Draegert, D. A., Stone, N. W. B., Curnutte, B., and Williams, D. (1966). Far infrared spectrum of liquid water. J. Opt. SOC.Am. 56, No. 1. 20. Bertram, F.-W. (1965). Bestimmung der bodennahen Wasserdampfgehaltes der Luft und seines vertikalen Gradienten aus Absorptionsmessungen in nahen Infrarot. Beitr. Phys. Atmosphare 38, No. 314. 21. Farmer, C. B., and Houghton, J. T. (1966). Collision-induced absorption in the Earth’s atmosphere. Nature 209, No. 5030. 22. King, J. I. F. (1965). Modulated band-absorption model. J. Opt. SOC. Am. 55, No. 11.

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23. Tillman, J. E. (1965). Water vapor density measurements utilizing the absorption of vacuum ultraviolet and infrared radiation. In “Humidity and Moisture: Measurement and Control in Science and Industry”, Vol. 1. Holt, New York; Chapman & Hall, London. 24. Plyler, E. K., and Griff, N. (1965). Absolute absorption coefficients of liquid water at 2.95 microns, 4.7 microns and 6.1 microns. Appl. Opt. 4, No. 12. 25. Berry, D. J., Farmer, C. B., and Lloyd, D. B. (1965). Atmospheric transmission measurements in the 4.3 microns CO, band at 5200 m altitude. Appl. Opt. 4, No. 9. 26. Kyle, T. G., Murcray, D. G., Murcray, F. H., and Williams, W. J. (1965). Absorption of solar radiation by atmospheric carbon dioxide. J. Opt. SOC. Am. 55, No. 11. 27. Murcray, D. G., Murcray, F. H., and Williams, W. J. (1965). Comparison of experimental and theoretical slant path absorptions in the region from 1400 to 2500 cm-l. J. Opt. SOC.Am. 55, No. 10. 28. Calfee, R. F., and Gates, D. M. (1966). Calculated slant path absorption and distribution of atmospheric water vapor. Appl. Opt. 5 , No. 2. 29. Stauffer, F. R., and Walsh, T. E. (1966). Transmittance of water vapor - 14 to 20 microns. J. Opt. SOC.Am. 56, No. 3. 30. Georguiyevsky, Yu. S. (1966). Installation for the investigation of the atmospheric spectral transparency with high resolution. Proc. Acad. Sci. USSR, Ser. Atmosphere Ocean 2, No. 1. 31. Rodgers, C. D. (1967). Some extensions and applications of the new random model for molecular band transmission. (Private communication.) 32. Furashov, N. I. (1966). Investigation of the longwave infrared radiation absorption by atmospheric water vapor. Opt. Spectr. (USSR) (English Trans/.) 20, No. 3. 33. Drayson, S. R. (1964). Atmospheric slant path transmission in the 15 micron CO, band. Technol. Rept. ORA Project 05863, Ann Arbor. 34. Bazhenov, V. A., Ivanova, R. N., and Miroshnikov, M. M. (1966). Determination of H,O, CO, and 0, in various atmospheric thicknesses. Proc. Acad. Sci. USSR, Ser. Phys. Atmosphere Ocean No 3. 35. Carlton, H. R. (1966). Humidity effects in the 8-13 micron infrared window. Appl. Opt. 5, No. 5. 36. Golubitsky, B. M., and Moskalenko, N. I. (1968). Measurement of spectral absorption in water vapour bands. Proc. Acad. Sci. USSR, Ser. Phys. Atmosphere Ocean 4, No. 1. 37. Golubitsky, B. M., and Moskalenko, N. I. (1968). Measurements of the CO, spectral absorption in the conditions of the artificial atmosphere. Proc. Acad. Sci. USSR, Phys. Ser. Atmosphere Ocean 4, No. 1. 38. Naumov, A. P. (1968). On the method of determining the water content of the atmosphere at the radio waves absorption measurements near 1 = 1.35 cm. Proc. Acad. Sci. USSR, Ser. Phys. Atmosphere Ocean 4, No. 2. 39. Tubbs, L. D., Hathaway, C. E., and Williams, D. (1967). Further studies of overlapping absorption bands. Appl. Opt. 6, No. 8. 40. Kislyanov, A. G., Nikonov, B. N., and Strezhneva, K. M. (1968). Experimental investigation of the atmospheric absorption dependency at the wave of 4.1 mm from the height above sea level. Proc. Acad. Sci. USSR. Ser. Phys. Atmosphere Ocean 4. No. 3.

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41. Golubitsky, B. M., and Moskalenko, N. I. (1968). Spectral absorption functions in the H,O and CO, vapor bands. Proc. Acad. Sci. USSR, Ser. Phys. Atmosphere Ocean 4, No. 3. 42. Golubitsky, B. M., and Moskalenko, N. I. (1968). Spectral transmission measurement and calculation in the N,O bands in the near infrared region. Proc. Acad. Sci. USSR, Ser. Phys. Atmosphere Ocean 4, No. 3. 43. Burch, D. E., Gryvnak, D. A., and Patty, R. R. (1967). Absorption of infrared radiation by CO, and H,O. Experimental techniques. J. Opt. SOC.Am. 57, No. 7. 44. Sanderson, R. B., and Ginsburg, N. (1963). Line widths and line strengths in the rotational spectrum of water vapor. J. Quant. Spectr. Radiative Transfer 3, 435. 45. Buznikov, A. A., and Kozyrev, B. P. (1965). Investigation of the absorption by the atmosphere of the radiation of a lightly heated absolutely black radiator. Eng.-Phys. J. 9, No. 1. 46. Mireles, R. (1966). Determination of parameters in absorption spectra by numerical minimization techniques. J. Opt. SOC.Am. 56, No. 5 . 47. Kostyanoy, G. N. (1966). On the approximative expression of the transmission function for the diffuse radiation. Trans. Central Aerolog. Obs. No. 70. 48. Yurguenson, A. P. (1966). On the spectral absorption of longwave radiation in the atmosphere. Trans. Arctic Res. Inst. 277, 11. 49. Goldman, A., and Oppenheim, U. P. (1966). Integrated intensity of the 6.3 micron band of water vapor. Appl. Opt. 5 , No. 6. 50. Beritashvili, B. Sh., Brounshtein, A. M., and Kazakova, K. P. (1966). On the dependence of the integral transmission function of the atmosphere upon the temperature of black radiation. Trans. Main Geophys. Obs. No. 184. 51. Shekhter, F. N. (1966). Spectral and integral transmission functions of longwave radiation. Trans. Main Geophys. 06s. No. 184. 52. Zander, R. (1966). Moisture contamination at altitude by balloon and associated equipment. J. Geophys. Res. 71, No. 15. 53. Saiedy, F., Jacobowitz, H., and Wark, D. Q. (1967). On cloud-top determination from Gemini-5. J. Atm. Sci. 24, No. 1. 54. Simmons, F. S. (1966). Band models for nonisothermal radiating gases. Appl. Opt. 5 , No. 11. 55. Kaplan, L. D. (1966). The absorption of solar radiation by COz. Probl. Mereorol. Stratosphere Mesosphere (Paris), 307. 56. Clough, S. A., and Kneizys, F. X. (1965). Ozone absorption in the 9.0 micron region. Rept. AFCRL-65-862, Phys. Sci. Res. Papers, No. 170. 57. Clough, S. A., and Kneizys, F. X. (1966). Coriolis interaction in the v1 and vg fundamentals of ozone. J. Chem. Phys. 44, No. 5 . 58. Lane, J. A. (1966). Far infrared spectrum of liquld water. J. Opt. SOC.Am. 56, No. 10. 59. Dave, J. V., and Mateer, C. L. (1967). A preliminary study on the possibility of estimating total atmospheric ozone from satellite measurements. J. Arm. Sci. 24, No. 4. 60. Hoover, G. M., Hathaway, Ch. E., and Williams, D. (1967). Infrared absorption by overlapping bands of atmospheric gases. Appl. Opt. 6, No. 3. 61. Robinson, G. D. (1966). Some determinations of atmospheric absorption by measurement of solar radiation from aircraft and at the surface. Quart. J. Roy. Meteorol. Soc. 92, No. 392.

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62. Zuyev, V. Ye., Lopasov, V. P., and Sonchik, V. K. (1967). Experimental investigations of water refraction complex index in the spectral region of 2.5-25 microns. Proc. Acad. Sci. USSR, Ser. Phys. Atmosphere Ocean 3, No. 1. 63. Seeger, W. (1966). Linienabsorption des Wasserdampfes der hohen Atmosphare. Illeteorol. Rundschau 19, No. 5 . 64. Hall, J. T. (1967). Attenuation of millimeter wavelength radiation by gaseous water. Appl. Opt. 6, No. 8. 65. Malkmus, W. (1967). Random Lorentz band model with exponential-tailed S-' lineintensity distribution function. J. Opt. SOC.Am. 57, 323. 66. Zhevakin, S. A., and Naumov, A. P. (1965). Some problems connected with the calculation and measurement of the absorption of the millimeter and submillimeter radio waves in the atmospheric water vapour. Proc. Higher Educ. Inst., Radiophys. 8, No. 6. 67. Dianov-Klokov, V. I., and Malkov, I. P. (1966). On the absorption band intensification during the passage of light through a cloud layer. Proc. Acad. Sci. USSR, Ser. Phys. Atmosphere Ocean 2, No. 5 . 68. Rodgers, C. D. (1968). Some extensions and applications of the new random model for molecular band transmission. Quart. J. Roy. MeteoroI. SOC.94, No. 399.

CHAPTER 5 1. Pivovarova, 2. I. (1967). Direct solar radiation income upon the walls of buildings. Trans. Main Geophys. Obs. No. 193.

2. Pivovarova, 2. I. (1967). Solar radiation variations according to the data of surface measurements. Trans. Main Geophys. Obs. No. 193. 3. Budyko, M. I., and Pivovarova, 2. I. (1967). Influence of eruptions upon the solar radiation income on the Earth's surface. Meteorol. Hydrol. No. 10. 4. Kondo, J. (1967). Analysis of solar radiation and downward longwave radiation data in Japan. Sci. Rept. Tohoku Univ., Ser. 5 , Geophys. 18, No. 3. 5. Dyer, A. J., and Hicks, B. B. (1965). Stratospheric transport of volcanic dust inferred from solar radiation measurements. Nature 208, No. 5006. 6. Angstrom, A., and Rodhe, B. (1966). Pyrheliometric measurements with special regard to the circumsolar sky radiation. Tellus 18, No. 1 . 7. Angstrom, A. K., and Drummond, A. J. (1966). Notes on solar radiation in mountain regions at high altitude. Tellus 18, No. 4. 8. Bishop, B. C., Angstrom, A. K., Drummond, A. J., and Roche, J. J. Solar radiation measurements in the high Himalayas (Everest region). (1966). J. Appl. Meteorol. 5, No. 1 . 9. Drummond, A. J., Hickey, J. R., Scholes, W. J., and Laue, E. G. (1967). Multichannel radiometer measurement of solar irradiance. J. Spacecraji Rockets, 4, No. 9. 10. Marr, G . V. (1965). The penetration of solar radiation into the atmosphere. Proc. Roy. SOC.Ser. A . 288, 531. 11. Norris, D. J., (1966). Solar radiation on inclined surfaces. Solar Energy, No. 2. 12. Occhipinti, A. G . (1966). Analysis of the empirical relations between visible solar radiation, the solar altitude and transparency of the atmosphere. Univ. San Paulo, Brazil (Rept).

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13. Hinterregger, H. E. (1965). Absolute intensity measurements in the extreme ultraviolet spectrum of solar radiation. Space Sci. Rev. 4, No. 4. 14. Searle, N. Z., and Hirt, R. C. (1965). Ultraviolet spectral energy distribution of sunlight. J. Opt. Soc. Am. 55, No. 11. 15. Belinsky, V. A., and Semenchenko, B. A. (1966). Ultraviolet radiation at different heights with cloudless aerosol atmosphere. Trans. Main Geophys. Obs. No. 184. 16. Lof, G. 0. B., Duffie, J. A., and Smith, C. 0. (1966). World distribution of solar radiation. Solar Energy 10, No. 1. 17. Gates, D. M. (1966). Spectral distribution of solar radiation at the earth’s surface. Science 151, No. 3710. 18. Raghavan, S., and Yadar, B. R. (1966). Depletion of solar radiation by particulate matter in the atmosphere - a study with special reference to New Delhi. Indian J. Meteorol. Geophys. 17, No. 1. 19. Bossolasco, M., Cicconi, G., Dagnino, I., Elena, A., and Flocchini, G. (1965). Solar constant and sunspots. Geophys. Pura Appl. 62. 20. Kondratyev, K. Ya., Nikolsky, G. A., Andreyev, S. D., and Badinov, I. Ya. (1967). Direct solar radiation up to the height of 30 km and stratification of the extinction component in the stratosphere. Appl. Opt. 6, No. 2.

CHAPTER 6 1. Plass, D. (1967). Eine photographische Methode zur Messung der Intensitatsverteilung der Himmelstrahlung mit einem Kugelspiegel. Optik 25, No. 2. 2. Plass, G. N., and Kattawar, G. W. (1967). Calculations of reflected and transmitted radiance for Earth’s atmosphere. Contr. No. AF 19 (628) - 5039, Sci. Rept. No. 6, Southwest Center for Advanced Studies, Dallas, Texas. 3. Kattawar, G. W., and Plass, G. N. (1967). Radiance and polarization of multiple scattered light from haze and clouds. Contr. No. A F 19 (628) - 5039, Sci. Rept. No. 7, Southwest Center for Advanced Studies, Dallas, Texas. 4. Plass, G. N., and Kattawar, G. W. (1968). Influence of single scattering albedo on reflected and transmitted light from clouds. Appl. Opt. 7, No. 2. 5. Bullrich, K., Eiden, R., and Nowak, W. (1966). Sky radiation, polarization and twilight radiation in Greenland. Geophys. Pura Appl. 64, 220. 6. Bullrich, K., Blattner, W., Conley, T., Eiden, R., Hanel, G., Heger, K., and Nowak W. (1968). Research on atmospheric optical radiation transmission 1 December 1966-30 November 1967. Contr. F 61052-67-C 0046, Mainz/Johannes-GutenbergUniversitat. 7. Feigelson, Ye. M. (1967). Interaction between radiation and cloudiness. In “Dynamics of Large Scale Atmospheric Processes.” Nauka, Moscow. 8. Sekera, Z. (1966). Inversion methods. NCAR Techn. Notes No. 11. 9. Sekera, Z. (1967). Determination of atmospheric parameters from measurement of polarization of upward radiation by satellite or space probe. Zcarus 6, No. 3. 10. Drobyshevich, V. I. (1967). Connection between the statistical characteristics of the cloudiness fields and the shortwave outgoing radiation. Proc. Acad. Sci. USSR, Ser. Phys. Atmosphere Ocean 3, No. 10. 11. Hudson, R. D., Elliot, D. D., Clark, M. A., and Chater, W. T. (1967). Satellite

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measurements of the Earth’s radiance from 1600 to 3200 Angstroms. Trans. Am. Geophys. Union 48, No. 1. 12. Marmo, F. F., Engelman, A., and Miranda, H. A. (1967). Ultraviolet satellite observations of noctilucent clouds. Trans. Am. Geophys. Union 48, No. 1. 13. Dave, J. V., and Mateer, C. L. (1966). Determination of ozone parameters from measurements of the back-scattered radiation. NCAR Techn. Notes No. 11. 14. Dave, J. V., and Furukawa, P. M. (1966). Intensity and polarization of the radiation emerging from an optically thick Rayleigh atmosphere. J. Opt. Soc. Am. 56, No. 3 . 15. Minin, I. N. (1967). Optical model of the Mars’s atmosphere. Astron. J. 44, No. 6. 16. Henderson, S. T., and Hodgkins, D. (1963). The spectral energy distribution of daylight. Brit. J. Appl. Phys. 14. 17. Fraser, R. S. (1966). Effect of specular ground reflection on radiation leaving top of a planetary atmosphere. In “Thermophysics and Temperature Control of Spacecraft and Entry Vehicles.” Academic Press, New York. 18. Marchuk, G. I., and Mikhailov, G. A. (1967). On the solution of the problems of atmospheric optics by the Monte-Carlo method. Proc. Acad. Sci. USSR, Ser. Phys. Atmosphere Ocean 3 , No. 3 . 19. Marchuk, G. I., and Mikhailov, G. A. (1967). Results of the solution of some problems of atmospheric optics by the Monte-Carlo method. Proc. Acad. Sci. USSR, Ser. Phys. Atmosphere Ocean 3, No. 4. 20. Sholokhova, Ye. D., Fedorova, E. O., Lobanova, G. I., and Bartkievich, V. S. (1967). Balloon measurements of the brightness of a day cloudless sky (the 1-6 micron region). Proc. Acad. Sci. USSR, Ser. Phys. Atmosphere Ocean 3, No. 4. 21. Smoktiy, 0. I. (1967). Multiple light scattering in the homogeneous sphericallysymmetrical planetary atmosphere. Proc. Acad. Sci. USSR, Ser. Phys. Atmosphere Ocean 3, No. 3. 22. Smoktiy, 0. I. (1967). On the brightness determination of the non-homogeneous spherically-symmetrical planetary atmosphere. Proc. Acad. Sci. USSR, Ser. Phys. Atmosphere Ocean 3, No. 4. 23. Lloyd, J. W. F., Silverman, S., Nardone, L. J., and Cochrun, B. L. (1965). Day skylight intensity from 20 km to 90 km at 5500 A. Appl. Opt. 4, No. 12. 24. Krasnopolsky, V. A,, Kuznetsov, A. P., and Lebedinsky, A. I. (1966). Ultraviolet spectrum of the Earth according to the measurements from the KOSMOS-65 satellite. Geomagnetizm i Aeronomiya 6, No. 2. 25. Hovis, W. A., Jr., and Tobin, M. (1967). Spectral measurements from 1.6 to 5.4 microns of natural surfaces and clouds. Appl. Opt. 6, No. 8 . 26. Wolff, M. (1967). Precision limb profiles for navigation and research. J. Spacecraft Rockets 4, No. 8. 27. Harihara Ayyar, P. S. (1966). Hemispherical transport of volcanic dust inferred from diffuse radiation measurements. Indian J. Meteoroi. Geophys. 17, No. 4. 28. Ivanov, A. I. (1967). Spectrophotometric investigations of the scattered light intensity. Probl. Astrophys. Arm. Opt., Trans. Astrophys. Inst. Acad. Sci. Kazakh SSR 8. 29. Snoddy, W. C. (1966). Irradiation above atmosphere due to Rayleigh scattering and diffuse terrestrial reflection. In “Thermophysics and Temperature Control of Spacecraft and Entry Vehicles.” Academic Press, New York. 30. Kondratyev, K. Ya. (1967). Meteorological investigations from manned space vehicles. Proc. 17th Internatl. Astronaut. Federation Congress, Warsaw. Polish Scientific Publishers, Warsaw.

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CHAPTER 7 1. Borhidi, A., and Dobosi, Z. (1967). Spatial distribution of the surface’s albedo over the territory of Hungary. Zdcjaras No. 3. 2. Mukhenbern, V. V. (1967). Albedo of the globe’s surface. Trans. Main Geophys. Obs. No. 193. 3. Korzov, V. I., and Krassilshchikov, L. B. (1967). Aircraft measurements of the scattering functions of the underlying surface’s brightness. Trans. Main Geophys. Obs. No. 203. 4. Kropotkin, M. A., Kozyrev, B. P., and Zaitsev, V. A. (1966). Infrared reflection spectra of sea and fresh water and some water solutions. Proc. Acad. Sci. USSR, Ser. Phys. Atmosphere Ocean 2, No. 4. 5. Pontier, L., and Dechambenoy, C. (1965). Mesure du pouvoir reflecteur monochromatique de I’eau sous incidence normale entre 1 et 38 microns. Ann. Geophys. 21, No. 3. 6. Coulson, K. L., Bouricius, G. M., and Gray, E. L. (1965). Optical reflection properties of natural surfaces. J. Geophys. Res. 70, No. 18. 7. Kozo, H., and Motoaki, K. (1965). On the albedo of radiation of the sea surface. J. Oceanogr. SOC.Jap. 21, No. 4. 8. Kozo, H., and Motoaki, K. (1965). On the reflection of light by roughened water surface. J. Oceanogr. Soc. Jap. 21, No. 4. 9. Barry, R. G., and Chambers, R. E. (1966). A preliminary map of summer albedo over England and Wales. Quart. J. Roy. Meteorol. SOC.92, No. 394. 10. Gayevsky, V. L., Rabinovich, Yu.I., and Reshetnikov, A. I. (1966). Some results of measuring the albedo of the stratus in the spectral region of 8-12 microns. Trans. Main Geophys. Obs. No. 196. 11. Chapursky, L. I. (1966). Experimental investigations in the spectral brightness characteristics of clouds, atmosphere and the underlying surface in the wave length interval of 0.3-2.5 microns. Trans. Main Geophys. Obs. No. 196. 12. Chapursky, L. I. (1968). Absolute spectral brightness of clouds and the underlying surface in the near infrared spectral region. Trans. Arctic Antarctic Res. Znst. 284.

CHAPTER 8 1. Mukhenberg, V. V., and Strokina, L. A. (1967). Absorbed radiation distribution over continents and oceans. Trans. Main Geophys. Obs. No. 193. 2. Mukhenberg, V. V. (1967). Spatial structure of an absorbed radiation field. Trans. Main Geophys. Obs. No. 209. 3. Borzenkova, I. I. (1967). On some regularities of vertical geographical zonality. Trans. Main Geophys. Obs. No. 193. 4. Borzenkova, I. I. (1967). On the influence of local factors upon the radiation income in mountainous areas. Trans. Main Geophys. Obs. No. 209. 5. Antropova, U. I. (1967). Some results of the calculations of daily amounts of total radiation income to slopes and data on the albedo of snow. Proc. Tashkent Geophys. Obs. 2. 6. Sekihara, K., and Suzuki, M. (1967). Solar radiation and duration of sunshine in Japan. Papers MeteoroI. Geophys. (Tokyo) 17, No. 3.

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7. Hanson, K. J., Von der Haar, T. H., and Suomi, V. E. (1967). Reflection of sunlight to space and absorption by the earth and atmosphere over the United States during spring 1962. Monthly Weather Rev. 95, No. 6. 8. Pivovarova, 2. I. (1966). Map of total radiation and net radiation over the territory of the USSR for the period of IGY. Nauka, Moscow. 9. Pivovarova, 2. I. (1967). Evaluation of the total income of shortwave radiation upon the walls of buildings. Trans. Main Geophys. Obs. No. 209. 10. Hydrooptical investigations (collection of papers). Trans. Znst. Oceanol. 77. 11. Tyler, J. E. (1965). In situ spectroscopy in ocean and lake waters. Jo. Opt. SOC.Am. 55, No. 7. 12. Lijf G. 0. B., Duffie, J. A., and Smith, C. 0. (1966). World distribution of solar radiation. SoIar Energy 10, No. 1. 13. Ivanov, A. P. (1968). On the principles and methods of carrying out the hydrooptical investigations. Proc. Acad. Sci. USSR, Ser. Phys. Atmosphere Ocean 4, No. 3. 14. Goverdovsky, V. F., and Melkov, F. I. (1968). Statistical characteristics of signals from water surfaces and ice formations registered by aircraft actinometric equipment. Trans. Arctic Antarctic Res. Znst. 284. 15. Gavrilova, M. K., and Kan, R. V. (1965). Solar radiation heat income in Yakutia from the actual observations data. Bull. Yakutia Univ. 16. 16. Lebedev, A. N. (1966). On the total radiation variability in Africa. Trans. Main Ceophys. Obs. No. 192. 17. Light regime, photosynthesis and the productivity of the forest. (1967). Nauka, Moscow.

