Advances in Geophysics, Volume 12

ADVANCES IN GEOPHYSICS VOLUME 12 Contributors to This Volume L. V. BERKNER P. CALOI JOHNA . DUTTON R. JOHNSON DONAL...

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ADVANCES IN

GEOPHYSICS

VOLUME 12

Contributors to This Volume L. V. BERKNER P. CALOI JOHNA . DUTTON R. JOHNSON DONALD E. B. KRAUS L. C. MARSHALL P. MELCHIOR CHESTERW. NEWTON

Advances in

GEOPHYSICS Edited by

H. E. LANDSBERG Institute for fluid Dynamics ond Applied Mothematics University of Maryland, College Park, Maryland

J. VAN MIEGHEM Royal Belgian Meteorological Institute Uccle, Belgium

Editorial Advisory Committee BERNARD HAURWITZ WALTER D. LAMBERT

ROGER REVELLE R. STONELEY

V O L U M E 12

1967

Academic Press

New York and London

COPYRIQHT

0 1967, B Y ACADEMIC PRESSINC. ALL RIQHTS RESERVED

NO PART OF THIS BOOK MAY B E REPRODUCED IN A N Y FORM

B Y PHOTOSTAT, MICROFILM, OR ANY OTHER MEANS, WITHOUT WRITTEN PERMISSION FROM THE PUBLISHERS.

ACADEMIC PRESS INC. 11 1 FIFTH AVENUE NEWYORK,NEWYORE10003 United Kingdom Edition Published by ACADEMIC PRESS INC. (LONDON)LTD.

BERKELEY SQUARE. LONDON,W.l. Library of Congreaa Catalog Card Number 52-12266

PRINTED IN THE UNITED STATES OF AMERICA

LIST OF CONTRIBUTORS

L. V. BERKNER, Southwest Center for Advanced Studies, Dallas, Texas P. CALOI,Istituto Nazionale d i Geojisica, University of Rome, Rome, Italy

JOHNA. DUTTON, Department of Meteorology, The Pennsylvania State University, University Park, P e n wylvania DONALD R. JOHNSON, Department of Meteorology, The University of Wisconsin, Madison, Wisconsin

E. B. KRAUS,Woods Hole Oceanographic Institution, Woods Hole, Massachusetts

L. C. MARSHALL,Southwest Center for Advanced Studies, Dallas, T e x m P. MELCHIOR,Royal Observatory of Belgium, University of Louvain, Brussels, Belgium CHESTERW. NEWTON, National Center for Atmospheric Research, Boulder, Colorado

V

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FOREWORD With the approach of a new sunspot maximum in 1968 a momentous decade in geophysics will pass into history. It encompassed the lnternational Geophysical Year, the Year of International Cooperation in Geophysics, and the International Year of the Quiet Sun. There were also a number of special programs, not connected with the solar cycle, such as the International Indian Ocean Expedition and the Upper Mantle Project. A tremendous impetus has also been given to geophysics by the beginnings of space exploration. It is quite clear that the level of activity will remain high and, in many fields, the effort will be several orders of magnitude larger than that of ten years ago. New ventures are just starting, among them the International Hydrological Decade, which will, it is hoped, provide stimulus to fields which have yet to participate in the upswing. The World Weather Watch is yet another large-scale experiment that will soon become reality. The most gratifying part of these undertakings is the international COoperative aspect. For better or for worse mankind is tied to this planet, and knowledge about its hazards and resources is essential for everybody’s survival irrespective of nationality. Geophysics is thus in the forefront of the sciences essential to our future. Accomplishments so far have not been mean and some of them are again reflected in the articles presented here. Several of these are directly connected with the efforts described above. We are grateful t o the authors for their cooperation and to our advisory committee for valuable suggestions. Articles in the forthcoming volumes will concern fields not covered in prior volumes of this series. H. E. LANDSBERG

J. VAN January, 1967

vii

MIECIHEM

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CONTENTS LIST OF CONTRIBUTORS .......................................... FOREWORD ....................................................

v vii

Current Deformations of the Earth’s Crust

P . MELCHIOR 1. Introduction ................................................. 2 . Expression of the Deformations Produced by a n External Potential on a Point a t the Earth’s Surface .............................. 3 . Form of the Luni-Solar Potential ............................... 4 . Expansion of the Potential in Its Principal Waves ................ 5 . Deformations of an Elastic Semi-Infinite Body Limited by a Plane Surface ..................................................... 6 . Study of an Elastic Sphere Subjected to Deformation ............. 7. Measuring Instruments ....................................... 8. Some Results of Recent Observations ........................... References .....................................................

2 5 14 23

28 35 43 60 76

O n The Upper Mantle

P. CALOI Introduction ................................................. The Earth’s Crust ............................................ Asthenosphere and Pa. Sa Waves .............................. The “20” Discontinuity” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 . Attenuation and Mixed Zones .................................. 6 . Free Oscillations of the Earth and Its Outer Shell ................ 7. Earth’s Internal Movements and Volterra’s Theory ............... References .....................................................

1. 2. 3. 4.

80 87 142 167 175 181 189 203

Wind Stress along the Sea Surface

E . B . KRAUS 1. Introduction: The Transfer of Kinetic Energy .................... 2 . The Atmospheric Boundary Layer .............................. 3 . Waves a t the Interface ........................................

ix

213 215 224

X

CONTENTS

4 . The Transfer of Momentum from Waves to Currents . . . . . . . . . . . . . . 5. Conclusions and Questions ..................................... Appendix ...................................................... List of Symbols ................................................. References .....................................................

241 245 247 251 253

Severe Convective Storms

CHESTERW . NEWTON 1. Introduction .................................................257 2 . General Thunderstorm Structure ............................... 259 3. Modes of Convection .......................................... 262 4 . The Severe Thunderstorm Environment and Its Modification...... 270 5 . Thunderstorms in a Sheared Environment ....................... 273 287 6 . Storm Movement ............................................. 7 . Squall Lines ................................................. 291 8. Severe Weather Manifestations. ................................ 296 9. Conclusion .................................................. 301 List of Symbols ................................................. 303 References ..................................................... 303 The Rise of Oxygen in the Earth’s Atmosphere with Notes on the Martian Atmosphere

L . V . BERKNER and L . C. MARSHALL 1 . Underlying Principles ........................................ 309 2 . Oxygenic Concentration in the Primitive Atmosphere of the Earth 310 3. Surface Oxidation in the Primitive Atmosphere ................. 315 4. Ecology for Photosynthetic Oxygen Production in a Primitive 317 Terrestrial Atmosphere ....................................... 5. The First Critical Level-02 +0.01 P.A.L. .................. 319 6. Identification of First Critical Level with Opening of Paleozoic Era 320 7 . The Second Critical Level-02 -+ 0.1 P.A.L.-the Late Silurian . . 321 8. Oxygenic Levels in the Late Paleozoie and Ensuing Eras . . . . . . . . . 321 9 . Further Refinement of the Model .............................. 323 10. Estimates of the Composition of the Martian Atmosphere and Surface .................................................... 324 11 . Life on Mars ................................................ 328 12. A General Theory of Origin and Planetary Atmospheres . . . . . . . . . . 320 References ..................................................... 330

C0N TEN TS

xi

The Theory of Available Potential Energy and a Variational Approach to Atmospheric Energetics

JOHN A . DUTTONA N D DONALDR . JOHNSON 1. Introduction ................................................. 2 . An Exact Theory of the Concept of Available Potential Energy .... 3. Applications to Observational Data ............................. 4. Variational Methods in Available Energy Theory ................. 5 . Contributions t o the Amount of Available Potential Energy and Its Relationship to Other Quantities ............................... 6. The Dynamics of the General Circulation ........................ 7 . Conclusion .................................................. List of Symbols................................................. References .....................................................

398 412 430 431 434

AUTHORINDEX ................................................. SURJECT IXDEX .................................................

437 443

334 341 373 389

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ADVANCES IN

GEOPHYSICS

VOLUME 12

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CURRENT DEFORMATIONS OF THE EARTH'S CRUST

.

P Melchior Royal Observatory of Belgium. University of Louvain. Brussels. Belgium

Page 2 4 2 . Expression of the Deformations Produced by a n External Potential on a Point at the Earth's Surface .......................................... 5 2.1. Basic Data ........................................................ 5 2.2. Elastic Deformations of the Earth .................................... 7 8 2.3. Components of the Deformation Tensor ............................... 2.4. Deviations of the Vertical with Respect to the Earth's Crust ............ 10 2.5. Variations of Gravity ................................................ 11 2.6. Cubic Dilatations .................................................. 12 2.7. The Case of a Homogeneous Incompressible Earth ...................... 13 3 Form of the Luni-Solar Potential ......................................... 14 3.1. Amplitude of the Perturbations aa a Function of Latitude . . . . . . . . . . . . . . . 14 3.2. Characteristics of the Three Kinds of Tide in the Various Components . . . 3.3. The Precession and Nutations Deduced from Tidal Forces . . . . . . . . . . . . . 4 Expansion of the Potential in Its Principal Waves .......................... 23 4.1. Analysis and Prediction of Earth Tides ................................ 23 4.2. Distribution of Deformations around a Given Point of the Earth's Globe . . 25 4.3. Deformations Relating t o the Semidiurnal Sectorial Forces . . . . . . . . . . . . . . 26 5 Deformations of an Elastic Semi-infinite Body Limited by a Plane Surface ..... 28 5.1. Evaluation of the Effects of Deformation .............................. 28 6.2. Termawa's Problem ................................................ 30 6. Study of a n Elastic Sphere Subjected to Deformation ........................ 35 6.1. The Two Deformations .............................................. 35 6.2. Application t o a Plane Surface Bearing an Alternating Series of Parallel Mountain Chains and Valleys .................................... 38 7. Measuring Instruments .................................................. 43 7.1. Clinometers ........................................................ 43 7.2. Gravimeters ....................................................... 60 7.3. Difficulties Encountered in Interpretation of Gravimeter Result parison With the Results of Horizontal Pendulums . . . . . . . . . . 7.4. High-Precision Equipment Needed for First-Order Stations . . . . . . . 8. Some Results of Recent Observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 8.1. Indirect Effects of Tides .......................... . . . . . . . . . . . . . . . 60 8.2. Direct Observations of Current Deformations of the Earth's Crust . . . . . . . . . 60 8.3. Hydrological Effects: Deformations of the Ground due to Variations of the 62 Level of the Meuse at Sclaigneaux ................................ 8.4. Atmospheric and Hydrological Effects in Leveling .................. 69 8.5. Atmospheric Effects ................................................ 74 8.6. Frequency Dependence of Flexure .................................... 74 8.7. Particle Accelerators in Nuclear Physics ............................... References ............................................................... 76

.

........................................................... 1.1. Measurement Units Selected ..........................................

1 Introduction

. .

.

1

2

P. MELCHIOR

1. INTRODUCTION

For some years now study of local or regional stability of the earth's crust has been increasing in importance for a number of reasons. I n relation to progress in geophysics, well-coordinated research on recent crustal movements should lead t o new discoveries of great significance. Also, highly stable zones must be selected for high-precision instrumental measurements of periodic (tidal) deformations of the earth, and such measurements will reveal many local anomalies for which an explanation must be given. For the physicist and engineer the introduction of new instruments to meet increasingly raised technological demands poses the critical problem of stability in relation to a site already chosen or t o be selected. The European Center for Nuclear Research (CERN) 30-GeV synchrotron a t Geneva has already raised delicate metrological problems, and the construction of a new 300-GeV synchrotron will impose even stricter demands, since the physicists require a stability of 0.1 mm over a 300 meter range, of less than one-tenth of a second of arc. The erection of stabilized platforms for satellite studies has raised stability problems of the same order. The first problem is, therefore, to establish the amplitude of the random type of transient deformations normally to be expected as well as what exceptional deformations are likely to occur in specific regions, what are their amplitudes, and which regions are primarily affected. A map of stable regions is a requirement. We shall consider the deformations produced by hwum forces, which can be divided into two groups: 1. The total of external forces exerted by the moon and the sun on the entire mass of the earth and produced by the attraction potential of these celestial bodies. 2. Surface forces applied on a local or regional scale: the referenceisusually to transient pressures exerted on the crust by masses of air (fluctuations of atmospheric pressure) or water (oceanic tides, rainfall or snow, and floods). The basic formulas for calculation of their effects will be given without detailed proofs, which relate to the mathematical theory of elasticity and may be found in the original papers cited. Furthermore, high mathematical precision is scarcely poseible and not really necessary because of the complex stratified structure of the earth's crust and the limited number of actual precise measurements which are presently available. Luni-solar external forces are responsible for the phenomenon of earth tides, elastic deformation of the globe produced by the attraction of these forces.

CURRENT DEFORMATIONS OF THE EARTH’S CRUST

3

If a perfect state of telluric rigidity existed we could, by the use of highly sensitive instruments, observe small-scale periodic deflections of the vertical (with an amplitude of about 0.02 sec of arc) and small-scale fluctuations of gravitational acceleration (with an amplitude of 2 x lo-’ or about 0.2 mgal). The variation of these perturbations and their amplitudes a t each instant may be computed for the various components to the required degree of precision from information relating to the orbits of the earth and the sun, and the lunar and solar mass values involved. These tidal forces underly the phenomena of precession and nutation of the earth’s axis of inertia originating in the inequality of the principal moments of inertia of our planet (polar flattening and possible equatorial ellipticity). However, the earth undergoes deformations because i t is not an ideal body, but has physical characteristics which are functions of highly complex (and still little understood) laws which form the subject matter of rheology, taken in conjunction with concepts of elasticity, viscosity, plasticity, etc. These telluric deformations, which are subject to regularly varying forces clearly identical with those of the lunisolar forces producing them, are bound to affect the amplitude and probably also the phase of the phenomena which we measure; moreover, they find expression in variable internal stresses and periodic cubic dilatations. Therefore, the purpose of measurements is to compare observed phenomena with analogous phenomena computed for an ideal, nondeformable globe. The basic elements for geophysical studies are the ratio of amplitudes and phase differences of the various fundamental waves as revealed within the limits of instrumental precision. It should be noted here that this phenomenon of earth tides is the only phenomenon of earth’s deformation for which we are able to calculate a priori the forces at work. Since Kelvin’s researches the principle of comparison has been the method employed in study of earth tides. Nevertheless, significant progress has been made in theoret,ical research in the past several years owing to the work of Jeffreys and Vicente and of Molodensky. Their important conclusions will be discussed later, but it is apposite to mention that observed geophysical characteristics are no longer compared with an absolutely rigid standard model of the earth, but with models that approximate far more closely to reality, i.e., models with a fluid core designed in accordance with the most recent seismological findings. These currently advanced theories have enabled us to predict that the globe will behave differently according to thetype of deformation involved. Description of the geometric properties of deformations and of the resulting observational conditions is a prerequisite for explanation.

4

P. MELCHIOR

1.1. Measurement Units Selected The unit of angular measure employed by astronomers is the sexagesimal second of arc. This unit is quite suited to its purpose, since the root-meansquare error of a meridian observation is of the order of O"2, while ephemerides and astronomical almanacs normally give positions t o one-hundredth of a second of arc. The practical unit used by geodesists for gravimetric surveys is the milligal (1 gal = 1 cm2/sec2, go 21 982 gal), which is approximately lo-' of gravity acceleration. The phenomena with which we are concerned are on a different scale. Our purpose is to reveal slow movements of the earth's crust expressed in warpings of a few hundredths of a second of arc and fluctuations of g of the order of one-tenth of a milligal. It will be shown that the sensitivity of the instruments used for the purpose is around O"0003 and 0.001 mgal. For ease of exposition we shall therefore select two new basic units as follows: 1 mseca (millisecond of arc) = O"OO1 1 pgal (microgal) =

gal

iO-9g

For reasons of theory, some authors prefer the radian for angular measurement; the conversion is O"OO1 =4.848 x 10-9radian

FIQ.1.

5

CURRENT DEFORMATIONS OF THE EARTH’S CRUST

2. EXPRESSION OF THE DEFORMATIONS PRODUCED BY AN EXTERNAL POTENTIAL ON A POINT AT THE EARTH’S SURFACE 2.1. Basic Data

Let us assume that the earth is a sphere of radius a and adopt the following system of spherical coordinates where r is the radius vector, 0 the colatitude, and h the longitude (reading positively toward the east). A local reference trihedron to which the observations are related and which relates to the above system is then defined a t a point P situated on an equipotential surface V = C ( r ) . The axes of this trihedron are Pz in a vertical direction, oriented toward the zenith, and hence running counter to gravity (from which g = - a V / a r ) ; Pz tangent to the meridian and orientated to the south (in the sense of increasing colatitudes) (0 = (7r/2) - cp); and P y tangent to the parallel and orientated to the east (see Figs. 1 to 3).

P

FIG.2.

It is evident from Fig. 2 that

&=ad0 (2.1)

dy = a sin 8 dh dz = dr

Let some external force f with a potential W be applied to a point P . Its components in the three local reference directions are then given by the change of potential in each:

G

P. MELCHIOR North Pole

- } D;ir~ion of the

FIG.3. f =--

aw ar

aw

fe = a a8

fx

=

aw a sin 8 ah

The direction of the vertical is the resultant of the forces applied to the point under consideration, and hence of the composition of this perturbational force with gravity. Deviation of the vertical is then given by its components

n1 2: tg n1 (2.3)

=--= aw

along the meridian

1 aw along the prime vertical n2 2: tg n2 =-ag sin 0 ah

while the variation of gravitational acceleration is given by fr=---

(2.4)

aw ar

The result is a deformation of the equipotential surface involved. If the radial deformation at the point R(r),the initial surface equation

4 is

V =C ( r ) becomes for P’(r

+ ()

v+

+w

+

tVv/a4 = C(r 6) (2.6) If this equation represents an equipotential surface its variable part may be

CURRENT DEFORMATIONS OF THE EARTH’S CRUST

7

equated to zero: (2.6)

t(av/az)+ w = o

Since a V / a z = -g by definition, the deformation of the equipotential surface is given by the equation

(2.7)

t = wig

2.2. Elastic Deformations of the Earth 2.2.1. Love Numbers. Love [ 11 introduced the two dimensionless numerical functions which bear his name to characterize the various aspects of the solid earth’s tide; Shida has shown that a third number is required to give complete expression to the phenomenon. Their meaning is very simple and each type of elastic deformation can be expressed by a combination of these numerical functions, which in turn are related to the distribution of rigidity, compressibility, and gravity inside the earth by tJhe relatively complex differential equations established by Herglotz [Z]. The key to Love’s theory is that the luni-solar perturbation potential may be adequately represented by a second-order spherical harmonic function W,, and that therefore all deformation induced by this potential in the earth may be represented by the same harmonic function with the coefficient appropriate to each stage of the event: This coefficient is one of the Love numbers or a simple algebraic combination of them. Thus, any displacement in the three local reference directions and any cubic dilatation produced a t the point P by forces arising from the secondorder potential W , can be expressed as follows:

(2.9) The coefficients supplied here arc, therefore, exclusively functions of radial distance, since we assume that hydrostatic equilibrium, with symmetrical distribut,ion of density, ridigity modulus, and compressibility around the center, exists below the layer of isostatic compensation. The heterogeneity of the upper mantle may be reflected in the Love numbers. Likewise, the potential caused by the deformation itself and the fluctuation in density which

8

P. MELCIIIOR

accompanies the cubic dilatation and surface displacement of matter can be expressed as follows:

V

(2.10)

= K ( r )W Z

For the earth’s surface where the observations are made we set the Love numbers:

H(a)= h

K ( u )= k

(2.11)

L(a)= 1

F ( 4= f where h is the ratio of the height of the earth tide to that of the corresponding static oceanic tide a t the surface, k is the ratio of the additional potential engendered by this deformation to the deforming potential, 1 is the ratio between the horizontal displacement of the crust and that of the corresponding static oceanic tide, and f is the ratio between the cubic dilatation and the height of the corresponding static tide at the surface. The number I? was introduced by Shida, and its importance was stressed by Hoskins in 1920. It was the author’s opinion that a fourth number f should be added. A knowledge of these numbers and their combinations is obtained by the use of various instruments, including horizontal pendulums, gravimeters, extensometers, and dilatometers.