CHAPTER 9 1. Kuhn, P. M., and Johnson, D. R. (1966). Improved radiometersonde observations of atmospheric infrared irradiance. J. Geophys. Res., 71, No. 2. 2. Kuhn, P. M., and Kox, S. K. (1967). Radiometric inference of stratospheric water vapor. J. Appl. Meteorol. 6, No. 1. 3. Zhvalev, V. F. (1967). On the methods for calculating longwave radiation fluxes in the atmosphere by means of electronic computers. Trans. Main Geophys. Obs. No. 203. 4. Zhvalev, V. F., Kondratyev, K. Ya., and Ter-Markariantz, N. E. (1967). On the calculation of the spectral values of the outgoing radiation and contrasts between the radiation temperature of the underlying surface and that of clouds in connexion with the problem of cloudiness detection from satellites. Trans. Main Ceophys. Obs. No. 203. 5. Saiedy, F., and Hilleary, D. T. (1967). Remote sensing of surface and cloud temperatures using the 899 cm-1 interval. Appl. Opt. 6, No. 5 . 6. Winston, J. S. (1967). Zonal and meridional analysis of 5-day averaged outgoing longwave radiation data from TIROS-IV over the Pacific sector in relation to the northern hemisphere circulation. J. Appl. Meteorol. 6, No. 3. 7. Winston, J. S. (1967). Planetary-scale characteristics of monthly mean longwave radiation and albedo and some year-to-year variations. Monthly Weather Rev. 95,No. 5 . 8. Raschke, E. (1966). Tropospheric water vapor content and surface temperatures from TIROS IV radiation data. NASA Contr. Rep.-595, Washington, D.C. 9. Raschke, E. (1967). A quasi-global analysis of the mean relative humidity of the upper troposphere. Tellus 19, No. 2.

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10. Raschke, E., and Bandeen, W. R. (1967). A quasi-global analysis of tropospheric water vapor content from TIROS IV radiation data. J. Appl. Meteorol. 6, No. 3. 11. Raschke, E., and Bandeen, W. R. (1967). A quasi-global analysis of tropospheric water vapor content and its temporal variations from radiation data of the meteorological satellite TIROS IV. Space Res. 7/2. North.-Holland Publ., Amsterdam. 12. Bandeen, W. R. (1966). Atmospheric water vapor content from satellite radiation measurements. NCAR Tech. Notes, No. 11. 13. Lethbridge, M. (1967). Precipitation probability and satellite radiation data. Monthly Weather Rev. 95, No. 7. 14. Huang Ching-Hua, Panofsky, H. A., and Schwalb, A. (1967). Some relationships between synoptic variables and satellite radiation data. Monthly Weather Rev. 95, No. 7 . 15. Wark, D. Q. (1966). Inversion problems. NCAR Techn. Notes, No. 11. 16. Wark, D. Q., et at. (1966). Indirect measurements of atmospheric temperature profiles. I. Introduction. 11. Mathematical aspects of the inversion problem. 111. The spectrometer and experiments. Monthly Weather Rev. 94, No. 6. 17. James, D. G. (1967). Indirect measurements of atmospheric temperature profiles from satellites. IV. Experiments with the phase I satellite infrared spectrometer. Monthly Weather Rev. 95, No. 7. 18. Wolk, M., Van Cleef, F., and Yamamoto, G. (1967). Indirect measurements of atmospheric temperature profiles from satellites. V. Atmospheric soundings from infrared spectrometer at the ground. Monthly Weather Rev. 95, No. 7. 19. Wark, D. Q., Saiedy, F., and James, D. G. (1967). Indirect measurements of atmospheric temperature profiles from satellites. VI. High altitude balloon testing. Monthly Weather Rev. 95, No. 7. 20. Lienesh, J. H., and Wark, D. Q. (1967). Infrared limb darkening of the Earth from statistical analysis of TIROS data. J . Appl. Meteorol. 6, No. 4. 21. Heidemann, M. (1966). Studie iiber die Verwendung von Fabry-Perot-Interferometern zur Auflosung des infraroten Wasserdampfspektrums in der hohen Atmosphare. Forschungsbericht W-60-02, Miinchen. 22. Kuers, G. (1966). Investigation of the infrared emission spectrum of the atmosphere and Earth. Annual Summary Sci. Rept. AF-61(052)-778, Miinchen. 23. Kennedy, J. S., and Nordberg, W. (1967). Circulation features of the stratosphere derived from radiometric temperature measurements with the TIROS VII satellite. J. Arm. Sci. 24, No. 6. 24. Bolle, H.-J. (1968). Satellite techniques for observing water vapor-height profiles. Miinchen (private communication). 25. Conrath, B. J. (1967). Inverse problems in radiative transfer: A review. Doc. X-62267-57, Goddard Space Flight Center. 26. Lebedinsky, A. I., Boldyrev, V. G., Tulupov, V. I., Kudinova, G. N., Levchenko, A. D., and Shvidkovskaya, T. E. (1967). The spectrum of the Earth's heat radiation according to the observation from COSMOS 45, COSMOS 65 and COSMOS 92 satellites. Space Res. 7/2. North.-Holland Publ., Amsterdam. 27. Girard, A. (1966). Analyse experimentale du contraste infrarouge entre la terre et l'espace. ONERA T.P. No. 350, Paris. 28. Zdunkowski, W. G., and McDonald, D. V. (1966). The distribution of infrared radiative flux densities and their divergence along the symmetry line of an idealized valley. Geophys. Pura Appl. 65, No. 3.

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29. Zdunkowski, W. G., Barth, R. E., and Lombarde, F. A. (1966). Discussion on the atmospheric radiation tables by Elsasser and Culberston. Geophys. Pura Appl. 63. 30. Zdunkowski, W. G., Nielson, B. C., and Korb, G. (1967). Prediction and maintenance of radiation fog. Techn. Rept. ECOM-0049-S1, Univ. of Utah. 31. Zdunkowski, W. G., and Lombarde, F. A. (1967). Discussion of the Moller radiation chart. Arch. MGB, Ser. B. 15, No. 1-2. 32. Warnecke, G. (1966). Synoptic applications of satellite-born infrared window measurements. NCAR Techn. Notes, No. 1 1 . 33. Warnecke, G . (1966). TIROS VII 15 micron radiometric measurements and midstratospheric temperatures. NCAR Techn. Notes, No. 1 1 . 34. Nordberg, W. (1966). Satellite radiation measurements in spectral regions. NCAR Techn. Notes, No. 1 1 . 35. King, J. I. F. (1966). Radiation physics. NCAR Techn. Notes, No. 11. 36. Teweles, S. (1966). Radiometer data in the 15 micron band. NCAR Techn. Notes, No. 11. 37. Zaitseva, N. A., and Kostyanoy, G. N. (1966). Meridional variation of the longwave radiation field in the atmosphere over the Pacific (from the weather ships’ data). Proc. Acad. Sci. USSR, Ser. Phys. Atmosphere Ocean 2, No. 12. 38. Kostyanoy, G. N. (1966). On the programming of radiation charts. Trans. Central Aerol. Obs. No. 74. 39. Boldyrev, V. G., and Sonechkin, D. M. (1967). Example of parallel analysis of cloud pictures and outgoing radiation fields (from satellite data). Meteorol. Hydrol. No. 5 . 40. Rodgers, C. D. (1967). The use of emissivity in atmospheric radiation calculations. Qua?. J . Roy. Meteoral. SOC.93, No. 395. 41. Gebhart, R. (1967). On the significance of the shortwave CO, absorption in investigations concerning the CO, theory of climatic change. Arch. Meteorol. Geophys. Bioclimatol., Ser. B 15, No. 112. 42. Glahn, H. R. (1966). On the usefulness of satellite infrared measurements in the determination of cloud top heights and areal coverage. J. Appl. Meteorol. 5, No. 2. 43. Twomey, S. (1966). Indirect measurements of atmospheric temperature profiles from satellites: 11. Mathematical aspects of the inversion problem. Monthly Weather Rev. 94, No. 6. 44. Liventzov, A. V., Markov, M. N., Merson, Ya. I., and Shamilev, M. R. (1966). Investigation of the angular distribution of the thermal radiation of the Earth into space at the launch of a geophysical rocket on 27 August, 1958. Space Investigations 4, No. 4. 45. Bazhulin, P. A., Kartashev, A. V., and Markov, M. N. (1966). Investigation of the angular and spectral distribution of the Earth’s radiation in the infrared spectral region from the COSMOS-45 satellite. Space Investigations 4, No. 4. 46. Hovis, W. A., Jr. (1966). Optimum wavelength intervals for surface temperature radiometry. Appl. Opt. 5, No. 5 . 47. Dorian, M., and Harshbarger, F. (1967). Measurement of the atmospheric spectral radiance at 75 km in the near infrared. Appl. Opt. 6, No. 9. 48. Plechkov, V. M. (1968). Preliminary results of the determination of water content of the atmosphere from the measurements of its thermal radioemission near A = 1.35 cm. Proc. Acad. Sci. USSR, Ser. Phys. Atmosphere Ocean 4, No. 2.

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49. Gorodetsky, A. K., Gurvich, A. S., and Migulin, A. V. (1967). On the underlying surface temperature determination from the outgoing radiation measurement in the 8-12 micron region from an aircraft. Proc. Acad. Sci. USSR, Ser. Phys. Atmosphere Ocean 3, No. 6. 50. Gorodetsky, A. K., and Filippov, G. F. (1968). Surface measurements of the atmospheric and the underlying surface’s radiation in the spectral region of 8-12 microns. Proc. Acad. Sci. USSR, ser. Phys. Atmosphere Ocean 4, No. 2. 51. Lebedinsky, A. I., Tulupov, V. I., Sofronov, Yu. P., Andrianov, Yu. G., and Karavayev, I. L. (1967). Statistical characteristics of the Earth’s radiation into space in the 7-26 micron wave length band. Geomagnetizm i Aeronomiya 7 , No. 3. 52. Lebedinsky, A, I., Andrianov, Yu. G., Karavayev, I. L., Sofronov, Yu. P., and Tulupov, V. I. (1968). Correlations of the spectral intensity of the Earth’s infrared radiation into space. Geomagnetizm i Aeronomiya 8, No. 1. 53. Lebedinsky, A. I., Boldyrev, V. G., Tulupov, V. I., Barkova, G. N., and Lelyakina, T. A. (1968). Meteorological interpretation of the outgoing radiation spectra registered from the COSMOS satellites. Geomagnetizm i Aeronomiya 8, No. 1 . 54. Glasko, V. B., and Timofeyev, Yu. M. (1968). Use of the regularization method for the solution of the problem of atmospheric thermal sounding. Proc. Acad. Sci. USSR, Ser. Phys. Atmosphere Ocean 4, No. 3. 55. Nijlisk, H. J. (1967). Some aspects of the amendment of the theoretical calculations of atmospheric thermal radiation. Investigations in the radiat. regime of the atmosphere. Trans. Atrn. Phys. Inst. Acad. Sci. Estonian SSR, Tartu. 56. Nijlisk, H. J., and Noorma, R. J. (1967). On the spectral distribution of the thermal radiation intensity and fluxes in the free atmosphere. Investigations in the radiative regime of the atmosphere. Trans. Atm. Inst. Acad. Sci. Estonian SSR, Tartu. 57. Lenoir, W. B., Barrett, J. W., Koehler, R. F., Jr., and Pape, D. C. (1967). Observations of microwave emission by molecular oxygen in the lower stratosphere. Trans. Am. Geophys. Union 48, No. 1 . 58. Lorenz, D. (1966). The effect of the longwave reflectivity of natural surfaces on surface temperature measurements using radiometers. J. Appl. Meteorol. 5, No. 4. 59. Zaitsev, V. A., and Lomakin, A. N. (1968). Radiothermal emission of ice cover in the 3-10cm range. Trans. Arctic Antarctic Res. Znst. 284. 60. Dmitriyev, A. A., and Yevnevich, T. V. (1966). Modelling of the problem of river temperature determination from satellites. Proc. Acad. Sci. USSR, Ser. Phys. Atmosphere Ocean 2, No. 8. 61. Gurvich, A. S., and Time, N. S. (1966). On the variations of absorption and brightness temperature of the atmosphere. Proc. Acad. Sci. USSR,Ser. Phys. Atmosphere Ocean 2 , No. 8. 62. Kostyanoy, G. N., and Kurilova, Yu. V. (1966). On the radiation properties of cloudiness. Trans. Central Aerol. Obs. No. 10. 63. Doronin, Yu. P., and Semenova, I. V. (1966). On the interpretation of the outgoing radiation fluxes. Trans. Arctic Antarctic Res. Znst. 277, 5 . 64. Bellman, R. E., Kagiwada, H. H., and Kalaba, R. E. (1966). Numerical inversion of Laplace transforms and some inverse problems in radiative transfer. J. Atrn. Sci. 23, No. 5 . 65. Barret, A. H., Kniper, J. W., and Lenoir, W. B. (1966). Observations of microwave emission by molecular oxygen in the terrestrial atmosphere. J. Geophys. Res. 71, No. 20.

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66. Johnson, D. R. (1966). The effect of bias and random radiometersonde temperature errors in the estimation of atmospheric downward, upward, net and equivalent infrared irradiance. J. Geophys. Res. 71, No. 24. 67. Kondratyev, K. Ya., Styro, D. B., and Ashcheulov, S. V. (1968). Experimental investigations of thermal radiation of an overcast sky. Probl. Arm. Phys. No. 5 . Leningrad State University. 68. Panin, B. D. (1967). On the determination of atmospheric water content from the outgoing radiation of the earth-atmosphere system in the spectral region of 6.0-6.5 microns. Proc. Acad. Sci. USSR, Ser. Phys. Afmosphere Ocean 3, No. 2. 69. Bernhardt, F., and Phillips, H. (1966). Die raumliche und zeitliche Verteilung der Einstrahlung, der Ausstrahlung und der Strahlungsbilanz im Meeresniveau. Abhandl. Met. Dienst DDR 10, No. 77. 70. Tannhauser, I. (1967). On the influence of atmospheric transmission functions on infrared measurements in planetary atmospheres. NASA Res. Grant, Meterol. Inst. of the Univ. Miinchen. 71. Charnell, R. L. (1967). Longwave radiation near the Hawaiian Islands. J. Geophys. Res. 72, No. 2. 72. Guirdyuk, G. V. (1966). Measurement of the effective radiation of the sea’s surface with a thermoelectric balancemeter. Trans. ConJ Marine Meteorol. Gidrometeoizdat, Leningrad. 73. Kozlov, V. P. (1966). Numerical reconstruction of the temperature vertical profile from the outgoing radiation spectrum and the optimization of the measurement method. Proc. Acad. Sci. USSR, Ser. Phys. Atmosphere Ocean 2, No. 12. 74. Yamamoto, G., Tanaka, M., and Kamitani, K. (1966). Radiative transfer in water clouds in the 10-micron window region. J. A f m . Sci. 23, No. 3. 75. Malevsky-Malevich, S. P. (1966). On the connection between the upward fluxes of longwave radiation and the surface’s temperature. Mefeorol. Investigations No. 15. Nauka, Moscow. 76. Novoseltzev, Ye. P. (1966). On the degree of blackness of the upper and middle layer clouds. Trans. Main Geophys. Obs. No. 196. 77. Kondratyev, K. Ya., Novoseltzev, Ye. P., and Ter-Markaryantz, N. Ye. (1966). Determination of the underlying surface and cloud temperature from meteorological satellites. Trans. Main Geophys. Obs. No. 196. 78. Lazarev, A. I., and Timokhin, V. I. (1966). Atmospheric thermal radiation scattered by aerosol layers. Rept. Acad. Sci. USSR 169, No. 5 . 79. Rakipova, L. R. (1967). On the statistical connections between the outgoing radiation and the meteorological parameters of the atmosphere. Proc. Acad. Sci. USSR, Ser. Phys. Afmosphere Ocean 3, No. 8. 80. Specht, H. (1967). Uber die Absorptions- und Emissions-strahlung der atmospharischen Ozonschicht bei der Wellenlange 9.6 microns. Mirt. Max-Planck-Znst. Aeron. No. 29. 81. Marlatt, W. E. (1967). Remote and in situ temperature measurements of land and water surfaces. J. Appl. Meteorol. 6, No. 2. 82. Jensen, C. E., Winston, J. S.,and Taylor, V. R. (1966). 500 mb heights as a linear function of satellite infrared radiation data. Monfhly Weather Rev. 94, No. 11.

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101. Kondratyev, K. Ya., and Timofeyev, Yu. M. (1967). Some questions of thermal atmospheric sounding. Proc. 17th Internatl. Astronaut. Federation Congress, Warsaw. Polish Scientific Publishers, Warsaw. 102. Kondratyev, K. Ya., Zhvalev, V. F., and Ter-Markariantz, N. E. (1967). On the possibility of using the infrared pictures of the Earth for the tracking of the sea current dynamics, the detection of jet streams and noctilucent clouds. Proc. Acad. Sci. USSR, Ser. Phys. Atmosphere Ocean 3, No. 11. 103. Kondratyev, K. Ya., Gayevsky, V. L., Guseva, L. N., Zhvalev, V. F., Novoseltzev, Ye. P., and Ter-Markariantz, N. E. (1968). Meteorological interpretation of the infrared pictures of the Earth from space. Meteorol. Hydrol. No. 4. 104. Kondratyev, K. Ya., Ashcheulov, S. V., and Styro, D. B. (1967). Investigations in the spectra of atmospheric emission. Trans. 6th Conf. Actinometr. Atm. Optics. Nauka, Moscow. 105. Kondratyev, K. Ya., and Timofeyev, Yu. M. (1968). Inverse problems of satellite meteorology. Geophys. Pura Appl., 54. 106. Kondratyev, K. Ya., Tikhonov, A. N., Glasko, V. B., and Timofeyev, Yu. M. (1968). Application of regularization methods in inverse problems of atmospheric optics. Proc. 18th IAF Congress, Madrid, (1967). Pergamon Press, New York. 107. Kondratyev, K. Ya., Styro, D. B., and Ashcheulov, S. V. (1966). On the possibility of determining the absorption coefficients of water vapor and other components from atmospheric thermal radiation. Probl. Atm. Phys. No. 4. Leningrad State University. 108. Kondratyev, K. Ya., and Ashcheulov, S. V. (1967). Aircraft measurements of the thermal radiation of the Earth and the atmosphere. Trans. 6th Conf. Actinometr. Atm. Optics. Nauka (Moscow).

CHAPTER 10 1. Allison, L. J., Nicholas, G. W., and Kennedy, J. S. (1966). Examples of the meteorological capability of the high resolution infrared radiometer on the NIMBUS-1 satellite. J. Appl. Meteorol. 5, No. 3. 2. Allison, L. J., and Warnecke, G. (1966). Synoptic interpretation of the TIROS-I11 radiation data recorded on 16 July, 1961. Bull. Am. Meteorol. SOC.47, No. 5. 3. Allison, L. J., and Thompson, H. P. (1966). TIROS-VII infrared radiation coverage of the 1963 Atlantic hurricane season with supporting television and conventional meteorological data. NASA T N D-3127, Washington. 4. Aizenshtat, B. A. (1967). Heat balance and the microclimate of damp intermontane valleys. Trans. Middle Asian Hydromet. Res. Inst. USSR No. 35, 501. 5. Davies, J. A. (1967). A note on the relationship between net radiation and solar radiation. Quart. J. Roy. Meteorol. SOC.93, No. 395. 6. Elterman, L. (1966). Aerosol measurements in the troposphere and stratosphere. Appl. Opt. 5, No. 11. 7. Funk, J. P., Deacon, E. L., and Collins, B. G. (1966). A radisonde radiometer. Pure Appl. Geophys. (Milan) 64. 8. Bratchenko, B. M. (1966). On the connection between net radiation and the latitude and altitude of terrain. Trans. Kazakh Hydromet. Res. Inst. No. 25.

856

Additional Bibliography

9. Berland, T. G. (1966). Contemporary state of the investigation in the radiation climate. In “Contemporary Problems of Climatology.” Gidrometeoizdat, Leningrad. 10. Belov, V. F., Zaitseva, N. A., Kostyanoy, G. N., and Shlyakhov, V. I. (1967). Actinometric sounding of the atmosphere over the Atlantic Ocean. Meteorol. Hydrol. No. 4. 11. Belov, P. N. (1967). Net radiation maps and the possibilities of their use in synoptical practice. Meteorol. Hydrol. No. 4. 12. Jensen, C. E., and Astling, E. G. (1967). Net radiation and net longwave radiation at Copenhagen 1962-1964. Arch. Meteorol. Geophys. Bioclimatol., Ser. B. 15, No. 1/2. 13. Jensen, C. E., Winston, J. S., and Taylor, V. R. (1966). 500 mb heights as a linear function of satellite infrared radiation data. Monthly Weather Rev. 94, No. 1 1 . 14. Adem, J. (1967). On the relation between outgoing longwave radiation, albedo, and cloudiness. Monthly Weather Rev. 95, No. 5. 15. Arking, A., and Levine, J. S. (1967). EBrth albedo measurements: July 1963 to June 1964. J. Atm. Sci. 24, No. 6. 16. James, D. G., Limbert, D. W. S., and McDougall, J. C. (1966). A radiometer sonde. Meteorol. Mag. 95, No. 1127. 17. Jurica, G. M. (1966). Radiative heating in the troposphere and lower stratosphere. Monthly Weather Rev. 94, No. 9. 18. Kondratyev, K. Ya., and Dyachenko, L. N. (1967). Comparison of the calculational results of the outgoing radiation with the data of measurement by means of actinometric radiosondes and meteorological satellites. Trans. Main Geophys. Obs. No. 203. 19. Kostyanoy, G. N., and Tarassenko, D. A. (1967). On the variation of the longwave radiation field near the tropopause. Trans. Central Aerol. Obs. No. 73. 20. Moller, F. (1967). Eine Karte der Strahlungsbilanz des Systems Erde-Atmosphare fur einen 14-tagigen Zeitraum. Met. Rundschau 20, No. 4. 21. Nordberg, W. (1967). Satellite studies at the lower atmosphere. Space Sci. Rev. 1, No. 516. 22. Nordberg, W., McCulloch, A. W., Foshee, L. L., and Bandeen, W. R. (1966). Preliminary results from NIMBUS 11. Bull. Am. Meteorol. SOC.47, No. 11. 23. Linton, R. C. (1967). Earth albedo using Pegasus thermal data. AZAA Thermophys. Specialist Conf., New Orleans. AIAA Paper No. 67-332. 24. Levine, J. S. (1967). The planetary albedo based on satellite measurements taking into account the anisotropic nature of the reflected and backscattered solar radiation. Master’s Thesis, Grad. School of Arts and Sciences, New York Univ., New York, N.Y. 25. Pearson, B. D., and Neel, C. B. (1967). Albedo and earth radiation measurements from OSO-11. AIAA Paper, No. 330. 26. Raschke, E. (1965). Auswertungen von infraroten Strahlungsmessungen des meteorologischen Satelliten TIROS 111. Teil 11. Beitr. Phys. Atrnosphiire 38, No. 3-4. 27. Raschke, E., Moller, F., and Bandeen, W. R. (1967). The radiation balance of the Earth-Atmosphere system over both polar regions obtained from radiation measurements of the NIMBUS I1 meteorological satellite. Doc. X-622-67-460, Goddard Space Flight Center, Greenbelt, Maryland. 28. Raschke, E., and Pasternak, M. (1967). The global radiation balance of the EarthAtmosphere system obtained from radiation data of the meteorological satellite NIMBUS 11. Doc. X-622-67-38, Goddard Space Flight Center, Greenbelt, Maryland. 29. Strokina, L. A. (1967). On the comparison of the calculated and observed values of the net radiation of the oceans. Trans. Main Geophys. 06s. No. 193.

Chapter 11

857

30. Rao, C. R. Nagaraja, and Sekera, 2. (1967). A research program aimed at high altitude balloon-borne measurements of radiation emerging from the earth’s atmosphere. Appl. Opt. 6, No. 2. 31. Rasool, S. I., and Prabhakara, C. (1966). Heat budget of the Southern hemisphere. In “Problems of Atmospheric Circulation.” Spartan Books, New York. 32. Singer, S. F. (1965). Survey of weather satellite achievements. Proc. 14th Internatl. Astronaut. Federation Congr., Paris 2. Polish Scientific Publishers, Warsaw. 33. Suomi, V. (1966). General circulation. NCAR T e c h . Notes, No. 11. 34. Shaw, J. H., McClatchey, R. A., and Schaper, P. W. (1967). Balloon observations of the radiance of the earth between 2100 cm-l and 2700 cm-l. Appl. Opt. 6, No. 2. 35. Vitsenko, G. V. (1967). Net radiation of the vertical surfaces of buildings. Trans. Main Geophys. Obs. No. 209. 36. Vetlov, I. P. (1967). Meteorological observations from satellites. Meteorol. Hydrol. No. 4. 37. Voloshina, A. P. (1966). “Heat Balance of the Surface of Mountain Glaciers in Summer.’’ Nauka, Moscow. 38. Smith, W. L., Horn, L. H., and Johnson, D. R. (1966). On the relation between TIROS radiation measurements and atmospheric infrared cooling. J. Appl. Meteorol. 5, No. 4. 39. Tepper, M. (1967). Space technology developments for the World Weather Watch. Bull. Am. Meteorol. SOC.48, No. 2. 40. Vassy, A., and Vassy, E. (1965). Possibilite de mesurer la temperature moyenne de la stratosphtre par satellite. Proc. 14th Internatl. Astronaut. Federation Congr., Paris, 1962 2. Polish Scientific Publishers, Warsaw. 41. Warnecke, G. (1967). Satelliten und Meteorologie. Ann. Meteorol. No. 3. 42. Yefimova, N. A. (1967). Net radiation map of a dampened surface. Trans. Main Geophys. Obs. No. 209. 43. Andreyev, V. D., Klinov, F. Ya., Mamayenko, G. Ye., Ruzheinikova, Yu. V. (1967). Actinometric measurements in the lower 300 m atmospheric layer by means of a high-altitude mast of the Institute of Applied Geophysics. Trans. Inst. Appl. Geophys. USSR No. 10. 44. Katayama, A. (1967). On the radiation budget of the troposphere over the northern hemisphere. 11. Hemispheric distribution. J. Meteorol. SOC.Japan Ser. II 45, No. 1. 45. Katayama, A. (1967). On the radiation budget of the troposphere over the northern hemisphere. 111. Zonal cross-section and energy consideration. J. Met. SOC.Japan Ser. II 45, No. 1. 46. Kondratyev, K. Ya., and Dyachenko, L. N. (1967). Comparison of the results of calculations of the outgoing radiation with the measurement data obtained by means of actinometric radiosondes and meteorological satellites. Trans. Main Geophys. Obs. No. 203. 47. Kondratyev, K. Ya., Yessipova, Ye. N., and Nikolsky, G. A. (1967). Radiation fluxes and the attenuation components of direct solar radiation in the troposphere and stratosphere. Trans. 6th Conf Actinometr. Atm. Optics. Nauka, Moscow. 48. Kondratyev, K. Ya. (1967). Space meteorologique. Atomes No. 5 . 49. Kondratyev, K. Ya., and Nikolsky, G. A. (1968). Direct solar radiation and the aerosol structure of the troposphere and stratosphere during the IQSY. Trans. Con$ Results IQSY, Nauka, Moscow.

858

Additional Bibliography

50. Kondratyev, K. Ya., and Nikolsky, G. A. (1968). Direct solar radiation and aerosol structure of the atmosphere from balloon measurements in the period of IQSY. Sveriges Met. och Hydrol. Inst. Meddel. Ser. B, No. 28.

CHAPTER 11 1. Belov, P. N., and Kurilova, Yu. V. (1967). Some possibilities of using radiation data from satellite in the synoptical analysis. Meteorol. Hydrol. No. 7. 2. Belov, P. N., and Alpatova, R. L. (1967). Some results of the numerical experiments connected with radiative flux divergence calculation at the numerical weather forecast. Trans. Hydrometeorol. Center, USSR 11. 3. Berkovich, L. V. (1967). On the parallel calculation of the radiative and turbulent flux divergences in the numerical weather forecast. Trans. Hydrometeorol. Center, USSR 11. 4. Lindzen, R. S. (1967). Physical processes in the mesosphere. In “Dynamics of Large Scale Atmospheric Processes.” Nauka, Moscow. 5. Yelisseyev, A. A. (1967). On the possibility of the direct measurement of the radiative flux divergence in the atmosphere. Trans. Main Geophys. Obs. No. 205. 6. Yelisseyev, A. A. (1967). Receiver for measuring radiation changes in the air temperature. Trans. Main Geophys. Obs. No. 205. 7. Dmitriyev, A. A., and Vinogradskaya, A. A. (1967). On the radiation and advection role in the local temperature forecast. Meteorol. Hydrol. No. 12. 8. Rodgers, C. D. (1967). The radiative heat budget of the troposphere and lower stratosphere. MIT Rept. No. A2. 9. Winston, J. S. (1966). Radiative heating. NCAR Techn. Notes, No. 11. 10. Lieske, B. J., and Stroschen, L. A. (1967). Measurements of radiative flux divergence in the Arctic. Arch. MGB, Ser. B 15, No. 1/2. 11. Brooks, F. A. (1966). Spectral-interval radiation exchange between air layers and the stratified atmosphere. Proc. Heat Transf. FIuid Mech. Inst. Stanford Univ. Press, Stanford, California. 12. Avaste, 0. A. (1967). Solar radiation flux divergence in the atmosphere and the total radiation flux at the sea’s surface. Investigat. Radiat. Regime Atm., Inst. Atm. Phys., Acad. Sci. Estonian SSR, Tartu. 13. Kostyanoy, G. N. (1966). On the direct measurements of the radiative flux divergence in the free atmosphere. Trans. Central Aerol. Obs. No. 70. 14. Sasamori, T., and London, J. (1966). The decay of small temperature perturbations by thermal radiation in the atmosphere. J. Atm. Sci. 23, No. 5. 15. Goody, R., and Belton, M. J. S. (1967). Radiative relaxation times for Mars. A discussion of Martian atmospheric dynamics. Kitt Peak Nat. Qbs., Contr. No. 199. Planet. Space Sci. 15, No. 2. 16. Atwater, M. A. (1966). Comparison of numerical methods for computing radiative temperature changes in the atmospheric boundary layer. J. Appl. Meteorol. 5, No. 6. 17. Krakow, B., Babrov, H. J., Maclay, G. J., and Shabott, A. L. (1966). Use of the Curtis-Godson approximation in calculations of radiant heating by inhomogeneous hot gases. Appl. Opt. 5, No. 11. 18. Joseph, G . H. (1966). Calculation of radiative heating in numerical general circulation models. Dept. of Meteorol., UCLA, Techn. Rep. No. 1.

Chapter 11

859

19. Walshaw, C. D. (1966). Remarks on the computation of radiative heating rates. I n “Les Probltmes Meteorologiques de la Stratosphtre et de la Mesosphtre.” Presse Universitaire de France, Paris. 20. Ryazanova, L. A., and Trubnikov, B. N. (1966). On the role of the radiative flux divergences in the formation of the thermal regime of the stratosphere. Trans. Central Aerol. Obs. No. 69. 21. Hering, W. S., Touart, C. N., and Borden, T. R., Jr. (1967). Ozone heating and radiative equilibrium in the lower stratosphere. J. Arm. Sci. 24, No. 4. 22. Kondratyev, K. Ya., Badinov, I. Ya., Gayevskaya, G. N., Nikolsky, G. A., and Shved, G. M. (1966). Radiative factors of the heat regime and dynamics of the upper atmospheric layers. In “Les Probltmes Meteorologiques de la Stratosphtre et de la Mesosphtre.” Presse Universitaire de France, Paris. 23. Kondratyev, K. Ya., and Shved, G. M. (1967). The part of radiation in the heat regime of the atmospheric layer from 30 to 100 km. Space Res. VII, North.-Holland Publ., Amsterdam. 24. Kondratyev, K. Ya., Nijlisk, H. J., and Noorma, R. J. (1968). On the spectral distribution of the radiative flux divergences in the free atmosphere in the spectral region of 1 2 4 . 4 4 microns. Proc. Acad. Sci. USSR, Ser. Phys. Atmosphere Ocean 4, No. 10.

APPENDIX 1 The Function P(w, u)

&

2-00

-

oo o\

1.00

i.m

1.40

;.a0

0

im

0.00

0.10

o a

0.30

0 . a

0.50

0.m

0.m

0.80

o.90

5.00

888

854

848

841

m2

gP

812

006

800

rn

787

781

774

768

761

755

5.10

as4

mo

844

837

828

m9

808

802

7a

789

789

777

?70

764

757

751

5 .a0

858

844

838

831

822

813

802

796

790

783

777

764

758

751

745

T .30

853

838

a3

825

816

807

777

771

758

752

74s

845

831

8a

81e

809

800

790 783

784

5.40

796 m9

'171 765

m

770

784

758

T51

745

738

139 '132

-4.80

E38

823

817

mo

769

762

756

750

743

'137

730

724

809

802

m4

781 773

775

815

767

781

7s

748

742

735

729

722

820

806

800

794

7T5

?64

752

745

739

733

726

n3

707

3

1.80

811

m

79l

784

766

755

758 749

m

ne

5.70

801 7R3 7a4 775

792

829

743

Z36

730

724

n 7

711

704

6%

k

4.90

800

786

mo

m

764

795

744

v.38

732

725

7lQ

713

706

700

693

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5.00

786

m

767

760

751

742

731

725

no

ne

706

SEO

674

774

760

754

747

738

729

ne

n 2

683

674

667

651

759

746

740

733

724

ns

m4

898

699 655

693

9.20 6.30

679

673

666

653

647

743

730

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n 6

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6%

6E7

6el

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662

656

649

660 643

636

630

3.40

728

n4

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632

683

672

666

7013 692 675 660

700 687

687

3.10

653

647

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634

628

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615

5.50

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634

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678

669

650 644

652

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633 619

620

614

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633

639 626

627

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533

620

613

606

600

593

587

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5e8 575

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600

594

587

580

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581

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5.60

700

700 686

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639 626 613

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8 4

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693

* R'

APPENDIX 1 (continued)

1-00

1-10

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749

714

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745

740

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739

734

2.30 4.40

733

726

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718

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701

4.50

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1.50

1.60

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733

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735

729

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72& 721

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727

722

71 7

711

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mi

696

691

697

692

687

686 882

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723

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712

767

701

596

691

686

681

676

672

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716

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704

693

701

695

630

585

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675

670

666

694

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668

663

659

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670

555

650

655

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580 672

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557

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634

6%

647

651 643

536

631

685

679

674

659

653

658

653

648

643

638

634

634

629

625

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618

614 6G1

2.20

2.30

578

4.80

532

587

682

676

6 70

665

560

654

540

644

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5.90

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576

571

565

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654

649

b23

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$58

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615

610

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611

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578

574

624

519

614

608

602

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592

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5a

577

572

567

562

558

3.40

5: 9

x 4

533

593

587

582

577

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565

562

557

552

547

543

5.50

5 x

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573

573

568

5 8

557

552

548

543

538

533

529

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3E2

5T7

572

565

550

555

550

544

539

535

530

525

516

5.m

550

554

359

533

547

542

537

531

526

522

51T

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520 507

503

555

55:

546

540

534

523

524

519

514

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499

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5 4

se

533

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501

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485

481

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5.00

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5.m 3.90

636

2.40 2-50

5.o~

1.00

i.20

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1.m

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0.10

0.20

0.30

0.40

0.50

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0.70

0.80

0.90 00 Q\

h,

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555

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328 51s

508

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54e

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576

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5=

551

544

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520

514

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500

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2.20

-.SG 5.m -2.EC 2.60

5.X

-1 .oo i-. i o 1.20

i .m -

562

519

543

537

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545

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51 S

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504

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461

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442

438

432

426

421

510 497

-4€'3 475

470

461 445 430

440

415

410

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427

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392

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380

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36

360

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365

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354

349

345

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340

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401

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394

405

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365

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354

347

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337

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315

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320

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311

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m2a2

% 6

277

274

270

287

= 2 8 1 264 260

,EO -11.90

TJ2

2.83

2Ec

277

274

253

264

261

258

255

253

250

247

245

242

239

262

255

253

249

246

242

238

236

233

230

228

226

224

222

218

215

206 lpo

204

232

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178

176

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1.30

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231

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225

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219

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212

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1e4

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4

194

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1.10

1.20

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1 A O

1.50

531

526

521

515

509

504

517 504 491

512 499

507 494

501

48E

495 483

486

481

475

469

431 478 464

2.40

477

472

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456

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486 4Tv 459 446

2.5c

464

459

454

448

443

450

445

44C

434

429

438 124

433 41 9

430 41 5

425 41 C 395

419 404 381

41 4 399

5.X

435 4% 405

403 395 380

404 390 375

i.oo

388

383 367

373 357

366

364 34s 332

359 343 327

335 339 323

3c

346

335 32C

331 31 6

31 4 297

310 293

306 2ac

303 207

299

304

352 338 318 3C1

286

282

279

276

272

259

256

z.00 5.10

2.20

-2ao -

z.60

-

2.70 2.EC

i.io i .a

i 30 i .XI

-

1.50

-

T.60

400

335 31 6

330 31

378 362 345 326 3c7

297

2513

289

372 355

272

350

270

341 322

256

274 253

i.60

236

233

230

267 247 227

1.90

213

21 1

209

207

c.co

92

190

182

c.1c

16e

167

185

1.70

-

250

384

166 1 8

I

264 244 225

224

205

2 0

184

182

161

160

261 242

1-60

499

258

24c

1-70

1.a

1-90

2.W

494

489

484 471 456 444 431

479

481 456

476 463

455 442

449

426

423 409

41 4 399

366 371

256

437

394 381 366

253

238

235

222

220

2:8

222

201

:a

180

17D

159

158

177 157

2.10

474

2.20

2.30

2.40

2.50

469

465

461

457

452 439 412

447 434 421 408

143

466 453 439

461 448 434

428

422

41E

413

404 390

399

403 331

393 387

394 3E2

390 378

367

353

371 357

376 362 346

372

376

406 395 380 367 353

358 344

3%

350

340

336

342 327 312

330

333

323 308

318

326 31 1

322 3c 7

304

329 314 3CC

297

296 280

293

269

285

282

277

273

27c

267

294 27;) 2 8

253 247 230 212

260 244 227 209

256 240

253

250

246 230

224 207

237 221 205

a 4 219 203

1513

191

189

187

185

217 2c1 163

176

174

173

1n

159

156

155

:54

152

151

168 150

36l

2g3

25c 233 215 1%

385

456 443 430 416

42%

430 41 7

404

167 14E

APPENDIX 1 (continued)

-.oo

-1.80

0.00

0.1 0

0.20

0.30

0.40

1% 133 110

157 132 109

15s 131 108

1 s

1 s

135 112

160 134 111

130 108

129 107

8 8 8 8 7 0 7 0

88 70

87 70

1

i.zo

r .a

166 138

ia

114

164 137 113

7.60

0.20 0.30 0.40

im 141 115

167 139 115

0.50 0.50

9l

m

a

90

90

89

€3

72

7 2 7 2

R

n

n

n

0.70

56 56 4 3 4 3 32 32

56

56

5s

43 32

56 43 32

55

43 32

56 43 32

5 5 5 5

0.90

56 43 32

4 3 4 3 32 32

43 32

1.00

23

2 3 2 3

23

29

23

29

2 3 2 3

23

0.50

o.m

0.60

151 128 107

150 127 106

149 127 106

8T

8

70

69

43 32

87 70 55 43 32

23

23

2

0.80

0.90

148

147

126

126

1C5

1oJ

*

3g,

k7

0.m

55

6

8

6

8 68

69

5 5 5 5 5 5 42 42 32 31 31

43 32

;12

2

2

2

2

2

2

2

6

-

APPENDIX 1 (continued)

1.00

1.10

1.20

1.30

0.20 0.30 0.40

146 125 104

1 45

125 104

144 1 24 1C4

143 123 103

0.50 0.50 0.70 0.60 0.30

85

85

E5

69

69

69

55

55

55

42 31

42 31

42 31

1 .oo

22

22

22

2.30

2.40

2.w

134 116 99

133 115 98

132 114 97

131 113 97

82

82

w

67

67

54

53

42 31

53 42 31

67 53 42 31

81 66

42 31

67 53 42 31

81 67 53 42 31

22

22

22

P

22

1-80

1.90

9.00

2.10

139 119 102

13e 118 101

137 118 101

136 117

loo

135 117 100

84

8 3 8 3

83

82

6 8 6 8 54 54 42 42 31 3:

68

54

42 31

68 54 42 31

83 66

42 31

22

22

2 2 2 2

22

22

1-40

1.50

1.60

142 122 103

141 121 1 C3

140 120 la2

84

84

69 55 42 31

69

1.70

2.20

ak K

54

53

42 31

w

APPENDIX 2 The Funcrion d P ( w , u) log rn log w

5.00

5.m

3.50

z.00

5.10

Loo

87

87

06

83

82

81

87

87

86

83

82

el

i.oo 1 .a

-

86

85

84

80

84

84

1.4 -

82

82

73

1.60

80

80

83 82 80

81 80

i.a

n

77

-1.83

76

76

76

2.30

-

00

m m

2.40

5.20

'i.60

G.70

a.80

80 80

78

76

79

76

73

n n

68

78

79

78

76

74

72

69

67

79

78

TI

m

P4

n

69

67

78

77

76

75

73

71

66

n

76

75

74

73

n

69

68 66

77

75

74

73

n

69

67

65

74

74

73

71

69

67

85

63 69

62

76

n

75

72

73

72

70

68

66

64

62

61

75

74

74

n

n

m

69

67

65

63

1 .Qa

74

73

i3

70

70

69

68

66

65

63

61 bl

GO 59

0.M) 0.04 0.08

73

72

72

69

68

68

67

65

64

62

60

58

72 70

n

m

68

68

67

66

64

63

a

59

57

m

69

67

67

66

65

63

62

60

58

56

0.12

69

69

8e

66

66

65

64

62

tn

59

57

55

0.16

67

67

66

64

64

63

62

61

60

58

58

54

0.20

63

65

63

63

62

61

60

59

57

55

53

0.24

66 64

64

63

61

61

60

59

58

57

55

54

52

0.28

62

62

61

59

53

55

57

56

55

54

52

50

0.32

60

60

53

57

57

56

55

54

53

52

51

49

z .oo

1.84

-1.92

z.20

68

64

@

*

2

a

w' N

APPENDIX 2 (continued)

logx

5.90

-1.00

i.oo

65

62

5.00

65

62

-i.oo

64

1 .P

1.40 1.60

-

-

-1.40

i.m

45 45

42

42

40 40

38

48

50

47

44

41

39

53

50

47

44

41

52

49

46

43

40

46

45

42

49

46

44

48

4s

44

51

48

40

47

,42

33

49

46

45 44 43

43

50

41

38

48

45

43

4c

47

44

42

39

45

43

41

45

42

40

44

41

39

36

43

40

38

36

45

42

39

37

35

17

44

41

30

36

34

4e

43

45

37

35

33

58

55

51

48

56

55

51

61

57

53

54

60

57

8

59

56

61

57

54

51

i- .eo

59

55

52

t .04

58

54

51

-1.92 1 .Q6

50

54

5?

53

56

52

0 .oo

55

51

0.04

54

X

0.08

53

43

0.12

52

4e

0.16

51

47

0.20

5c

45

0.24

4:

G .28

0.32

1.88

~~

i.eo

1.30

-

-

1.50

i.m

1.10

i.eo

-

f.90

0.00 34

37

36 36 35

39

37

35

32

38

36

34

32

39

37

35

33

31

41

38

33

31

38

34

32

30

37

36 36 35

34

41

33

32

30

37

35

33

31

29

36

34

32

31

29

38

35

34

32

30

28

37

35

33

31

30

28

38

36

34

33

31

29

27

37

35

34

32

30

29

27

34

33

32

30

28

26

34

32

31

29

28

26

33

31

30

29

n

25

32

30

29

26

26

24

3i

29

29

27

25

23

38

34 32

*

w

P k

R’ h)

00

m

4

00

m W

APPENDIX 2 (conrinued)

log w

-

4.00

3.00

-

3.50

2.00

-

E

2.10

2.20

2.30

55

54

2.50

5.m

Lrn

53

52

50

49

47

2.40

2.80

0.36

58

58

57

56

56

0.40

56

56

55

54

53

53

52

50

49

4e

47

45

0.44

54

54

53

52

52

52

50

49

48

47

46

43

0.46

52

52

51

50

50

49

48

47

46

44

43

41

0.52

5c

50

49

47

47

46

45

44

43

42

41

39

0.56

4e

48

47

45

45

44

43

42

41

40

39

38

0.60 0.64 0.a

45

45

44

43

43

42

41

40

39

38

37

36

43 40

43

42

41

41

40

39

38

38

37

36

35

4c

39

37

37

36

35

35

34

33

37

37

37

36

35

34

34

33

32

31

30

0.76

34

34

34

38 36 33

38

0.72

33

32

32

31

31

30

29

28

0 .Bc

31

31

31

30

30

29

29

28

20

27

27

26

0.84

28

28

26

27

27

27

26

26

26

25

25

24

0 0.92

25

25

25

25

25

25

24

24

23

P

22

2:

23 21

23 21

23

23 20

23

23

22

22

21

2l

20

19

2l

20

20

19

19

19

18

18

17

19

19

19

18

18

18

17

17

17

16

16

.=

0.96 1

.oo

15

;P

51

3g N

APPENDIX 2 (continued) log m

-

1.10

i.m

iao

log w

h.oo

1.00

0.36

44

41

33

36

34

0.40 0.44

43

40

38

35

41

3e

36

34

0.48

39

37

35

0.52

37

35

0.56

36

0.60 0. In

i.50

i.ao

1.40

i.eo

i.m

32

30

28

27

25

25

23

33

31

26

25

24

22

30

26

25

24

23

21

33

31

29

29 28 27

27

32

25

24

23

22

20

33

31

29

28

26

24

22

2l

19

34

32

30

28

27

a

23

P 22

n

20

18

34

32

30

28

26

25

24

22

21

20

10

17

33

31

29

27

25

24

22

21

20

19

18

16

0.68

31

29

2s

26

24

23

n

20

19

18

17

15

0.72

29

27

24

24

22

2l

20

19

18

17

16

14

0.76

27

25

24

23

2l

20

18

17

17

16

15

13

0.a 0.a 0.85 0.92 0.96

25

24

22

a

19

18

16

1s

15

14

13

12

23

22

20

19

17

16

15

14

13

13

12

11

.w

1.40

0.00

-* ii’

1

21

25

19

18

16

1s

14

13

12

12

11

10

13

18

17

16

15

14

13

12

11

11

10

10

I?