2.3. Components of the Deformation Tensor I n spherical coordinates the components of the elastic deformation tensor are expressed by the relations: 85, e =ar

1 asg egg=--+-

r

(2.12)

ae

S,

r

9

CURRENT DEFORMATIONS OF THE EARTH'S CRUST

Since the displacements in question are extremely small (the angular variations do not exceed 0"1)measurements in the local reference axes may be equated to measurements made in a system of spherical coordinates whose origin is at the earth's center:

u=sr,

(2.13)

v=sg,

W=SA

(which amounts to equating the angle to its sine). Let us introduce values of u, v, and w expressed in terms of the Love numbers and set:

and

W,

= r2S,

where S, is a surface spherical harmonic function. The six components of the tensor then become: (2.14) (2.15)

h err = 2 - W,+ ag egg

h =--'ag1 a2w a62 + w,

1 a2w, (2.16) eAA =-agsin20 ax2 + (2.17)

egh =

1 aw, h w, c e cos 8- a0 + ag

21 a2w, 21 -- -cote-awz ah ag sin 8 aeah ag sin 9

aw, + I aw, + 1' aw, - 1 gnaw, -e ah agsin e ah g s i n e ah sin eg2 ah 1 aw, h aw , +raw, aw, erg = -+ -- 1-g i ag ae ag ae g ae g2 ae h

(2 18) (2.19)

eAr = --

agsin

The deformation along any one direction of the direction cosines is given by (2.20)

+

d = aI2err a,2eee

+ a:eu + a1afe,e +

a3eeA

+ a3

( a 1 ,a , , a3)

a1eAv

Considerable interest attaches to (2.21)

i a

sin 8--

ae

10

P. MELCHIOR

which approximately t o the second order represents the horizontal areolar deformation, i.e., the increase of the horizontal surface per unit of surface or of surface expansion. This expression is considerably simplified by application of the fundamcntal property of harmonic functions, namely, of having a zero Laplacian. I n polar coordinates:

(2.22) With i = 2, equation (2.21) then becomes a t the earth's surface:

w,

C ( a )= 2 ( h - 32) -

(2.23)

ag

Equations (2.14) to (2.19) will be used t,o intterpret the results yielded by an array of extensometers. Six extensometers will clearly be needed to obtain a complete representation of the phenomenon. Expression (2.23) is particularly notcwortliy because it reveals the simple combination ( h - 31), which may also be experimentally determined by t w o suitably arranged extensometers. Moreover, the difference E~ - E ~ permits direct isolation of the numbcr 1:

(2.24)

(sin ~2 - E:$ = 7

2.4. Deviations

ag sin 0

of

oao'a' W , - cos 0 -aa0W ,

-

sec O -

the Verticul with Respect to the Earth's Crust

I n deformation of the earth, especially of its crust where observations arc made, our reference levels are mobilc and pendiiliim mcasureinent,s of deviations from the vertical cannot yield an expression of tjhe typc given by equation (2.3),since the question is no longer one of measuring t,he anglc between the vertical and some fixed direction which might be an arbitrary reference level on the recording instrument, but of nieasiiring tthe anglc between t,he vertical and the ,wnwcLl to the deforvned surface. The eqiiat#ionof the deformed surface is (2.25)

1' = U

+u =

4-h(W,/g)

0,

The attracting potential W , is given by

(2.26)

fnd w, =2r:' (3

COi% - 1)

where 2 is the zenithal tfistnncc of the pert,urbing body. This fuiiotion is tt zonal sphcricnl function whose axis is the direcbtiori linking t Iic: ccnttr of t.he earth and the cent,er of the perturbing body. The deformed surfacc is

CURRENT DEFORMATIONS OF THE EARTH'S CRUST

11

ellipsoidal and the angle i between the normal to the ellipsoid and the radius vector is the ground tilt. Let us identify equation (2.25)with the equation of an ellipsoid of revolution of mean radius n and ellipticity e :

(2.27) The major axis of the ellipsoid extending toward the celestial body corre. (2.26)into consideration, we obtain sponds to = ~ 1 2Taking

+

,

-3fham e =qr3

(2.28)

As demonstrated by Fig. 3, the angle i is given by

.

(2.29)

1 dr

a =--

sin +cos

=-ez

rd*

+

reading the angle in the sense of increasing a,h

(2.30) The deformation in itself modifies the quantity kW, and the total deflection of the pendulum is therefore

(2.31) while the deflection of the reference line is, in accordance with equation (2.301,

h

aw,

+Gae

(2.32)

since the zenithal distance is a negative linear function of colatitude. Observation provides the expression

(1+k-h)--

(2.33) The factor y

= (1

+k

-

1

aw,

ag

ae

h) may be determined by a horizontal pendulum.

2.5. Variations of Grarity Deformations of the earth will likewise modify expression (2.4) since the crust will involve the measuring instrument in its movements and because the earth potential is itself modified b y a quantity W ' .

12

P. MELCHIOR

The potential of the deformed earth a t the point P’(r

+ 5 ) is

vt = v + t(av/ar) + W, + wi

(2.34) consequently,

(2.35)

dg = g -go = - ((a2vlar2)- aw,lar - awi/ar

The additional potential W’ arising from the deformation is of the form S,/a3 and, consequently,

(2.36) while

(2.37)

with the immediate result that (19 = - (1 + h - ip)a w , p

(2.38) since

2 W,/r = a W2pr

(2.39) The factor 6 = (1

+ h -#k)

may be determined by gravimeters.

2.6. Cubic Dilatations

Deformations are accompanied by cubic dilatations because the earth is not strictly incompressible. We have written in (2.9)

(2.40) and have designated by f the value of F ( r ) at the surface ( r = a),f being the fourth Love number. It is of interest t o examine how this new number is related to the previous numbers. For this purpose let us write the expression of the divergence in spherical coordinates: + +

(2.41)

div s =

(r2B.,sin 6)

a

a

+rsA)) ae ( m e sin 6) + ah

so that if the approximation of a spherical earth is accepted, i.e., if g is independent of 6 :

CURRENT DEFORMATIONS OF THE EARTH'S CRUST

(2.42)

i a div s = - - ( r 2 s i2ar

)

13

1 asx +--r sini ~ aa (sin o 0 so) + r sin 0 aA

We also have

Once again, applying the basic theorem of harmonic functions (2.22) for i = 2 and also developing the first term of (2.43). we have a t the surface (r = a ) :

(2.44)

D(a)= 2 ( 2 h - 31)-w2 + ag

or

(2.45) This expression may not be simplified unless the densities and the deformations are homothetic relative to the center: h'jh = g'/g = - 2 / r , and equation (2.44)is then written:

(2.46)'

D(a) = (4h - 61)W2/ag

2.7. The Case of a Homogeneous Incompressible Earth This case is of no practical interest but may establish certain orders of magnitude. Kelvin's theory demonstrates that we should have

3 5

k =-h 3 1 =-h

10

=:+g) -1

h

(1

The factors become

y=l+k-h=l-ih 6 = 1 + h - - Q k = l +i'ch

f = 4h - 61 = +$ h 1 Disregarding a term containing m a factor multiplies 2, which is of the order of 0.05.

7~ = $[(w2a3)/jM]-

= 1/200, that

still

14

P. MELCHIOR

and with g = 981 cm/secZ

p = 5.5 a = 6.370 x 10' cm

this yields the accompanying tabulation P 2 x 4 x 8x 12 x 18 x

10" 10" 10" 10" 1011

h

Y

6

f

1.61 1.19 0.78 0.58 0.48

0.36 0.52 0.69 0.77 0.82

1.16 1.12 1.08 1.06 1.05

3.54 2.62 1.72 1.28 1.01

Observations yield y % 0.7

8 z 1.19

which is adequate demonstration that the heterogeneity of the globe should clearly be taken into consideration. 3. FORM OF

THE LIJNI-SOLAR POTENTIAL

3.1. Amplitude of the Perturbations (ICU a Function vf Latitude It has been shown that the horizontal and vertical components derive from the potential fm a' wz~--(3cos'z-l) 2 rs

(3.1)

where z is the zenithal distance of the perturbing body a t the point under consideration, a is the earth's mean radius, and r the distance between the centers of mass of the earth and the moon. In other words, the horizontal and vertical components of the attractive force involved are

2"

=-gm

(3.3)

-aw' aa

o3

- sin25

3

=gm(;)

(1 -3cos22)=-grn 2

(g)3(cos2r+;)

CURRENT DEFORMATIONS OF THE EARTH’S CRUST

15

Thcse formulas clearly lead us to set:

C,=-grn(;) 3 2

(3.4)

3 = -2G a

where G‘ is Doodson’s tidal constant, whose numerical value for the moon is:

G = 26,206 cm2/sec2 I t follows that

C,

(3.5)

= 82.26 pgal = 0.83956 x

10-7g0

and that, t,herefore, the numerical coefficient expressing the angular deviation from the vertical is

C,,

(3.6)

= arc sin C,/go=arc

sin (0.83956 x

and the value of this extremely small angle is readily obtainable from sin 1” = 1/206,265

Thus,

C,,

(3.7)

= O”0173172

which applies to the case where the moon is the perturbing body. The solar constant G corresponding to the lunar constant G is 0.46051. It follows that the amplitude factors for the sun are

C‘,

= 37.88 pgal

C‘,. = O”0077647 As the factor (cos 22 + g), which governs gravity fluctuations, varies

+$

between and -8 of the two units, the total variation of g may be 2 x (82.26 37.88) pgal = 240.25 pgal, whereas the deviations of the vertical may reach 2 x (O”0173 O”0077) = O”05 since sin 22 may vary between -1 and + l . The radial deformations of the geoid are expressed by the relation (2.7), here written, by virtue of (3.1):

+

+

S = W,/g = G/g(cos 22 or, for the moon

Glg = 26.7 cm and for the sun G ‘ / q= 12.3 cm

+ )I

c

aa

AnnflMMP

..

-loopqol-uP-,

vvvvvu

Brussels Grovlmeter Askanio No 145

20mxco-M

SClalpnwux (Namurl Horizonla1 pendulum Verboanderi - Mclchlar No I

2 0 mseco-

I961 January

I

2

3

4

I

I

I

I

5

6

7

8

9

I0 II

I2

I3 14

15

I6

17

18

19 20 21

2 2 23 24 25 26 27 28 29 30 31

FIG.4. Recordings of the earth tides simultaneously in three components: vertical-Askania NS, EW-Verbaandert-Melchior horizontal pendulums at Sclaigneaux.

gravimeter at Brussels; horizontal

CURRENT DEFORMATIONS OF THE EARTH’S CRUST

17

+

The total oscillations of the geoid surface may therefore be put a t 2 x (26.7 12.3) cm = 78 cm. Since the experimental value of h is 0.48, the actual deformation of the earth’s crust can reach 36 cm. It should once again be noted that the term (cos 2z 4) implies that the fluctuations of g and the geoid level are not symmetrical with respect t o the value and the position in the undisturbed state. This is illustrated by the curves shown in Fig. 4, which were recorded in Belgium. The instantaneous amplitude of the various phenomena to which we have alluded is strictly dependent on the relative position of the point of observation and the perturbing bodies. To obtain a clearer view the coordinates (latitude and longitude) of the point and the lunar and solar orbital constants should be inserted in the mathematical expressions given above. Introduction of the concept of the (lunar or solar) potential, from which all manifestations of earth tides are derived, makes it possible t o restrict the mathematical treatment to study of this single function from which to deduce by appropriate derivation the relative characteristics for each tidal effect investigated.

+

Pole

FIG.5. Triangle of position of spherical astronomy.

The expression (3.1)of the potential W e is not, however, manageable because it includes a local coordinate z of the perturbing body involved. Therefore, m e shall substitute conventional equatorial coordinates ( H is the hour angle, S is declination) and the astronomical coordinates (A, cp) of the observation point serving us as a triangle of posit,ion as defined in spherical astronomy (Fig. 5 ) :

(3.8)

cos z

= sin

4 sin S + cos 4 cos 6 cos H

FIG.6. The three kinds of tide.

CURRENT DEFORMATIONS OF THE EARTH’S CRUST

19

The potential is then formulated as follows:

3 W , = G{ C O S , ~ COS~S cos 2 H

(3.9)

+ sin 2 4 sin 28 cos H + 3(sin24- g)(sin28 A)} -

We owe this decomposition of the potential into three terms to Laplace, who was the first to call attention t o its remarkable meaning and geometric characteristics. These three terms represent the three types of spherical harmonic functions for a second-order surface (Fig. 6): In Fig. 6a the first of these functions has as nodal lines (lines where the function is zero) only the meridians: those situated 46” to either side of the meridian of the perturbing body. These lines divide the spherical surface into four sectors where the function is alternately positive and negative. The areas where W is positive are those of high tides (I> O), the negative areas are those of low tides (6 < 0). This function is termed the sectorial function, the period of the corresponding tides is semidiurnal, and their amplitude has a maximum a t the equator when the declination of the perturbing body is zero. Polar tidal amplitudes are zero. Laplace called tides of this type “mades de troisieme espkce” (tides of the third kind). It should be noted that variations of mass distribution a t the earth’s surface subject to the sectorial distribution do not modify either the position of the pole of inertia or the major moment of inertia C (which determines the speed of rotation of the earth). I n Fig. 6b the second function has as a nodal line a meridian (90” from the meridian of the perturbing body) and a parallel, namely, the equator. It is a tesseral function which divides the sphere into areas which change sign with the declination of the perturbing body. The corresponding tidal period is diurnal and the amplitude is maximum a t latitude 45”Nand 45”s when the declination of the perturbing body is maximum; the amplitude is always zero at the equator and the pole. Laplace referred to tides of this type as “mar6es de deuxieme esp6ce” (tides of the second kind). The variations of mass distribution a t the surface of the earth following the tesseral distribution produce positional oscillations of the inertial pole but not of the major moment of inertia C . The resultant perturbing potential of polar motion has the same form. This distribution corresponds to the precession-nutational couple which, acting on the earth’s equatorial bulge, tends to tilt the equatorial plane against the ecliptic. A diurnal tesseral wave of the harmonic development of the tidal potential corresponds to each stage in the development of the precession-nutational couple. The effect of this couple may be that the fluid core rotates relative to the mantlc.

20

P. MELCHIOR

In Fig. Gc the third function, which is dependent only on latitude, is a zonal function; its nodal lines are the parallels +35" 16' and -35" 16'. Since it is only a squared sine function of the declination of the perturbing mass, its period will be fourteen days for the moon and six months for the sun. These are Laplace's tides of the first kind. The variations of the mass distribution a t the surface of the earth conforming to the zonal distribution do not produce any drift of the incrtial pole, but do affect the major moment of inertia C . We may therefore expect some fluctuations in the earth's speed of rotation corrcsponding to the periods given above. They can be effectively detected by a Bureau of Time Standards equipped with precision instruments (PZT). The equipotential surface will be lowered 28 cm a t the pole and raised 14 cm a t the equator, The effect of this permanent tide is a slight increase of the earth's oblateness. (The difference between the major and minor axes of the terrestrial ellipsoid is 21.37 km.) 3.2. Characteristics of the Three Kinds of Tide i n the Various Componen,ts Let us now examine the time variation of projections of the gravity vector on the axes of a trihedron with its origin a t the observation point, the axis 0, directed toward the center of the earth, the axis 0, directed to the south, and and the axis 0, directed tto the west. The azimuths are counted positive from south toward the west. By virtue of equation (3.9), we have

+C ( t 2 sin2v)(# 2 sin%) +C sin 2~ sin 26 cos H +C cos 2H

F, =g

-

-

COS'~I COS%

F,

=C

sin 22 cos A

Fw = C sin 22 sin A Now cos z =sin q~ sin 6 sin z cos A

= -cos

+ cos q cos 6 cos H

?sin 6 +sin

sin z sin A

= cos 6

9, cos 6

cos H

sin H

which yields subsequently (3.10)

P8 = C{ 8 sin 2q(4 - sin%) - cos 29, sin 26 cos H

+

1- sin 29, cos26 COB 2 H ) (3.11)

FW= C{sin 9, sin 26 sin H

+ cos q~ cos26sin 2 H )

CURRENT DEFORMATIONS OF THE EARTH’S CRUST

21

These components of the force of attraction can be derived directly from the expression (3.9) of the potential by derivations dldr, dlacp, a/cos V, ah. It will be noted that the east-west component does not contain any longperiod terms, which was evident a priori since these terms are derivatives of a zonal harmonic function (see Fig. 6). The accompanying tabulation summarizes geographic factors, and will be of value in calculation of the theoretical amplitudes of various waves:

Components:

Vertical North-south East - west

Long period

i( 1 - 3 sinztp) 3 sin 2p 0

Waves Diurnal sin 2p cos 2p sin p

Semidiurnal cos2p

4 sin 2tp cos p

Figure 7 depicts the amplitude of the five principal waves in all three components as functions of latitude. Only the diurnal horizontal component is nonzero a t the polcs, its amplitude obviously being the same in all azimuths (sin = - cos 2 q ) , as is obvious. Therefore, the foot of the vertical describes a circle with a period of one day.

3.3. The Precession and Nutations Deduced from Tidal Forces I t is evident on examination of formulas (3.10) and (3.11)or Fig. 7 that if the earth is an ellipsoid of revolution all tJhecouples resultsingfrom the forces B’s and F , cancel except, by virtue of oblateness, the couple resulting from the F,-diurnal components. Calculation of this couple leads directly to the development of precession-nutation formulas with a nutation term corresponding to each diurnal wave of the tesseral tide. The following integration will suffice to demonstrate this:

where T = a(l - E sin2q), E = ( a - c)/a (oblateness), H = t - o! - A, t is Greenwich sidereal time, FsT= G cos 29, sin 2 6 cos H , in accordance with (3.10)and A is the principal equatorial moment of inertia. + The resultant couple R gives rise to tilt of the equatorial plane against the plane of the ecliptic, the angular velocity fi of which is related to the classical expressions of nutations 6 4 by the relation

(3.13)

h = - B sin t + tj sin e cos t

(in this case 6 is the inclination of the ecliptic).

22

P. MELCHIOR

FIG.7. Amplitude variation of the principal waves as a function of latitude. Zones where earth tide measurements are presently functioning are shown in gray. There is no station in the Southern Hemisphere. (a)Vertical component, note Caracas (rp 10"); (a) north-south horizontal

-

CURRENT DEFORMATIONS OF THE EARTH’S CRUST

23

Equation (3.13)indicates that nutations are motions of the axis of the figure of the earth relative to an inertial trihedron practically fixed by alignment with celestial bodies, while tides are related to the earth itself which rotates at an angular velocity of 2 r / t . It would therefore be possible to produce a table relating diurnal tidal waves to nutation terms by subtracting from the tidal frequency a frequency t =t, h (where h is the mean solar longitude). These comments are important if we are to understand the bearing which the theories of Jeffreys and Vicente [3] and Molodensky [4] have on the dyna.mic effects of the earth’s liquid core [5, 281.

+

4. EXPANSION OF THE POTENTIAL IN ITS F’RINCIPAL WAVES

4.1. Analysis and Prediction of Earth Tides The analysis and prediction of earth tides, as of oceanic tides, is based on expansion of the potential. I n 1883, G . H. Darwin published an expansion which was much used, and which was of great value in its time. However, as observations were accumulated, systematic residues soon appeared, indicating the need t o extend the expansion. Moreover, Darwin’s expansion was not purely harmonic : The coefficients and arguments contained as constants terms which were in reality slightly variable (Darwin took the lunar orbit and not the ecliptic as his reference). A purely harmonic expansion had been advanced in 1874 by Ferrel, but it was restricted to the principal terms. In 1921, A. T. Doodson published a purely harmonic expansion based on Brown’s theory of the moon. All modern methods of harmonic analysis are based on Doodson’s expansion. The arguments of the tidal components (which are very numerous, even when restricted to the principal ones) are expressed as a function of six independent variables, namely: mean solar time mean longitude of the moon mean longitude of the sun h longitude of the lunar perigee P N’ = - N where N is the longitude of the ascending lunar node P, longitude of the perihelion 7

S

Mean solar time is expressed as a function of mean lunar time by the relation 7 + s = to h = sidereal time, i.e., t = r s - h. The sign of the longitude of the ascending lunar node has been changed in the interests of uniformity, since it is the only one of the six variables which augments toward the east.

+

+

24

P. MELCHIOH

The periods of variation of the six variables under consideration are givcn in Table I with the corresponding hourly speeds. The same table also gives t81w various lunar periods as functions of these variables. Table 11, on the other hand, gives the five longitudes s, h, p , N , and p , as functions of mean solar time. This leads t>oan expansion in a very large number of waves, of which only the six principal waves will be mentioned here as the only ones capable of leading to valid geophysical conclusions. The reader is referred to Melch ior [a] for greater detail. TABLEI. Periods of variation of six variables with corresponding hourly speeds. Hourly speed= 10 = t t = 10 T =1

- h Mean solar day

+h

-8

Sidereal day Mean lunar day

15" 090000 15" 04 10686 14' 4920521

1.000000m.s.d. 0.997270 m.s.d. (t - 4 m ) 1.035050 m.s.d. = 24 hr 50.47 min

27.321582 day Period of variation of the declinations 365.242199 day (oscillations in longitude)

a

Tropic month

0" 5490165

h

Tropic year

0" 0410686

P

0 ' 0046418

8.847 year

"

0" 0022064

18.613 year

PS

0" 0000020

20,940 year

0" 5512229

0" 5079479

27.21222 day Oscillations of the moon in latitude 27.55455 day Interval between two passages of the moon a t the perigee 29.53059 day Return of the lunw phases 31.812 day Evection period

8

-N

8

-p

Mean draconitic month Mean anomalistic month

8

-h

Mean synodic month*

o0 5443747

8-2h$p=(8-p)-2(h-p)

o"4715211

h-p,

0" 0410667

h-P

0" 0364268 1" 0168958

t(8

- h)

Period of revolution of the mean perigee of the Moon Period of revolution of lunar nodes Period of revolution of the solar perihelion

365.25964 day Mean anomalistic year 411.78471 day 14.76530 day Period of variation

a The hourly speed IJ is obtained observing that the moon's revolut,ion takes 27.3 days. There is thus every day a lag of 24 hr/27.3 = 50.47 min. b The synodic month (29 days) gives nearly the commensurability of the lunar and solar periods and will give the fundamental interval for an harmonic analysis.