16

IS

14

13

12

11

11

10

10

0

8

15

14

13

12

11

10

10

9

9

8

8

7

N

00

m W

00

4 0

APPENDIX 3 Dependence of Atmospheric Mass on the Visible (Measured) Solar Height (deg)

90

1

.ooo

40.5

1 .S8

65

1.004

40.O

13 5 3

ec

1.015 1.035

75

74 73 72

n

23.8

P .6

38.O

1.040 1.045 1.052

38.5

37.5

1.640

23 .O

1.ose 1.064

37.0

1.658

22.8

63

1 .an

6e

1.078

35.O

67 65 65

1.086

54

1.112

34.5 34 .o 33.5 33.0

62

1 .lP 1.132 1.143 1.154

32 .o 31 .5 31 .O

62 51 4C

1.a94

1.103

2.447 2.466 2.486

16.7

3.445

16.6 16.5

2.506 2 2526 2.546 2.567

1414

3.465 3.485 3.505

24 .O

1. a 1 .go4 1 .a

36.5 36.0 35.5

70

3.407 s.4a

2.410 2.428

1.571)

39.5 39.0

16.9 16.8

24 .I 24.2

23.4

P.2

i .6m

22.6

1.698

22.4

2.m 2.610

1 .no

22.2

2.82

1.740

22 .O

:.me

21.8 21 .6 21.4 21.2

I .a31

21 .o

2.654 2.677 2.700 2.7% 2.7G 2.773

20.8

2.738

1.762

32.5

1 .e56 1.882 1.910 1 .La7

1.754

2c .6 20.4

16.3

3.528

4.631 4.870 4.9IO 4.950

11.3 11.2 11.1 11.o

6.6 6.5

8.1 9

6 .4 6.3 6.2

8.41

4.932 sm4

s.ow

6.1

8.W

5.120

6 .O

8.90

5.9 5.8 5.7

9.03

5.6 5.5

9.45 9.59

5.4

9.74

16.2 16.1 16.0

3.546 3.567

3.588

10.9

5.164

15.9 15.8 15.7 15.6 15.5 15.4 15.3 15.2 15.1 15.0

3.610

10.8

s.Bse 3.m4

10.7 10.6 10.5 10.4 10.3

5.210 5.256 53 0 3 5.351

14.9

3.a0

3.676 3.699 3.722 3.745 3.768

8.90

3.53

6-65

+ w

10.2

10.1

9.17 9.30

5.399

5.448 5.438

5.3

9.90

5.2 5.1

10.06

3.732

1o.c

5.549 5 .Bo

3 -816

9.2

5 .a5

5.0

10.22 10.40

9.8

5.n 5.76

4.9

10.57

2.824

2.850

11.7 11.6 11.5 11.4

9.7

30.S 90.o

.=

2G 3

2.677

14.8

2.904 2.918 2.s2 2. E46

14.7 14.6

2.031

2c .O 19.9 19.8 1E.7 lC.b

2.044

13.5

1

59

1 .!66

56

1.178

57

1.131

29.6

2.OG7

56

1.2G5

29.6

2.019

55

1 .!a

29.4

29.2 29.C 28.8 29 .d 28.4

54

1.235

53 52

1.251 1.257

51 9G

1.285 1.304

4J.5

1.314

42 .O

1.324

27.6

42.5

1.334 1.344

27.2

46.5

1.355 1 .366 1.378

46.0

1 .?&9

4& .O

47.5 47 .O

45.5

1.401

1.935

2.260 2.275 2.W9 3.004

9.6 9.5 9.4 9 03

5.e 5.87

9.2 9.1 9.0

6.05

14.2 14.1

3.367 3.993 4.020 4.047

14.0

4.075

14.5 14.4 14.3

3.863 3.890 3.a5 3.941

4.3 4.2

8.9

6.24

4.1

11.97 12.20

8.8

6.31 6.37

4.0

12-44

19.3

13.2 13.1 19.0

3.C19

z.2

2.m 2.096. 2.153

3.034 3 .c49

13.9 13.8

4.1C3 4.131

8.7 8.6

28 .G

2.123

16.9 16.8

3 .C64

4.159 4.100

8.S 8.4

2.137

19.7

3 .G95

4.218

27.6

2.151

16.6

3.110

13.7 13.6 13.S 13.4

27.4

ia.5 :s.4 18.3

3.1 28

13.1)

37.0

2.165 2.1EO 2.1 35

8.3 8.2 8.1

3.142 3.159

13.2 13.1

26.8

2.21 1

1662

26.6 26.4

12.226

t6.1

3.1 75 3.132

2.242

16.0

3.2C9

13.0 12.2 12.8

6.44 6.51 6.58 6.66 8.W

3.9 3.8

1L.54 11.75

12.89

=* Pa

g w

12.94

3.7

13.5

3.6

13.48

3.5

13.76 14.06 14.37

6.81

8.0

6.88

3.4

7.9

3.3

3.2 3.1

14.63 15.C2

3.0

15.36

4.4H

7.8 7.7

6.M 7.05 7.13

4.4336

7.6

7.21

4.372

10.94

4.4

19.4

4.278 4.309 4.340

11.13 11.33

6.18

2.056

4.248

5.99

10.75

4.7 4.6 4.5

6.11

2.069

3.C79

5.m

4.8

M

APPENDIX 3 (continued)

h

m

h

45.8

1.413

26.2 26.0 25.8 25.6

44.5

44.0 43 .S 43 .O 42.5 42.0 41.5 41 .O

1 .426 1.438 1.451 1.464 i .4m

m

25.2 25 .O

22 5 8 2.274 2.2m 2.306 2.322 2.339 2.357

24.8 24.6

2.374 2392

25.4

1.4z

1.507 :.522

h

m

h

m

17.9 17.8 17.7 17.6 17.5 17.4 17.3 17.2 17.1 17.C

3.226 3.243 3.260

12.7 12.6 12.5 12.4 12.3 12.2 12.1 12.0 1 1 .9 1 1 .8

4.469 4.503 4.537 4.572 4.607 4.643 4.673 4.7l6 4.753 4.792

3.m

3.296 3.314 3.332 3.350 3.369 3.3m

h

7.5 7.4 7.3 7.2 7.1 7.0 6.9 6.8 6.7

m

h

m

2.5 2.0 1 .s 1 .o 0.5

17.3 19.8 P.9 27.0 32.3

0

39.7

7.30 7.39 7.48 7.57 7.67 7.77 7.87 7.97 8.08

*

3g, R’ W

873

Appendix 4

APPENDIX 4 Solar Spectral Zrradiance outside the Earth's Atmosphere at the Mean Sun-to-Earth Distance (So = 1390 W t r a ) Wavelength

SA L

SA, _ .

AA

Wavelength

P OD01

0.225 0.230 0.235 0.240 0.245 0.225

0.250 0.755

0.260

- 0.m - 0.230 - 0.235 - 0.240 - 0.245 - 0.250 - 0.250 - 0.255 - 0.250 - 0.265 - 0.270

0.9

0.065

0.325-- 0.350

0.36

0.026

03 4

0.024

0.33

0.024

0.38

0.026 0.027

0.365 0.370

- 0.360 - 0.35 - 0.370 - 0.375

1 .TI

0.127

0.350

0.47

0.034

0.375

0.83

0.045

0.380

0 -85

0.061 0.ow

0.305

0.085

0.335

0.375

0.285 0.270

- 0.275

1.12 1.18

0.250

- 0.275

4.25

0.306

1 .15

0.083

0.275 0.280

0.285 0.230 0.295 0.273

0.300 0.305

- 0.280 - 0.285 - 0.290 - 0.235 - 0.300 - 0.300 - 0.3G5

1.45

0.330

0.4W

2.03

6.17

0.44

5.66

0.41 0.44

7.06

0 .s1

6.26

0.45

- 0,375

31.27

2.25

- 0.300 - 0.385 - 0.330 - 0.335

6.99

0.50

5.93

0.43

5.96 6.32

0.43 0.42 0.45

31.05

2.z

9.27

C .67

9.52

0.66

9.66

0.69

9.50

0.68

- 0.400 - 0.400 - 0.405

5 .R5

47.40

3.41

0.49

0.61

0.52 0.67

0.223

6.56 9.38 10.31

0.3C8

0.445

- 0.450

10.87

0.77

1.373

0.425

- 0.450

47.41

3.41

10.75 10.85

0.77

10.51

0.76

0.155

2.88

0 207

3.10

0.223

10.73

0.772

G.230

- 0.325

030C

- 0.325

19-09

- 0330

5.w

0.41

5.73

0.41

0.450 0.455

5.49

03 9

0.460

5.74

0.41

0.465

5.66

0.41

0.470

- 0.335 - 0.340 - 0.345 - 0.350

2 8 . s

%

0 .oI)

2.15

0.315 0.320

0.325 0.330 -0.335 0.340 0.345

nx-1

9.53

0.104

- 0.315

- 0.320

SO wm-2

- 0.410 0.410 - 0.415 0.415 - 0.420 0.420 - 0.425 0.400 - 0.425 0.425 - 0.430 0.430 - 0.435 0.435 - 0.440 0.440 - 0.445

0.31C

0.310

SAh

6.07

0.405

3.20 3.58 3.95 4 -08 4.28

-

- 0.355

0.350 0.355 0.380

03 6

SA,

AA

SO

0.257 0.284

- 0.455

- 0.450 - 0.465 - 0.470

- 0.475

0.74

0.W

10.56

0.76

1o.n

0.77

874

Appendix 4

APPENDIX 4 (continued) Wavelength

$Ah

0.450 0.475

- 0.475 - 0.480

- 0.4e5 0.4% - 0.430 0.490 - 0.495 0.415 - 0.500 0.475 - C.500 0.5CO - 0.505 0.5CS - 0.510 0.510 - 0.515 0.515 - 0.520 0.520 - 0.525 0.4EO

0.500

0.525 0.530

0.535 0.540 0.545 0.525

- 0.525

53.18

3.82

0.620

0.630 0.640

10.80

0.78

10.55

0.76

10.10

0.73

10.90

0.74

10.25

0.74

52.00 9.80 9.80

3.74

0.70 0.70

9.65

0.69

9.60

0.69

9.65

0.59

48.50

3.49

9.75

0.70

9.e5

0.7l

9.90

0.71

- 0.5%

9.80

o.n

9.m

0.70

- 0.550

49.15

3.153

- 0,530 - 0.535 - 0.540 - 0.545

- 0.555 - 0.560 0.560 - 0.564 0.565 - 0.570 0.570 - 0.575

0.550

9.74

0.70

0.550

9.58

0.69

9.52

0.68

9.30

0.m

9.57

0.69

0.550

- 0.575

47.91

3.45

0,575

- 0.580

9.57

0.69

0.590

- 0.5E5 - 0.590 - 0395

0.595

O&?O

0.5eo 0.585

-

9.52

0.68

9 $52 9.48

0.68 0.68

9.35

0.67

0.575

- 0.600

47.44

3.41

0.600

- 0.610 - 0.620

17.93

1.29

0.610

17.56

1.26

0.600

0.850 0.690 0.670 0.680

0.690

- 0.530 - 0.650

- 0.640

- 0.650 - 0.680 -

0.670

- 0.680 - 0.630

- 0.700

0.650 -0.700

0.700 0.710

0.720 0.7300.740

0.7co 0.750

0.760

0.770 0.780 0.730

SAh -

so

AX

17.19

1.24

16.83

1.21

16.44

1.IS

85.95

6.18

16.07

1.16

15.70

1.13

153 4

1.10

14.97

1.a3

14.62

1.05

75.70

5.52

14.27

1 .a3

- o.no - 0.720 - 0.730

13.95

1 .oo

133 7

0.B

0.740

13.23 12.90

0.35 c -93

$7.92

4 .m

12.57

0.90 0.a

- 0.50 - 0.750 - 0.760 - 0.770 - 4.780 - 0.730 - 0.800

- 0.800 0.8CO - 0.810 0.81 0 - 0.82'2 0.820 - 0.830 0.W

12.26 12.00

0.05

11.67 11.40

03 4 0.82

59.90

4.3!

11.13

0 .80

10.5

0.76

10.63

0.76

0.830

10.40

0.73

0.840

10.15

0.73

53.19

38 2

9.92

o.n

- 0.840 - 0.850 0.800 - 0.850 0.850 - 0.880 0.860 - 0.870 0.870 - 0.850 O.et50 0.E90

0.850

- 0.900 0.890

- 0.m

9.69

0.m

9.47

0.68

9.26

0.67

9.06 47.40

0 .a5

3.41

875

Appendix 4

APPENDIX 4 (continued) Wovelength

s,

A

AA

Wovelength

AX P

50

P

0.900

s,,

Wrn-'AA.'

- 0.910

- 0.920 - 0.930

%

8 .R5

0.64

e .66

0.62

8.47

0.61

8.23

0.60

8.12

0.58

2.300

50

- 2.400

0.960

- 0.990

7.47

0.54

0.990

-

7.32

0.53

ae.14

2.74

7.19

0.52

- 2.503 2.000 - 2.500 2 2 4 0 - 2.600 2 . m - 2.700 2.700 - 2.800 2.8W - 2.900 2.9W - 3.000 2.500 - 3.000 3.000 - 3.100 3.100 - 3.230 3.200 - 3.300 3.300 - 3.400

7.06

0.51

3.400

6.94

0.50

6.82 6.70

0.49 0.48

34.71

2.50

6.58

0.47

- 3.500 3.500 - 3.600 3.600 - 3.700 3 . m - 3.m 3.800 - 3.9X

8.46

0.46

3.900

@.OlO

0.920 0.930

0.940

o.pc0 0.950 0.960

0.970

- 0.940 - 0.950 - 0.950 - 0.960 - 0.970 - 0.980 1.ooo

- 1.000 I .ow - 1 .CQO 1.010 - 1.020 1.020 - 1.mo 1 .a30 - 1.040

0.950

1.040

1.m

1.050

-

1.050

- 1.050 -1 .w

42.39

3.04

7.95

0.57

7.78

0.56

7.62

0.55

- 1.070 1 .em - 1 .om 1.060 - 1 .ow

6.35

9.46

6.23

0.45

1.092

6.12

0.44

1 .OM)

-

1.100

- 1 .loo 1.100 - 1.200 1,200 - 1.300 1.300 - 1 .400 1.400 - 1.500 1.100 - 1.500 1.050

1.500 1.600 1.700

-

-

3.000

4.000

%

6.02

0.43

5.21

0.37

36.02

2.59

4.53

0.33 0.28

3.95 3.47 3 .cB

0.25 0.22

2.70

0.19

17.71

1.27

2.39 2.13

0.172

1 .w

0.137

1 .m

0.122

1.53

0.1 10

0.153

9.65

0.894

1.38

0.099

1.24

0.W9

1.12

0.W

1.02

0.073

0.m

0.067

5.69

0.409

31 .74

2.28

4.200

- 4.000 - 4.100 - 4.2'20 -430

55.16

3.97

4.300

4.400

0.61

0.044

45.P

3.25

4.400

- 4.500

0.57

0.041

36.38

2.62

29.34

2.1 1

3.45

0.248

0.52

0 -037

0.48 0.-

0.m2

186.11

1l.W

3.500 4.000 4.100

-

4.500

- 4.5CG - 4.600

4.600

4.7W

4.000

-

0.64

0,060

0.75

0.054

0.68

0.049

0.035

1.700

19.61

1 .41

4.000

G.41

0.030

1 .eco

16.22

1.17

13.53

0.97

11.37 84.53 9.61

0 .e2 6.00

4.90G 4.W 5.000

5.000 5.000 !!.OX!

0-38 2.24 2.55

0.027 0.161 0.183

7.000

1.34

0.093

8.18

C .59

1.75 0.n

0.126 0 .C52

7.00

0.50

- 1.900 1.900 - 2.000 1.500 - 2.wo 2.000 - 2.100 2.100

- 3.500

Wrn-*AA-'

- 4.800 -4.W

1.600

1 .8W

2.200

2.400

-

s,,

S,

0.200

- 2.300

P .m

1.72

0.69

4.700

6.002 -

7.0 11 .O

11.0 30.0

l c m to 3Gm

510-!'

4

10-13

876

Appendix 5

APPENDIX 5 Solar Irradiance at Sea Level on an Area Normal m

= 2,

to the Sun for

So = 1322 Wm-=

.301

.1n

.61

1168

1-11

.302

.342

.62

1165

1.12

69.9

1.62

194

303

.647

.a

1176

1.13

98.3

1.63

189

126

1.61

198

.304

1.l6

.64

1175

1.14

lag

1.64

184

.305

1.91

.a

1173

1.15

216

1 .a

173

.306

2.89

.66

1156

1.16

271

1.66

163

.307

4.15

.67

1160

1.17

328

1.67

159

.308

6.11

.68

1149

1.18

346

1

.309

8.38

.69

978

1.10

344

1 .do

139

.m

1108

1.20

373

i.m

132

.a

145

.310

11 .o

311

13.9

.n

1 om

1.21

402

1 .n

124

312

17.2

.72

832

1.22

431

1.72

115

.313

2:

.o

.73

955

1.23

420

1 .73

105

.314

25.4

.74

1041

1.24

387

1.74

97.1

315

30.0

.75

867

1.25

328

1.75

83.2

.316

34.0

.76

566

1.26

31 1

1.76

58.9

.317

39.8

P77

968

1.27

381

1 .n

38.0

18.4

.318

44.9

.?a

9c7

1.28

382

1.78

319

49.5

.W

923

1 .a

346

1 .m

,320

54.0

.a

857

1.30

264

1.80

.a .a

do8

1.31

208

1 .el

801

1.32

168

1.82

115

1 .83

.33

101

.83

863

1.33

.34

151

.84

858

1.34

58.1

1 .a4

55

188

.8J

t?aQ

1.35

18.1

1.85

5.m ,920

-

--

877

Appendix 5

APPENDIX 5 (continued)

1.36 1.37

.36

233

.86

813

.37

279 336 337

.87

798

614 51 7

1.38

480

1.40

.660

-

-

1.86 1.87

1.88

-

-

*40

470

-88 .89 90

.41

672

.Ql

375

1.41

1.91

1.91

.42

7"3

.92 .83

258

1.42

3.72

1.92

2.34

7.53

1.83

3.m

.94

278

1.44

13.7

1 .w

.95

487

1.45

P.8

1.95

-98

504

1.46

30.5

1.96

31.7

.97

633 645 643 630

1.47

45.1

1.97

37.7

1.48

83.7

1 .Qa

1.49

128.

1.50

157

1.99 2.00

B2.6 I .50 2.66

.38

.39

.< .42

.45 A6 .47 .48

.49 .50

.51 .52

7e7

911 1,006

.

149

1.39

1.43

-

1.89

1 .m

.705

5.30 17.7

1080 1 I38 1187 1210 1215

1 .oo

1206

1.01

620

1.51

187

2.01

19.0

1199

1.02

1.52 I .53

209 21 P

2.02

47.6

2.03

.98

.W

.53

1188

1.03

610 601

.54

1198

1.04

592

1.54

226

2.04

55.4 54.1

.55

1190

1.05

551

1.55

P I

2.05

38.3

.58

1182

56.2

519

1.08

512

1.58

1.09

514

1.59

21 7 213 209 205

2.06

1178 1168 1161

I .06 1 .o'I

1.56

.57

1167

1.10

252

1.80

202

.58 .59 -60

526

1.57

2.07

77 .O

2.08

88.0

2.09

86.8

2.10

85.6

2.1 1

84 .4

2.12

83.2

2.13

zp.7

2.14

-

APPENDIX 6 Energy Distribution in the Spectrum of Solar Radiation for a Perfectly Clear and Dry Atmosphere P -

P Po

=I

Po

cal/cm2min)

P

- =0.13

= 0.25

Po

A+ m- I

0.29- 0.40 0.40 0.51 0.51 0.61 0.61 0.71 0.71 0.81 0.81 0.91 0.91 s.00 1.00 1.20 1.20 1.40 1.40 1.60 1.80 1.60 1.80 2.00 2.00 5.00

-

0.29 - 5.00

80 255 240 214 1 74 139 101 173 116 76

m=2

m=3

m.4

m=l

m=2

m=3

45

26 166 190 187 164 135

16 133 169

126 301 259 222 178 141 102 1 74 117 76

105

204 247 217 177 140 1 02 174 117 76

89 289 237 21 1

53

37

a5 21 4 201 170

137 100 172

99

114 76

17l 113 76

53

53

53

37 117

37 11 7

37 117

1775

t641

1766

1628

177 160

133 97 170 111 76 52 37

m=4

m-l

n=2

m=3

TI

136

31 1

121 300

282

254

224

140 101 173 I1 6 76

2s4 226 203 1 74 139 101 173 116 76

220 im 141 102 1 74 4’7 76

53

53

53

53

53

37 118

37 118

37 118

37 118

am

1891

1861

-

-

175

117

118

37 118

1 534

1448

1 904

1847

1 ?95

1747

1525

1440

1834

1837

1786

1 743

im 141 102 174 117 76

-

m=4

IL29

-

APPENDIX 6 (continued) P -

P -

= 006

0.29- 0.40 0.40 0.51 0.51 0.61 0.61 0.71 0.71 0.81 0.81 0.91 0.91 t -00 1.00 1.20 1.20 1.40 1.40 1.60 1.60 1.80 1.80 2.00 2.00 5.00

-

0.29

-

5.00

-

= 004

143

133

125

119

147

138

31 4

310

3c5

P64

253

316 256

313 262

21 9

z 5

223

132 31G 257 221

;n

253 2?2 178

301 247 216

IE

1%

1 73

:m

178

141

2225

:41

111

14:

;4:

1c2

It2

1c2

174

:74

;74

I17

117

117 75 53 37

1 c2 174 117 76 53 37

141 102 1 74 ?I7 76 53 37

141 102 1 74 1:7 76

118

:1e

1EX

ien

76

73 53

53 57 ;I€

17 1lf

19i3

:3%

= 0.02

PO

PO

PO

:32 174

1 :7 76

143 31 4

137 312

268

147 31 '5 266

263

261

?26

225

2?4

171

IW

:79

:42

142

142

222 179 142

:M

102

1:2

174 117 76

1 74

11.7 76

1 74 117 76

53

53

53

53

37 118

37 118

37 118

37 118

I963

1952

1942

1930

126

153

X6

31e

253

21 9 178 141 1 c2 1 74 117 7'5

53

53

37

37

:18

118

118

53 37 11 8

1352

1933

1916

1QGO

1:2 1 74

117

m

APPENDIX 7 Calculation of the Intensity of Direct Solar Radiation in an Ideal Atmosphere A * CL

So,in

% of

So

03 C22-024

0 02

0 4

0 07

1 4

0 24

0,11

2.2

C.26 0.28 0.30 0.32

- 0 26 - 0.28 - 0.30 - 0.32 - C.34

0.42 0.44 C.46

0.48

0.50 C.52

6

8

10

00

0

1.03

20.4

3 -8

0.7

0.1

c .o

0.0

0.0

0 .o

0 .o

1.54

30.5

12.4

5.0

2.0

.G.8

0.3

0.1

0.c

0 .o

1.67

33.1

17.4

9.3

5 .O

2.7

1.4

0.7

c .2

0.1

1 .E2

35.0

2: .9

13.3

6.1

4.9

3.0

1.8

0.7

0.2

1.74

34.5

23.1

15.5

10.4

7.c

4.7

3.2

1.4

0.6

c.40

3.E

17e.5

w .8

43.0

25.6

15.4

3.4

5.8

2.3

0.9

C .42

2.67

52.3

3E .2

27.6

2C .G

:4;5

10.5

7.6

4-0

3

0.44

2.70

53.5

41 .o

31.4

24.1

18.5

i4.a

10.9

6.4

2.1 3.8

0.46

3.1

61.4

49.2

39.4

31.6

25.3

20 ;3

16.3

10.5

6.7

4

0.48

3.:

61.4

51 .O

42.3

35.1

29.2

24.2

2tj.l

13.8

9.5

2.9

57.4

48.8

41.6

35.4

x -2

55 .?

22 .e

19.9

t!&

2.6

55.4

48 .O

41.6

36.0

31.2

27.C

23.4

17.6

13,2

2.7

53.5

46 .E

41 .o

35.9

31.4

27.5

24.1

18.5

14.2

2.8

55.4

43 .O

43.4

38.4

34.0

30.1

26.6

20.8

16.3

2.7 2.7

53.5 53 ;5

47.4 48.1

42 .O

37.2 36.6

33 .O 34.3

23.2 31 i 3

25.9 2861

20.3

43.2

15.9 16.2

- 0.50

- 0.52 - C.54

- 0.56 0.56 - 0.58 0. 58 - 3.6C 0. 60 0.62 - 0 . a c. - C.65 c. 55 - C . G

0.54

0.82

F A

c. 6E

5

6.1

-C.40

0.4C

4

14.3

0.38

-

3

0.72

0.36 0.22

2

0.31

- C.36 - 0.3e

0.Z4

I

- G.7c

22.8

2.5

493

44.7

40 :4

365

35 ;O

23 if3

8669

22 ic

1e10

"4

<7.5

43.8

3c .4

27.2

34.3

31.5

29.1

24.7

a .9

2.4

47.5

A .3

41.3

36.5

35.9

33.5

31.2

27.1

23.5

2.2

43.5

i1.2

36.9

36.8

34.6

32.9

31 .l

27.8

24.8

2.1

41.6

39.6

37.7

36.0

34.3

32.7

31.2

28.3

25.7

* R'

r

.^

_ . 7 .

- c.7:

C.72 -:0.74

- C.76

C.74

- 0.78 c.78 - C.80 C.4C - 0.8C C.EC - 0.85 G.65 - C.93 0.90 - 0.95 0.35 - 1.00 1-00 - 1.50 1.x - 2.00 0.76

2.cc 2.50

3-00 3.50 4.cc

5.cc 6.CC

-

5: 7.5

453.5

400.5

354.4

33.c

37.3

$6.3

30.6

zB.l

37.e

35.3

35 .c

33.3 32.6

31 .9

1 .9

34.7 33 .t

31.5

30.4

1 .e

35,E 33.7

34.6 32.9

33.7 32 .I

32.6

31 .o 30.5

31 .O 29.7

30.2 29.0

28.3 28.6 27.6

I .7

4El.l'

1.7

33.7

32.9

32.1

4E .3

267.E

55.7

761.5

3 -8

75.2

3.4

67.3

3 .O

53.4

2.E 16.7

55.4

73.6 53.3 58.7 54.8 329.4 124.0 53.5 26.3 14.5 8.3 8.9 4.6 2.4 2.8

330.7

6.27

124.1

2.m

2.70

53.3

3.iO

1.33

26.3

0.73 0.42

14.5

0.45

8.9

0.23

4.6

0.12

2.4

0.14

2.8

- 3.50 - 4.0C - 5.&0 - 6.00 - 7.00

7.00

5x23

2 .c

E7.€

- i.72

c.7i

8.3

280.3

224.3 25.8 26.3 27.1 26.2

31.3 31.4 931.5

30.7

30.0

29.3

28.0

26.7

613.5

554.6

503.9

42c.9

356.4

72.4

n .o

69.7

68.4

67.1

64.6

62.1

65.3 58.0 54.3 328.1 123.9 53.5

64.3

63.3

62.4

61.5

19.7

57.9

57.3

56.6

55.9

55.2

53.9

52.6

53.8 326.8

53.3

52,8

225.5

324.2

52.3 322.9

317.7

123.7

123.6

123.6

123.5

51 3 320.3 123!2

53.5

53.5

53.5

53.5

53.5

26 .3

26.3

26.3

26.3

26.3

26.3

26.3

14.5

14.5

14.5

14.5

14.5

14.5

14.5

8.3 8.3 4.6 2.4 2.8

8.3

8.3

8.3

E .3

8.3

8.9

8.9

8.9

8.9

8.9

4.6

4.6

4.6

4.6

4.6

2.4

2.4

2.4

2.4

2.4

2.8

2.8

2.8

2.6

2.8

8 -3

8.9 4.6 2.4 2.8

50.3

~

*

3

123.0 53.5

4

C.EO

0.22

-

-

42.12

a33.4

100.02

i3EO.l

~~2.3 1762.6

823.3 16E.6

818.4 1525.4

813.4

808.6

1 142 -3

1372.6

1313.5

1217.5

1114.2

803.8

794.3

784.9

Absorption by perrnonent gases

16.0

12.0

13.0

14.C

15.0

16.0

18.0

13.0

1512.4

1428.3

1357.6

1297.5

1135.5

1035.2

Rodiont intensitv ot the level

00 00 c

882

Appendix 8

APPENDIX 8 Mean Transparency Coefients at Different Atmospheric Mass Numbers for the Real Atmosphere

8

5

4

3

2

1.5

I

0,5

0.a9

0.930

0.925

0.920

0.NO

0.906

O.'rnO

0.932

0.922

0.917

0.911

O.po1

0 .A96

0.890

0.Fl.S

0.3%

0,915

0.910

0.203

0.892

0.M

0.m

o .e74

0.970

O.ilo8

0.902

0 .ex5

0.883

0.876

0.870

0.801

0.914

0.901

0 .a95

O.FB7

0 .874

0.867

0.W

c.e54

0.E8

0 .P34

0.887

0.879

0.865

0 .@5€

0.650

0.843

0.902

O.0RP

0.W

0.67l

0.C46

0 .&98

0.840

0.632

0 .cab

O.Rj1

0 .A72

0.662

0347

0 .&I3

O.WG

0.822

o.em

0.074

0.8%

0 .a54

0.W8

0 .a23

0 .Em

0.012

O.EfA

0.867

0.m~

0.F4'1

0.822

C .820

0.P10

0.8n1

0.698

0.678

@.W>O

0.850

0 .u38

0 .m

0.810

0.800

0.711

0.PZ

0.%2

0.042

0.80

0.811

0.730

0.m

0.X8

0.P45

0 .la4

0 .Fm

C .oc2

G .800 0.m

0.7Ho

o .r:ip

0.631?

0.81 4

0.7D

0.m

0.770

0.770 0.E39

0 .R53

O.M1

0.820 0.m9

0.KO

0.784

0.772

0.7~0~

0.748

0.t47

0.E24

0.812

0.7!8

0.775

b.702

0.750

0.738

0 .Ml

0.81 7

0 .PO4

0.W7

0.7GG

0.752

0.740

0.m

.m

0.610

0.W7

0,TE~l

0.757

0.742

0.730

0.718

0329

0.803

0.75%

0.i73

0.74b

0.73

0.720

0.707

.em

0.794

0.782

0.765

0.733

0,724

0.710

0.0w

0.817

0.703

0.774

0.756

0.730

0.7l4

0.700

0.m7

0.61 1

0.782

0.766

0.748

0.721

0.705

0.690

0.676

o.em

0.775

0.759

0.743

0.712

0.996

C.6w)

C.055

0.799

0.758

0.751

0.732

0.703

0.688

0.070

0.555

0 0

0.792

0.761

0.744

0.724

0 A94

0.677

0.160

0.644

0.m

0.754

0.736

0.71'3

0.585

0.0'57

@.&so

c.634

0.780

0.747

0.729

0.708

0.676

0 .858

0.840

0.6%3

0.774

0.740

0.721

0.700

0.667

0.648

0 .mo

0.613

0.7~3

0.733

0.7l4

0 .OR2

0.058

0.C38

0.620

0.602

0.752

U .726

0.708

0.884

0.549

0.629

0.610

0.5'92

0.756

o.no

0.03

0.678

0.640

0.5XJ

0.600

05e2

0.750

0.7l2

0.892

0.667

0.831

0.01 0

0.5'&

0.1171

0.744

O.Tc5

0.684

0.C53

G.031,

0.500

0.JdU

0.560

G .7J6 0.732 0.7?1

0.0&3

0.677

0.651

0.613

0.591

0.570

03 0

0.691 0.W4

0.669

0.604 0.595

0.582

0.560

0.540

0.582

0.643 0.m5

0.m

0.577

0.654

0.627

0.51!6

0.7l4

0.670

0.647

0.819

0.m

0 .a39

0.702

0.m

0.896

0.849

0.708

0.572

0.550

0.529

0.540

0.518

0.578

0.562 0.553

0.570

0S C B

C.511

o..pr,9

0.533

0.5m

0.498

0.532

0.602

0.5a

0.534

0.510

0.45

0 A24

0.5J4

0.591

0.924

0.500

0.477

Appendixes 9 and 10

APPENDIX 10

883

884

Appendix 11

APPENDIX 11 Absorption Coefficient k and Real (n,) and Imaginary (ni)Parts of Index of Refraction (Water)

x

104

2.20

1.293

2:tl

1.292

.@26

1.5

10-7

17

1

23

2.35

.CCOSdt

1.328

,067

4.80

1.282 1 .z16 1.270

30

0041

1-34 1.2 2.77

2.25 2.30

1.290

.021

:000290 .OW272 .COO304 .00042l

1.206

1.328 1.327

.020

.85

.a

.oo .fx 9

.a

-

10-7 10-7 10-7

2.40

.=

*Qs

1.327

.3l%

2.76 ' 1 0 4

2.421 2.50

.97

1.327

.a

9.55

104

2.55

1.260

1.oo

1.326

2.02

1.325 1.325 1.324 1.3235 1.323

104 1.10 104 1.07 ' lo4 104 1.66 7.32 104 9.9 104 9.74 104 0.a 104 8.8 104

2.60

1 .05

.3s ,131 .128 .190 .800

I .06 1.10 1.15 1.19 1.20

1.2s

1 .ma 1.30

I .35 1.40 1.45 1 .so

1.ti5

1.OM

1.321 13 2 0 1-320 1.319

1C 8

ta18 1.317 1.316

1.63

1.316

1 .&I 1.315

1.75

1.80 1.as

1.90 1.a4

1.95

1.05

1.323 1.322 1.322

1.50

1.70

"I

10-7

1w3m

-76

k

3.3

.006

.75

"2

1.49

1.330 1.323 1.329

.70

x

k

"2

1.315 1.314 1.312 1.311 1.309 1.307 1.307

.m .Pa

2.70 13. 26.0

-

-

11.17 29.0

80.5 114.0 110

5.0

1.m

69.3

10-4

3.40

.OooO66 .0000637 .000089l .0001146 .0001399 .001217 .00176 .001107

.001851

1.303

3.00'

.oooo789

83 97 99

5.26

3.20 3.30

.0OOog70

9d

3.10

lob

5.1 5.0 5.1 5 6 .4

3.00

14.48'

.00C2C65

.001189

S.5

-

.0001184

8.0

2.73

2.96

9.6 6.2

.0m2

61

41 1 468 431 x.8 229 276

2.625 2.65 2.70 2.w) 2.90

17.3

42

I .252 1.242 1.234 1.2l7 1.188 1.m 1.333 1.365 1.413 1.501 1.489 1.470 1.443 1.423 1.402 1.372 1.358 1.349 1.341 1.338 1.336 1331 1.318

-

lod lod

16 15.3

3.50 3.60 3.75 3.a 4.00 4 .so 4.68

4.80 5.0

109 131

.OOlPbe

.002048

.oom .002763

192

.004125

Qm

.01 9Bo5

3008

-06702

6784

.15656

7360

.1 7337

7040

.1W7

5594

,13800

3700 1650

. O W .04333

638

.018p8

334

.00930 .GO567 .00355 .W338

198

119 111 151

.ma .01472 .01736

.01647 .01225

.coos9 .01208 -03226

885

Appendix 1 1

APPENDIX 11 (continued)

x

“2

k

2.00

1 2304

68

.001082

6.0

1313

2138

2.05

1d C 2

41

.ocogrm

8-05

1.324

2326

2.1 0 2.15

1.300

26 19

.M04345 .Oc10325

6.4 6.5

1.347 1.338

7.0

1.3s

618

I

.2w

852

7Q4

.a

.ow3 .03456

19

1

20

1.722

25

1 .m

1877

90

I .652

1411

7.5

1.3m

579

8.0

1 .293

568

8.5

567

9 .o

1.286 1.289

.03603 .a760

566

.MOM

93

1

9.5

1.245

579

.04377

ss

1 A15

In 1

10.0

1.2l4

.05316

40

1.577

1194

2450

.a

10.5

1.185

668 g16

.06m

42

1.567

In1

11

1.151

1166

.ton

so

1.390

12m

11.5

1

1672

1.609

1.180

n91

.lSO .2092

52

12

60

1 .m

12.9

1.19

2451

.2438

Q

1.636

75 83

1

.la

.mo

245 939

13

1.220

2EZl

.a18

13.5

1.2%

14

1

2980 3215

3202 6562

100

1 .Em

665

,4298

117

1.942

54C

.408a

150

1.995

412

2800

,3355

152

1 .Qm

2 m Y

3,239

Po

2.025

.m

15

1330

9601

16

1.a0

381

.a

17.5

1

18

1.535

18.5

1 .do6

1.801

m

326

4

886

Appendix 12

APPENDIX 12 Absorption Coefficient k and Real (n,) and Imaginary (ni) Parts of Index of Refraction (Ice)

x

k

"2

A

"I

-

1.55

1.297 1 .298 1.295 1.294 1 .294 1 234 1 -294

.it 8.3 10-7 -25 1 . G . 1 0 4 .34 2.8 * 1 0 4 .30 2.5 * lo4 2 1 1.84' 10-6 .32 2.93' lod .96 9.16' 104 1.42 I .41 * 1.20 1.24' lo4 I .84 2.04' 10-5 10.2 1.18- lo4 .16.e 5 . s 10-4 53.0 6.41' lo4 47.4 5.83' 10-4

1.80

1.28

29.9

3.130- 1 0 4

1.65

1.293

s.1

1 .m

1.2925

1.75

I .a2

14.2 10.2

3.03' lo? 1.92' 1.42- lo4

1 .eo 1 .a

1.292 1.292

1 .so

1.292

1.95 2 .oo 2.05 2.10

1.291 1.2a

7.9 4.3 25.9 71.7 101 .O 85.8 5'2.1 30.5 17.6 1?.9

1.13' lo4 6.33' 1 0 4 3.91- 1 0 4 1.11' lo3 1.61- lo3 1.40' lo9 6.35' lo4 0.76' 104 3.08' 2.13' lo4

2.65 2.8 2.85 2.9 2.95 3 .O 3.05 3.075 3.1 3.15 3.2 3.25 3.3 3.35 3.4 3.45 3 -5 3 .xi 3.6 3.8 3.9 4.O

10.8

1.98- 1 0 4

4.1

1.356 1.340 1.32'1 1.316

10.7

3.50'

4.2

1 .SO7

0.S

I

.oo

1.03 1.05 1.10

1.302 1.302 1.301 1.301 1.300

1 .IS

1.2j9

1.20 1.25 1.30 1.40 1.45 1.50

1.2Gs

1.52

2.15

2.m 2.25 2.30

2.s

1.289

1 .?a8 1 .!Be 1 .s2 1.978

1 .?75 1.270

i04

2.40 2.45 2.50 2.56 2.60 2.625

k

7

29.9 36.2

5.7l * 1 0 4 lo4 7.07 7.m 10-4 8.60 lo4 8.02 ' lo4 8.35 lo-' 1.43 103 ,0129 .0353 .1014

"2

1.258 1.247 1 .z35 1.220 1.206 1.200

41.8 39.O 39.8

1.193

67.8

1.152 1.140 1.132 1 .I25 1.190

t .lo2 1.225

1.m 1-54? 1.557 1.550 1 .so

1.515 1.490 1.445 1.422 1.408 1.395

40.1

550 1557 4394

-

.

-

-

7690

.1r05

9522

.2273 .3178

13093 14008 13184 9979

,3428

-3252 .2502

6135 3479

.1562

2381 I649 1135

,0625 .0440 .0307 .0226

824

586 458 367 270 335 389

.OW

.0163 .012Q .0105

.0082 .0104

.0124

459

,0150

525

.elm

887

Appendix 12

APPENDIX 12 (continued)

x 4.3 4.4 4.3 4.6 4.7

"2

k

1 .ma

636 836

5.6

1.288 1.280 1.273 1.2M 1 .a8 1.252 1.247 1.241 1 .a6 1.231 1.221 1.226 1.226

5.7

1.226

5.6

1.227 1 .a2 1.2% 1 .a5 1.54 1 .a2 1 .za 1.226 1.z?5 1 .zw 1.222 1 .m 1.221 1 .a1 1.221

4.8 4.9 6.0

5.1 5.2 5.3 5.4 5 .5

5.9 6.0 6.05 5.1

9.2 6.3 6.4 0 .S

6.6 6.6

6.7 6.8 6.9

7.0

",

A 7.1

1.221

M 7

7.2

i .22l

Tu)

1 .a

609

I .219

580

1.217 1 -213

5; 0

.03w 3.69' 3.52' 10-2 3.alo-=

1.132

4iO

9.10'

10-2

1. l a

820

4.:s.

10-2

.OW0

7.5

.028?

376

.0213 .0173

8.0 8.1 9.O 9 -8 10

377 334

.014,'

321

-0130 .013:1 .0152

360 408 479

548

690 918 11% 1293 1341

fl,

-0218

783

322

k

.Me?

921

454

"2

.013;1

10.2 1o.s

.0617

11 11.5 12 12.5 13 13.1 15 17.5

.064C

20

.OM3

2s 30 35 40 14 52 63

.0173

.021 CI .m44 A313 .0424 ,0526

1323 1 2-

.om 6

1154

.OW

lorn

.w!6s

1 om

.M51

lot3

.W9

1085

.OS4

1.142 1.195 1.290 1.392 1.432 1.565 1 A12 1 A13 1.550 1.486 1.456 1.425 1.427 1.440 1. a1 1.490 1 .ma

1 .sm

520

720

6.02' 10-2 9.64' 10-2 12.20 1.14- 10-1 1260 1 .a'10" 1.19' 10-1 1190 1 0 ~ ~ 0 1.08' 10-1 870 9.35' 10-2 640 7.62' los 230 3.47' 10-2 160 2.56' 10-2 150 2.38' 10-e 223 51.25' 10-2 401) 1 .I 1' 10" 780 2.42' 10-1 1650 .6W 050 .2690 60G ,3008

toid0

1039

.m54

83

1.620

33C

UEZ)

.0535

1 .bso

126

ua

,0508

100 117

90

881

.0401

1.690 1 .m 1 .no

150

15e

.0473 .0453

.n80 . l a .027e

25

.02@

25

.0901

This page intentionally left blank

AUTHOR INDEX

Numbers in parentheses are reference numbers and indicate which of an author’s work is referred to. Numbers in italics show the page on which the complete reference is listed.

A Aagard, R. L., 697 (98, 99), 698 (98), 787 Adel, A., 117 (92), 134 (167), 137 (lo@, 147 (167), 155, 156, I59 Adem, J., 856 Angstrom, A., 55 (9, 83, 113 (91), 155, 197 (53), 200 ( 5 9 , 215, 845 Aizenshtat, B. A., 855 Albrecht, F., 468 (21), 533 Alexandrov, B. P., 43 (19), 48 Alisov, B. P., 676 (41), 784 Allen, C. W., 246 (44), 357 Allison, L. J., 765 (162), 790, 855 Alpatova, R. L., 858 Altovskaya, N. P., 191 (42a), 192 (42a), 214 Ambach, W., 526 (103b, 103c), 537 Ambartzumian, V. A., 20 (8), 23 (8), 30 (8), 47, 521 (93), 536 Anderson, A., 842 Andreyev, S. D., 5 (22), 6 (22), 48, 117 (169), 150 (132), 157, 159, 623 (107), 651, 846 Andreyev, V. D., 857 Andrianov, Yu. G., 852 An-Ti, Chai, 842 Antipov, B. A., 158 Antropova, U. I., 848 Archangelskaya, V. A., 246.(58), 357 Arking, A., 856

Armstrong, B. H., 841 Arnulf, A., 108 (70), I54 Artzbashev, E. S . , 416 (24), 417 (24), 450 Asakura, T., 557 (20), 647 Ash, S. U., 623 (106, 107), 651 Ashbel, D., 482 (44),534 Ashburn, E. V., 413 (2), 449 Ashcheulov, S. V., 5 (22), 6 (22), 48, 117 (169), 150 (132), 157, 159, 617 (103, 104), 619 (103), 651, 853, 855 Astling, E. G., 724 (127), 755 (158), 756 (158), 757 (158), 788, 790,856 Atlas, D., 186 (29), 214 Atroshenko, V. S . , 208 (65), 216 Atwater, M. A., 858 Avaste, 0. A., 150 (122), 157, 376 (13, 409, 417 (26), 450, 858 Averkiev, M. S., 167 (8), 213, 261 (75, 76), 324 (129, 130), 327 (129, 130), 347 (151), 352 (163, 164), 358, 360, 361, 362, 373 (ll), 409, 468 (22, 23), 533

B Babcock, H. D., 227 (9), 258 (9), 355 Babrov, H. J., 858 Badinov, I. Y.,5 (22), 6 (22), 48, 117 (169), 150 (132), 157, 159, 456 (7), 532, 623 (107), 651, 686 (69), 701 (69, l l l ) , 714 (69), 785, 787, 846, 859 Baker, D. J., 837

889

890

Author Index

Band, H. E., 728 (129a), 788 Bandeen, W. R., 751 (155), 752 (155), 753 (155), 758 (162), 759 (162), 789, 790, 850, 856 Baner, K. G., 686 (81), 786 Barashkova, E. P., 376 (14, 15), 380 (15), 409, 422 (37a), 444 (37a), 449 (37a), 451, 461 (16), 533, 565 (52), 567 (52), 568 (52), 578 (52), 648, 813 (32), 834 Barashkova, Y. P., 837 Barkova, G . N., 852 Barret, A. H., 852 Barrett, J. W., 852 Barry, R. G., 848 Barteneva, 0. D., 187 (36a), 212 (71), 214, 216, 532 (109), 537, 838 Barth, C. A., 841 Barth, R. E., 851 Bartkievich, V. S . , 847 Baryshev, V. A., 765 (169), 790 Bastin, J. A., 93 (19), 152, 624 (110), 651 Bater, M., 854 Bauer, E., 559 (35), 647 Bazhenov, V. A., 843 Bazhulin, P. A., 726 (128), 788, 851 Belayeva, I. P., 686 (74), 714 (24), 786 Beletzky, F. A., 332 (135), 361, 561 (40), 563 (40), 566 (40), 648 Beliayeva, I. L., 424 (37c), 451 Belinsky, V. A., 289 (102), 359, 846 Bell, E. E., 624 (111, 112), 651 Bellman, R. E., 852 Belov, P. N., 856, 858 Belov, V. F., 150 (117), 156, 191 (40a), 214, 416 (24), 417 (24), 450, 686 (73), 714 (73), 786, 856 Belton, M. J. S., 858 Benedict, W. S., 91 (10, ll), 152 Bennett, H. E., 108 (46), 153, 598 (88), 624 (1 13), 650, 651 Bennett, J. M., 108 (46), 153, 598 (88), 624 (1 13), 650, 651 Berezkin, V. A., 524 (99), 537 Berg, H., 483 (43), 534 Beritashvili, B. Sh., 844 Berkovich, L. V., 858 Berland, M. E., 464 (18), 465 (18), 466 (20), 467 (20), 533, 561 (43), 569 (57), 570

(58), 573 (58), 574 (58), 577 (58), 605 (92, 93), 648, 650 Berland, T. G., 338 (142), 340 (143), 361, 406 (39), 410, 463 (16a), 469 (24), 471 (28, 29), 474 (29), 533, 569 (56), 570 (58), 573 (58), 574 (58), 577 (58), 578 (56), 648,674 (38), 675 (38), 676 (38), 728 (138), 741 (138), 746 (138), 784, 789, 856 Bernhardt, F., 853 Berry, D. J., 843 Bertram, F.-W., 842 Beyers, N. J., 822 (46), 834 Bickert, A., 827 (62), 835 Bignell, K., 108 (65), 154, 624 (115), 651 Billings, W. D., 414 (16), 450 Birukova, L. A., 223 (5), 355,358,463 (17), 533 Bishop, B. C., 845 Black, J. N., 482 (40, 41), 534 Blackwell, M. J., 837 Blattner, W., 846 Blau, H. H., Jr., 840 Block, L. C., 728 (129a), 788 Bocharov, E. I., 151 (136), 157 Bogel, A., 347 (152), 361 Bogdanov, P. G., 347 (153), 361 Bogdanova, N. P., 605 (97), 606 (97), 650 Bogdanovich, G. P., 487 (67), 535 Boiko, P. N., 367 (2), 408 Boldyrev, V. G., 645 (162, 163), 654,850, 851, 852, 854 Bolle, H.-J., 108 (57), 154, 557 (19), 624 (122, 123, 124, 125, 126, 127,128,129), 647, 652, 842, 850 Bolsenga, S. J., 841 Bolz, H. M., 38 (17), 48, 567 (53, 648, 650 Borden, T. R., Jr., 859 Borhidi, A., 848 Borisenkov, E. P., 750 (149), 752 (149), 759 (149), 765 (167a), 168), 766 (167a), 777 (168), 779 (168), 789, 790,823 (53), 835 Boroboy, A. G., 200 (56), 215 Borzenkova, I. I., 848 Bossolasco, M., 657 (22), 783, 846 Bossy, L., 347 (150), 361 Bouricius, G. M., 848

Author Index Braslavsky, A. N., 649 Bratchenko, B. M., 855 Breen, R. G., 86 (4), 91 (4), 152 Brewer, A. W., 150 (131), 157, 700 (101, 102), 714 (120), 787, 788 Bricard, J., 108 (70), 154 Brinkmann, R. T., 841 Brooks, D. L., 108 (64),154, 807 (39), 818 (39), 819 (39), 834 Brooks, F. A., 533, 858 Brooks, J., 150 (128), 157 Brooks, L. F., 795 (2), 798 (2), 832 Brounstein, A. M., 158, 844 Brown, W. L., 837 Bruinenberg, A., 796 (3,832 Bryant, H . C., 841 Buchenkov, V. A., 837 Buck, A. L., 838 Budyko, M . I., 447 (34), 448 (43), 451, 569 (56), 578 (56), 648, 657 (15, 24), 674 (38), 675 (38), 676 (38), 728 (134, 137, 140), 739 (137), 750 (140), 783, 784, 788, 789, 845 Buettner, K. J. K., 854 Bullrich, K., 108 (162), 159, 171 (12), 182 (12), 213, 381 (25), 409, 838, 846 Burch, D. E., 95 (142), 103 (143), 108 (43, 59, 61, 67, 7 9 , 110 (43), 111 (43), 125 (43), 141 (59), 153, 154, 155, 158,624 (114), 651,844 Burdecki, F., 482 (42), 534 Burgova, M. P., 121 (96), 122 (96), 155, 456 (7, 8, 9), 532 Buring, R. F., 524 (loo), 537 Burroughs, W. J., 854 Businger, J. A., 692 (89), 698 (loo), 786,787 Buznikov, A. A,, 837, 844

C Calfee, R. F., 843 Callendar, G., 140 (109), 156 Cameron, R. M., 854 Carlton, H . R., 843 Carrier, L. W., 171 (13a), 213, 840 Cato, G. A., 893, 840 Chacko, O., 482 (59,534, 573 (64, 71), 649 Chambers, R. E., 848