CURRENT DEFORMATIONS OF THE EARTH'S CRUST

25

TABLE 11. Exprossion of the mean longitudes measured in the ecliptic from the instantaneous mean vernal point.= 8 =270" 43659 +481,267" 89057 T +0.00198T2 +0.000002 T3 h=279" 69668+36,000" 768922'+0.00030T2 p=334" 32956+4,069" 03403T -0.01032T2-0.00001T~ N=259" 18328- 1,934" 14201T f0.00208T2 +O.OOOOO2T3 = - N ' pJ' =281" 22083fl" 7190211 +0.00045T2+0.000003T~

a Variable T:in Julian centuries of 36,525 mean solar days (unity) from 31 Dec. 1899 mean noon of Greenwich (origin). b In astronomical practice we take for the beginning of a solar year the moment when the mean longitude of the sun is 280" (longitude measured from the mean corresponding equinox and augmented of the corresponding part of aberration). C The T term has a negative sign because all longitudes are augmenting in the eaat direction except N. In the computations we urn N' = -N .

These principal waves are Three semidiurnal waves:

M, 8, N,

lunar (period 12" 2!jm 14') (argument 27) solar (period 12 hr) (argument 27 2s - 2h) lunar elliptic (i.e., due to the eccentricity of the lunar orbit) (period 12" 39"' 30') (argument 2T -s p)

+

+

Three diurnal waves:

I<, luni-solar with a period of exactly one sidereal day (23h66m4') (argument T + s = t h) corresponding to the luni-solar precession 0, lunar (25"49"' 10') (argument t h - 2s) corresponding to the bi-

+

+

monthly nutation of argument -2s P , solar of period 24h3m548 (argument t - h) corresponding to the semiannual solar nutation of argument -2h Given a knowledge of the orbital elements of the earth and the moon and the ratios of the mass of the moon and the sun to the earth, the amplitude of each of t>hesewaves may be calculated as a function of the latitude of the observation point.

4.2. Distribution of Deformations around a Given Point of the Earth's Globe General expression has been given to the deformation along the direction of any one of the direction cosines (u,,u 2 , a3)by expression (2.20) which contains the six components of the tensor of the deformations (2.14) t o (2.19) expressed in spherical coordinates. On insertion of the expression for W , given by the expansion (3.9) it is possible to calculate the amplitude of the deformation as a function of the

26

P. MELCHIOR

azimuth in which it is t o be measured and as a function of the latitude of the point where the measurements are made. Here an important circumstance should be emphasized. When attempting to represent the components of the tidal force producing variation of the gravity factor it was sufficient to take the theoretical value derived from the form of the luni-solar potential. Effectively measured components are proportional to the theoretical values and it is this proportionality factor, which is quite close to unity, that is the object of investigation, as we have emphasized in the introduction. When, however, one wishes to measure the deformation itself (with an extensometer) there is no theoretical value for comparison because the deformation is, by definition, zero for a rigid earth. This is shown by the presence of Love numbers as factors in the components of the tensor of the deformations [expressions (2.14)to (2.19)]. When it is desired, therefore, t o represent the distribution of the deformations produced around a point by an external potential, plausible, if arbitrary, values of h and 1 must be selected. To simplify the calculations we shall here adopt h =0.58, 1 =0.05, and Qlqr 2 42 x 10-O.

4.3. Deformations Relating to the Semidiurnal Sectorial Forces Let us consider the potential corresponding t o the fundamental wave M z :

W (M,)

=G

cos2p,COB 27 = G sin20 cos27

where p, is the latitude, 8 is the colatitude, and Simple calculation yields

7

is the lunar time.

(a) in the plane tangential to the earth’s surface:

a,=o

a:=l-ag

and

G

h sin%

+ 2-2G

qr

+ 2) + cos%-

{ag(cos 28 - cos28

G

- 6 - 1 a z u 3 c ~ ~esin 27 9r ( b ) in the meridian plane: a3 = 0

and

a: = 1 - u,2

2}

27

CURRENT DEFORMATIONS OF THE EARTH’S CRUST

G +( h + l)ala2sin 28 gr

(c) i n the plane of the prime vertical: a2=0 af=1-a3

2

and

[

G d(M,) = -142

- ai)sin28

G +2 z a:(cos28

I

- 2 ) cos 27

gr

gr

G

+

2 - ( h l ) a , a3sin 8 sin 27 9r Let us consider the distribution of the deformations when the moon is in the local meridian ( T = 0). The following numerical values were adopted -

G G - h = 2 4 x lop9 -l=2 x gr gr The results are depicted in Fig. 8, where the units are expressed in loFg. A similar calculation may be made for tesseral diurnal forces, and the wave 0,has been depicted in Fig. 8. (4.1)

N

N

N

I

N

I I I

I

I

.WE GROUND (0)

GROUND (b)

FIG.8. Distribution of the deformations a8 a function of the azimuth at the earth’s surface: ( a )wave Ma and (a) wave 0.

28

P. MELCHIOR

5. DEFORMATIONS OF AN ELASTIC SEMI-INFINITE BODY

LIMITEDBY

A

PLANE SURFACE

5.1. Evaluation of the Effects of Deformation Study of this problem is a first approximation to evaluation of the effects of deformation relating to local overloading of the earth's crust. The phenomena of concern are: ( a ) variations of atmospheric pressure, (b) tidal variations of sea level, ( c ) river flooding, and (d) accumulation of snow or ice. In 1878, Boussinesq [6], who employed the logarithmic potential, gave a solution for a homogeneous flat plate. Given a homogeneous mass dM of height h and density p on a sector do, we have: (5.1)

= ph da

dM

(5.2)

da=rdrde

(5.3)

dV

= hr

drde

The attraction exerted by this mass on the original point is

f dM (5.4)

=-

T2

da =fph7

and the deviation of the vertical due t o this attraction is (5.5)

The load per unit of area is (5.6)

P = gph

and we write (5.7)

dP = P da =gph d a

Therefore, since P is a force acting normally to the surface of the body, the displacements to which it gives rise are given by the solution of Boussinesq

CURRENT DEFORMATIONS OF THE EARTH'S CRUST

where \r= (6.9)

29

ss s Prdxdy=

rdP

(5.10)

from which

w=-because Ar

=

+ 21r, from which

(6.12)

The component here of concern is the vertical deformation represented by the displacement w. It is immediately evident that (5.13)

and since our equipment is situated almost a t ground level, we have z that

= 0, so

(5.14)

To obtain the slope of the ground we differentiate with respect to x and, taking equation (5.7) into consideration, we obtain: (5.15)

The deviation of the vertical occasioned by the same mass is given by equation (5.5),which is similar in form to equation (5.15), and so the slope assumed by the ground under the effect of the load is written in the form:

30

P. MELCHIOR

m

1 g2 - -- = 11.93 x 10" l-471f

-

12 x 10"

This demonstrates that the vertical displacement of the crust is proportional a t each point to the gravitational potential of the load and a function of the elastic constants (A, p) of the medium. Let us consider the following two extreme cases. (a)

h = co, an incompressible crust, for which we have: IDA

(b) A

=p,

192 1 =----a 45rf P

1

= wl-a

P

a highly compressible crust, which yields: 392

3 1 a=-wl-a 471f 2P 2 P 1

wtl=--.-

Numerical examples are given in the accompanying tabulation. Modulus of rigidity of the crust

p p

=8

Incompressible crust

x 10"

= 3 x 1011 p = 1 x 1011 (gneiss)

WA=2U W A = 4U W A = 12U

p = 0.18 x 1011 p = 0.14 x 1011 (Japanesetuff)

WA

Highly compressible crust

W A = 87u

= 80a

5.2. Terazawa'a Problem The solution given by Boussinesq may be applied by dividing the loaded surface into circular sectors centered on the point a t which the measurements are made and summing the effects for each sector. Terazawa [7]studied the total effect of a circular area of radius a loaded by a mass of density p. Taking the center of this area as the origin 0 and taking as the axis 0,the normal interior to the earth's surface treated as a plane (an acceptable approximation here), he designated the pressure normal to the surface z = 0 by

CURRENT DEFORMATIONS OF THE EARTH'S CRUST

31

and since the loaded area was circular he was naturally led t o employ the Bessel functions Ji. Terazawa assumed :

Z ( k ) =k

(5.16)

joQ

S(X)J~(~X)X~X

and demonstrated that the vertical displacement is given by

dk ' jm Z(k)e-"J0(kr)dk - A + 2 p jm Z(k)e-kzJo(kr)T 211 2P@+CL)

(5.17) w = - -

0

0

He considered two special distributions: 1. A uniform load

f(r)=-gph (5.18)

=O

forr
>a

which yields

Z(k ) = - gphu J (ka)

(5.19)

and 15.20)

Operating with the fundamental theorems of Bessel (Neumann) functions, he obtained after several changes of variables

(No= (5.21)

(31,

+

2p)aq2(+el o1- ql)

= - 2p(A + p )

where

32

P. MELCHIOR

2. A n elliptical dome The profile of the load is represented by the ellipse

(5.22)

z2/b2

+ ,‘/a2 = 1

whose axes are a and b. The normal pressure is then of the form

(5.23)

P=O

forraa

and we have

(5.24)

Z ( k ) = - gpb a

{

sin k a - ka cos ka k2a2

which need only be inserted in the previous expressions to obtain

1. Application to a Stratified Medium. Nishimura [ 8 ] extended Terazawa’s method to a medium consisting of two superposed layers, the upper layer of thickness H with the elastic constants (A, p ) resting on a half-space with the constants (Alp‘). In the simplified case A = p, A’ = p’, and where the thickness H of the surface layer is slight in relation to the radius of t8heloaded area, the expression obtained for pressure distributed in accordance with an elliptical dome is

(5.26)

for r ,< a

This then yields a t the center ( r = 0) and if p‘

(6.27)

3ypha 7gph +-H 32p 12p

W==-

=2p:

CURRENT DEFORMATIONS OF THE EARTH'S CRUST

whereas for the homogeneous medium H

=0

and p'

=p

33

yield a t the center:

(5.28) which duplicates the first term (since p'/p was taken to be 2.) Therefore, in the case of a stratified medium the deformation will be in the proportion pip' to the deformation of a homogeneous medium augmented by a correction term independent of the loaded area, which will however be small since H has been assumed to be small relative to a. Numerical examples. Let us assume in all cases, in cgs units,

f = 6.6 x

9 21 lo3

( a ) Cylindrical loading Let us select

a = 200 x lo5,

h = lo2,

r = 300 x

lo5

p=l X = p = 6 x 10" Terazawa's solution yields r i a = 1.5

q1 = 0.00255

aq2(+el w1 - ql)= 0.864 from which cr = O"0024 W B = O"0069

which is

m, z 3cr as in the corresponding solution of Boussinesq. ( b ) An elliptical dome The same load distributed in an elliptical dome yields a very similar result: a = O"0023 w g = O"0066

( c ) Rectangular load This problem was investigated by Steinhauser to evaluate the down-warping produced by the snow load on the Alps [9]. The distribution of the load is

34

P. MELCHIOR

depicted in Fig. 9 where a = 1 0 0 x 10"cm b = 360 x lo5cm h=

h

-

p

= 0.3

p = 3 x 10"

4x102cm

which represents a total load of 10" tons.

I

I2 Fra. 9 . Hypothetical distribution of the snow load on the Alps adopted by Steinhaueer.

I n the axes employed in the figure, the equation of the surface is given by

and the load is

P=

P

=0

gph(a2- z2) a2

for

-a \ <x < a

-b
for the outer domain.

Steinhauser obtained the accompanying table of deformations. center Distance x :

0

a12 50 km

3a/4 75 km

a 5a/4 3a/2 3a 1 Oa 100 km 125 km 150 km 300 km 1000 km

Vertical diaplacement, cm

3.99

3.56

3.08

2.67

2.23

1.97

1.22

0.38

36

CURRENT DEFORMATIONS OF THE EARTH’S CRUST

I

o=100km

20

30

-

I’

1

FIQ.10. Deformation of the soil under the effect of the snow load in the Alpsobtained by Steinhauser.

( d ) A stratijed medium As in case ( a ) let us assume gpha = 2 x 10l2

and

H

= 10

x lo5

from which

gphH z 10” and

w = 0.33

+ 0.09

when for H = 0 w = 0.66. 6. STUDY OF AN ELASTIC SPHERE

SUBJECTED TO

DEFORMATION

6.1. The Two Deformations There are now two problems to examine: 1. To study the deformation of a sphere subjected to a normal pressure exerted by a given mass. The distribution of the masses will be expressed by a spherical harmonic function Wi and the components of the surface force will be of the form:

2. To study the deformation of the same sphere under the gravitational effect occasioned by the same distribution of the masses. This effect flows from the potential W i . The two problems have been examined by Thomson (Lord Kelvin) and Tait (Natural Philosophy, articles 737 and 834).

30

P. MELCHIOR

The solutions proposed are as follows for a homogeneous elastic sphere ( p , A, p ) of radius a : The components ( a , p, y ) of the deformation are of the form

The solution for problem 1 is

El

=

i(i

+ 2)(A+ p) - P ,

+ 1)(2i + 3)(A+ p), 2(2i + 1)kp ;(A +p)W + 1)P (2i + l ) k p

F , = (i

2(i - 1)kp

(6.2)

G, = The solution for problem 2 is

with 03-41

k =Mi

+ + Il(A + p ) -(2i + 1)p

Assuming

I = 2(i

(6.5)

+ 1)2 + 1

we write

(6.6)

A

+p

= [k

+ (2i + l)pI/I

Darwin similarly separated the terms dependent on compressibility from those independent of it. He obtained:

with

(6.8)

po = (a2 - 3 2 )a- wi + 2(2i + l)SW, ax

CURRENT DEFORMATIONS OF THE EARTH’S CRUST

37

On the basis of these results i t is a simple matter to obtain the six components of the strain tensor whose classical form is

with da

8 =-

dx

dy + dj3 - +dy dz

(cubic dilatation)

Let us assume

r=a+hSi

(6.10)

is the equation of a spheroid representing the earth’s outer surface. The weight of the surface inequalities is -gphSi and the solution of the first problem is obtained by setting in (6.1)and (6.2)

(6.11)

ri

W , ( l ) = -gph-8. a’

a

For the second problem we may set in (6.1) and (6.3) (6.12)

3 ri wi (2) = gph -. Si 2i + 1 a’

which is the expression of the potential of the layer hSi in the interior of the sphere. The equations are in a form suitable for combination, which is expedient because the two effects occur simultaneously. The over-all solution then becomes

(6.13) with

(6.14)

w.=

2(i -1) 8 gph - Si 2i+ I a2

--

It may be noted that

W , ( 2 )= --

22

3

+ 1 W,(l)

38

P. MELCHIOR

and

3 W,(2)= --Wz(l) 6 When 6’ is colatitude, the expression of a surface zonal harmonic (Legendre polynomial) is . i(i - 1) cosv - cosi-% 4(1!)!

sin’%

- 2)(i - 3) + i ( i - 1)(i 4’(2!)’

c0s”4e

sin4e

- ...

or

cos2e+-i2(i 4!- 2)2 sin+%

C O S ~ B-

...

Setting (6.15)

p2=y2+z2,

P

s i n e = -r,

X

case=-r

we obtain (6.16) Darwin has given the detailed expansion of the strains corresponding to this case.

6.2. Application to a Plane Surface Bearing an Alternating Series of Parallel Mountain Chains and VaUeys If in the previous expansions the radius of the sphere and the order i were assumed to be infinite, we are now concerned with a plane surface and a series of mountain and parallel valleys. Let z be the depth beneath the surface. Let us set in the meridional plane = O . We then have (6.17)

p =a-z

Let us stipulate a t the limits i

= 00:

(6.18)

=r

s = i/a = i/p

This yields

(6.20)

sa = cos sx

wi=aiepEzcos sx

CURRENT DEFORMATIONS OF THE EARTH'S CRUST

39

Therefore,

+hSi = a + h cos sx

r =a yield8

(6.21)

Z=

-hcossx

The wavelength of the relief wave is 1 = 2 ~ 1 s This . relief corresponds to a cylindrical distribution of axis 0,.We must moreover introduce into Wi the factor which allows for the height of the mountains and for gravity, a factor which is t o be found in equation (6.14) and which, for i = 03, becomes -gph/ai. We then have

W i= - gph e-sz cos sx

(6.22)

Darwin obtained the following expression of the displacements for the case of an incompressible medium :

(6.23)

p=0

+

(it is verified that 0 = aa/az a/3/ay

N,

au

=2 p -

ax

+ ay/az = 0). The strains are

= - gphzse

--Iz

cos sx

N, =O (6.24)

N,

= 2~

aY = gphzse az

T,=T,=O

cos sx

E)

T p = P ( z+ -

= gphzse

-8z

sin sx

The basic equations of elasticity, which are here reduced to

a N , aT, - + ~ + X O = O

aT, aN,

a z + z +z,=o

40

P. MELCHIOR

may be used to verify that

aw,

a wi

X , = --,ax

0-

aZ

The tilt assumed by the surface is given by (6.25)

Numerical example. Let us assume that we are concerned with a n oceanic wave for which

I = lO'(100 km),

h = 102(1meter),

with g=103,

p=l,

p = l 11~

We deduce

mmr

=@ 2 0.5 x

2 O"1 (1" N 1/200,000)

2P

The maximum vertical deformation is proportional to wavelength. This explains, moreover, why isostatic readjustment takcs place the more rapidly, the greater the extent of the inequality of relief under consideration. On the other hand, as will readily be conceded, the tilt acquired is independent of it. Slichter and Caputo have studied the deformations produced by the pressure exerted by a polar cap [lo]. Caputo concluded this study by taking into acount the gravitational effect due to the presence of the surface masses thus added, and by extending the calculation to the case of an axisymmetric distribution [11a]. The equations of elastic deformation are simplified because the effect of a polar cap does not have a component dependent on longitude, and in polar coordinates (0 is colatitude, 1 is longitude) they become:

with

41

CURRENT DEFORMATIONS OF THE EARTH'S CRUST

+--r sini 8 86a (so sin 0)

i a

D = -{Zaar -( r (6.27)

[see equation (2.42)]

ia 1 aso 2wE=--(,my) --r ar

r

ae

for which, as is customary, solutions of the type (6.28)

s, = R l ( r ,n)P,,(cos 0)

(6.29)

se =

R2(r,n)

ap,

(COB

e)

ae

are sought, where the P,,(cos 6 ) are Legendre polynomials and two relations in R, and R, are obtained. Furthermore, D and

w,

satisfy the equation

r(u?rD/d?) - n(N

(6.30)

+ l ) D =0

whose solution is = [Ern

+ Pr-n-l]Pn(cos 0)

(6.31)

D

(6.32)

w L= [Gr" +Hr-"-']aP,/ae

The conditions a t the limits prescribe that the displacements and the strains should be finite, independent of longitude, and continued across the surface of the core ( r = r I ) Moreover, the strain a t the outer surface ( r = r 2 ) should be equal to the strain applied. This is a step function whose value is Po in the two polar caps: 0 < 0 < w and T - w < 0 < 7r and zero everywhere else. Finally, shearing stress is zero a t the surface of the core and pressure is constant Thc strains are of the well-known form:

AD + 2p (as,/ar)

P,,

=

P,,

=o

(6.33)

Srirfacc displacements have been calculated for four values (4", 8", 16", and 25") of the diameter of a cap exerting a pressure of 10' dynes/cm2 (1 bar), adopting the following values for the earth model:

r2 = 6.371 x 10' cm

A,

rl =0.545r2

A, = 8 x 10"

= 14/11 x

10"

12

p2 = 10

11.1

=o

42

P. MELCHIOR

The calculations were continued to Legendre polynomials of order 140; the results are illustrated by Fig. 11.

10cm

45’

90’

FIG. 11. Curves fi*omCaputo.