891

Chandrasekhar, S., 31 (23), 47 (23), 48, 178 (21), 213 Chaney, L. W., 854 Chang, D., 634 (143), 653 Chapelle, E. L., 418 (28), 450 Chapursky, L. I., 701 (105), 787, 848 Charnel], R. L., 853 Chasmar, R. P., 10, 83 Chater, W. T., 847 Chayanov, B. A., 191 (42), 214 Chayanova, E. A., 839, 840 Chekirda, A. Z., 854 Cheltzov, N . I., 301 (lll), 302 (lll), 360, 464 (19), 533 Chernigovsky, N. T., 404 (37), 406 (37), 410, 426 (41), 451, 657 (27), 671 (34), 783, 784 Chikirova, G. A., 657 (9, 661 (5), 783 Chin Wan-cheng, 822 (47), 834 Chizhevskaya, M. P., 668 (31), 671 (32), 682 (42), 784 Chubb, T. A., 243 (36), 356 Chudaikin, A. V., 246 (58), 357 Chumakova, M. S., 560 (36), 565 (51), 604 (91), 647,648, 650 Cicconi, G., 657 (22), 783, 846 Clark, J., 827 (61), 835 Clark, M. A., 847 Clarke, D. B., 686 (83), 786 Clough, S. A., 844 Cochrun, B. L., 847 Coffeen, M . E., 227 (9), 258 (9). 355 Collinbourne, R. H., 837 Collins, B. G., 855 Collis, R. T. H., 839 Condit, H . R., 455 (3), 532 Conley, T., 846 Conover, J. H., 758 (161), 790 Conrath, B. J., 758 (162), 759 (162), 790, 850 Coulson, K. L., 848 Courvoisier, P., 50 (3), 65 (3), 83 Cox, A., 841 Craig, R. A., 820 (40), 834 Croom, D. L., 644 (159), 653 Culberston, M. F., 93 (20), 117 (20), 128 (20), 139 (20), 152, 551 (ll), 646 Curcio, J. A., 840

892

Author Index

Curnutte, B., 842 Curtis, A. R., 30 (12), 48 Czepa, O., 419 (31), 451, 560 (38), 647

D Dagnino, I., 657 (22), 783, 846 Danilchenko, V. Y., 340 (143), 361 Danzer, K. H., 108 (162), 159, 381 (25), 409, 838 Darkow, G. I., 707 (112), 787 Das, S. R., 456 (5), 532 Dave, J. V., 197 (Sla, 51b), 215, 842, 844, 847 Davies, J. A., 855 Davis, P. A., 104 (29), 105 (29), 153, 557 (22), 647, 750 (151), 763 (164), 764 (164), 789, 790, 810 (21), 825 (56), 827 (56), 833, 835 Daw, H. A., 108 (79), I55 Dayeva, L. V., 414 (lo), 417 (lo), 418 (lo), 450 Deacon, E. L., 855 Debaix, A., 134 (168), 147 (168), 159, 624 (118), 651 de Bary, E., 381 (24, 25), 409, 841 Dechambenoy, C., 838, 848 De Coster, M., 573 (65), 649 Deirmendjian, D., 407 (41), 410, 559 (33), 647 Delterne, R. W., 559 (34), 647 Detwiler, C. R., 246 (50), 249 (SO), 357 Dianov-Klokov, V. I., 845 Dietze, G., 840 Dirmhirn, I., 413 (8, 9), 415 (8, 9), 417 (8, 9), 450, 657 (14), 676 (14), 679 (14), 750 (150), 783, 789 Divari, N. B., 841 Dmitrievsky, 0. D., 150 (116, 117), 156 Dmitriyev, A. A., 551 (8, 9), 646, 852, 858 Dobbins, R. A., 841 Dobosi, Z . , 848 Dobson, G. M. B., 150 (131), 157 Dogniaux, R., 347 (154), 361,482 (38), 534 Dorian, M., 851 Doronin, Y. P., 765 (167a, 168), 766 (167a), 777 (168), 779 (168), 790, 852 Dovgyallo, Ye. N., 838

Draegert, D. A., 842 Draper, J. S . , 838 Drayson, S . R., 843, 854 Dreyfus, M. G., 643 (156), 653 Driving, A. Ya., 841 Drobyshevich, V. I., 846 Drozdov, 0. A., 676 (41), 784 Drummond, A. J., 482 (36), 534,532 (110), 537, 845 Drury, V. I., 368 (4), 409 Duffie, J. A., 846, 849 Dunkelman, L., 245 (42), 357 Duntley, S. Q., 521 (96b), 537 Dutton, J. A., 686 (81), 786 Dyachenko, L. N., 422 (37a), 444 (37a), 449 (37a), 451, 573 (67), 649, 728 (135, 136), 729 (135), 731 (135), 733 (135), 789, 856, 857 Dyer, A. J., 845

E Eddy, J. A., 179 (22a), 213 Edwards, B. N., 839 Edwards, D. K., 108 (a),153 Efimova, N. A,, 574 (72), 577 (72), 649, 657 (15), 673 (37), 677 (37), 783, 784 Eggers, D. F., 108 (42), 153 Egle, K., 414 (17), 450 Eguchi, M., 150 (123), 157 Eiden, R., 381 (25), 409, 840, 846 Eisenstadt, B. A., 561 (42), 562 (42), 616 (42), 648, 657 (2, 4, 7), 658 (2), 660 (2, 4), 664 (7), 665 (7), 680 (2), 682 (43, 44), 683 (2), 699 (2), 700 (44),742 (2), 782, 783, 784 Eisner, L., 624 (111, 112), 651 Elbert, D., 178 (21), 213 Eldrige, R. G., 838, 842 Elena, A., 846 Eliatberg, M. E., 144 (144), 158 Elliot, D. D., 847 Elliot, W. P., 813 (33), 834 Elnesr, M. K., 482 (45), 534 Elsasser, W. M., 93 (20), 98 (13), 108 (37), 117 (20), 128 (20), 139 (20), 152, 153, 551 (10, ll), 646 Elterman, L., 244 (38a), 357. 855

893

Author Index Elyashevich, M. A., 86 (l), 151 Engelman, A., 847 Epstein, E. S., 137 (106), I56 Esipova, E. N., 686 (72), 701 (72), 714 (72), 786 Espinola, R. P., 840 Estoque, M. A., 829 (69), 835 Evans, W. E., 839 Evdokimova, N. M., 686 (71), 701 (71), 714 (71), 785 Eufimov, N. G., 563 (48, 49), 564 (48, 49), 565 (48, 49), 569 (49), 648

F Falkenberg, G., 650 Faraponova, G. P., 289 (103), 359,686 (63, 64, 84), 691 (64),765 (171), 785, 786, 790 Farmer, C. B., 842, 843 Fedorov, Ye. K., 854 Fedorova, E. O., 186 (29a), 214, 842, 847 Fedorova, M. P., 485 (64, 65, 66), 489 (66), 535, 686 (48, 69), 700 (48), 701 (69), 714 (69), 784, 785 Fenn, R. W., 187 (30a), 214, 697 (97), 711 (97), 787 Fessenkov, V. G., 252 (63), 265 (79), 358, 838 Feugelson, E. M., 120 (93, 126 (93, 155, 191 (40),208 (65), 212 (70), 214, 216, 386 (29), 388 (29), 393 (29), 395 (29), 397 (33), 410, 804 (12), 829 (71), 833, 836 Filipovich, 0. P., 30 (13), 48 Filippov, G. F., 852 Fioccio, G., 839 Fivel, D. J., 95 (12), 96 (12), 152 Fleagle, R. G., 829 (70), 835 Fleischer, R., 657 (20), 783 Flocchini, A. E., 657 (22), 783 Flocchini, G., 657 (22), 783, 846 Foitzik, L., 83, 209 (67), 216, 263 (78), 283 (78), 358, 837, 840 Fomichev, A. A., 727 (129), 788 Forsch, L. F., 434 (56), 452 Foshee, L. L., 856 Foskett, L. W., 158

Foster, N. B., 158 France, W. L., 108 (59, 61), 141 (59), 154 Fraser, R. S., 847 Fridzon, M. B., 837 Fried, D. L., 838 Friedman, H., 246 (53, 54), 357 Fritschen, L. G., 695 (91, 92), 696 (91), 786 Fritz, H., 648 Fritz, S . , 301 (113), 303 (113), 360,421 (36), 422 (36), 440 (66), 442 (66), 443 (66), 451, 452, 482 (48, 49), 534 Funk, J. P., 419 (30), 450, 804 (7a), 813 (27, 28, 29), 832, 833, 834, 855 Furashov, N. I., 843 Fursenkov, V. A., 150 (118), 157 Furukawa, P. M., 842, 847

G Gaigerov, S. S . , 686 (49), 784, 807 (15, 16), 808 (15, 16), 833 Gainulin, I. H., 121 (96), 122 (96), 155 Galin, M. B., 829 (79), 836 Galperin, B. M., 313 (119), 360, 377 (18), 378 (18), 399 (33, 409, 410, 475 (30), 476 (30), 477 (30), 533, 561 (41), 576 (73), 648, 649 Gandin, L. S., 774 (175), 775 (175), 790 Caring, J., 107 (31), 108 (47), 153 Garrett, D. L., 246 (50), 249 (50), 357 Gates, D. M., 108 (53), 150 (125, 126, 127), 154, 157, 559 (31), 561 (31), 647,843, 846 Gavrilova, M. K., 404 (38), 410, 478 (32), 533, 849 Gayevskaya, G. N., 614 (69), 686 (48, 55, 67, 68, 70), 700 (48), 701 (67, 68, 70), 714 (55, 67, 68,70), 784, 785, 813 (25), 829 (25), 8.33, 859 Gayevsky, V. L., 43 (18), 48,422 (37a), 444 (37a), 449 (37a), 451,558 (27), 562 (49, 648, 686 (45, 46,47), 695 (43, 700 (45, 46, 47), 705 (43, 706 (47), 784, 848, 855

Gear, A. E., 93 (19), 152, 624 (110, 651 Gebhart, R., 851 Genin, V. N.. 158

894

Author Index

Geodakian, 0. A., 505 (72), 535 Georgi, J., 258 (69), 270 (69, 83), 358 Georguiyevsky, Yu. S., 843 Gergen, J . L., 763 (166), 790 Gershun, A. A., 434 (57), 452, 524 (99), 537 Gerson, N. S., 1 (20), 48 Gibson, F. W., 839 Giddings, J . C., 418 (28), 450 Gille, J., 829 (77), 836 Gillford, F., 187 (36), 214 Ginsburg, N., 844 Girard, A., 850 Girault, P., 686 (82), 786 Glahn, H. R., 851 Glasko, V. B., 852, 855 Glazova, K . S., 191 (40), 208 (65), 214, 216 Glovatsky, D. N., 727 (129), 788 Godson, W. L., 103 (16), 104 (16), 152 Gotz, F. W . P., 454 (2), 532 Goisa, N. I., 573 (68), 649, 657 (21), 686 (50), 783, 784 Goldman, A., 844 Golikov, V. I., 209 (68), 211 (68), 216 Golikova, 0. I., 413 (3, 6), 449 Golubova, T. A., 517 (83), 536,666 (28,29), 667 (29), 784 Golubitsky, B. M., 843, 844 Goody, R. M., 5 (21), 29 (lo), 30 (12), 48, 99 (14), 103 (161), 108 (34, 39, 54), 117 (39), 138 (107), 152,153,154,156, 159, 624 (119), 651, 823 (50, 51), 829 (66, 75, 76, 77), 834, 835, 836, 858 Gorbatova, E. S., 837 Gorchakov, G. I., 838, 839 Gordov, A. N., 345 (146), 352 (146), 361 Gorlenko, S. M., 332 (136, 137), 361 Gorodetsky, A. K., 852 Goverdovsky, V . F., 849 Goysa, N . I., 837 Grams, G., 839 Gray, E. L., 848 Gray, L. D., 624 (130), 652, 841 Gray, T. Y., Jr., 765 (167), 790 Green, A. E. S., 108 (73), 154, 155, 841 Greenfield, J . R., 209 (66), 216 Greenfield, R. S., 822 (47), 834

Griff, N., 843 Griggs, M., 108 (60), 154, 155 Grishchenko, D . L., 433 (53), 438 (53), 452 Grishchenko, M. N., 352 (165, 166), 362 Grishechkin, V. S., 456 (9), 532 Grosh, R. J., 829 (78), 836 Grum, F., 455 (3), 532 Grunow, I., 317 (122), 360 Gryvnak, D. A., 108 (67), 141 (59), 154, 844 Guermoguenova, 0. A., 839 Guermoguenova, T. A., 841 Guirdyuk, G. V., 853 Guliayev, B. I., 514 (80), 536 Gupta, M. G., 757 (160), 790 Gurvich, A. S., 644 (158), 653, 840, 852 Guseva, L. N., 235 (24a), 356, 532 (109, 11l), 537, 855 Gushchin, G. P., 147 (112, 113), 156

H Hanel, G., 846 Hales, J. V., 757 (159), 790, 804 (13), 833 Halev, M., 751 (155), 752 (155), 753 (155), 789 Hall, J. T., 845 Hall, L. A., 728 (131), 788 Hamburg, P. Y.,347 (148), 361 Hammermesh, B., 151 (133), 157 Hand, J. F., 150 (115), 156, 482 (50),534 Hanel, R. A., 751 (156), 789 Hanson, K. J., 701 (103), 787, 849 Harihara Ayyar, P. S., 847 Harris, E. D., 839 Harrop, W . J., 108 (53), 154 Harshbarger, F., 851 Hartmann, W., 317 (121), 360 Harvey, J., 108 (49), I53 Hathaway, C. E., 843, 844 Hebermehl, G., 837, 840 Heger, K., 840, 841, 846 Heidemann, M., 850 Henderson, D., 757 (159), 790, 804 (13), 833 Henderson, S. T., 847 Henschel, K., 573 (60), 648

895

Author Index

Herbold, R. J., 150 (126), 157 Hering, W. S., 859 Herman, B. M., 821 (43), 834, 839 Herman, R., 91 (10, l l ) , 152, 453 ( I ) , 532

Herzing, F., 148 (114), 149 (114), 156 Hewson, E., 301 (114), 304 (114), 360 Hickey, J. R., 845 Hicks, B. B., 845 Hilleary, D. T., 643 (156, 157), 653, 849

Hinterregger, H. E., 109 (163), 159,246 (46, 52), 357, 728 (131), 788, 846 Hinzpeter, H., 263 (78), 283 (78, 97), 83, 358, 359 Hirt, R. C., 846

Hitschfeld, W., 186 (29), 214, 827 (57, 61), 835

Hlopov, B. V., 727 (129), 788 Hmelevtzev, S. S., 197 (50, 51, 52), 215 Hodgkis, D., 847 Hodkinson, J. R., 209 (66), 216 Hoehne, W. E., 839 Holcke, T., 317 (124), 360 Holland, A. C., 838 Holm, T. S., 561 (44),648 Holmes, R. W., 517 (83a), 536 Holstein, T., 91 (9), 152 Honey, R. C., 839 Hoover, G. M., 844 Hopfield, R. F., 838 Horn, L. H., 755 (1581,756 (1581,757 (158), 790, 857

Houghton, J. T., 150 (119, 131), 157, 629 (134), 637 (145, 146), 652, 653, 700 (101, 102), 701 (104, 107), 787, 821 (41, 42), 827 (57), 834, 835, 842 House, F. B., 750 (154), 754 (154), 755 (154), 789 Hovis, W. A., Jr., 847, 851 Howard, J. N., 107 (31), 108 (43, 47, 48, 63, 76), 110 (63), 111 (631, 118 (76), 125 (63), 153, 154, 155 Howell, H. B., 839, 841 Huang Ching-Hua, 8-50 Hudson, R. D., 847 Hulburt, E. O., 420 (33), 451 Hvostikov, I. A., 172 (15), 213

i Indichenko, I. G., 413 (7), 450 Irvine, W. M., 842 Ivanoff, A., 521 (96a), 536 Ivanov, A. I., 838, 847 Ivanov, A. P., 840, 849 Ivanov, K. I., 519 (84, 85, 88), 536 Ivanov, L. A., 514 (79), 536 Ivanova, R. N., 843 Ivanov-Kholodny, G. S., 246 (58), 357 Iyer, N. V., 573 (71), 649

J Jacobowitz, H., 839, 841, 844 James, D. G., 850, 856 James, G., 643 (157), 653 Jensen, C. E., 853, 856 Jerlov, N. G., 520 (89, 90), 536 Jizmagian, G. S., 841 Johnson, D. R., 701 (106a), 787, 849, 853, 857

Johnson, F. G., 798 (5a), 822 (44), 832, 834

Johnson, F. S., 246 (43, 49), 250 (49), 258 (43, 49), 357 Johnston, R., 260 (71), 358 Jones, F. E., 10, 83 Jones, G. O., 93 (19), 152, 624 (110), 651 Joseph, G. H., 858 Judd, D. B., 456 (4), 532 Jurica, G. M., 856

K Kabanov, M. V., 158, 197 (48, 52), 200 (48, 56, 57, 58), 215, 841 Kagan, R. L., 205 (62), 216, 557 (21), 647, 657 (24), 707 (1 13), 777 (176), 283, 787, 790

Kagiwada, H. H., 852 Kalaba, R. E., 852 Kalitin, N. N., 133 (102), 156, 220 (4), 221 (4), 223 (4), 231 (4, 15), 232 (15), 233 (IS), 252 (61, 62), 292 (105), 310 (IlO), 331 (134), 335 (140), 355, 357, 359, 360, 361, 379 (19), 409, 420 (32), 451,

896

Author Index

458 (15), 459 (15), 460 (15), 461 (15), Kisseleva, M. S., 150 (116, 117, 118), 156, 462 (15), 467 (15), 478 (15), 479 (15), 157, 842 480 (33, 34), 507 (7.9, 509 (73, 533, Kivganov, A. F., 854 534, 535 Klemm, St., 854 Kamitani, K., 558 (25), 647, 853 Klinov, F. Ya., 857 Kamiyama, K., 150 (123), 157 Kluchnikova, L. A., 829 (74), 836 Kan, R. V., 849 Kneizys, F. X . , 844 Kano, M., 482 (53), 534 Knepple, R., 573 (66), 649 Kaplan, L. D., 18 (3), 47, 101 (15), 105 Knestrick, G. L., 840 (15), 108 (42), 152, 153, 633 (139, 140, Kniper, J. W., 852 141), 652, 844 Koehler, R. F., Jr., 852 Karol, B. P., 527 (105), 537 Kogan, S. Y., 191 (40), 208 (65), 214, 216 Karavayev, I. L., 852 Kohanenko, P. N., 158 Kamey, J. L., 839 Kolmakov, I. B., 840 Karpova, V. M., 854 \Koltzov, v. v., 533 Kartashev, A. V., 726 (128), 788, 851 Kondratyev, K., Ya., 5 (22), 6 (22), 9 (1, Kartzivadze, A. I., 347 (156), 362 2), 20 (2), 21 (4), 30 (13), 116 (22), 47, Kasatkin, A. M., 723 (122), 788 48, 107 (156), 117 (169), 119 (93), 121 Kasten, F., 113 (90), 114(90), 115 (90), 155 (96), 122 (96), 126 (98, 99, loo), 130 (99, 170), 131 (loo), 137 (105), 142 Kastrov, V. G., 150 (117), 156, 261 (72, (110), 143. ( l l l ) , 150 (132), 152, 155, 73, 74), 262 (74), 263 (74), 277 (90), 350 (161), 351 (161), 358,359,362,381 156, 157, 158, 159, 205 (63, 64), 206 (22), 409, 429 (51), 452, 686 (49, 51, (64),209 (64),216, 368 (9), 370 (9), 52, 53, 54), 687 (54), 689 (54), 690 (51), 376 (17), 377 (17), 409, 414 (10, 11, 784, 785, 807 (15, 16), 808 (15, 16), 12, 13, 14), 416 (13, 14), 417 (10, 11, 12, 13, 14), 418 (lo), 420 (13, 14), 423 833 Katayama, A., 557 (20), 647, 857 (38), 429 (50, 52), 435 (50), 436 (60, 61, 62), 437 (61, 62), 450, 451, 452, Kattawar, G . W., 838, 846 Katulin, V. A., 686 (84), 786 456 (7, 8, 9), 482 (58), 485 (58, 59, 60, 61, 62, 63, 64,65), 492 (64), 531 (108), Kazachevskaya, T. V., 246 (58), 357 Kazachevsky, V. M., 367 (2), 408 532, 534, 535, 537, 546 (4, 5 ) , 551 (15, Kazakova, K. P., 158, 844 16), 557 (23), 561 (a), 579 (16), 580 Kazdan, R. M., 836 (78), 585 (80, 81, 82), 592 (85), 599 Keister, M. P., Jr., 838 (82), 607 (99), 610 (10, 617 (103, 104), 619 (103), 620 (105), 623 (106, 107), Kellog, W. W., 818 (38), 834, 854 624 (132), 628 (133), 629 (133), 646, Kennedy, J. S., 763 (165), 790, 850, 855 647, 648, 649, 650, 651, 652, 686 (55, Kerker, M., 186 (29), 214 Khmelevtsev, S. S., 840 67, 68, 69, 70, 71, 72), 701 (67, 68, 69, 70, 71, 72), 714 (55, 67, 68, 69, 70, Khromov, S . P., 297 (106), 359 71, 72), 728 (134, 135, 136), 729 (135), Khvostikov, I. A., 243 (33, 34), 356 731 (135), 733 (135), 750 (148, 149), King, J. J. F., 108 (37), 153, 629 (135, 136, 752 (149), 759 (148, 149), 765 (167a, 137, 138), 631 (138), 652, 842, 851 168), 766 (167a), 777 (168), 779 (168), Kirillova, T. V., 422 (37b), 424 (39), 451, 785, 786, 788, 789, 790, 796 (3), 801 524 (loo), 537, 563 (47), 578 (74, 7 9 , 603 (90), 648, 649, 650, 657 (7, 9, 11, (6), 804 (9), 813 (9, 25, 34, 35), 829 (25), 832, 833, 834, 841, 842, 846, 847, 12), 664 (7), 665 (7, 9, ll), 666 (12), 674 (40), 701 (110), 783, 784, 787 849, 853, 854, 855, 856, 857, 858, 859 Kislyanov, A. G., 843 Kondo, J., 845

897

Author Index Koocherov, N. V., 56 (4), 83, 235 (20), 356, 563 (47), 648 Koptev, A. P., 427 (44), 440 (a),442 (44), 443 (44), 451, 686 (79, 714 (79, 786 Kopylov, N. M . , 313 (118), 360 Korb, G., 113 (90), 114 (90), 115 (90), 155, 851 Korb, H., 386 (30), 410 Korff, S. A., 151 (133), 157 Korolev, F. A., 86 (2), 93 (2), Koronatova, T. D., 191 (40), 208 (65), 214, 216 Korsak, R. S., 278 (92), 359 Korzov, V. I., 848 Koshelev, B. P., 197 (51, 52), 215 Kostyanoy, G. N., 538 (2a), 646, 686 (76, 77, 78), 695 (76, 77, 78), 705 (76), 707 (77, 115, 116), 714 (76, 77), 721 (121), 786, 787, 788, 844, 851, 852, 856, 858 Kovaleva, E. D., 573 (69,70), 649,804 (lo), 833 Kovetnikova, L. A., 839 Kox, S. K., 849 Kozhevnikov, N. I., 104 (28), 153 Kozlov, V. P., 638 (150), 653, 853 Kozlyaninov, M . V., 838 Kozo, H., 848 Kozyrev, B. P., 36 (16), 48, 108 (69), 154, 686 (84), 786, 837, 840, 844, 848 Krakow, B., 858 Krasikov, P. N., 569 (57), 605 (94,95), 648, 650 Krasilnikov, V. A., 187 (31), 214 Krasnopolsky, V. A., 847 Krassilshchikov, L. B., 413 (3, 4, 5 , 6), 449, 848 Krat, V. A., 165 (6), 188 (37, 38), 212, 214 Kraus, H., 657 (16), 783 Kreitz, E., 567 (54), 648 Krinov, E. L., 368 (5, 7, 8), 409, 413 (I), 422 (l), 449 Kroening, J. L., 839 Kronke, R. H., 840 Kropotkin, M . A,, 837, 848 Kruglova, A. I., 482 (37), 534 Krug-Pielsticker, U., 289 (104), 359 Krylov, P. A., 252 (66), 358

Kudinova, G. N., 850, 854 Kudriavtzeva, L. A., 150 (117), 156, 368 (9), 370 (9), 409, 436 (60), 452, 503 (68), 535 Kuers, G., 850 Kuhn, P. M., 108 (72), 154, 559 (29, 30), 647, 692 (84a, 85, 86, 87, 88, 89), 693 (87), 695 (90), 698 (loo), 701 (106a), 707 (1 12, 114), 710 (90), 786, 787, 849 Kuiper, G. P., 243 (31), 246 (49, 356, 357 Kunde, V . G., 645 (164), 654, 758 (162), 759 (162), 790 Kung-Chin-pen, 109 ( l a ) , 159 Kuprevich, N. F., 218 (3), 355 Kurilova, Yu. V., 852, 854, 858 Kurpayeva, Y u . V., 854 Kurtener, A. V., 43 (19), 48 Kushpil, V. I., 456 (6), 532 Kuzmin, P. P., 352 (167), 353 (167), 362, 427 (43), 434 (43), 451, 515 (81), 528 (107), 529 (107), 530 (107), 536, 537, 608 (loo), 650 Kuznetsov, A. P., 847 Kuznetzov, E. S., 120 (94), 155, 203 (60), 204 (61), 216, 538 (1, 2), 646, 792 (l), 832 Kuznetzov, V. P., 266 (80), 267 (80), 358 Kuznetzova, M. A., 191 (40),208 (65), 214, 216 Kyle, T. G., 837, 842, 843

L Labs, D., 246 (47), 357 Laktinov, A. G., 216 Landsberg, H. E., 480 (39, 534 Lane, J. A., 844 Langer, R. M., 108 (66), 154 Larmore, L., 108 (71), 154 Laue, E. G., 845 Lauscher, F., 435 (58, 59), 452 Lavrentyev, M. M . , 639 (152), 653 Lazarev, A. I., 853 Lazarev, D. N., 533 Lebedev, A. N., 849 Lebedeva, K. D., 837 Lebedinsky, A. I., 542 (3), 646, 727 (129), 788, 847, 850, 852

898

Author Index

Lelyakina, T . A., 852 Lenoble, J., 521 (95, 96), 536 Lenoir, W. B., 852 Lenz, K., 365 (l), 367 (l), 408 Leovy, C., 829 (65), 835 Leslie, F. E., 150 (129, 130), 157 Lethbridge, M., 850 Leupolt, A., 840 Levchenko, A. D., 850 Levine, J. S . , 856 Liao-Huai-che, 109 (164), 159 Lichtman, D. L., 657 (7), 664 (7), 665 (7), 777 (176), 783, 790, 836 Lienesch, J. H., 623 (108), 644 (160), 651, 653,850 Lieske, B. J., 813 (31), 834, 858 Ligda, M. G . H., 839 Lileev, M. V., 319 (127), 360 Limbert, D. W. S., 856 Lindholm, F., 86 (6), 91 (8), 152,482 (56), 534 Lindsay, J. C., 246 (57), 357 Lindzen, R. S., 829 (67), 835, 858 Linke, F., 373 (12), 374 (12), 375 (12), 409 Linton, R. C., 856 Liventzov, A. V., 724 (123), 725 (123), 788, 851 Livingston, P. M., 841 Livshitz, G. Sh., 188 (38a), 214, 838, 839 Lloyd, D. B., 843 Lloyd, J. W. F., 847 Lobanova, G. I., 847 Lof, G . 0. B., 846, 849 Lohnquist, O., 585 (83, 84), 650 Loh, L. T., 854 Lomakin, A. N., 852 Lombarde, F. A., 851 London, J., 728 (139), 745 (139), 746 (139), 747 (139), 752 (139), 789, 810 (19, 20), 833, 858 Longley, H., 301 (114), 304 (114), 360 Loposov, V. P., 845 Lopukhin, E. A., 504 (69, 70, 71), 505 (70), 506 (70), 508 (70, 76, 77), 509 (70), 510 (70), 511 (77), 535, 686 (56, 57, 58, 79, 80), 689 (56), 706 (57, 48), 785, 786

Lorenz, D., 852 Lubimova, K. S., 526 (103a), 537 Lugin, N. P., 235 (19), 356 Lugina, K. M., 422 (37a), 444 (37a), 449 (37a), 451 Lundholm, D. V., 98 (24), 152 Lvova, E. M., 686 (59), 785

M MacAdam, D. L., 456 (4), 532 McCaa, D. T., 842 McClatchey, R. A., 108 (62), 154,624 (130), 652, 857 McCulloch, A. W., 856 McDonald, D. V., 850 MacDonald, J. E., 112 (88), 155, 559 (32), 647 McDonald, J. H., 301 (113), 303 (113), 360 McDonald, R. K., 559 (34), 647 MacDonald, T. H., 482 (49), 534 McDougall, J. C., 856 Mackiewicz, M., 482 (54), 534 Maclay, G. J., 858 Makarevsky, N. I., 300 (109), 360,507 (74), 508 (74), 535 Makarova, E. L., 150 (121), 157 Makhotkin, L. G., 280 (94, 95, 96, 96a), 281 (94), 282 (94), 289 (99), 290 (99), 310 (95), 359, 380 (20, 21), 409 Malevsky-Malevich, S. P., 422 (37b), 451, 701 (108, 109, llO), 787, 853 Malkevich, M. S., 191 (40), 208 (65), 214, 216, 605 (98), 637 (147, 148), 638 (147), 639 (147), 645 (161), 650,653,686(84), 765 (170, 171, 172), 786, 790,829 (73), 831 (73), 836 Malkmus, W., 108 (45), 153, 845 Malkov, I. P., 765 (171), 790, 845 Maltzev, Y . V., 185 (23c), 213 Mamayenko, G. Ye., 857 Mamikonova, S. V., 349 (160), 362 Mamontova, L. I., 227 (lo), 297 (106), 355, 359 Manabe, S., 811 (22,23), 829 (22), 833 Mandelstamm, L. I., 171 (14), 223 Mandelstamm, S. L., 243 (32), 244 (32), 356

Author Index

Mani, A., 482 ( 5 3 , 534, 573 (64, 71), 649, 701 (106), 787 Manier, G., 113 (90), 114 (90), 115 (90), 155

Manolova, M. P., 368 (9), 370 (9), 409, 482 (58), 485 (58, 59, 60, 61, 62, 63), 534,535 Mantis, H. T., 711 (118), 788 Marchuk, G. I., 624 (131), 643 (131), 652, 847 Margenau, H., 86 (3,152 Markov, M. N., 724 (123, 124, 125, 126), 725 (123), 726 (124, 126, 128), 727 (124, 126), 788, 851 Marlatt, W. E., 853 Marmo, F. F., 847 Marr, G. V., 109 (165), 159, 845 Marshunova, M. S., 559 (28), 574 (28), 647, 657 (27), 671 (33, 34), 783, 784 Masov, I. P., 182 (23b), 213 Masterson, J. E., 839 Matee, C . L., 847 Mateer, C . L., 844 Matroshina, T. D., 137 (105), 156 Matulavicene, V. I., 415 (18), 450 May, E. C., 854 Mayot, M., 158, 827 (59), 835 Medvedev, V. S . , 246 (58), 357 Megaw, E. C . S., 187 (33), 214 Melkov, F. I., 849 Merrit, E. S . , 634 (143), 653 Merson, Ya., I. 724 (124, 125, 126), 725 (123), 726 (124, 126), 727 (124, 126), 788, 851 Mevers, G. E., 838 Mezenov, A. V., 108 (69), 154, 840 Miers, B. T., 822 (46), 834 Migeotte, M., 108 (74), 155, 839 Migulin, A. V., 852 Mikesell, A. H., 187 (313, 214 Mikhailin, I. M., 841 Mikhailov, G. A., 847 Mikhailov, V. V., 456 (9), 532 Mikirov, A. E., 179 (22b), 211 (69), 212 (69), 213, 216, 245 (41), 357, 840 Miller, D. H., 415 (21), 450, 728 (132), 788 Miller, S. I., 841

899

Milne, E. A., 24 (9), 47 Milyutin, S . N., 839 Minin, I. N., 200 (54), 215, 847 Miranda, H. A., 841 Mireles, R., 844 Mironova, Z. F., 414 (10, 12, 13, 14), 416 (13, 14), 417 (10, 12, 13, 14), 418 (lo), 420 (13, 14), 450, 456 (7), 532 Miroshnikov, M. M., 843 Mirtov, B. A., 245 (40), 357 Mirzoyan, L. V., 238 (25), 356 Mocker, H., 526 (103c), 537 Moller, F., 108 (82), 113 (89, 90), 114 (90). 115 (90), 155, 386 (30), 410, 551 (6, 7, 7a), 646, 740 (143), 789, 805 (13a, 14, 14a), 811 (22), 829 (22, 72), 833, 836, 856 Morikofer, W., 258 (67), 358 Mokievsky, K. A., 524 (101), 525 (101), 537 Monin, A. S., 765 (170), 790 Monteith, J. L., 444 (69), 452, 573 (62), 649, 657 (17), 783 Moore, S . E., 91 (10, ll), 152, 227 (9), 258 (91, 355 Moriguchi, M., 150 (123), 157 Morozkin, A. A., 750 (149), 752 (149), 759 (149), 789 Morozov, V. M., 245 (39), 357 Morris, R. T., 414 (16), 450 Moskalenko, N. I., 843, 844 Motoaki, K., 848 Mugge, R., 551 (6), 646 Muller, H.-G., 696 (93, 94, 95), 711 (93), 786, 787 Mukhenberg, V. V., 444 (70), 445 (70), 452, 499 (67a), 535, 657 (6). 662 (6), 672 (6), 783, 848 Mullamaa, Y.R., 437 (63a, 63b), 452 Murcray, D. G., 150 (126, 128, 129, 130), 157,843 Murcray, F. H., 150 (128, 129, 130), 157, 843 Murcray, W. B., 624 (120), 651 Murgatroyd, R. J., 823 (50, 52), 825 (52), 834, 835 Murk, H., 279 (85, 86), 281 (86), 283 (83, 358, 359

900

Author Index

Mussaelian, Sh. A., 854 Mustel, E. R., 20 (8), 23 (8), 30 (8), 47

N Nagel, M. R., 108 (46), 153, 598 (88), 650 Nardone, L. J., 847 Narovlansky, G . l., 551 (9), 646 Naumov, A. P., 843, 845 Nebolisin, S . I., 317 (126), 318 (126), 360, 457 (14), 458 (14), 533 Neckel, H., 246 (47), 357 Nedovesova, L. I., 126 (98), 156 Neel, C. B., 856 Nekrasov, Y . I., I58 Neporent, B. S., 94 (25), 113 (25, 86, 87), lSO(116, 117, 118), 152, 155, 156, 157, 842 Nesina, L. V., 424 (39), 451 Nesmelova, L. I., I58 Neuburger, M., 301 (112), 303 (112), 360 Neupert, W. M., 728 (130), 788 Neven, L., 108 (74), 155 Newell, R. E., 827 (63), 835 Newkirk, G., Jr., 179 (22a), 213, 839 Nicholas, G . W., 855 Nicolet, M., 258 (68), 259 (68), 260 (68), 347 (150), 358, 361 Nielsen, A. H., 91 (23), 152 Nielson, B. C., 851 Nijlisk, H. J., 126 (99, loo), 130 (99), 131 (loo), 143 (l l l ), 144 (141), 156, 158, 551 (16), 556 (18), 557 (23, 24), 579 (16), 647, 804 (9), 813 (9, 35), 833, 834, 852, 859 Nikitinskaya, N. I., 235 (23), 239 (23), 356 Nikolayeva-Tereshkova, V. V., 194 (46a), 215 Nikolsky, G. A., 246 (56), 357, 686 (55, 67, 68, 69, 70, 71, 72), 701 (67, 68, 69, 70, 71, 72), 714 (55, 67, 68, 69, 70, 71, 72), 785, 786, 846, 857, 859 Nikonov, B. N., 843 Nikonov, V. B., 218 (2), 240 (27), 355, 356 Nilson, T., 427 (49), 451 Noorma, R. J., 804 (9), 813 (9, 35), 833, 834, 852, 859

Nordberg, W., 758 (162), 759 (162), 790, 850, 851, 856 Norris, D. T., 845 Northend, C. A., 839 Novikova, N. K., 352 (167), 353 (167), 362 Novoseltzev, E. P., 413 (6), 449, 466 (20), 467 (20), 533, 853, 8.55 Nowak, W., 846 Nugent, L. J., 171 (13a), 213,839

0 Obukhov, A. M., 182 (32, 35), 214, 637 (149), 653 Occhipinti, A. G., 845 Odishaw, H., 243 (37), 356 Oetjen, R., 624 (112), 651 Ogneva, T. A., 657 (7), 664 (7), 665 (7), 783 Ohring, G., 815 (36), 817 (36), 818 (36), 820 (40), 834 Okujava, A. M., 527 (106), 528 (106), 537 Onishi, G., 110 (84, 85), 117 (84), 155, 803 (71, 832 Oppenheim, U. P., 844 Orvig, S . , 657 (18), 671 (35), 761 (35), 783, 784 Oser, H., 187 (30a), 214 Osipov, B. A., 823 (53), 835 Osterberg, C., 137 (106), 156 Otto, A. N., 414 (12, 13, 14), 416 (13, 14), 417 (12, 13, 14), 420 (13, 14), 450 Owens, J. C., 838 Ovchinsky, B. V., 521 (97), 537

P Pakhomova, L. A., 686 (77), 695 (77), 707 (77), 714 (77), 765 (171), 786, 790, 837 Palmer, C . H., Jr., 108 (77), 153, 155 Panin, B. D., 853 Panofsky, H. A., 850 Pape, D. C., 852 Pasternak, M., 856 Patalahin, I. V., 150 (117), I56 Patty, R. R., 844 Paulsen, H. S., 562 (46), 573 (46), 648,837 Pavlov, V. Ye., 839

901

Author Index Pavlova, E. N., 224 (6), 258 (6), 355 Pearson, B. D., 856 Pelevin, V . N., 517 (83b), 536 Penndorf, R., 823 (49), 834 Perelman, A. Y., 214, 838, 839 Pereslegin, S. V., 854 Perevertun, M . P., 415 (19), 450 Persano, J., 758 (162), 759 (162), 790 Petrova, L. V., 804 (12), 833 Pevzner, S. I., 531 (108), 537 Philipps, A. L., 347 (159), 362 Philipps, D. L., 640 (153), 642 (153), 653 Phillips, H., 853 Piaskowska-Fesenkova, E. V., 188 (39), 189 (39), 190 (39), 192 (39), 214, 265 (79), 358 Piatovskaya, N. P., 413 (9,440 (67), 449, 452, 686 (60, 61), 713 (60, 61), 785 Pisiakova, N. M., 526 (103), 537 Pivovarova, Z. I., 335 (139), 361,422 (37a), 444 (37a), 449 (37a), 451,514 (80), 536, 560 (37), 647, 657 (19), 783, 845, 849 Pkhalagov, Yu. A., 840 Plass, D., 846 Plass, G. N., 22 (5, 6, 7), 47, 95 (12), 96 (12), 97 (22), 99 (22), 100 (22), 103 (22, 140), 105 (17, 26), 106 (30, 33), 108 (68, 80,81,83), 117(81), 130(80),152, 153, 154, 155, 158, 159, 815 (37), 834, 838, 839, 846 Plechkov, V. M., 851 Pleshkova, T . T., 407 (40), 410, 657 (19), 783 Plyler, E. K., 843 Podolskaya, E. L., 610 (101), 650, 796 (4), 832 Pohl, W., 696 (96), 787 Pokasov, V. V., 840 Pokhunkov, A. A., 828 (64), 835 Poliakova, E. A., 197 (47), 212 (71), 215, 216, 227 (ll), 235 (22), 242 (22), 355, 365 Poliakova, M . N., 297 (107, 108), 298 (107, 108), 299 (108), 359,360 Pollack, J. B., 842 Polli, S., 527 (104), 537 Polyanova, Ye. A., 838 Pontier, L., 838, 848

Popham, R. W., 633 (142), 652 Popov, A. K., 829 (79), 836 Popov, S. P., 234 (18), 235 (18), 241 (18), 356 Popova, L. V., 414 (12), 417 (12), 450 Prabhakara, C., 759 (163), 760 (123a, 163). 761 (163), 788, 790, 851 Preobrazhensky, L. Y., 438 (46), 451 Pressman, J., 822 (49, 825 (45). 834 Prokh, L. Z., 813 (30), 834 Prokofjev, V. K., 83 Prokofyeva, I . A., 132 (101), 147 (101), 156 Pronin, A. K., 414 (15), 450 Prudhomme, A., 108 (50), 153 Purcell, J . P., 243 (30), 246 (50), 249 (50), 258 (43), 356, 357 F'yldmaa, V. K., 840

Q Queney, P., 838 Quenzel, H., 624 (124, 127), 652, 842

R Rabinovich, Y. I., 193 (43, 215, 235 (24a), 356, 848 Rachkulik, V. I., 424 (37c), 451 Raghavan, S., 846 Rakipova, L. R., 853 Rao, C. R. Nagaraja, 851 Rao, P. K., 724 (127), 750 (152, 153), 752 (157), 788, 789, 854 Raschke, E., 849, 850, 856 Rasool, S. I., 759 (163), 760 (123a. 163), 761 (163), 788, 790, 851 Rauner, Y. L., 667 (30), 784 Rautian, G. N., 457 (ll), 533 Raymond, F. W., 840 Razumov, I. K., 347 (149), 348 (149), 361 Razumova, T. K., 246 (58), 357 Regula, W., 108 (41), 153 Reifenstein, E. C., 840 Reines, F., 151 (133), 157 Reshetnikov, A. I., 848 Riabova, E. P., 347 (157), 362 Riazanova, L. A., 261 (75, 76), 358 Richardson, W. H., 517 (83a), 536

902

Author Index

Roach, W. T., 108 (39, 40), 117 (39), 153, 714 (119), 788 Robinson, G. D., 440 (68), 443 (68), 444 (68), 452, 551 (13), 646, 657 (23), 783, 812 (24), 813 (24), 833, 844 Rodgers, C. D., 804 (8), 827 (58, 60), 832, 835,843,845, 851,858 Rodhe, B., 845 Rodionov, S. F., 168 (9), 213, 224 (6), 242 (28), 258 (6), 355, 356 Ronicke, G., 710 (117), 788 Romanova, L. M., 386 (31, 32), 391 (32), 410 Rosenberg, G. V., 178 (22), 194 (46a), 213, 215, 243 (29), 356, 521 (94), 536, 686 (84), 765 (170, 171), 786, 790, 838, 840,841 Ross, Y. K., 376 (16), 409, 427 (48, 49), 451 Rubinstein, E. S., 676 (41), 784 Rudenko, S. I., 829 (79), 836 Rudnev, N. I., 667 (30), 784 Rusin, N. P., 340 (144), 342 (144), 343 (144), 361, 426 (42), 428 (42), 451, 475 (31), 482 (31), 533, 671 (36), 784 Ruzheinikova, Yu. V., 857 Ryazanova, L. A., 859

S Safonova, G. A., 144 (144), 158 Saiedy, F., 5 (21), 108 (54, 65), 48, 152, 154, 624 (115), 651, 844, 849, 850 Sakai, H., 153 Sakali, L. I., 657 (21), 783 Sakharov, M. I., 512 (78), 513 (78), 536 Sanderson, R. B., 844 Sandormirsky, A. B., 191 (42a), 192 (42a), 214 Sapozhnikova, S. A., 158,657 (3), 659 (3), 660 (3), 661 (3), 782 Sasarnori, T., 108 (35, 36, 58), 126 (35, 36), 128 (58), 129 (35, 36), 131 (35, 36, 58), 153, 154, 598 (87), 650, 858 Sastri, V. D. P., 456 (5), 532 Sato, T., 347 (158), 362 Sauberer, F., 565 (50), 567 (50, 5 9 , 648, 657 (13, 14), 676 (13, 14), 783

Savelyev, B. A., 200 (58), 215, 841 Savikovsky, I . A., 686 (62), 785 Savinov, S. I., 278 (91), 359, 360, 399 (34), 410 Savostyanova, M. V., 230 (14), 235 (14), 236 (14), 237 (14), 355 Schaper, P. W., 857 Schilling, G. F., 818 (38), 834 Schmidt, K.-H., 579 (77), 649 Schonmann, E., 454 (2), 532 Scholes, W. J., 845 Schiiell, W., 193 (43), 200 (43), 215, 258 (67), 358, 573 (65), 649, 837 Schulze, R., 457 (12), 482 (46), 533, 534 Schwalb, A., 850 Schweizer, W., 728 (131), 788 Scolnik, R., 245 (42), 357 Searle, N. Z., 846 Sedacheva, T. P., 158 Seeger, W., 845 Seeley, J . S., 150 (119), 157 Seidel, H., 840 Seits, W . S., 98 (24), 152 Sekera, Z., 841, 846, 851 Sekihara, K., 482 (53), 534, 848 Selyukov, N . G., 837 Sernenchenko, B. A., 846 Semenchenko, I. V., 421 (39, 451, 838 Senenova, I. V., 852 Senderikhina, I . L., 205 (64), 206 (64), 209 (64), 216 Serova, N. V., 701 (109), 787 Severny, A. B., 20 (8), 23 (8), 30 (8), 47, 218 (2), 355 Shablovskaya, V. A., 506 (73a), 535 Shabott, A. L., 858 Sharnilev, M. R., 724 (124, 125, 126), 726 (124, 126), 727 (124, 126), 788, 851 Sharonov, V . V., 259 (70), 358, 368 ( 5 , 6), 409, 532 (112), 537 Shaw, C. C. 150 (126), 157, 55? (31), 561 (31), 647 Shaw, J. H., 108 (75), 115 (139), 155, 158, 159, 624 (109, 114, 121), 629 (134), 651,652,842,857 Shekhter, F. N., 126 (97), 156, 551 (14, 17), 647, 804 (10, ll), 829 (68, 74), 833,835,836,844

Author Index Shepelevsky, A., 313 (118), 360 Sheppard, P. A., 108 (65), 154, 624 (115), 651 Shifrin,K.S.,35(15),42(15),48, 150(122), 157, 171 (lo), 172 (lo), 179 (lo), 185 (24, 25, 26, 27), 186 (10, 26, 28), 187 (30), 197 (47), 200 (54), 209 (68), 211 (68), 213, 215, 214, 216, 418 (27), 427 (27), 450, 532 ( l l l ) , 537, 558 (26), 605 (96, 97), 606 (97), 647, 650, 839, 840 Shiliakhov, V. I., 686 (65, 66), 699 (65), 705 (65), 706 (65), 785 Shishko, A. F., 674 (39), 784 Shishlovsky, A. A., 83 Shklovsky, I. S., 246 (51), 357 Shlyakhov, V. I., 482 (37), 534, 856 Shneerov, B. E., 744 (147), 789 Sholokhova, Ye. D., 847 Shubzova, V. G . , 324 (131), 326 (131), 360 Shuleikin, V. V., 521 (98), 537 Shuster, G. I . , 727 (129), 788 Shved, G. M., 30 (14), 48, 823 (54, 5 9 , 827 (54, 5 9 , 835, 858, 859 Shvidkovskaya, T . E., 850 Silverman, S., 91 (10, l l ) , 152, 847 Simmons, F. S . , 844 Singer, S . F., 851 Singleton, E. B., 95 (142), 108 (59), 141 (59), 154, 158 Singleton, F., 823 (52), 825 (52), 835 Sitnik, G. F., 150 (121), 157, 246 (48), 248 (48), 357 Sitnikova, M. V., 424 (37c), 451, 452, 482 (52), 534 Sivkov, S. I., 263 (38), 268 (82), 270 (84), 271 (84), 278 (82), 287 (98), 288 (98), 293 (82), 294 (98), 295 (82), 296 (98), 297 (107, 108), 298 (107, 108), 299 (108), 317 (123), 357, 358, 359, 360, 381 (23), 409, 438 (64), 440 (64), 452, 469 (26, 27), 533, 615 (102), 650 Skopintzev, B. A., 519 (86, 87, 88), 520 (86), 536 Skvortzov, A. A., 424 (40), 442 (40), 451, 657 (8), 665 (8), 783 Sloan, R., 624 (109, 121), 651 Smart, C., 182 (23a), 213