For large diameter caps the gravitational effect due to the presence of the masses forming the overload becomes significant and reduces the calculated deformations [11b] . The use of this type of expansion has previously been treated with justifiable skepticism since, as H. Bouasse states (Verges et Plaques, Paris, Delagrave, 1927, p. 288): “The problem of plates, beloved of mathernaticiana, is the type of difficult problem whose solutions do not increase our knowledge of the properties of solids. I n this reapect what is signified by solutions approximated by complicated equations in connection with experiments that cannot be carried out with precision? If the measurements confirm the theory we are no better off, and if they are at variance, this will be put down to experimental errors, approximations in calculation, and the heterogeneity of the material.” We are a t present in a better position since we now possess data processing machines that enable us to seek numcrical solutions with very high precision

CURRENT DEFORMATIONS OF THE EAECTH'BCRUST

43

and that can be programed with a very large number of experimental parameters to describe the heterogeneous constitution of the subsoil. It is even more necessary to know this with precision and to measure these parameters suitably. I t is in no sense certain that rock in situ behaves in the same way as specimens submitted to laboratory tests. The proliferation of measuring stations would therefore seem to be the only actual way of making progress, but it is to be expected that large data processing machines will in the future provide greater interpretation power. It should, moreover, be noted that when one wishes to study the major effects produced by oceanic tides which displace large masses of water, a cylindrical distribution no longer corresponds to reality. I n fact, there is no symmetry of revolution around the vertical of a n amphidromic point where, on the contrary, low tides and highs tide are diametrically opposed. 7. MEASURINGINSTRUMENTS

An apparatus which amplifies the component to be detected should be used to measure each aspect of crustal deformation. We shall therefore have a range of equipment which may be classified as follows: 1. Clinometers, used to make angular measurements in the horizontal components. 2. Gravimeters, used to measure changes in gravity, i.e., changes in the vertical component of the perturbations. 3. Extensometers, used for direct measurement, in accordance with their orientation, of the various components of the strain tensor. 4. Dilatometers, used to reveal cubic dilatations in the crust. We shall not here recount the history of these various instruments, for which we would refer the reader to a more specialized work [ 5 ] , but we shall briefly describe the instruments that are currently the most precise and most widely employed, paying particular attention to the essential problem of precise calibration.

1.1. Clinometers 7.1.1. The Horizon.ta1 Pendulum. The horizontal pendulum yields the greatest measurement accuracy a t tshepresent time. This instrument is illustrated by Fig. 12, which shows a Verbaandert-Melchior pendulum [12]. A rigid support (an assembly of triangles) is fixed t o a baseplate on two adjusting screws situated at right angles, whose purpose is leveling. Two fine wires suspended from two supporting points situated practically in the same vertical carry the horizontal arm (the Zollner type of suspension). This arm, weighted with a mass of approximately 10 gm, is free to oscillate around a fictitious axis of rotation which passes through the two points of

44

P . MELCHIOR

attachment of the wires to the support and which may be made as nearly vertical as is desired by adjusting the leveling screw known as the fine adjustment screw (the screw a t the bottom of the figure).

FIQ.12. The Verbaandert-Mslchiorhorizontal pendulum.

I n practice the angle i between the axis of rotation and the vertical is of the order of 5". The return force acting on the center of gravity of the arm is thereby reduced to g sin i,i.e., t o a very small quantity, which becomes comparable to the luni-solar attraction itself. Let us now imagine that the local vertical runs from 02 to 02' (Fig. 13), and that under these conditions the zenith follows the elementary arc ZZ' = d on an arc of the great circle whose plane is normal to the pendulum arm. Under the influence of this deviation of the vertical thc pendulum is disequilibrated; in returning t o a stable equilibrium the pendulum weight is displaced from m to m' and its center takes up a position in the new vertical plane of the axis A A ' . The arc of the great circle Z'A defines this new vertical plane of AA' on the sphere; the extension of Z ' A now passes through the center of ni'. Since the arc d and i and the angle p are elements of tlir elcmentary right-angled triangle Z Z ' A with its right angle a t 2, we may write: sin i = t a n d cot p where, by virtue of the smallness of the angles p = d / i , where p is expressed in radians.

CURRENT DEFORMATIONS OF THE EARTH'S CRUST

45

The angle i characterizes the sensitivity of an ideal pendulum, the smaller i becomes, the greater is the deviation of the pendulum arm following a given deviation d from the vertical. The factor l/i thus characterizes the amplification of the pendulum; if a value of 1" has been arrived a t for i, the amplification factor will be 206,265.

On the other hand, the free period of the pendulum arm which is

To = 2 4 / g in vertical suspension, here becomes

T I = 2 7 4 1 / g sin i and we therefore have

i 21 sin i = T:/Ti2 The instrumental constant Tomay be precisely measured before suspending the arm by making it oscillate vertically around its point of intersection with the axis passing through the points of attachment of the wires t o the frame. A priori determination of this intersection is, however, very difficult. This procedure reduces the problem of calibrating the pendulum to measurement of Ti, thus simplifying it. A considerable correction must, however, be introduced to allow for the return torque arising from the torsion of the wires. The fictitious angle E which has to be added to the angle i is approximately 5"5 for the pendulum in Fig. 12. We shall have occasion to return subsequently to a high-precision method of calibration perfected by Verbaandert [13]. In the Verbaandert-Melchior pendulum shown in Fig. 12 the base stand is

46

P. MELCHIOR

an aluminum plate treated to resist corrosion. The three points of support consist of a fixed rectangular foot and two stainless steel screws with an 0.5 mm thread situated 27 cm from the foot. These leveling screws are operated by slow mechanisms situated under the plate, whose flexible controls are better seen in Fig. 14. The angle i is simply adjustable by the fine adjustment

FIG.14. Installation of a Verbaandert-Melchior horizontal pendulum at the underground station a t Warmifontaine.

CURRENT DEFORMATIONS OF THE EARTH'S CRUST

47

screw to a value of 4"5, which corresponds to an oscillatory period of 80 sec ( i E = 10"). The other screw, which is called the drift screw, is used for precise adjustment of the azimuth of the arm. For highly sensitive control the two screws should clearly be placed a t a right angle. The support is a quartz tetrahedron fixed to the baseboard by pressure. The pendulum arm is a 12 cm quartz tube weighted with a 10 gm quartz weight. Under these conditions the suspension wires are also of quartz fixed to the various points of attachment by autogenous welding. This procedure has the great advantage that it permits simple adjustment of the length of the wires and the reduction of all initial torsions a t the time of attachment of the wires. These are important factors in ensuring stability and high sensitivity of the apparatus. It should be noted that high thermal stability is a feature of quartz. It is a disadvantage of metal apparatus that the wires have to be inserted in clips, which does not give good definition to the point of attachment and does not eliminate initial torsion. Quartz wires with a diameter of 0.04 mm (40 p ) were employed. Another advantage of quartz is the high breaking strength of fine wires. It should be noted that the lower wire should clearly bear double the tension. A flat mirror mounted on the mobile arm transmits the light-spot of a projector via a lens with a focal length of 5 meters. The reflected lightspot slowly scans the slit of a photographic recorder when the arm turns (Poggendorf's method).

+

7.1.2.Calibration of Verbaandert-Melchior Pendulums. The greatest amplitude of the relative deviations of the earth and the vertical is usually less than O"05. In the case of pendulums for which the short sides of the base right angle triangle are approximately 0.25 meter this angle corresponds to relative variations in the height of the three ground points on which the three leveling screws of less than 0.05 p rest. In reality these values of 0.05 p are relative to the greater amplitude of the phenomenon under investigation, and fractions of this value are to be measured on the photographic record of the movements of the pendulum. Thus, a 1 mm variation of the ordinate, which can be visually estimated on the recording, derives on average from a deviation from the vertical of not more than O"001, which corresponds to oscillations of the level of the points of support of approximately a millionth of a millimeter, or 15 A which is of the same order of magnitude as the molecular unit of quartz. I t is noteworthy that these movements are still perceptible although they are in a range of distance which falls below the discriminating power of ordinary electron microscopes. A suitable method of calibrating the apparatus is to submit it to artificial

48

P. MELCHIOR

inclinations of known amplitude. Verbaandert's device is a hollow steel capsule whose upper surface is 5 nim thick. Mercury is injected into the internal cavity under pressure. Variation of this pressure produces bulging of the capsule which is accurately measured with a n interferometer functioning with the green light of mercury. Each capsule must be carefully calibrated since the deformations arc very small: of the order of 0.273 p (half the wavelength of the mercury ray) for a difference of Hg level of approximately 40 cm. This calibration iu precise to within 0.1 yo If this capsule is placed under the "drift screw" of a horizontal pendulum as shown in Fig. 12 and if the Hg pressure is varied so as t o produce an elevation of A/2 =0.273 p, the frame of the apparatus (273 mm in length) will be inclined by O"2. All that needs to be done to calibrate the instrument is to produce periodic variations of pressure by fixing the mercury level to a revolving arm. This produces a known artificial tide and it is then sufficient to measure the response of the pendulum and its characteristic period to establish its calibration factor K , which is defined for a standard focal distance of 5 meters by the relation

K

=

ST^

where s represents the sensitivity of the pendulum in milliseconds of arc per millimeter on a photographic record made a t a distance of 5 meters.

7.1.3.The Im%d?ation of Horizontal Pendulunas in Underground Stations. It is obvious that the conditions under which a pendulum is installed will have as much bearing as the qualities of the apparatus itself on the quality of the results that it is hoped to obtain. The apparatus must rest on the rock with the minimum transition. The rock itself should be sufliciently homogeneous, and temperature a t the point of installation should be absolutely constant. For this reason a depth of a t least 30 meters is needed to obtain reliable records. Most stations are installed at depths of more than 50 meters. We may point out by way of example that if a temperature difference of 0.003"Cshould accidentally occur between the drift screw and the other two adjusting screws of the apparatus, this would result in a parasitic deviation of approximately 5% of the maximum amplitude of the earth tide. This is ample justification for all the precautions taken in installation. Moreover, it has been demonstrated by calibrations of VerbaandertMelchior pendulums that, with an operating period of 70 sec and a focal distance of 5 meters, the record gives in practice: 1 mm = O"OO1

which corresponds to a difference of level of only 15 A (1.5 ten-millionths of

CURRENT DEFORMATIONS OF THE EARTH’S CRUST

49

a millimeter)! A precision of the order of 0.3 mm is effectively obtained in reading records; i.e., variations of level of 5 A, which is of the order of magnitude of molecular dimensions, are disclosed. This result could be obtained only by a new installation method. The experience of others (Toniaschek, Jobert) also led to rejection of artificial pillars for the installation of apparatus of this type. In particular, concrete and cement should be rejected as foundation elements by virtue of their instability, their creep, and their sensitivity to humidity and humidity variations. Areas where the rock is most homogeneous and strongest are selected in the rock walls [ E l , and niches for installation of the pendulums are chiseled out without the use of an explosive (Fig. 14). This procedure avoids all cracking of the rock that might adversely affect the stability of installation. Moreover, it has the effect of “embedding” the apparatus in the rock wall, which suppresses in large measure the sometimes disastrous effect of air currents capable of producing thermal changes. Three stainless steel cylinders (3 cm in diameter and 5 cm in height) fitting into three precisely drilled holes in the rock were sealed in position to act m a bond between the pendulum and the rock. The bond was effected by a thin coating of thermosetting resin. The thickness of this adhesive laterally enveloping the steel cylinders was approximately 4 mm, and the lower part of the steel cylinders was directly in contact with the rock, which exactly fitted its slightly conical end. The upper part of these steel cylinders is cup-shaped and the cups are filled with inert liquid octyl sebacate of zero vapor tension which perfectly protects the supporting points of the apparatus. The geophysical results obtained with these instruments form the subject of Section 8. Here we shall deal with the measurements which characterize the instrumental stability as strictly defined. It was found by placing up to six different pendulums for a total period of 6 years in the same niche that all the instruments drifted with time in the same sense and at a regularly decreasing rate. This drift, which may rightly be attributed to an expansion effect of the rocks, is characteristic of the niche. For some niches it could initially be as much as O”02 per day a t the outset. A t the present, after 3 to 4 years, it is of the order of O”002per day [14]. Once stabilized, niches possess the great advantage of affording guaranteed shelter for very long term studies and three fixed reference points for the apparatus. The geophysical aspect of the drift may be considered as soon as stabilization seems established, taking the precaution of duplicating the apparatus in each component. It has been found that the sensitivity of the pendulums is very stable with time. For example, a pendulum installed a t Warmifontaine (pendulum No. 23)

50

P. MELCHIOR

maintained a self-period of 90.48 sec (a sensitivity of O"000755 per mm) for almost a year with a total fluctuation of 2.9 sec and a mean dispersion of f0.25 sec. The root-mean-square measurement error reduced progressively to O"00015.

7.2. Gravimeters Only a very brief account will be given of the principles of the vaiious gravimeters since, unlike horizontal pendulums, they are well known and have been described in the many specialized works on gravimetry. There are two types of gravimeter: static and astatic. Figure 15 is a n internal diagram of the Askania G.S. 11 gravimeter, which &..,.

FIG.16. Askania gravimeter, interior view.

is a static linear gravimeter. Two curved springs coiled in opposite directions end functioning in torsion support the beam L weighted by a mass M .

CURREKT DEFORMATIONS OF THE EARTH’S CRUST

51

If a is the initial angle of twist of the springs needed to compensate the value of g and to maintain the arm horizontal, and if 7 is the rotational moment for a flexure of a unit angle, we obtain (7.1)

MgZ cos a = r(a

+ a)

In practice one always works in a limited zone for which a z 0. For a = 0, we have

If we differentiate (7.1) and take (7.2) into consideration, we obtain Aa = 4Asls)

There is, therefore, a proportionality between the angle of twist and the gravity variation. For an initial angle of twist of 360” z 1.3 x loGsec of arc, we obtain

A y = 1.3”lmgal. To obtain a resolution of 1 pgal one must be able to measure at least an angle of O”0013, or a mass displacement of 6 x lo-’ mm or 6 A which is a thousandth of the wavelength of yellow light, which is not even the spacing of two atoms in a metal [15]. It is noteworthy that the threshold for angular measurements is the same as for horizontal pendulums. To reach this precision a light-spot, which has been passed through a slit and a lens, is doubly reflected on the mirror mounted on the beam of the apparatus, passes back through the lens and falls on two differentially mounted photocells which receive the same amount of light in the zero position. Any inequality between the amount of light received by the two cells will give rise to a voltage difference and will produce a current of corresponding direction and strength which passes through the coils of a sensitive galvanometer and produces deviation of another light-spot reflected by the mirror suspended from this galvanometer. This is the spot visually observed on the graduated dial of the gravimeter. The recording equipment consists essentially of a highly sensitive galvanometer, a photoresistance follow-up recorder, and a slowly revolving drum carrying the recording paper. The high sensitivity galvanometer is connected in parallel with the normal galvanometer fitted to the gravimeter. The light beam that falls on the mirror of this second galvanometer is reflected by four plane mirrors so as to form an optical arm of 1 meter. The movements of this spot are followed by a cadmium photocell mounted on a carriage whose movements it controls (follow-up).The

52

P. MELCHIOR

carriage also carries a sapphire which scratches the waxed paper of a mechanically rotated drum. Continual recording of gravity variations is ensured by the provision of a roll of waxed paper which turns on the drum a t 6 mm an hour for 31 to 32 days. Astatic gravimeters are so assembled that the spring counterbalancing gravity is seen from the point of rotation of the beam at a n angle u approximately 90". Under these conditions, if the spring is of zero length when at rest, the sensitivity of the instrument is then theoretically infinite. Zerolength springs are obtained by quite simple procedures. In practice high sensitivity is achieved by affecting the value of the angle u. The North American, LaCoste-Romberg, Frost, Worden, and other gravimeters conform to this principle. The zone of linearity is particularly extended in the LaCoste-Romberg gravimeter. The tide-recording model is a very complicated apparatus [l6]. The light beam reflected by the mirror on the beam passes through a slit and falls on a photocell which detects all movement by variation of the illumination. A servosystem and selsyn generators have been employed to make a self-recording follow-up system. Figure 16 shows the basic principles of the drift-free photoelectric optical system. A beam of light from the source of illumination b is reflected from the

-n

FIO,16. LaCoste-Romberg gravimotor with photoelectric reading system.

CURRENT DEFORMATIONS OF THE EARTH'S CRUST

53

mirror a on the gravity-responsive system through the double slits c which is so adjusted that cqual amounts of light pass through thc upper and lower slits when the gravity-responsive system is at its null point. As the system moves away from the reading line there will result an increase in illumination through the slit in the direct,ion of motion and a decrease in illumination through the other slit. The disk d (shown in Fig. 16) is rotated to interrupt the light beams passing through the two slits, so that half of the time light passes through the upper slit and half of the time through the lower slit before it falls on the photocellf. The lower light beam may pass through openings 1 and 1' and the upper light, beam may pass through openings 2 and 2' as the disk rotates. The light from the two beams is focused as an image of the light source on the photocell f by the lens e . A photosensitive detector h, used to provide a phase reference, is located in such a manner that illumination from the source g will fall on it except when the portion of the disk d between R and R" passes between h and g. The motor m rotates the disk d in such a manner that the radius R'r' always coincides with the dark section between the double slits c , and the light path gh always passes between the radii R and R". From this description it can be seen that when the gravity-responsive element is a t its null position, the light intensity falling on the photocell is constant because the upper and lower beams are equal. Also, as the gravityresponsive element moves away from the null position in either direction, the light intensity varies and the magnitude of the variation is proportional t o the deflection from null. The direction of the deflection is determined by the phase of the light variations. A photocell h is used to provide a phase reference and could, of course, be replaced by a commutator or other device. The signal obtained from the phase detector drives the servosystem which controls the position of the gravity-responsive system with respect to its null point. The servo turns a selsyn generator which in turn operates a recorder, and thus a self-reading system is obtained. The most delicate part of the gravimeter is the spring which has to provide the force counterbalancing gravity. If one wishes to maintain g precise t o within 0.01 mgal the force of the spring must remain constant to lo-' and if one accepts a drift of 0.1 mgal/hr the force of the spring must then remain constant to lo-' approximately for a period of 1 hr. For 30 years it was thought pointless .to make this demand of a metal spring since the effects on it of temperature, fatigue, and elastic reactions are of such an order of magnitude that no one had thought that these conditions could be satisfied. Now after operation for several weeks in a thermostatically controlled room the drift may be reduced to some 0.1 mgal per month or approximately 0.1 pgallhr, and the force of the spring is therefore constant to approximately lo-"! This has been possible because steel with a thermoelastic coefficient of

64

P. MELCHIOR

between +10 x lo-" and -10 x 10-o/"C has been produced in the last two decades, whereas the coefficient for ordinary steel is -200 x 10-o/oC. Small portable thermostats can equalize external temperature differences of 10" to 20°C to one hundredth so well that temperature within the gravimeter remains constant to within 0.01"C or almost to 0.001"C. The instrumental drift of gravimeters is very perceptibly linear but its ultimate fluctuations are largely dependent on external conditions: surrounding temperature, power supply of the thermostats, and operation of contact thermometers. The beam must also be protected from changes of air density related to variations of atmospheric pressure, wherefore the moving parts are hermetically sealed. In addition, the mass is balanced in volume in relation to pressure variations by one or more floats whose position on the arm may be varied to "compensate" the beam as exactly as possible. This compensation is effective to better than 0.5 pgallmbar for Askania gravimeters. It may be taken to be perfect for LaCoste-Romberg gravimeters. It may be checked by placing the instrument in a pressure chamber in which pressure may be varied at will.

7.3. Diflculties Encountered in the Interpretation of Gravimeter ResultsComparison with the Results of Horizontal Pendulum Since the beginning of the International Geophysical Year in 1967 many recordings of periodic deformations of the crust have been made a t various points of the globe with gravimeters and with horizontal pendulums [18, 201. For instance, we may mention that the International Center for Earth Tides has assembled the results of 3280 harmonic analyses: 1187 in the vertical component (gravimeters), 1070 in the horizontal north-south component (horizontal pendulums), and 1023 in the east-west horizontal component (horizontal pendulums). Although the precision of the two types of instruments is practically the same relative to the amplitude of the tides from the point of view of random errors, the same may not be said of systematic errors, whose influence seems to affect the results so far obtained with gravimeters to such a point that it may seem highly risky to advance any explanation of the results obtained. Consequently, in particular, two gravimeters situated in the same spot yield systematic differences in their results that are of thc same order as the differences between stations. On the other hand, horizontal pendulums are very accurate and several different apparatus in the same spot yield the same results even for a short series of obscrvations, as demonstrated by Table 111. In the author's opinion the origin of these dispersions for gravimeters is

CURRENT DEFORMATIONS

or THE

55

EARTH'S CRUST

that the operator cannot correctly determine sensitivity. It should be necessary to know the sensitivity of pendulums and gravimeters a t every instant,. So far there are no published results of the tests which have already been carried out, and the basis that has to be taken is that of sporadic indirect determinations (once or twice a month) between which interpolation is carried out (linear or better). 4.5 P Q O l

4 0 pqai

I

1956

,

3 5 ppl 1960

1961

I

1962

I

1963

O"0015

95 O"OO10

P B

O"0005 1960

I

1961

I

I

I962

I

1963

I 1964

1965

,

1966

FIG.17. Variation of the sensitivity of the instruments with time. The scale of the ordinates expresses the sensitivity per millimcter on the recording paper. In order to make gravimeters and horizontal pendulums comparable the scales adopted are such that the A12 wave at Brussels is represented by the same amplitude expressed in centimeters.

Figure 17 demonstrates that, although this procedure may involve some imprecision in intcrpreting the recordings of horizontal pendulums, it is patently bad for gravimeters. Moreover, the calibration method itsself is unreliable, once again especially for gravimeters. The following is a brief description of the procedures adopted.

(a) Gravimeters The calibration operation is carried out in two stages. The beam of the gravimeter is provided with a correction spring uhose extensions may be regulated by a micrometer screw accessible on the upper plate of the apparatus. The visual micrometer with which this screw is fitted permits an estimation of its rotation with a precision of the order of 1 yo (Fig. 15).

56

P. MELCHIOR

A first operation, in principle definitive, is to calibrate the scale of this niicrometer in gravity units. For this purpose the gravimeter is displaced along a known base and a t each point along the course the beam is adjusted to its null position by the correction spring. Since gravity differences along this course are known in general to be precise to within yo 0.1, similar precision is obtained in calibration of the spring. A second method is to add very small precisely known masses to the mass of the beam (the ball system of the Askania gravimeter) and compensate the effect by adjusting the spring under investigation. Useful control is provided by the use of both methods. The second, repetitive operation is a monthly calibration of the sensitivity of the galvanometer associated with the gravimeter by several consecutive forward and backward movements of the light-spot (7 to 8) deplacing the beam with the spring calibrated in the previous operation (Fig. 18). It is

then a simple matter to transfer from one calibration to the other and make microgals correspond to centimeters of displacement on the recording paper. Unfortunately this handling is not conducive to good functioning of the apparatus. It is in itself a major disturbance to the recording apparatus and it is often found that the gravimeter reacts abnormally a t the time of the first displacement, so that the first measurement of a series often has to be rejected. The root-mean-square error of the measurement arising from a single

CURRENT DEFORMATIONS OF THE EARTH’S CRUST

57

displacement of the beam is 1.2 yo.It is therefore approximately h0.5 % of the arithmetic mean of a complete calibration operation (7 to 8 displacements). If the sensitivity variation of an Askania gravimeter in the course of a year is represented by a parabola of third or fourth degree, the dispersion of monthly measurements relative to this parabola is approximately 1.8 yo.

( b ) Horizontal pendulums The three-stage calibraton operation has already been described. High precision has been obtained owing to Verbaandert’s invention of the dilatable capsule. The probable error of determination of K for an instrument is 0.25 yo on the average. There is little difference in the values of K obtained for some sixty different horizontal pendulums. With a minimum of 5.9and a maximum of 6.4 the range of variation is 10 %. Once the value of h’is known, the apparatus is installed and its characteristic period is measured each month by chronometer. The precision is 0.2 sec for periods of the order of 70 sec. We therefore have

ds/s = dK/dK

+ 2(dT/T)< 1%

Variation of s with time is slight and regular, and hitherto linear interpolation between monthly determinations has been accepted. If this variation is represented by a parabolic formula of third or fourth degree, the dispersion of the monthly measurements relative to this parabola is approximately 0.5 yo. It is obvious that these methods are merely expedient and that a permanent system for control of the sensitivity of the apparatus should be produced for exact work. This system will be incorporated in equipment produced for first-order stations.

7.4. High- Precision Equipment Needed for First-Order Stations Two basic improvements have been proposed [181: 1. reduction of the frequency “frame of reference” from an hour to a minute, or a t least to a frequency of 10 min; and 2. superimposition on the tidal spectrum of three artificially produced lines enclosing the two families of lines of semidiurnal and diurnal tides (Fig. 19). The first condition has already been in~plementedfor the Uccle gravimeter [19] and is now to be implemented for a pair of horizontal pendulums a t Warmifontaine.

58

P. MELCHIOR

i

I

2

10' 3rn T

= 2r/w

FIG.19. Representation of the line spectrum of the earth tide at Warmifontaine in the east-west component with tho addition of three lines (A7,A19, A37).

The second improvement may be readily achieved as follows:

( a ) For horizontal pendulums By permanently installing a Verbaandert dilatable capsule with a casing of double the usual thickness beneath the drift support of the pendulum. The mercury level should be actuated by an upward and downward movement regulated by a small Kelvin wave generator providing three waves of equal amplitude (Fig. 20).

(b) For gravimeters By forming a condenser with the mass of the mobile beam, as Tomaschek and Lecolazet have previously done, but electrically summing the three sinusoidal waves of equal amplitude exactly as for horizontal pendulums. The periods of the three artificial waves introduced t o provide a permanent check on the sensitivity of the apparatus should be so selected as to avoid all resonance or commensurability between them and with the tidal waves. We suggest periods of 7, 17, and 37 hr, which are quite unlike any known astronomical and geophysical periods. The tidal curve will be systematically modified under these conditions. As an example we have calculated what would have been the effect on the curve recorded in the east-west component a t Warmifontaine in June 1965 of injecting a set of three artificial waves with individual amplitudes of O"005 into the capsule. With the thick capsule now available an appropriate effect could

CURRENT DEFORMATIONS OF THE EARTH’S CRUST

59

be obtained by giving a diameter of 5 cm to the driving wheel of the Kelvin machine . It is obvious that the combination of artificial waves for introduction into the pendulum or gravimeter recording should be made as rigorous as possible so that each superimposed line should be of the maximum fineness. With this in mind, provision will be made for all equipment a t Warmifontaine to be guided by a Rohde and Schwarz transistorized quartz crystal clock powered by battery in case of failure of the power supply. There will be a control potentiometer on each of the axes of rotation of the motors of the small Kelvin tide generators and on the last driving wheel whose rotation will give the sum of the three movements. The quartz crystal clock will simultaneously control the three driving motors of the components of the Kelvin wave generator and will cause the potentiometer readings (of the spot followers and the potentiometer of the summation drive wheel) made by a three-digit transistorized electronic voltmeter to be recorded once a minute on punched tape. In addition, the positions of the three-phase control potentiometers on the axes of rotation of the motors of the Kelvin wave generator will be punched once an hour.

Fro. 20. Kelvin wave generator for permanent calibration of horizontal pendulums.

60

P. MELCHIOR

This will provide all possible controls, and the punched tapes may be analysed directly in the data processing machine. Combinations of special commands may be used to extract the three artificial waves from the recording and to retrace reliably the development of apparatus sensitivity throughout the frequency band relating to the tides. The sensitivity curve will then be used to transform the recorded values to milliseconds of arc or microgals, after which spectral analysis or analysis by combinations of ordinates may be made.

8. SOME RESULTSOF RECENT OBSERVATIONS

8.1. Indirect Effects of Tides

A few characteristic results of earth tide measurements have already been described in Section 7. A complete description is to be found in another very detailed work [5]. The new apparatus now available makes observation of earth tides a simple matter, and it is not difficult to obtain curves on the recording equipment whose amplitude reaches 5 to 6 om. The indirect effects of oceanic tides are the main perturbing phenomenon in earth tides. There is nothing essentially new to add to the description already given in Advances in Geophysics (vol. 4, pp. 411-416, 1958), and the matter will not be further considered here. The same applies t o extensometers, which are instruments for which development has been slow and the interpretation of records is acknowledged to be difficult (ibid., pp. 418-421). Later in this section we shall show by a series of examples what is to be expected of the new gravimeters and horizontal pendulums in a study of the slow deformations currently occurring in the earth’s crust. This study is still in its infancy and several years of experience will undoubtedly be needed before conclusions may be drawn. Moreover, this study is a t present in a difficult situation, particularly in relation t o the effects of atmospheric pressure, because of the paucity of good stations. 8.2. Direct Observations of Current Deforniations of the Earth’s Crust The great difficulty in this type of observation arises from the fact that t>hc measuring instruments usually have a self-drift of an instrumental nature that it is difficult if not impossible to calculate. Interpretation of the curve thus becomes highly problematical if reliable separation of wh:tt is instrumental from what is geophysical is not possible. In this respect noteworthy results have been obtained by installing horizontal quartz pendulums in niches cut in the rock. The most important feature is the decrease in the drift with time for niches

61

CURRENT DEFORMATIONS OF THE EARTH'S ORUST

employed immediately after they have been cut. All apparatus placed in such a niche drift in the Sam direction and in comparable amounts. This obviously corresponds to the phenomenon of relaxation of the rocks after cutting of the niche and demonstrates that the purely instrumental drift is very slight for this type of instrument. The observed drift is in general of the order of 3 mseca per day or 1" per year. The accompanying tabulation gives an example of decrease of the drift observed with five different instruments successively placed in the same niche.

Station: Sclaigneaux I Pendulum No. 4 4 4 4 9 9 30 30 42 13 a

Epoch

26 Nov. 1959-4 Jan. 1960 14 Jan. 1960-7 Mar. 1960 18 Mar. 1960-9 Apr. 1960 23 Apr. 1960-11 July 1960 21 July 1960-10 Feb. 1961 27 Feb. 1961-25 Mar. 1962 27 Apr. 1962-23 Aug. 1962 31 Aug. 1962-15 Oct. 1962 8 Dec. 1962-5 Mar. 1963 26 Apr. 1963-19 July 1963

Niche B N

NS component a1

40 54 23

+21.59 +21.76 +19.61 16.48 +8.27 +7.52 +8.68 +5.99 f10.50 +2.21

80

205 392 119 46 88

85

+

Here N means the number of days in the period.

* Here a1 means the daily linear drift expressed in milliseconds of arc (O"OO1). In effect the angle u measured by our instruments is the difference between the tilt i of the ground and the deviation E of the vertical, to which a possible instrumental drift d is added

It is very difficult to comprehend the part played by deviations from the vertical in this phenomenon. It should be slight since astronomical observations could detect them directly if they exist. The only precise indices that are available are furnished by the stations of the International Polar Motion Service and they only note an extremely weak quantity in relation to the amplitudes detected by comparative levelings, which may attain an order of magnitude of O"O2 per year in some instances, but are generally less. It is therefore certain that the tilt values of the order of 16" or 9" per year obtained a t the Russian station a t Kondara (Ostrovsky, Bakrushin, and Mironova) are unreal and can only be attributed to an instrumental fault or

62

P. MELCHIOR

to poor installation of the equipment. Once again the use of brick or concrete pillars for these measurements is contraindicated.

8.3. Hydrological Effects: Deformations of the Ground due to Variations of the Level of the Meuse at Scluigneaux [14] Figure 21 depicts the situation of the station a t Sclaigneaux in relation to the Meuse, which is a river 120 meters wide a t this point and 500 meters distant from the station.

S

/ N

FIG.21. Position of the Sclaigneaux station in relation to the Meuse: S, station; E , entry to the gallery; SA-SB, extreme radii of the sectors for which the attraction of the water in the Meuse was calculated.

It is fortunate that flooding of the Meuse is not frequent, since it has a clearly defined effect on the recordings. This is the combined result of two effects: (1) the attraction of the additional water mass, and (2) the downwarping of the ground under the weight of this mass. It is a fairly simple matter to calculate the attraction effect by dividing the river into circular sectors with their apex a t the station, Aeast = f p h (cos 0, - cos o,)(log r2 - log rl)

q sin 1"

(8.1) AsrJuth = f p h (sin 0,

y sin 1"

-sin 0,)(10gr, - 106r , )

where the angles 0 are read off from the north direction. The calculation was extended for a radiusof 2 km on each side of the station. Figure 21 demonstrates that this may appear to be sufficient: since the farthest zones yield negligible results, as was verified by calculation. The total effect established is

CURRENT DEFORMATIONS OF THE EARTH'S CRUST

Aeast = 0.1202 mseca (8.2)

63

for a meter of water

Asouth= 0.1494 mseca

It is very difficult to calculate deformation of the ground in any very precise manner. Nevertheless, given the local nature of the force applied to a very restricted space and the slight distance of the station (500-700 meters), it may be considered that only the uppermost layers are affected by compression. These arc precisely the layers to which it is very difficult to apply the calculation. I n fact, as is evident from Fig. 22, the dolomite bed subjected to deformation by the pressure of the water of the Meuse is a wedge which, by virtue of the Landen fault, is unrelated to the superjacent layers and rests completely on the subjacent layers. This dolomite is greatly fissured and water-saturated. NORTH

8

I

C

SOUTH

V8seon dolamlie

2 Psommite (Fomennlon) hematlie bed at base 3 Frosnion (limestone) 4 Silurian a Landen fault b

c

Venn fault Sepulchre p11s n o

2

FIQ.22. Geological situation of the Scleignaeux station.

This process has led to dissolution of the limestone, leaving behind beds of clay. It is likely that there are one or more clay joints that take up water a t the time of a flood and yield to pressure. The phenomenon observed should therefore correspond t o plastic flow of this water-saturated clay and the dolomite beds should only return slightly to position after disappearance of the overload, as after a marked reduction in the level of the river. One would expect a hysteresis loop when the ground is unloaded. This is not so and, except in one case (Fig. 23b), return to the initial state has not been observed. It may logically be thought that the ground behaves differently according to whether there is compression or decompression. Furthermore, it may be noted from Fig. 23 that successive floods do not produce deformation strictly proportional to their amplitude. This may be dependent on the state of the ground: water penetration, temperature, etc. This demonstrates that indirect effects may vary with time a t a coastal station, provided that the surface geological structure plays an important part in this instance. The results are summarized in the accompanying tabulation.

64

P. MELCHIOR

Rising flood water

Amplitude meters

+ +

North deviation

East deviation

+

0"12(P.1 ) +0"07(P.1) +0"07(P.9) +0"07(P.9) -

1m20 5 Dec. 1960 2m60 31 Jan. 1961 +2m50 1 Dec. 1961 13 Feb. 1962 1m Falling flood water 24 Sept. 1962 -2m 15 Jan. 1963 -2m60

+

-

-0"05(P.13) -0"05(P.42)

Ground temperature at a depth of 2 meters

+8" +5" +7"

+roe

+

14' +3"

A tilt O"07 would correspond to depression of the ground by 0.2 mm a t 500 meters which sets a lower limit t o the order of magnitude since flexure is clearly not a linear function of distance. The complex and fissured structure of the dolomite bed as shown in Fig. 22 scarcely justifies any further excursion into theoretical calculations, The figures representing pendulum drifts were constructed by eliminating from the observed curve the theoretical tide relating to an amplitude factor equal to that yielded on average by harmonic analysis and different for diurnal and semidiurnal waves. It will be noted that there is a slight delay of the pendulum drift, i.e., of the deformation of the ground relative to commencement of the flood. The amplitude of the deviations is then of the order of 200 times the attractive force of the active mass. The solution given by Boussinesq for deformation of a flat plate yields for the slope of the surface.

where

a, = g2/4nf = 12 x 10" Conceding that the upper crust is highly compressible, one may set h = p. As a result if p = 10" cgs (the rigidity of gneiss)

awlas = 1

(8.4)

8 ~

which will yield a total deviation from the vertical of (8.5)

A

+ a m p s= 1

9 ~

A value of p of the order of 10"' cgs must be assumed for the magnitude of

65

CURRENT DEFORMATIONS OF THE EARTH'S CRUST

I I N

I

I

S

10 12

1961

30 I 31 I I 2

10.2 (b)

3- 75 2' 50

I I

b05

uw 1962

I59

209

309 110

259

510

10 10

i

(CI

FIQ.23. Effect of the Meuse flood at Sclaigneux in ( a ) December 1961, ( b ) February 1961, arid (c) September 1962.

66

P. MELCHIOR

the effects observed a t Sclaigneaux. A similar value has previously been given by R.Takahashi (Japanese tuff). If the load were to be proportional to the flexure, the mean rigidity of the distorted bed could be deduced by formula (8.3),even if this onlyestablished the order of magnitude owing t o the complication of the outline presented in Fig. 22. Similar observations have been made at the Dourbes station, which is situated a t a depth of 60 meters near a small river, the Viroin. It has been noted that the drifts of the four pendulums installed a t Dourbes are affected by variations of the level of the Viroin in unexpected proportions; this once again testifies to an effect due to the impregnation of extremely porous ground (limestone). This demonstrates that hydrology has a large part to play in high-precision measurements and that an overriding advantage will be obtained by avoiding water courses, even of a secondary nature, when installing observing stations. This matter must be seriously considered before interpreting the results of geodetic leveling. Ground subsidence has been attributed notably to pumping in aquifers in industrial zones or in zones with a high population density. Astronomers are well acquainted with the case of the latitude variation observation station a t Chardjui in the U.S.S.R. which had to be abandoned because of considerable disturbances to the direction of its vertical caused by fluctuations of the bed of the river Amu Darya. In this respect it is of interest to study the behavior of aquifers by installing ordinary tide gages in wells or boreholes. These aquifers respond like manometers to compressions and dilatations of the crust. It is also of some interest to note the existence of true tides even in the center of continents [21]. The latter are merely the reflection of the cubic dilatations described in Section 2 by formulas (2.40) to (2.46). These tides have the following characteristics which are criteria permitting of their classification as deformations of the elastic crust to the exclusion of all other phenomena: 1. The amplitude ratios of the various waves are those of celestial mechanics. 2. The wave phases are a t 180°, since a high theoretical tide corresponds to a dilatation and consequently to a low tide of the water and conversely. No other information can be extracted from these wells because we do not know either the volume of the water affected by the cubic dilatations, or the porosity of the containing beds and so i t is impossible to construct the theory of the apparatus. Although measurements have certainly been made in many wells, there have only been nine instances so far that have provided data for harmonic analysis. The most striking result is that there would appear to be a relation between the amplitude of the tide and the depth of the ground water table, as the accompanying tabulation demonstrates.

67

CURRENT DEFORMATIONS OF THE EARTH’S CRUST

Depth of the ground A(Mz)cos%$ water table in meters

Station

BasBcles, Belgium Carlsbad, New Mexico, U.S. Milwaukee, Wisconsin, U.S. Sontra, Germany Iowa, U.S. Duohov, Czechoslovakia Heibaart, Belgium Turnhout, Belgium Kiabukwa, Congo ~~~~~

~

39 86 120 120 252

600 1200 2175 B 2400”

~

1.11 0.63 2.00 2.88

2.06 3.45

3.00 3.79 7.67 _

_

_

_

~

~

Here the depth of the ground water was estimated from the geothermal degree and water temperature. a

Here consideration has been given to the amplitude of the principal semidiurnal lunar wave M , “reduced to the equator” (divided by cos2+)to make all the stations comparable by correcting for the theoretical effects of latitude. 8.4. Atmospheric and Hydrological Effects in Leveling

Of the numerous causes of tilting of the earth’s surface, particular interest is attached t o flexure arising from loads due to atmospheric pressure. Bontchkovskii has observed tilting of the surface of the earth arising from variations in the distribution of atmospheric pressure [23]. The period of these variations of tilting equals the life of high pressure formations, i.e., is approximately ten days. Leontiev has noted that this corresponds to the duration of a ground survey by forward and back traverse [22]. This author carried out first-order leveling along the Syzran-Astrakhan (1050.3 km) and Priyutnoye-Astrakhan (389.4 km) lines to establish the relation between variation of atmospheric pressure and differences of forward and back traverses. He selected a central point on the synoptic chart for each part of the line leveled and took atmospheric pressure values for a distance up to 1500 km, employing 900 synoptic charts from which he took approximately 8000 atmospheric pressure values. Mean atmospheric pressure values to each side of the central point were deduced for each day of the observations and for each leveling direction. Leontiev calculated the differences

for each day of measurement, where Ab(,irectand Ab,,,,

give the difference

~

68

P. MELCHIOR

of mean prcssures for a given sector between the direct and back traverse. Figurc 24 shows the comparison with the measurement results. The curves exhibit a quite marked correlation. Large waves whose length is of the order of 100 km are revealed in the course of these curves. It is only a t the end of tlie line that there is no directly observable dependence betwcen the curves. The author explains this by the higher lcvel of ground water in the Volga valley and by variations of this level.

2

B

400 km

FIG.24. Change in the distribution of atmospheric pressure end differences of surveys in opposite directions.

Nevertheless, the coefficients of correlation are fairly weak because variation of atmospheric pressure also modifies the other sources of error in highprecision leveling which are not related to tilting of the earth's surface. Moreover, the value of the modulus of rigidity p in the formula for the upper layers of the earth's surface varies greatly. According to Bontchkovskii and Lettau [23, 241 the modulus of rigidity a t a depth of the order of 4 to 5 metcrs is 0.3 x 10" dynes/cm2. At the depth of the leveling marks, however, the value of the modulus of rigidity should be greatly reduced owing to variation of density and humidity in the ground. Thus, according to Lettau, p = 0.02 x 10'' dyneslcm2 for a depth of less than 3 mm. Mean values of p calculated in accordance with tho formula are 0.1 x 10" dynes/cm2 for the Syzran-Astrakhan line. The value obtained merely indicates the order of magnitude of rigidity at these depths.