903

Smirnov, A. S., 182 (23b), 213 Smith, C. O., 846, 849 Smith, H. J. T., 93 (19), 152,624 (IlO), 651 Smith, R. A., 83 Smith, R. C., 838 Smith, W. L., 854, 857 Smoktiy, 0. I., 847 Smoliakov, P. T., 342 (149, 361 Snoddy, W. C., 847 Sobolev, V. V., 20 (8), 23 (8), 30 (8), 47 (24), 47, 48, 202 (59), 216 Sofronov, Yu. P., 852 Sokolova, V. S., 235 (21), 356 Sonchik, V. K., 158, 845 Sonechkin, D. M., 851 Sosnin, A. V., 840 Spankuch, D., 837, 840, 841 Specht, H., 853 Spurr, S. H., 417 (25), 450 Spytkin, A. V., 421 ( 3 3 , 451 Sreedharan, C. R., 701 (106), 787 Srinivason, V., 701 (106), 787 Stair, R., 226 (7, 8), 258 (7, 8), 260 (71), 355, 358 Staley, D. O., 692 (87), 693 (87), 695 (90), 710 (90), 786, 808 (18), 833 Staras, H., 187 (34), 214 Staude, N. M., 162 (1, 2), 165 (5, 7), 212 213 Stauffer, F. R., 153, 843 Steen, R. R., 839 Steinhauser, F., 317 (125), 360 Stevens, D. W., 813 (33), 834 Stewart, K. H., 728 (132), 788 Stone, N. W. B., 842 Strange, J., 751 (155), 752 (155), 753 (155), 789 Stranz, D., 374 (13), 409 Strezhneva, K. M., 843 Strickler, R. F., 81 1 (23), 833 Strobehn, J. W., 841 Strokina, L. A., 657 (24), 783, 848, 856 Strong, J., 137 (104), 156 Stroschen, L. A., 813 (31), 834, 858 Stroud, W. G., 751 (156), 789 Stull, V. R., 105 (26), 108 (68, 80, 81), 117 (81), 152, 154, 155 Stupnikov, N., 224 (6), 258 (6), 355

904

Author Index

Styro, D. B., 617 (103, 104), 619 (103), 620 (105), 623 (lM), 651, 813 (34), 834, 853, 855 Subbotina, Z. Y.,289 (101), 359 Sulakvelidze, G. K., 527 (lo@, 528 (lo@, 537 Sulev, M. A., 837 Suomi, V. E., 559 (30), 647, 692 (87, 88), 693 (87), 707 (112, 114), 786, 787, 849, 857 Surh, M. T., 854 Sutherland, G., 140 (109), 156 Suzuki, M., 848 Swensson, J., 108 (74), 155 Swinbank, W.C., 573 (63h 649 Szeicz, G., 657 (17), 783

T Takacz, L., 482 (51), 534 Takahashi, K., 557 (20), 647 T a m , I. E., 173 (17), 213 Tanaka, M., 558 (25), 647, 853 Tanner, C. B., 692 (89), 786 Tannahauser, I., 853 Tarassenko, D. A., 856 Tarnizhevsky, B. V., 333 (138), 334 (138), 336 (138), 361 Tashenov, B. Y.,838 Tatarsky, V . I., 637 (147), 638 (147), 639 (147), 653 Tavartkiladze, K. A., 573 (61), 649 Taylor, J. H., 108 (38), 153 Taylor, U. R., 853, 856 Temnikova, N . S., 384 (27), 386 (27), 410 Teller, M., 857 Ter-Markariantz, N. E., 419 (29), 429 (50, 52), 433 (54, 55), 434 (54, 53,435 (SO), 436 (61, 62), 437 (61,62,63), 438 (63), 440 (63, 65), 441 (63, 65), 450, 451, 452, 849, 855 Ternovskaya, K. V., 297 (107), 298 (107), 359 Teweles, S., 851 Thekaekara, M . P., 357 Thishstun, W . R., I58 Thompson, A. H., 808 (17), 833

Thompson, H. P., 855 Tihanovsky, I. I., 178 (18, 19, 20), 213 Tihomirov, V. S., 416 (22, 23), 450 Tikhonov, A. N., 639 (151), 653, 855 Tikhov, G. A., 368 (3, 4), 409 Tillrnan, J. E., 843 Tillotson, L. C., 841 Time, N . S., 852 Timofeeva, V . A., 520 (91, 92, 92a), 521 (91), 536 Timofeyev, M. P., 657 (7, 9, lo), 664 (7), 665 (7, 9, lo), 783 Timofeyev, Y. M., 107 (156), 152, 158, 624 (132), 652, 841, 842, 852, 854, 855 Timofeyeva, V. A., 839 Timokhin, V . I., 853 Tobin, M., 847 Tooming, H., 424 (47), 427 (47), 428 (47), 451, SO6 (73), 535 Toporetz, A. S., 83 Torletzkaya, V . V., 350 (162), 352 (162), 362 Tornheim, K. A., 562 (46), 573 (46),648 Toropova, T. L., 150 (120), 157 Toropova, T. P., 235 (24), 237 (24), 238 (24), 356 Totunova, G. F., 121 (96), 122 (96), 155 Touart, C. N., 859 Tousey, R., 243 (30, 35), 246 (43, 50, 55), 249 (50), 258 (43), 356, 357 Trifonova, G. I., 191 (42a), 192 (42a), 214 Trofimov, A. V., 526 (102), 537 Trubnikov, B. N., 859 Tubbs, L.D., 843 Tulupov, V . I., 727 (129), 788, 850, 852 Tverskoy, P. N., 163 (3), 212, 266 (81), 277 (81), 358 Tvorogov, S. D., 98 (154, 155), 105 (149), 118 (149), 144 (149), 197 (51, 52), 158, 199 (149), 215 Twomey, S., 641 (154), 642 (154), 653,839, 841, 851 Tyler, J. E., 517 (83a), 536, 838, 849 Tzarevskaya, A. A., 413 (4), 449 Tzutzkiridze, Y.A., 384 (24), 385 (26), 410, 482 (47), 534

905

Author Index

U Ukraintzev, V. N., 313 (118), 315 (120), 316 (120), 328 (120), 329 (120), 360, 469 (25), 533 Ulger, K., 381 (25), 409 Ulmitz, E., 373 (12), 374 (12), 375 (12), 409 Unsold, A,, 86 (3), 152 Utkina, Z . M., 836

V Vaisala, V., 600 (89), 650, 822 (48), 834 VanCleef, F., 850 Vand, V., 182 (23a), 213 Vand de Hulst, H. C., 171 (ll), 182 (ll), 213 Vanderwerf, D. F., 159 Van Mieghem, J., 656 (l), 699 (I), 742 (l), 782 Van Wijk, W. R., 695 (92), 786 Vasilevsky, K. P., 94 (25, 32), 95 (32), 113 (25, 32, 86, 87), 152, 153, 155 Vasilyeva, M. A., 592 (85), 650 Vassilevsky, A. M., 837 Vassy, A., 150 (124), 157, 840, 857 Vassy, E., 150 (124), 157, 857 Venkateshwaren, S . P., 482 (55), 534 Vershinin, 0. Ye., 36 (16), 48 Vetlov, I. P., 857 Viebrock, H. J., 701 (103), 787 Viezee, W., 104 (29), 105 (29), 153 Vigroux, E., 108 (Sl), 134 (166, 166a, 168), 147 (168), 154, 158, 159, 624 (118), 651, 827 (59), 835 Vikulina, Z . A., 649 Vinnikov, K. Y., 739 (142), 741 (144, 145, 146), 742 (144, 146), 744 (146), 749 (146), 789 Vinogradskaya, A. A., 858 Viskanta, R., 829 (78), 836 Vitsenko, G. V., 857 Voitko, N. V., 854 Volkenstein, M. V., 172 (16), 213 Voloshina, A. P., 857 Volz, F., 193 (a), 215 Vonder Haar, T . H., 849

von Essen, K. J., 840 Vowinckel, E., 482 (36), 534, 671 (35), 761 (35), 784

W Waldram, J. M., 194 (46), 215 Walsh, T. E., 843 Walshaw, C . D., 107 (18), 108 (78), 137 (78), 138 (78), 107), 152, 155, 156, 804 (8), 827 (58), 832, 835, 859 Wark, D. Q., 623 (108), 633 (142), 641 (155), 643 (157), 644 (160), 651, 652, 653, 844, 850 Warnecke, G., 765 (167), 790, 851, 855, 857 Watanabe, K., 135 (103), 137 (103), 156, 243 (30), 356 Watson, W. W., 86 (5), 152 Weickmann, H. K., 697 (97), 711 (97), 787 Weinberg, V. B., 347 (147), 361 Weldon, R. G., 413 (2), 449 Westphal, J., 624 (116), 651 Wexler, H., 482 (39), 534, 790 Wexler, R., 108 (5 9 , 154, 657 (25), 757 (25), 783 Wierzejewski, I., 50 (3), 65 (3), 83 Williams, D., 95 (142), 103 (143), 108 (43, 59, 61, 63, 67), 110 (43, 63), 111 (43, 63), 125 (43, 63), 141 (59), 153, 154, 158, 624 (109, 121), 651, 842, 843, 844 Williams, W. J., 150 (129, 130), 157, 843 Williamson, E. J., 624 (117), 651, 701 (104), 107), 787 Wilson, A. W., 714 (120), 788 Wilson, H., 246 (43), 258 (43), 357 Winninghoff, F. J., 724 (127), 788 Winston, J. S . , 750 (152, 153), 789, 849, 853, 856, 858 Woestman, J. W., 108 (56), 154 Wolff, M., 847 Wolk, M., 841, 850 Wood, R . C., 151 (134, 135), 157, I58 Woolley, R., 30 (ll), 48 Wright, P. J., 93 (19), 152, 624 (110), 651 Wyatt, P. J., 105 (26), 108 (80, 81), 117 (81), 130 (80), 152,155

906

Author Index

Wyrtki, K., 657 (26), 783 Wyszecki, G., 456 (4), 532

Y Yadar, B. R., 846 Yakovleva, E. A., 854 Yakovleva, T. A., 854 Yakushevskaya, K. E., 628 (133), 629 (133), 652,801 (6), 813 (25), 829 (25), 832,833 Yamamoto, G., 108 (35, 36, 52), 110 (84, 85), 117 (84), I26 (35, 36), 129 (35, 36), 131 (35, 36), 153, 154, 155, 551 (12), 558 (25), 573 (59), 598 (87), 623 (108), 634 (144), 644 (160), 646, 647, 648, 650, 651, 653, 803 (7), 813 (26), 832, 833, 842, 850, 853 Yampolsky, B. Y., 227 (12, 13), 355 Yanishevsky, Y. D., 49 (l), 83, 402 (36), 404 (36), 410, 524 (99), 537 Yarger, D. N., 821 (43), 834, 839 Yaroslavtzev, I. N., 231 (16, 17), 233 (16), 276 (89), 278 (93), 279 (93), 319 (128), 337 (141), 338 (141), 339 (141), 355, 359, 360, 361, 369 (lo), 384 (28), 404 (28), 405 (28), 409, 410, 415 (20), 416 (20), 417 (20), 427 (20, 45), 429 (20), 430 (20), 431 (20), 450, 561 (39), 565 (39), 648 Yastrebova, T. K., 837 Yatabe, Y., 150 (123), 157 Yates, H., 108 (38), 153 Yefimova, N. A., 506 (73b), 535, 857 Yegorenkova, G . S., 561 (41), 648 Yegorov, S. T., 644 (158), 653 Yelaghina, L. G., 151 (137), 157 Yelisseyev, A. A., 858 Yelovskikh, M. P., 21 (4), 47, 142 (110), 156, 649

Yessipova, Ye. N., 857 Yevnevich, T. V., 852 Young, C., 854 Young, J., 624 (112), 651 Yudin, M. I., 205 (62), 216 Yurguenson, A. P., 844

Z Zaitsev, V. A., 848, 852 Zaitseva, N. A., 851, 856 Zaitzev, G . A., 150 (116, 117), 156 Zakharova, A. F., 347 (155), 352 (155), 361, 514 (82), 517 (82), 536 Zander, R., 839, 844 Zavodchikova, V. G., 376 (17), 377 (17), 409 Zaytov, I. R., 413 (7), 450 Zdunkowski, W., 624 (127), 652, 757 (159), 790, 798 (5a), 804 (13), 832, 833, 850, 851 Zege, E. P., 841 Zeitin, G . H., 657 (7), 664 (7), 665 (7), 783 Zhevakin, S. A., 845 Zhvalev, V. F., 620 (105), 651, 813 (34), 834, 849, 855 Zolotarev, B. M., 842 Zubenok, L. I., 569 (56), 578 (56), 648,674 (38), 675 (38), 676 (38), 784 Zuyev, M. V., 561 (42), 562 (42), 616 (42), 648, 657 (2), 658 (2), 660 (2), 680 (2), 683 (2), 699 (2), 742 (2), 782 Zuyev, V. E., 98 (154, 155), 105 (149), 118 (149), 144 (144, 145, 146, 147, 148, 149), 158, 197 (48, 49, 51, 52), 199 (149), 200 (56, 57, 58), 215, 840, 841, 845 Zvereva, S. V., 289 (loo), 294 (loo), 295 (loo), 359

SUBJECT INDEX

A Absorbed radiation, see Radiation Absorption band, 98 ff statistical model of, 99f Absorption function, 16 ff Absorption of radiation, see Radiation Absorption spectroscopy, 144 f Absorptivity, 15 Actinometric instruments, 52 ff radiosondes, 692 f scales, 56 Aeroso1 attenuation, 194,243, 291 microstructure, 212 scattering, 179 ff Aircraft measurements instrumentation, 699 f radiation, 191 f Albedo, 411 f annual range, 430f balloon measurements of, 701 f daily variations, 416,427 f geographical distribution, 444 f integral in dependence on solar height, 426 measurements of, 62 clouds, 396 f, 440 f ice, 427 sea, 433 ff snow, 379,417,425 f soil, 423 f vegetative covers, 424 f water, 419,431 f

spectral, of the earth, 421 f surface, 16 Albedometer, 62 Angstrom’s compensation pyrheliometer, 52 Angstrom’s formula, 197 f, 570 f Angular intensity distribution, 37, 584 f Anisotropic scattering, 169 ff Annual totals and variations of radiation, see Radiation Atmospheric composition, 144 f Atmospheric gases absorption coefficient of, 15, 116 ff Atmospheric mass, 162 f Atmospheric optical thickness, 161 f, 200 Atmospheric spectral coefficient, 235, 263 Atmospheric thermal emission instruments for measuring, 77 f Atmospheric transparency, 234, 283 f annual variation of, 267 ff daily variation of, 283 f geographical variation of, 286 f Atmospheric windows, 116 ff Attenuation of radiation, see Radiation

B Balloon measurements instrumentation, 699 f radiation, 179 Berland‘s table, 574 Blackbody emission intensity of, 32 Boltzmann’s constant, 31

907

908

Subject Index

Boltzmann’s law, 26 Bouguer’s law, 17 Brightness, instruments for measuring, 65 f Broadening of lines natural, 86 due to collisions, 86 f due to Doppler effect, 86f Brunt’s formula, 572

C Carbon dioxide absorption by, 123f absorption spectrum coefficients, 124 Chappuis bands, 133 Charts, radiation Kondratyev and Nijlisk, 556 f Moller, 551 Shekhter, 551 f Yamamoto, 551,803 Chromosphere, 3 Circumsolar radiation, 274 f Cloud absorption by, 302 albedo, 391,396 attenuation by, 300 f effect of radiation on, see Radiation reflection by, 396 f size distribution, 210, 389 Coefficient absorption, 15 ff emission, 14 molecular conductivity, 829 scattering, 16 ff, 170 transparency, 235 f, 265 f standard spectral, 236 turbulent transfer, 830 Collisions frequency of, 29 Color temperature of, 4 Conductors thermal emission of, 40f Coronal gas, 3 Curtis-Godson approximation, 106 ff

D Daily totals and variations of radiation, see Radiation Dielectrics thermal emission of, 35 f Differential equations method of reduction, 203 f Diffusivity factor, 18 f Dipole emission, 173 Direct solar radiation, see Radiation Doppler profile, 92 f Downward atmospheric radiation, 7, 557 ff Dust effect on radiation income, 324 f, 599 f

E Edge effect, 55 Effective radiation, see Radiation Effective wavelength, 73 Einstein coefficients, 26 f Elsasser absorption band, 98 f coefficient, 14 Emission, line intensity, 91 Emission spectroscopy, 622 f Emissivity, 14 relative, 15 Energy, distribution of, effective radiation spectrum, 615 f solar spectrum, 218 f near the earth’s surface, 217 ff infrared, 231 f ultraviolet, 223 f visible, 229 f outside the atmosphere, 245 ff over wide intervals, 224 ff spectral, 7 f Eppley pyrheliometer, 63 f Equation radiation, see Radiation radiative flux divergence, 792 f transfer, 28, 43 ff, 539 f Equivalent line width, 97 Excited state, molecular, life time, 30 Exosphere, 3

F Filters, 72 f reduction factor, 74

909

Subject Index Flux of radiation, 6, 12 Forest net radiation of, 666 f radiation climate of, 512 f Funk's net radiometer, 68 ti

Global radiation, see Radiation Goody absorption band, 99 f Grass integral emissivity, 43

H Hartley bands, 132 Heat exchange, turbulent, 8, 30 Herzberg bands, 132, 134 Hopefield bands, 134 Huggins bands, 133

I Ice albedo, 427 attenuation coefficient, 521 trasparency, 526 f Illumination instruments for measuring, 65 Infrared radiation, see Radiation Infrared spectral region, 8,229 f absorption due to ozone, 135 f due to water vapor, 116 f radiation composition, 617 f spectrometer, 78 f Isotropic radiation, 19 ff

K Kastrov's formula, 278 Kirchhoff's law, 22 ff

L Line isolated, 86 f shape of, 86ff width of, 89 Line broadening, 86 ff Line intensity, 91

Linear molecules, 123 Linear law of absorption, 97 Linke's formula, 272, 585 Linke-Feussner actinometer, 59 Local thermodynamic equilibrium, 24 f Longwave radiation, see Radiation Lorentz profile, 91 Luminosity, 10

M Makhotkin's formula, 280 Maxwell's distribution, 26 Mean free. path, 29 Mesosphere, 823 f Metals absorptivity, 40 Mie's theory, 181 f Mihelson actinometer, 56 f Molecular scattering, 169 Moll-Gorczynski solarimeter, 63 Monochromatic brightness, 412 Monochromatic equilibrium, 25 Monthly totals of radiation, see Radiation Multiple scattering, 200 f

N Natural broadening, 86, 89 Natural illumination, 531 Net radiation, see Radiation Nonhomogeneous paths, 106 ff

0 Optical atmospheric thickness, 161 f Outgoing radiation, see Radiation Oxygen, absorption by in the far ultraviolet, 134 spectrum, 134 f Ozone, absorption by, coefficients for the ultraviolet, 132 f spectrum in the infrared, 135 f determination of the total content, 145 f

P Photosphere, 3 Planck's constant, 31 Planck's law, 30f

910

Subject Index

Polarization of light with Rayleigh scattering, 177 Pressure broadening, 94 f F'yrgeometers, 69 Pyrheliometric scale, 56, 258

Radiant energy, measurement, 49 ff Radiant flux, 12 intensity, 9 ff spectral distribution, 11 f Radiation absorbed, 7 absorption of, 85ff for equally spaced lines, 89 f for isolaled lines, 96 f for non homogeneous paths, 106f for overlapping lines pressure dependence on, 90, 115 ff selective, 86 ff diffuse angular distribution of, 368 f for clear sky, 366f for cloudy sky, 370 annual total, 406 f cloud effect on, 382f daily variation, 376 f measurement of, 60f monthly total, 402f spectral composition of, 363 direct solar absorption of, experimental data, 109 ff attenuation of absorption and scattering, 179,260 ff clouds, 300f ice, 526f water, 123,517 f annual variation, 309 f cloud effect on, 317 f daily variation, 317 f measurement of, 52 f balloon, 701 f monthly total, 331 f on slant surfaces, 342 f spectral composition, 245 f spectral distribution, 217 ff

effective, 7 cloud effect on, 599 formula for empirical, 565 f theoretical, 549 f measurement of, 66 f of slopes, 610f of the earth's surface, 7,559 f global annual total, 478 f annual variation, 474 f daily. total, 470 f daily variation, 461 f flux of, 457f geographical distribution of, 480 f measurement of, 60f balloon, 671 ff on slopes, 485f spectral distribution of, 453 f surface absorption of, 482 theoretical calculations, 463 f under vegetative covers, 502 f longwave, 116 f absorption of due to carbon dioxide, 123 due to ozone, 155 f due to water, 122 due to water vapor, 116 f outgoing, 742 ff satellite data, 750 f spectral composition, 728 rocket measurement of, 724 f measurement by aircraft, 191 f balloon, 179, 701 ff rocket, 179,226,250,724 f net, 7, 655 ff annual total, 668 f annual variation, 663 daily total, 662 daily variation, 657 f effect of forest, 666 effect of wetting on, 663 f formula for atmosphere, 656 earth-atmospheric system, 656 underlying surface, 539 measurement of, 66f

91 1

Subject Index balloon, 701 f of the atmosphere, 729f of the earth-atmosphere, 744 of slopes, 680 f vertical profiles of, in free atmosphere, 686 infrared rocket measurements, 724 f intensity of, 9 f reflected, 6, 540 scattering coefficient of, 16, 177 ff function of, 16, 170 f, 390 matrix, 409 on large particles, 180f on small particles, 171 f shortwave absorption of, 109ff measurement of, 52f thermal spectral distribution of, 617 f temperature of, 46,247 Radiation charts (See Charts, radiation), 551 f Radiation field, quantitative characteristics, 9ff Radiation income, planetary, 254 ff Radiative air cooling, 805 f Radiative equilibrium 815 Radiative flux divergence, 804 f Radiative heat exchange, 791 f Radiative heat totals, 327 ff Radiative heating, 805 ff Radiative transfer, 28 ff Radiative temperature changes, 794 ff Rayleigh atmospheric optical thickness, 238 f Rayleigh attenuation coefficient, 176 Rayleigh scattering, 171 f, 238 Reflectivity, 16 Refractive index, 163 ff Relative emissivity, 15

S Satellite data statistical methods for processing, 765 f Savinov formula, 312 f Savinov-hgstrom formula, 647

Savinov-Yanishevsky actinometer, 57 Scattering of radiation, see Radiation Schott filters, 73 Schulze net radiometer, 67 f Shortwave radiation, see Radiation Shuman-Runge continua, 134 Sky brightness, 179, 192, 369 f Solar constant, 252 f Solar corona, 3 Solar radiation, see Radiation Spectral distribution of radiation, see Radiation Spectral luminance of natural formations, 412 Spectral measurements, instrumentation, 77 f Spectrobolometric measurements data, 218 f Spectroscopic hygrometers, 151 Square root law, 97 Statistical band model, 99 f Stefan-Boltzmann’s law, 33 Stratosphere, 815 f Strong-line approximation, 99 Successiveapproximation, method of, 201 f Sun, 1 Sunspots, 3 Surface albedo, see Albedo Surfaces, natural, thermal emission of, 42 f

T Temperature effective, 4 of the earth-atmosphere system, 629 radiation, see Radiation stellar, 4 Terrestrial radiation, 538 f Thermal emission, 35 ff of conductors, 40f of dielectrics, 35 f of the earth’s surface, 4 2 f laws of, 22ff Thermal sounding Kaplan’s method, 633 f indirect, 624 f Thermodynamic equilibrium, 22 ff

912

Subject Index

Transitions, 25 Transmission function, 16 ff, 545 Transparency, 234 f Turbidity factor, 198,272 f, 293 f Turbidity index, 280 f

U Ultraviolet spectrum, 5, 8,223 f

V Vacuum ultraviolet, 8 Visible spectral region, 8 absorption due to water vapor, 109 Rayleigh attenuation coefficient, 176 f

W Water, absorption and attenuation by, 120 f absorption bands, 121 coefficients, 121 f

Water basins absorption and attenuation with, 517 f albedo, 431 f Water vapor absorption bands, 108 ff molecular structure, 109 scattering, 239 total content, 147f Waterstream pyrheliometer, 55 Wavelength color division, 8 Weak line approximation, 99 Wien’s displacement law, 34 Wolf number, 3 f

X X-ray emission, 250

Y Yanishevsky net radiometer, 68 f Yanishevsky pyranometer, 60 f