CURRENT DEFORMATIONS OF THE EARTH’S CRUST

69

Like atmospheric precipitation, evaporation and the thawing of snow produce movements of the earth’s surface, but the wetting of the soil by precipitation or by the thawing of snow should modify the modulus of rigidity and this may accentuate the flexure produced by variation of atmospheric pressure. Repeated leveling in 1951 on the Syzran-Astrakhan line revealed that Astrakhan had risen by more than 100 mm. First-order leveling from Urbakh to Astrakhan was first carried out in 1928 and the differences between the forward and back h v e r s e s in the first and second levelings from Urbakh t o Astrakhan were not more than 15 mm. The different distribution of atmospheric pressure a t the time of these levelings could not have produced such a large systematic elevation over a distance of approximately 500 km.Variation of the rhythmic regime of the atmosphere for these years had, however, caused the level of the Caspian Sea to fall by 2 meters and this had given rise to a deviation from the vertical and tilting of the earth’s surface. The elevation or the tilting of the earth’s surface should be greatest in the north-south direction owing to the shape of the Caspian Sea. According to Leontiev the elevation from Urbakh to Astrakhan cannot be completely explained by the fall in the level of the Caspian Sea. The general reduction of precipitation is felt throughout the Caspian lowland, whereas the decline in the level of the ground water produces subsidence of the earth’s surface that is particularly noticeable on a sector of the curve (Fig. 25) in the Astrakhan region.

FIG.25. Tilting of the earth’s surface in the Caspian Sea depression between 1928 and 1951.

Finally, Leontiev correctly remarks that the observed variations in the rate and sign of movements of the earth’s crust may be explained by invoking the effect of external loads.

8.5. Atmospheric Effects 8.5.1. A Remarkable Gravimetric Perturbation Observed at Brussels duriiig the Passage of a Cold Front Accompanied by Exceptional Variation of Barometric Pressure. On 18 July 1964, between 1700 and 2000 hr, a very pronounced cold front crossed central Belgium a t a rate of approximately 90 km/hr (gusts of

70

P. MELCHIOR

wind to 130 km/hr). Atmospheric pressure fluctuations reached 4 mm Hg, or 6.3/mbars. The pressure variation affecting the ground was, therefore, 5.4 gm/cm2 over an expanse of approximately 60 km centered on Brussels. The Royal Belgian Meteorological Institute obtained a recording a t Bruasels on a large Richard weight barograph (the record is reproduced in Fig. 26).

;I hours

18/7

1817

1917

19/ 7

756 754

752 750

FIG.26. Effect of the disturbance of 18 July 1954 at Brussels: upper curve, gravimetric recording; lower curve, barometric recording.

At this moment the Askania gravimeter No. 146 of the Belgian Royal Observatory was also functioning; the very remarkable record obtained is shown on the same figure. Unfortunately, a general power failure due to the violent storm interrupted recording for approximately 30 min. The accompanying tabulation is based on 10 characteristic points of the two curves. A more detailed account is scarcely possible (owing to the slow drum speed), nor would it be of real value. Numerical analysis of this tabnlation yielded the following results (see Fig. 27): coefficient of correlation,

CURRENT DEFORMATIONS OF THE EARTH'S CRUST

Point

UT hr min

Atmospheric pressure,

18 10 25 40 50 19 05 20 30 45 20 0 10

754.8 754.4 752.7 752.2 752.0 752.5 753.2 753.2 753.2 753.3

Gravimeter reading": in 0.1 mm in 0.1 pgal

A1 1 2 3 4 5 6 7 8 9 10

71

195 173 150 141 148 162 152 158 176 172

A9 362 253 140 95 130 199 150 179 268 248

* Sensitivity of gravimeter s = 49.32 pgal/cm. We took Ag = s x A1 - 600.

755

752

t

*I

t

10

20 30 40 Gravity variations+qal

FIG.27. Correlation observed at Brussels on 18 July 1964 between gravimetric and barometric records.

/?= mbar/pgal 1 mm of mercury = 7.929pga1, or

r = 0.869; angular coefficient,

(8.6)

1 mbar =

+ 5.96 pgal

72

P. MELCHIOR

8.5.2. The Wave S , due to Variatwns of Atmospheric Pressure. The result (8.6) is in amazingly good agreement with that obtained by Melchior and Brouet, on the assumption that the disagreement between the amplitude factors of the tidal waves M, and S, may be totally attributed to the effect of atmospheric pressure. These authors submitted their hourly values of atmospheric pressure at Uccle to the same outline of harmonic analysis that the present author applied to gravimetric measurements (application of the Pertsev filter followed by the method of Lecolazet). They reported that the lunar waves M, and 0,are practically nonexistent in variations of atmospheric pressure, that the wave K , is very weak, but that the wave S, is distinctly significant. This is normal if it is considered that the diurnal component of atmospheric pressure is related to heating of the atmosphere by the sun and therefore corresponds to a first-order sectorial distribution, whereas diurnal tidal waves correspond to a tesseral distribution. The only effect of the diurnal component of atmospheric pressure will be a slight displacement of the earth's center of gravity. The wave S,, on the other hand, is sectorial; as it is of second-order it does not influence the center of gravity and it is perceptibly affected by the contribution of the semidiurnal solar subharmonic in atmospheric pressure. But for the fact that the observations were disturbed by the indirect effects of oceanic tides which are precisely of semidiurnal type in the Atlantic, the proposal could be made that the efficiency 118 = Ag(pgal)/Ap(mbar)on the assumption that t,he correction arising for 6(S,) and K(S,) should be such as to yield

6(M,) =W,) K(M2)

=4 8 2 )

There are no known geophysical causes apart from indirect effects and the effects of atmospheric pressure that could account for a difference between these two factors. Using gravimeter No. 145,four years of recording a t Uccle [I41yielded:

S(M,) = 1.169 6(S,) = 1.221 a($,) - 6(M,) = 0.052

+

~(8 =~ -) 1'13 K ( M , )= + 0'15 ~(8,) - K(M,)= - 1'28 Harmonic analysis of atmospheric pressure yields:

CURRENT DEFORMATIONS OF THE EARTH’S CRUST

a(&,) - 6 ( M , ) = + 0.052 for

73

= 5.72

or

1 mbar =

(8.7)

+ 5.72 pgal

If this correction were to be retained it would imply a correction of +3” to the phase of the wave S,, which corresponds to the estimate previonsly made by Lecolazet from gravimetric observations at Strasbourg. In the author’s opinion results (8.6) and (8.7), which come from pressure waves of very different periods (approximately 4 hr and 12 hr), are conclusive. The coefficient arrived a t is, of course, the result of a combination of geophysical and instrumental reactions: (1) attraction of the air mass, (2) flexure of the ground bearing the instrument under the weight of the air, (3) variation of the telluric potential due to this flexure, ( 4 ) direct instrumental effect due t o imperfect compensatioc of the gravimeter beam-it seems however that this compensation is very good (Fig. 28)-and (5) deformation of

+

gn

8h30

9h

FIG.28. Response of Askania 145 gravimeter to a pressure variation of 300 mm of mercury in a pressure chamber.

the framework of the instrument under the effects of accompanying pressure and temperature variations. The effect may therefore differ from instrument t o instrument. The combined effect of ( 4 ) and (5) may be measured by placing the instrument in a chamber in which pressure may be artificially varied. When Lecolazet tested a North American gravimeter in this manner he found that a variation of 1 mm of mercury corresponded to an apparent variation of 3 pgal. It is also possible t o evaluate the attraction of the air mass responsible for the pressure variation on various assumptions. First, if this mass were t o be concentrated in a thin horizontal layer of thickness c and were assumed to be of infinite extent, it would exert an attraction

A , = - 21~j’pe

14 or, for P E (8.8)

P. MELCHIOR = 1 mbar

(f = 6.67 x lo-'):

A,

= 0.42 pgal

On the other hand, when Jobert [26] assumed that the variation of air density with altitude obeyed the law derived from the definition of the standard atmosphere (ICAN) he obtained (with H = 44.30km) an attraction for a 1 mbar variation of pressure that was only slightly dependent on the wavelength of the phenomenon and varied from 0.46 pgal for an infinite wavelength to 0.39 pgal for a wavelength of 280 km. Thus, the calculated attraction is scarcely dependent on the model selected and the result (8.8)may be accepted. To establish a further order of magnitude, let us recall that a 1 cm flexure of the ground would produce a gravity variation of +3 pgal as a result of variation of the altitude of the gravimeter. It may be seen that all the calculated orders of magnitude are compatible with observations.

8.6. Frequency Dependence of Flexure In a particular cam, Zadro [26] was able to study the frequency dependence of loading effects. The Trieste station is simultaneously subjected to (I)the direct effects of the earth tide, (2) the indirect effects of tides in the Adriatic, and (3)the effect of Adriatic seiches. The first two effects have the known tidal periods in the 12 hr and 24 hr bands, whereas the basic transverse seiche in the Gulf of Trieste has a period of 213 min. I n a first approximation Zadro adopts the theory of Boussinesq, assuming that the effect of the loading is /3 times the gravitational effect of the masses in movement. However, she introduces Maxwell's elastico-viscous postulate replacing the modulus of rigidity p by the operator p(1 +[1/(27ri f 7 ) l - l where T is the relaxation time ( T =q/p, q is viscosity). Spectral analysis was made of recordings of the NS and EW components and of atmospheric pressure and the level of the Adriatic. The ellipses described by the displacement vector were constructed for the four frequencies corresponding to the principal peaks in the power spectrum; their major axes were oriented in the same manner. The relaxation time established was approximately 113 min. The total effect recorded by the pendulums is here approximately four times the purely gravitational effect.

8.7. Particle Accelerators in Nuclear Physics The installation of the present large accelerators which are several hundreds of meters in size raises important problems of stability.

75

CURRENT DEFORMATIONS OF THE EARTH'S CRUST

To ensure that particles are not deflected from the target after several hundreds of thousands of circuits, the leveling of the electromagnets must be precise to within f0.05 mm for a diameter of the order of 200 meters. When this problem was raised a t CERN, which was installing a 25 thousand million electron volt synchrotron a t Geneva, the leveling of the magnets had to be precise to within kO.1mm of their true position in a circle with a radius of 100 meters. Decae made a special study of the stability of the upper layers on this occasion [27]. A series of shafts 20 meters deep and 15 cm in diameter were sunk t o the subjacent tertiary molasse; since the ground was very waterlogged these pits filled with water and a hollow copper sphere was floated in them and connected with the bottom of the shaft by an Invar wire attached with araldite. I t was thus possible to observe the movements of the moraine relative to the molasse. Decae demonstrated that the lower beds underwent periodic expansions and contractions with an amplitude of 0.5 mm per 500 meters (lo-') with a period of half a lunar month. This is illustrated by Fig. 29 showing how the processes in the soil were related to the processes a t f

1

June

July

~ ~ ~d5e , , po -M!2!l

I

August

o

I September I

o

0

October

o

1 I

I November I December o

0

o

A

Scale ~O.Imm/lOO m

-r.m.s

occurocy tO.O3mm/IOQrn

'

I\-

\ I

Heavy Rainfalls

I

'

1'""

- Movements of

Y

-1

IVI

I .

I TI I I I I the Ground

.

y

1

1 ' 1 ' 1 I I I I I

-

I957

I I l l

-

I

FIO.29. Movements of the ground in 1957 related to processes at depth.

depth. Repeated levelings each month for some two years demonstrated the existence of variations with an amplitude of 7 mm in 4 km, with the same period of half a lunar month. It is evident that this amplitude is not to be explained solely by earth tides and Decae suggests the possibility of

76

P. YELCHIOR

indirect effects of the Atlantic Ocean, demonstrating in support of this hypothesis that these deformations are not isotropic but are of NE-SW orientation. These ground movements should be checked by the recordings of horizontal pendulums since the amplitude of the phenomenon seems rather large. REFERENCES 1. Love, A. E. H. (1909). The yielding of the Earth to disturbing forces. Proc. Roy.SOC. (London)A82, 73-88. 2. Herglotz, G. (1905). uber die Elastizitat der Erde boi Beriicksichtigung ihrer variablen Dichte. 2. Math. Phya. 52, 275-299. 3. Jeffreys, H., arid Vicente, R. 0. (1957). The theory of nutation and the variation of latitude. Monthly Not. Roy. Aat. SOC.117, 142-161 and 162-173. 4. Molodensky, M. S. (1961). The theory of nutations and diurnal Earth tides. 40 Symposium International sur les Mardes Terrestres. Commun. Oba. Belg. 188 ( S . Qkoph. 58, 25-56). 5 . Melchior, P. (1966). “The Earth Tides,” 450 pp. Pergamon Press, Oxford. 6. Boussinesq, J. (1878). Equilibre d’dlasticith d’un sol isotrope sans pesanteur supportant diffdrents poids. Compt. rend. Acad. Sci. 86, 1260-1263. 7. Terazawa, K. (1916). On periodic disturbance of level arising from the load of neighbouring oceanic tides. Proc. London Phil. SOC.217, 35-50. 8. Nishimura, G. (1932). On the determination of a semi-infinite elastic body having a surface layer due to surface loading. Bull. Earthquake Rea. Inat. Tokyo Univ. 10, 1. 9. Steinhauser, F. (1934). Uber die elastische Deformation der Erdkrusto durch lokale BeladtUng mit besonderer Beriicksichtigung der Schneebelastung der Alpen. Qerl. Beilr. Qwphya. 41, 466-478. 10. Slichter, L. B., and Caputo. M. (1960). Deformation of an Earth model by surface pressures. J. Qeophya. Rea. 56, 4151-4156. lla. Caputo, M. (1961). Deformazioni di un modello della terra causate da masse aasisimmetriche. Zet. Qeod. Uniu. Trieate 62. l l b . Csputo, M. (1962). Tables for the deformation of a n Earth model by surface mass distribution. J . Qeophya. Rea. 67, 1611-1616. 12. Verbaandert, J., and Melchior, P. (1960). Les stations gdophysiques souterraines e t les pendules horizontaux de I’Observatoire Royal de Belgique. Oba. Roy. Belg. Monthly 7, 1-146. 13. Verbaandert, J. (1962). L’htalonnage des pendules horizontaux. Boll. Qeofia. Teor. Appl. T h t e 16, 419-446. 14. Melchior, P., and Brouet, J. (1964). Sur les diverses perturbations dans l’enregistrement des mardes terrestrea. 58me Symposium International sur les Mardes Terrestres. Commun. Obe. Belg. 236 ( S . Qkoph. 69,352-371). 15. Schulze, R. (1957). Das Askania-Gravimeter mit Registriereinrichtung.-Hinweise zur Inbetriebnahme. Commun. Oba. Belg. 114 ( S . Qdoph. 39, 15-21). 16. Clarkson, H. N. and La Coste, L. J. B. (1956). An improved instrument for measure. ment of tidal variations in gravity. Trans. Am. Qeophya. Un. 37, 266-272. 17. Clarkson. H. N., and La Coste, L. J. B. (1957). Improvements in tidal gravity meters and their simultaneous comparison. Trana. A m . Qeophya. Un. 38, 8-16. 18. Melchior, P. (1964). L’dvolution des iddes et des techniques d’observation dans l’dtude des mardes terrestres. Commun. Obs. Belg. 235 (S.Qkoph. 68).

CURRENT DEFORMATIONS OF THE EARTH’S CRUST

77

19. Melchior, P., and Plquet, P. (1963). L’automation dans les mesures de deformations

periodiques du globe terrestre: realisation d’un gravimbtre perforateur. Commun. 06s. Belg. 220 ( S . Ukoph. 65). 20. Bulletins d’observations des Mar6es Terrestres (1962-1965). Obs. Roy. Belg., 12 volumes. 21. Melchior, P. (1960). Die Gezeiten in unterirdischen Fliissigkeiten. Erdoel Kohle 13, 312-31 7. 22. Leontiev, G. Y. (1965). Les charges atmospheriques et hydrologiques temporaires Bur la surface de la Terre et leur influence sur le nivellement de haute pdcision. Translation: Bull. Inform. Madea Terreatrea 39. 23. Bontchkovskii, V. F. (1940). Inclinaisons de la surface de la Terre. Trwli Seiamolog. 99. 24. Lcttau, H. (1937). Das Horizontaldoppelpendel. 2.aeophya. 13, 26-33. 25. Jobort, G. (1960). Perturbations des Marees Terrestres. Ann. Uhphya. 16, 1-55. 26. Zadro, M. (1964). On the frequency dependence of the loading effects due to ocean tides and seiches. 58me Symposium Mades Terrestres. Commun. Oba. Belg. 236 (S. U h p h . 69, 372-380). 27. Decae, A. (1960). On some movements of the ground in Geneva. Ueophya. J . 3,112120. 28. Melchior, P. (1966). Determination experimental des effets dynamiques du noyau

liquide de la Terre dans les marees terrestres diurnes. Commun. Oba. Belg. S.B4 (8. Qeoph. 73).

This Page Intentionally Left Blank

ON THE UPPER MANTLE P. Caloi lstituto Nazionale di Geofisica. University of Rome

Page

...........................................................

1. Introduction 80 1.1. TheUpperMantle .................................................. 80 1.2. Problems of the Upper Mantle .................... 1.3. Composition of the Upper Mantle ..................................... 80 87 2 . The Earth's Crust ...................................................... 2.1. General ........................................................... 87 87 2.2. Use of the Body Waves ............................................. 2.3. Determination of Structural Faatures of the Earth's Crust by Studying 112 Surface Waves ..................................................... 142 3 Asthenosphere and Pa, Sa waves ......................................... 142 3.1. General ........................................................... 3.2. The Asthenosphere as Channel Guide of Seismic Energy . . . . . . . . . . . . . . . . . 143 145 3.3. Dynamic Features of the Asthenosphere Channel ....................... 145 3.4. Real Velocities of Pa, Sa Waves ...................................... 3.5. Channeling as a Phenomenon Concerning the Upper Mantle and the Crust 149 150 3.6. Channeling in Case of Deep Earthquakes .............................. 162 3.7. Discussion of Pa, Sa Waves .......................................... 3.8. The Problem of Viscosity ............................................ 154 3.9. Explanation of the Shadow Zone, Apart from Channeling . . . . . . . . . . . . . . . . 159 163 3.10. Shadow Zone and Channeling Zone ................................. 4 The "20' Discontinuity" ................................................ 167 167 4.1. Discussion of the "20" Discontinuity" ................................ 168 4.2. The Asthenosphere and the 20" Discontinuity .......................... 4.3. Consequences of the Existence of the 20" Discontinuity 171 5 Attenuation and Mixed Zones ............................................ 175 5.1. Introduction ...................................................... 176 5.2. Attenuation and Internal Friction .................................... 176 178 5.3. Attenuation and Mixed Zones ........................................ 6. Free Oscillations of the Earth and Its Outer Shell ........................... 181 181 6.1. Free Oscillations of the Earth ........................................ 182 6.2. Distribution of Earthquake Energy ................................... 6.3. Distribution of Seismic Energy between Body Waves and Surface Waves . 182 6.4. Possibility of Free Oscillations of the Outer Shell of the Earth . . . . . . . . . . . 183 183 6.5. Periods of Outer Shell of the Earth ................................... 6.6. Observation of Periods of Same Order as Those of Section 6.5. ........... 185 189 7 Earth's Internal Movements and Volterra's Theory ......................... 7.1. Polar Movements, Continental Shifts, and Subcrustal Currents ........... 189 7.2. Volterra's Theory of the Earth's Internal Movements . . . . . . . . . . . . . . . . . . . 192 References 203

.

.

..................

.

.

.

............................................................... 79

80

P. CALOI

1. INTRODUCTION

1.1. The Upper Mantle

It is well known that the term “mantle” signifies that region of the earth which extends from the MohoroviEiC! discontinuity (lower boundary of the crustal layers) down to the central core. The “upper mantle” delimits the outer part of the earth’s mantle, and can be regarded as a spheroidal shell lying from the MohoroviEii: discontinuity to a depth of nearly 950-1000 km. However, this is only a rough estimate without any pretensions of accuracy, and serves to distinguish the outer part of the earth, nearly 1000 km thick. It is obvious therefore that the crustal layers are included in a wide program of research, which aims a t clearing up the basic problems dealing with the physicochemical compositions of the upper mantle. Only a good knowledge of crustal features will facilitate research on the region immediately below. 1.2. Problems of the Upper Mantle Among the fundamental problems that need a more complete solution, apart from the earth’s crust, we should include those relating to the constitution of the asthenosphere as a channel guide of seismic energy and the so-called “20” discontinuity.” As related problems we may mention: origin of deep earthquakes, correlations between middle-depth earthquakes and active volcanism, “mixed” regions, possibility of independent oscillations of the upper mantle, convection currents, continental drift, possible expansion of the earth, secular and nonsecular phenomena, periodical and nonperiodical variations of the apparent vertical, magnetic and gravitational anomalies, heat flux,etc.

1.3. Composition of the Upper Mantle Research on the upper mantle will have made substantial progress when its chemical composition as well as its behavior with regard to elasticity, viscosity, plasticity, and internal friction are known. Some attempts in this direction have already been made. A short report of these will be given here, not in order to anticipate conclusions (these are indeed too premature a t the present state of knowledge), but to indicate which course we should follow after having clarified our knowledge of physical features, which shall primarily occupy our attention. The author regards the “upper mantle” (see Section 1.1) as the outer region of the eart8h,extending from the surface to a depth of about 950 km. It is well known that it is subdivided into three regions, called A , B , and C, respectively. Region A includes substantially the earth’s crust, limited downward by the MohoroviEii: discontinuity (depth ranging from 15

ON THE UPPER MANTLE

81

to about 60 km); region B is nearly 500 km in depth and includes the lowvelocity region; region C extends from about 500 to 900 km and is 400 km in depth. What causes the MohoroviEid discontinuity? It is widely believed that it is not related to some variation of the chemical composition of the earth’s materials, but rather to some change in the crystal structure owing to increase of pressure with depth (see Fermor, Holmes, Lovering, Kennedy, etc.). Against such an opinion Bullard and Griggs [l] cited three objections: ( 1 ) stability reaction dealing with comparison between temperature variation with depth and transition temperature variation with pressure is such that both continental and oceanic discontinuities cannot have the heavier material below them; ( 2 )depth variations of MohoroviEid discontinuity are less from place to place than would be expected from variations in heat flow; and (3) a discontinuity caused by a phase variation should be gradually distributed, hence without presence of a sharp transition. Therefore the problem is still open to discussion. Uncertainties concerning B and C layers are even greater. Of course data about the outer part of the earth, particularly of a geophysical and seismological nature, are available but information from these different sources is still vague. So far the only information that is very reliable concerns seismic wave velocities. Therefore all conclusions about classifications of substances forming the above-mentioned layers reflect the doubts that are still present both in seismological data and in our knowledge about the reactions of matter under changing conditions of temperature, pressure, etc. There are numerous references dealing with the argument. We mention only a few of these, more for emphasizing the great difficulties that are still t o be overcome than for establishing a list of values (which are indeed a matter of opinion). More detailed information on the B layer, including the asthenosphere, will be given in Sections 2 and 3. The greatest difficulties seem to arise with the transition between the B a n d C layer (relating to the “20” discontinuity,’’ see Section 4) and with the C layer. This region is very important in determining density and pressure, and affects findings on physical features of the other layers: The main problem about the upper mantle can be solved only after an explanation of the nature of the C layer which can be regarded as the key layer of the upper mantle. Birch’s research on physical features of this region-particularly of the C layer-is well known. I t starts from a distribution of densities and velocities in the mantle as shown in Fig. 1. If hydrostatic equilibrium conditions are fulfilled inside the mantle, the density variation in a homogeneous layer is expressed by the equation (Birch, 1952):

82

P. CALOI

where p is the density, p the pressure, r the distance from the earth’s center, T the temperature, a the thermal expansion coefficient, and K , the isothermal bulk modulus. Bullen [2] assumed that substances are of a homogeneous constitution in the B and D layers. But the ratio between bulk modulus K and density p increases in an anomalous way in the intermediate C layer. The

FIO. 1. Donsity distribution (after Birch).

interpretation of such an anomaly is one of the biggest problems about the upper mantle. Some authors ascribe it to changes of the constituents of this layer. For example, Bernal[3] thought such changes could be attributed to a rearrangement in the crystal state of olivine to the high pressure phase, while Birch [4]proposed that it could be attributed to changes in chemical coniposition. Shima [5] agreed with Birch’s conclusion. Studying the variation of the ratio bulk modulus/density on the assumption that the mantle is made up of ionic crystals, Shinia found that if the change were the polymorphic transition from low pressure phase to high pressure phase, the decrcase of‘ the gradient K / p would occur in the C layer: This is in contrast with observed results. Therefore, the change must bc a variation of chemical composition. Miki [6, 71, in a detailed analysis on the earth’s

ON THE UPPER MANTLE

83

mantle, came to the very opposite conclusion concerning both the composition and equilibrium. Referring t o equation ( l . l ) ,the thermal relation between isothermal incompressibility K , , and adiabatic incompressibility Ks is

where yu = Ks/p * aA/Cp ( A is the mean atomic weight, and C, is the specific heat a t constant pressure) is Griineisen's parameter. Following Miki, equation ( l . l ) ,in the various assumptions made by Williamson, Adams, and Bullen, cannot lead to valid issues. The quadratic distribution of density in the C layer is not an established fact as there are no reasons to consider that the conditions of homogeneity, adiabatic gradients of temperature and hydrostatic equilibrium are fulfilled in the same layer. Miki, following a stricter reasoning in which the variation of Griineisen's parameter with depth is also considered, found that the variation of seismic wave velocity with depth in the upper mantle can be very well explained on the assumption of a homogeneous medium, as well as the steep increase of seismic velocities in the C layer (see Fig. 2). On the hypothesis that the earth's mantle satisfies the hydrostatic relation

d p = -gp dr, from the equation

(where V is the molar volume), which is valid for increasing temperature with depths, we find

dr that is

because

Figure 3 shows the values of -K,d(log V ) / & and gp calculated by Miki from the observed values of Kslp and from the distribution of density p and force of gravity g with depth, Figure 3 clearly shows that inequality (1.5) is

84

P. CALOI

satisfied in the B and D layers but not in the C layer. In calculations it has been assumed that K , = K,; on the other hand, if we assume that T = 2000"K, o! = 2 x deg-', and yu = 1.5, i t follows that Tayo has a value of about 0.06. Equation (1.2) shows that the very large difference of inequality (1.5) on both sides in the C layer cannot be attributed to identification of K , with K,.

14 \

k

$ 12 *I5 5 2 9

2

10

8

6

4

0

1000

2000

3000

Depih (km)

FIQ.2. Distribution of volocities of soismic waves in mantle (after Birch). Solid lines show the values as deduced from tables (after Miki "71).

Therefore this difference is to be ascribed to the hypothesis that the C layer is in the state of hydrostatic equilibrium. Then Miki reached this surprising conclusion: The C layer is homogeneous and not in the state of hydrostatic equilibrium. Among the other issues of Miki's theory, this one seems to be the most interesting in relation to the existence of deep focus earthquakes in the C layer. The internal force in the

85

ON THE UPPER MANTLE

C layer acts toward the earth’s center and its intensity is twice as much as gravity (assuming a comparative maximum in the C layer). Thus, the earth below the D layer seems to be “strangled” by a spherical shell of about some hundred kilometers, roughly beginning at a depth of nearly 400 km. I n this spherical shell, strength has its greatest value in the middle and decreases toward both boundaries.

0 0

1000

2000

3000

fl+&kmj FIG.3. Distribution of pressure gradient and gp (after Miki [i’]).

It is quite obvious that the problem of the C layer is still wide open for discussion. If we consider chemical composition of the upper mantle, from the MohoroviEiC: discontinuity downward, we are confronted with even bigger uncertainties. Information about substances below the earth’s crust is necessarily indirect, and it is impossible to come to any conclusion without neglecting in some way our knowledge, i.e., regarding rock chemistry, meteorites, and solar atmosphere. At the moment our knowledge about the earth’s internal constitution deals almost only with physical parameters: seismic waves, density, moment of inertia, etc. Research into the chemical composition of the mantle has also been attempted. Shimazu [8] started from the hypothesis that the earth’s mantle,

86

P. CALOI

regarded as isothermic and incompressible, has reached chemical and hydrostatic equilibrium. That being the case, he studied the distribution of chemical elements on the basis of differences in their gravitational differentiation owing to different density. He calculated the equilibrium distribution of the Fe0-Mg0-Fe-Si0, system, and concluded that FeO increases with depth and reaches its maximum a t a depth of several hundred kilometers, and then it decreases toward the earth’s center. Shimazu saw in this behavior an explanation of the “20” discontinuity” origin and chemical properties. Regarding Bullen’s C layer, Shimazu [9] thought its transient character could be explained by the decomposition mechanism of olivine. At pressures and temperatures corresponding to nearly 600 km in depth, olivine may be in a state of almost critical stability, and a small perturbation energy is necessary to start decomposition reaction owing t o its endothermic character. Under the action of gravitational differentiation, upward squeezing of SiO, may take place there, with molar volume decrease, following the reaction: Mg,SiO, -+2MgO

+ SiO,?

The phase transition of SiO, from normal quartz to a high pressure phase (coesite) helps olivine decomposition to form an oxide at high temperature and pressure. Olivine phase would correspond to the B layer and oxide to the D layer. The squeezing of SiO, is also related t o the problem of crustal evolution, origin of deep focus earthquakes, and orogenic processes. Nishitake [lo] compared the elastic properties of some parts of stony meteorites with three different models of the mantle. The bulk modulus and the ratio between bulk modulus and density in stony meteorites having been calculated, Nishitake found their values considerably lower than those related to the three models of the mantle drawn from seismic observation. Following his line of thought, arguments exist for thinking the earth’s mantle is composed of minerals similar to meteorites. On the basis of earlier research carried out by the author on physical properties of dunite, Nishitake concluded that the mantle is mainly composed of dunite. Referring also to chemical characteristics of meteorites, a few geophysicists and geochemists think the earth’s mantle has a composition resembling chondrites. Elements in support of this hypothesis have been advanced by Ringwood [ l l ] and MacDonald [12].But objections and reservations have also been voiced in this argument. Gast [13],in a paper in which isotopic composition of strontium and K, Rb, Cs, Sr, and Ba concentration in rocks and meteorites were reexamined, pointed out that the isotopic abundance of SrH7 in the upper mantle and in the crust seems t o be less than in chondrites. Therefore, new limitations and uncertainties are existent in the chemical composition of the upper mantle. More recently Shimozuru [14] reviewed temperature distribution in the

ON THE UPPER MANTLE

87

upper mantle using the melting curve of eclogite regarded as the main component of this zone. This led him to the claim that the melting point is reached between 120 and 220 km of depth: This region nearly coincides with the “low-velocity zone” of the subcontinental upper mantle. Magmas (“rock magma”) could originate in this region. The great difficnlty of the problem dealing with the composition of the upper mantle is evident from this brief view. Only a further clarification of its physical-particularly, elastic-properties will ensure progress. The present state of knowledge about such properties now will be detailed.

2. THE EARTH’S CRUST

2.1. General We designate by the term “earth’s crust” the outer part of the earth, limited downward by the MohoroviEid discontinuity. Methods of study concerning this thin superficial layer of the earth are mainly based on interpretation of seismic records owing to sharp perturbations of the elastic field. The examination of the variations of other fields (i.e., gravitational, magnetic, and electrical) may help in the findings, but broadly speaking, it gives only general information. Seismic methods are of two different types: those that utilize body waves’ records, and those that utilize surface wave characteristics. The former, until a dozen years ago, dealt with vibrations caused by natural earthquakes only. Since the past few years big, conventional explosions or nuclear explosions as well as earthquakes are being employed as sources of elastic oscillations.

2.2. Use of the Body Waves 2.2.1. Methods. Not every method of seismological survey will be discussed, especially the classical “body waves” methods. These have been dealt with in many texts and specific publications, to which the reader may refer [15]. Indeed, it is more in accordance with the character of this review t o dwell upon critical comparison of the results of different methods, giving analytical exposition of new methods only or of the physicomathematical analysis of controversial points. The study of earthquake “body waves” is an essential part of research into the earth’s interior features. This is easily understandable, as various types of body waves are caused by refraction or reflection, corresponding t o those discontinuities whose existence and position we want to determine. Regarding the earth’s crust, the study is limited to earthquakes originating fairly close t o the earth’s outer surface (i.e., in the crustal layers) and close to

88

P. CALOI

seismic recording stations. Only under such conditions are records of waves obtained that have traveled in the crystal layers and in the matter immediately below, thus they may give us information about crustal features. Therefore, while the former condition does not a t all limit the possibility of using body waves methods for determining features of crustal layers (since most earthquakes originate from crystal layers), the latter does limit their use because of the uneven distribution of seismic stations over the world. Owing to this reason, so far we have reached the best results in some particular continental areas, while in some other-especially oceanic-areas we have to use different methods. Most of these methods as well as methods for determining the geographical coordinates of earthquakes utilize recording times of P waves. The reason for such a preference is to be attributed to the early arrival of P waves on seismograms; therefore, in most cases arrival time can be calculated with good accuracy. On the contrary, S waves are recorded on the middle part of a seismogram, so their beginning cannot always be very clear. We must realize, however, that S waves are generally much larger than P waves, and that their starting pulse can be found very easily; then methods that utilize P waves can also be extended to S waves. Research using both P and S waves, indeed, may be very useful in obtaining a confirmation of the results observed using one type of wave only [16]. Another characteristic of almost all methods for determining the earth’s crustal features using body waves is treatment of the earth’s crust as subdivided into constant velocity layers, which leads to assumption of a rectilinear propagation of seismic waves and use of rectilinear propagation laws. This hypothesis is broadly applicable t o all practical purposes, and layers are regarded as strata of constant average velocity, which are calculated from the travel-time curves. Seismic velocities, indeed, vary in the same layer, and discontinuities are zones in which there are sharp variations of velocity; then the knowledge of velocity variations with depth permits determination of the depth a t which such sharp variations occur, i.e., discontinuities and their depths. One of the methods for determining the earth’s crustal features aims a t finding out just such a function, or the variation of the factor k = (up v,)/ (vp -v8), which amounts to the same thing since, when the values of k a t various depths are known, we can easily find the values of corresponding velocities. These are correlated by the Poisson coefficient which is generally known a t various depths of the earth’s crust. There are several well-known methods for ascertaining discontinuities, which are based on reflection and refraction of elastic waves. These methods are so well known that they need no mention here [la]. However, it is worth

ON THE UPPER MANTLE

89

remembering that the methods for recording sharp variations of velocity with depth are probably t o be preferred. One of these methods was pointed out by the writer in 1942 [17] and has been applied to seventeen earthquakes in Central Europe. It permits the determination of focal coordinates of near earthquakes and the calculation of the parameter k = (v1 vz)/(v, - v2), where vl, v2 are Pg and Sg wave velocities, respectively. This method led to the results concerning a wide region of Central Europe summarized in Table I and in Fig. 4.

",-":

FIG.4. Values of v1 . v z / ( v l - v ~with ) depth (for rocks).

These data clearly show that the value of k increases irregularly with depth; a sharp variation takes place at about 18 km in depth, which confirms the existence of a discontinuity in the physical features of the earth's crust a t that depth. This discontinuity is present a t least in a wide region of the Alpine Range and Wiirttemberg. Further research into variations of k led to the conclusion that the earth's crust in Central Europe consists of two layers at least. This method is indeed laborious, but it could lead to very reliable results if it were systematically used (see also references).

2.2.2.Applications i n Europe and California. It is superfluous to recall the different methods that can be applied to reflected or refracted body waves corresponding to two or more layers on top of each other with surface limits parallel, or more or less inclined, with respect to the earth's surface. In south Central Europe, where the above-mentioned work has been intensively pursued because of the great number of seismic stations in this area, the earth's crust was a t first considered as formed of two layers only (see Gutenberg, Conrad, Hiller, and Caloi). Later a new type of P wave (and corresponding S wave) was found, which led t o a model of the earth's crust, formed by three layers [18]. The same results were obtained in other regions of the earth (i.e., in southern California).

TABLEI. Values of k = ( v l .u z ) / ( v l Earthquakes and Dates Bodensee (31 January 1935) Cansiglio (18 October 1936) Prealpi Carniche (8 June 1934) Hohenzollern-Alb (17 June 1937) Yverdon (1 March 1929) Hohenzollern (10 October 1933) Tubingen (19 June 1936) Bodensee (15 March 1936) Hohenzollern (1 January 1934) Schwarzwald (30 December 1935) Oberschwaben (27 June 1935) Oberschwaben (11 April 1938) Hohenzollern (2 August 1938) Ebingen (1 March 1939) Onstmettingen (6 August 1940) Sierre (25 December 1939) Hohenzollern (21 February 1933)

v2)

for 17 Central European earthquakes [17].

Epicentral coordinates 9" 4'.6 f 0'.7 12"21'.2 f 3'.1 12"26'.1 f 2'.3 9"13'.5 f 3'.0 7'.1 6"31'-1 8"54'-8 f 0'.8 9" 1'-2 f 0'.7 9'27'-4 f 5'.0 9" 4'.5 f 2 ' 3 8" 7'08 f 2'.7 9"30'*9 f 1'.3 9"32'.3 f 1'.3 9" 5'07 f 4'.7 9" 3'.3 f 0'.8 9" 5'.2 f 1'.3 7"33'-6 f 4'.4 9" 9'*6 + 1'.2

47"40'.4 f l'.l 46"10'.2 f 2'.7 46"20'.8 f 1'.4 48"13'.8 f 1'.3 46"46'.4 f 4 ' 3 48"15'-3 W.3 48"33'.8 f 0'.4 47'36'*4 f 2'.5 48'21'-3 f 0'.7 48'37'*2 f 2'.1 48" 1'*6 f 0'.9 48" 2'*0 f 0'.5 48'16'-1 f 1'3 48'12'- 8 f 0'.5 48"18'*1 f 0'.7 46'13'*8 f 6'.7 48"19'.6 f 0'.4

*

Depth (km)

k

15.8 f 5.6 18.0 f 41.0

8.204 f 0.15 8.160 f 0.102 8.025 =t 0.062 0.150 8.187 -j= 8.790 & 0.232 8.062 i 0.054 8.712 f 0.050 8.158 & 0.404 8.006 f 0.085 0.190 8.964 0.082 8.263 0.069 8.367 8.200 f 0.234 8.202 j= 0.046 8.250 f 0.072 7.883 f 0.609 11.207 f 0.18

0

17.6 f 5.7 28.9 f 18.8 6.5 f 1.2 22.8 1.2 12.3 f 6.8 0 33.9 & 6.6 18.6 f 3.9 16.0 f 2.1 6.5 f 7.9 2.1 9.0 4.7 f 5.0 0 61.1 f 2.5

*

**

TABLE 11. Principal data on the crust obtained by means of body waves of near earthquakes. Region and Authors Southern Germany (Gutenberg [152]) Tauri (Conrad) Schwadorf (Conrad [1521) South West Germany (Hiller) England (Jeffreys) [136] Wiirttemberg (Caloi [17]) Tyrol (Caloi [17]) Cansiglio, Pre-Alps of (Caloi [17]) Central Pre-Alps (Di Filippo-Peronaci [20]) Po Valley (Caloi [18]) Tuscan Apennines (Caloi[21]) Garfagnana (Caloi 1211) Gran Sasso (Di Filippo-Marcelli) Balkan (Peronaci [20]) coastal California intermediate (Gutenberg) zone, 1771) Sierra Nevada L

wPK

VSE

vp.1

Vp.2

5.55 5.4 5.6 5.0

3.39 3.3

7.1

8.1

6.29 6.47 6.4

7.83 8.12 8.1

4.32 4.7

7.8

4.35

10 20

8.0

4.4

13

40 35

45-50 45

3.98

8.34

4.42

11

3741

50-00

4.2

8.16 7.9

4.52 4.34

1816 25-30

20-23

30-34

5.4 5.5 5.7 5.7

3.3 3.2

5.46

3.24

6.09

6.92

5.1 5.4

3.08 3.01

6.09

6.94

5.3 5.46

3.3 3.01

5.49 5.6

3.3

3.36

3.6 3.48

3.57 3.63

6.6 6.38 6.30 6.0

d l is the depth of Conrad discontinuity. Here, da is the depth of the intermediate discontinuity. c Here, dM is the depth of the Mohorovifiid discontinuity. b

3.76

6.5 6.7 6.6

> I

a Here,

3.7

6.94 6.94

3.6

8.17 8.0

30

32

8.1 8.19

4.1

1@15

25

4.4

24-27 17.5

60 30 32

58 36 44 60

02

P. CALOI

The values found by analyses of near earthquakes in Central Europe and California are shown in Table 11. As to velocities, one can note that research into body waves leads to concordant results, apart from small regional differences, only if the waves have traveled in the granitic layer or in the upper part of the mantle; indeed, vPg and vSg velocities do not vary from the average values of 5.5 and 3.2 km/scc, and up,, and us, have average values of 8.2 and 4.4 kmlsec. In contrast, various values have been attributed to P*-wave velocities. It is certain that in certain regions we deal with two different types of waves (PI*and P2*)which have traveled through two intermediate layers between granite and mantle. Moreover, it is likely that many other values of P* waves will have to be reexamined on this basis. Data from Italy and California on the earth’s crustal structure are particularly interesting. Gutenberg [19] found that the earth’s crust in California consists of three layers of various thicknesses. Indeed, there along the coastline is a granitic layer about 17.5 km deep, followed by a first intermediate layer about 14.5 km deep, and a second intermediate layer about 4 km deep (total thickness of the crust about 36 km), while eastward, i.e., toward the continent, the second intermediate layer becomes steadily thicker (at first 11 km, then 30 km deep below Sierra Nevada). Therefore, the MohoroviEi6 discontinuity goes deeper and deeper, down t o 60 km deep below Sierra Nevada. Hence, we get a confirmation of the isostatic theory and of the existence of the “mountain roots,” that in this case consist mainly of the second intermediate layer. A similar result has been found in Italy, where data from the Apennines, the Po Valley, and the Alps have been examined. The crust below the Apennines has a total thickness of about 60 km (average values: granitic layer 25 km, gabbro-basaltic layer 35 km). The MohoroviEid discontinuity does steadily rise below the Po Valley, so that the crust in that area is about 30 km thick (sediments 3-6 km, granitic layer about 10 km, basaltic layer 7 km, and gabbro 11 km). Northward, below Prealpi Lombardo-Venete, the MohoroviEid discontinuity lies deeper again, and so does the second discontinuity (basalt-gabbro boundary); but the lower boundary of the granitic layer is as deep as before. The “roots” in this area consist mainly of the first intermediate layer, which becomes thicker and expands toward both higher and lower boundaries. Following Di FilippoPeronaci 11201, the granitic layer rises even more, up t o a thickness of only 5 km, below the western Alps. Furthermore, it is thought that also in those areas of the Alpine Rangc in which the granitic layer was earlier regarded as 3 5 4 0 km thick, it is actually 16 km thick, but a greater thickness (nearly 21 km) is attributed to the first intermediate layer that would constitute the very root of the Alps [21]. Total crustal thickness in northern Europe is probably less; the granitic layer is probably growing thinner toward continental boundaries (Fig. 5 ) .

93

ON THE UPPER MAN”JAE

....

-..-.

FIG.6.The crust from the Apennines to the Atlantic Ocean.

2.2.3. Earth’s Crustal Structure Inferred from Explosion Data 2.2.3.1. Premise. Bouasse, in his book entitled Skismes et sismographes [22], wrote the following about big explosion data: “La littkrature sur le sujet est vaste: elle prouve uniquement I’extrbme discordance des resultats.. .” And then: “. . . plus forte est l’explosion et plus faible la distance, plus la vitesse enregistr6.e est grande eeterk p r i h ; ce qui prouve simplement qu’on enregistre une phase plus voisine du vrai &but B mesure que croit I’amplitude du ph6nomkne.” And then: “Recommencer des experiences dont on sait depuis quarantecinq ans [he was writing in 19271 qu’elles donneront n’importe quoi, c’est perdre son temps. . . Au voisinage de l’explosion, ces ph6nom&nes,essentiellement locaux, sont dkterminks par le terrain: ti grrtnde distance on ne peut que retrouver les resultats fournis par les skismes: la nature nous en fournit une telle abondance qu’il est bien inutile d’en fabriquer d’artificiels: pour intenses qu’ils nous paraissent, ce sont jeux d’enfant dkvant les skismes naturels, au surplus avec un hypocentre trop exactement superficiel pour qu’ils se propagent i grande distances.” The necessity of a greater number of seismic stations was put forward in 1947 in a note entitled “The University Seismograph Problem” (Earthquuke

.

94

P. CALOI

Notes, Eastern Section, Seismological Society of America, Vol. XIX, No. 1-2, September-December), in order t o resolve the problem of regional structural anomalies. Therein it was stated that “details of these anomalies can be explained only by obtaining vastly more seismographic data than those available a t the present time. “Thus far seismologists have learned mainly that the earth has a core of about half its radius, which reacts to seismic waves like a fluid, and that there is a somewhat heterogeneous outer crust varying from 20 to 60 km in thickness. But too little is known of the details of crustal structure or the nature of the zone between the crust and core. Obviously, seismograph stations must be more numerous and more uniformly spaced if earthquakes are t o be located more accurately and if anomalous travel times are to be interpreted in terms of the earth’s structure. “To make further progress in the investigation of travel-time phenomena the charts of many individual earthquakes must be obtained and carefully scrutinized. They must be obtained with such a perfection of detail that the seismologist will have no difficulty in translating the data in terms of the earth’s structure. This has been accomplished thousands of times in seismic exploration of shallow geologic structure in the search for oil. There is no reason why the deeper structure of the earth cannot be similarly explored if the facilities are made available. The current programs of certain seismological groups call for the use of great quantities of unserviceable ammunition to create artificial earthquakes which will yield precise data on the deeper structure of the earth. But as no man-made explosion, not even an atomic bomb, can compete with nature’s own manifestations of power, the standard seismograph will always furnish the best picture of seismic wave transmission through the earth as a whole. Earthquakes are very frequently recorded on seismographs all over the world.” Therefore in 1947, the Eastern Section of the Seismological Society of America (at that time formed by J. T. Wilson, E. J. Walter, F. Robertson, D. S. Carder, W. T. McNiff, and F. Neumann) thought that research into the earth’s internal structure should be substantially based on natural earthquake recording. Dissenting opinions were voiced afterward, and some came to negative judgments on the results obtained from near earthquake study. Thus Steinhart and Meyer [23] listed the following deficiencies which they attributed t o near earthquake records: “(1)too few observations, (2) inadequate time control, (3) recording speed too slow, ( 4 ) poor range control, (5) inadequate knowledge of depth of focus, (6) ignorance of the source conditions, (7) nonuniformity of recording instruments, and (8) biased nature of station distribution.” The author prefers to report results obtained from explosion data before rendering a judgment on these different opinions.

ON THE UPPER MANTLE

95

2.2.3.2. Applications in Eurasie and America. Research into the earth’s crust through near earthquake recording has undoubtedly the disadvantage that earthquakes are not easily located in time and space. An inaccuracy in calculation of focal depth and time may affect travel-time determinations and therefore velocities and determinations of layer thickness, especially in the case of earthquakes of local origin. Ascertaining the precise facts has led seismologists to take advantage of the opportunities of using body waves obtained by explosions. I n this case, point and time of origin are well known. Use of explosions, moreover, has the additional advantage of carrying research into those areas in which there are no regular seismic stations or in which there is no possibility of recording near earthquakes, i.e., in the central areas of nonseismic regions. Researches on Eurasia. From explosions in Haslach, Black Forest (April 28-29,1948), Reich et al.[24] and Rothe and Peterschmitt [25] have found that in that area the earth’s crust consists of at least two layers: granite (20-21 km thick, following Reich et al.; 18 km thick, following Rothe and Peterschmitt), in which the Pg-wave velocity is nearly 5.9-6.0 km/sec; and gabbro-basalt (nearly 10 km thick) in which u p . = 6.5 km/sec. The MohoroviEi6 discontinuity should be located a t the average depth of about 31 km, lowering downward beneath the Alpine Range where its depth, calculated from earthquake data, is about 4 0 4 5 km. Values of granitic layer thickness are in accordance with those found by Caloi in Wiirttemberg from the data of seventeen earthquakes [17]. Moreover, the value of 20 km found by Reich [26], through the study of a reflection of explosion in Ulm, is also in accordance with that found by Caloi, as is the value of the depth of the MohoroviEi6 discontinuity (28 km), Fig. 6. The values obtained through several investigations in northern Germany are very interesting. Not far from Gottingen, Wiechert and Brockamp found a thickness of the granitic layer of only 8 km. Such a result indicates a gradual thinning of this layer toward the continental border. In the same area, Schulze and Fortsch [27] and Willmore [28] have studied the Heligoland explosion data and found that the Conrad discontinuity is only 9-14 km deep. The same authors have found the MohoroviEi6 discontinuity about 26 km deep (Schulze and Fortsch), and 30 km deep (Willmore). The latter, however, supposed that the crust in this region consists of one layer only, with 5.95 km/sec average velocity and lower boundary at 27 km depth. Many systematic investigations by explosions have been performed during the last ten years by the European Research Group for Explosion Seismology. Teams from France, Germany, Italy, and other countries belong to this Group. The evaluation of the numerous records hitherto obtained is still going on.

96

P. CALOI

0

50

25

125

100

75

150

h

175

Fro. 6. The crust in southern Germany (after RBich et al. [24]). h i s o f ncqatiy~ momaliu

Awj ofparifivr

gr&y

A h/Idonmz

-

0

10 4E 20

++

I

1

f

C

qravify snornalia

6rdn &adso

II

I

Pdvour

Doraflaira

+

B

D

-k

I VREh-disconhdy L

3 30

8 40 50

60 o

20

40

60

80

im km

FIG.7. West-East section across weatern Alps (after Fuchs el al. [29]).

97

ON THE UPPER MANTLE

I n every case the most important features that can be gathered from the results already published are as follows: (a)The surface boundary below the “granitic” layer appears everywhere a t relatively small depth. (b) At the western boundaries of the Po Valley, the “Ivrea discontinuity” sharply outlines the upper boundary of a dense body of basic rock which is responsible for an important positive gravity anomaly. This body, whose velocity is as high as 7.4 km/sec, has a depth of approximately 10 km (Fig. 7). ( c ) It is shown, without any doubt, that there is a root below the Alps, in conformity with the previous conclusions obtained by Gutenberg and Caloi in the study of near earthquakes. The value of 45-50 km, obtained by Caloi for the central-eastern Alps, as the mean depth of MohoroviEii: discontinuity, is generally confirmed [29,30]. Results show substantially that the MohoroviEid discontinuity slopes upward from mountain ranges to continental boundaries, as had already been shown from earthquake data (Fig. 8). Table 111 shows the main results from explosion data: If one looks at this

100

A

0- - -&-100

-

n I

_

.

-200

mgal

BOUGUER-

Anomalies

5Q

0

- 50 -1 0 0 - 1 50

CRUSTAL SECTION

FIG. 8. Magnctic AZ anomalies, Bouguer anomalies, and crustal section along a north-south profile in western Germany (after Fuchs el ~ l[29]): . V denotes the Variscan Mountain system, C is the Conrad discontinuity, and M the Mohorovihiddiscontinuity.

TABLEIII. Principal results from explosions. Region and Authors Schwanwald (Reich et d.[24] (RothB and Peterschmitt [25]) Ulm (Reich [26]) Northern Germany, Gijttingen (Wiechert, [26a]) (Brockamp [26b]) Northern Germany, Heligoland (Schulze and Fortsch [27]) (Willmore[28]) Central Alps, Lago Lagorai (Peterschmitt et ol. [30]) Western Alps, Gran Paradiso (Fuchs et al. [29]) Turkmenistan (Galperin-Kosminskaya[32]) Caspian Sea zone (Galperin-Kosminskaya[32]) Tennessee

Virginia Canadian Shield (Hodgson [37]) (Hodgson [37]) Western Transvaal (Willmore et al. [39])

0

Year

VPg

UP.

VPn

5.9

6.55

8.2

5.97

6.54

8.15

5.9

6.72

6.18 5.57 5.95

6.60 6.50

1965

5.9

6.35

8.2

1963

6.0

6.7

8.15

1959

6.0

6.6

8.0

6.6

8.0

1948 1950 1953

VSS

3.4

V Y

4"

dMb

20-21

31

VSn

3.65

20

30 28

10

1926 1931 1950 1949

1959 1953

6.1

1953 1953

6.24 6.23

1952

6.09 6.09

Here, dl is the depth of the Conrad discontinuity.

8.3 8.18 8.18

3.67

3.87

9 14 27

17

15

30-35 40-45 40 32 30-35

(7)

8.1 8.17

3.54 3.54

6.83

8.27

3.68 3.68

b

47 40-50

8.1

8.27

26 30

(3.9) 3.89

4.78 4.85 4.83 4.83

Here, dM is the depth of the Mohorovifiid discontinuity.

36

22

39 34

ON THE UPPER MANTLE

99

first group of results, one realizes the good agreement obtained for thickness of layers from P*- and Pn-wave velocities. On the contrary, the Pg velocities obtained from explosion data show values (ranging from 5.9 to 6.1 km/sec) a bit higher than those obtained from earthquake data. During large-scale seismic surveys in USSR, where very interesting data about features of particular areas have been obtained, vpg velocities generally below 6 km/sec have been found. First of all, the existence of mountain roots in the Alai-Transalai-Pamir Range [31] has been confirmed: Along this range two layers have been located from explosion data; P-wave velocities are 5.5 km/sec in the first layer (value in accordance with those obtained from earthquake data) and 6.4 km/sec in the second layer. The lower boundaries of these layers slope considerably down toward the Pamir: the Conrad discontinuity slopes from about 20 to 35-40 km in depth beneath the Translai Range; the MohoroviEid discontinuity from about 40 kni down to nearly 70 km beneath the mountain ranges. Research in the area of the Caspian Sea, carried out during the International Geophysical Year [32], led to locating three groups of waves corresponding to three discontinuities : sediments-granite, granite-gabbro, and gabbro-mantle; the velocities of the P waves are 6.0, 6.6, and 8 kmlsec, respectively. The first type of waves (6.0 kmlsec). however, has not been found in the Caspian depression itself: Therefore, in this area, the crust must consist of a sedimentary layer about 20 km thick and low velocities (3.5-4.0 km/sec) and of a gabbro-basaltic layer that extends from 20 t o 4 0 4 5 km in depth; the granitic layer must be absent or be so thin that the corresponding waves mingle with the other groups. All three types of waves, on the contrary, are present in the Epi-Ercinic Turkenian Shield and in the region between this shield and the Caspian Sea. A structural difference, however, takes place in these two zones, while beneath the shield the Conrad and MohoroviEid discontinuities are nearly horizontal and about 15 and 30-35 km deep, respectively, and sedimentary layers are only 2 km thick. In the transition zone, sediments become steadily thicker and the Conrad discontinuity slopes sharply from about 15 to 25-30 km in depth; the MohoroviEid discontinuity slopes downward to 40 km in depth. The sedimentary layers grow substantially thicker, and the granitic layer thinner, from the continental shield toward the Caspian Sea. The granitic layer’s upper boundary slopes considerably downward. Such a structure is typical of particular areas in which there has been, and still is, large sedimentation. A similar sloping downward of the granitic layer’s upper boundary and an increasing thickncss of sedimentary layers have been found by Caloi [21] from records of small explosions in the transition zone between the Alpine Range and the Po Valley: The granitic layer, that in several Alpine valleys is

100

P. CALOI

nearly a t the supcrface, in the Po Valley has its upper boundary 6 km deep beneath a thick sedimentary layer. This sharp step is correlated with the present shape of the Alpine geosyncline: The Alps, in fact, are related to the Po Valley as island arches are to oceanic trenches. Explosion records from the Asian border, in the Okhotsk Sea, Kurile Islands, and eastward are particularly reliable for finding general features of the crust [32] (see Fig. 9). Travel-time curves are different in different areas:

I

0

0

20

20

40

40

*.

20 40

b--4--4

FIG.9. Crustal section in the Okhotsk Sea zone (from continent to Kurile Islands). (1-1) Kolima promontory-Pacific Ocean, southeast from Iturup island; (11-11) Southern Sachalin-Okhotsk Sea-Iturup Island-Pacific Ocean. 1, water (1.5 kmlsec); 2, sedimentary layer ( 3 kmlsec); 3, granitic layer (5.2-6.4 kmlsec); 4, basaltic layer (6.5-6.8 kmlsec); 6, subcrustal layer (8 kmlsec); and 6, forcal zone (after Kosminskaya [32]).

Thus structural differences must exist. I n the oceanic shield east of the Kurile Islands, the travel-time curve of P waves shows two branches corresponding to P* and Pn, respectively; P* waves travel in the gabbro-basaltic layer a t a velocity of 6.6-7 km/sec; Pn travel below the MohoroviGi: discontinuity a t a velocity of about 8.0 kmlsec. In contrast, travel-time curves obtained for the Kurile Islands, the Kamchatka Peninsula coastline, and the northern part of the Okhotsk Sea, show three branches, corresponding to Po (associated with the sedimentary layers, v = 6 kmlsec, and with the granite layer, v = 6.0 km/sec), P*, and Pn waves, respectively. Arrival times in this second type of travel-time curve, moreover,

ON THE UPPER MANTLE

101

are 4-6 sec later than in the first type, seismic waves having traveled longer in low-velocity layers. Finally, in the southern part of the Okhotsk Sea and the Bering Sea, the travel-time curve shows two branches corresponding to P* and Pn waves; their arrival times have an average value between the curves of the first and second type. The three types of curves must be peculiar to three types of crustal structure-continental, intermediate, and oceanic-that differ in thickness or lack of granitic layer and total crustal thickness. North Atlantic. The granitic layer is also lacking in the area between Bermuda and the American coastline, in accordance with explosion data; indeed, Ewing et al. [33] have examined refracted P waves obtained from explosions in this area, and have pointed out that only one type of wave is present, which travels a t a velocity of 7.58 km/sec: It is not yet clear whether they are P* or Pn waves. At any rate, the granitic layer is lacking in the north-western Atlantic, as already pointed out by several studies using Love waves. From data thus far examined, explosions, apart from some different values of velocity in the granitic layer, seem to confirm the structural features found by use of other methods. Actually, results obtained in a few seismic surveys in North America have reopened the discussion about a multilayered crust, which had already been considered as a resolved question (except for some residual dissent [34]).

American Studies Tstel et al. [35], in fact, denied the hypothesis of a multilayered crust as a result of several seismic surveys in Tennessee, Maryland, Virginia, and California. Following these authors, it should be realized, first of all, that in travel-time curves there is too large a scattering of points t o be attributed only to the path of the waves through heterogeneous surface layers in different areas; a great heterogeneity is present a t any depth, therefore there is no reason to assume that the crust is formed by rock layers, with more or less sharp boundaries and with different average velocities. With regard to the depth of the MohoroviEi6 discontinuity (which is the only discontinuity that causes reflections), it is about 45-50 km deep in Tennessee, 40 km in Minnesota, 32 km in Maryland, and 30-35 km in Virginia. In California there is no evidence of a sharp transition t o highvelocity materials (8.1km/sec), and this is assumed to be caused by the vague and discontinuous nature of the buried stratification. But these results are not in agreement with those found by Gutenberg from earthquakes in California (see Section 2.2.2). In this region, apart from other intermediate stratifications, the MohoroviEii: discontinuity is clearly present and slopes down from the coastline toward the continent.

102

P. CALOI

Tuve et al. [36] deny the existence of a multilayered crust. Hodgson [37,38] arrives also a t negative results on crustal stratification, after a careful survey carried out with explosion data from mining zones in the Canadian Shield. Main phases in these records are P, (vp, = 6.246 kmlsec), S, (vs, = 3.544 km/sec), Pn (vPn = 8.176 kmlsec), and Sn (vSn= 4.85 kmlsec). Secondary arrivals, that are very complicated especially in the S group, could not be explained by refractions or reflections a t the discontinuities, but as caused by heterogeneity of matter forming the crust and by its various thicknesses. The lower boundary of the crust is undulating a t an average depth of about 36 km. This causes every phase to travel along several different paths. Moreover, the presence of waves reflected by the MohoroviEid discontinuity a t epicentral distances very near to the critical ones show that such a discontinuity is not so sharp that it could reflect waves a t a normal incidence. Following Hodgson, explosion records in the Canadian Shield can be explained by the hypothesis of a very heterogeneous layer. Willmore et al. [39], after having carried out a seismic survey in Western Transvaal, were in doubt about the existence of crustal stratification, too. During this survey the following values for the velocities have been obtained: vpg = 6.09 kmlsec, vsg= 3.68 kmlsec, vp, = 6.83 km/sec, us, = 3.89 km/sec, vpn = 8.27 km/sec, and vsn = 4.83 kmlsec. The Conrad and MohoroviEid discontinuities are about 22 and 39 km deep, respectively. The P* phase, however, is assumed to be caused by a gradual increase of velocity with depth and therefore the existence of the Conrad discontinuity is ruled out. The generalization of these results to consider the crust as formed by a single layer seems to be arbitrary; these results may be considered, a t most, valid in some particular areas only. Indeed, it does not seem reasonable that all preceding conclusions obtained from both earthquake and explosion data result from misinterpretation and misjudgment of records, as Tatel et al. [35]. believed. Limitations in recording discontinuities by explosions are more likely. It is, therefore, necessary to explain why results obtained from explosion and earthquake data are so different, both as regards velocity values in the granitic layers and the existence of intermediate discontinuities.

2.2.3.3.Influence of the type of discontinuity on elastic wave reflection. One of the reasons that led some authors (particularly those who studied explosions data) to deny the existence of discontinuities in the earth’s crust is the lack of reflected waves in records of explosions (it would be more correct to say: “the lack of reflected waves on the recording sets that are in common use in seismic surveys”). Such a lack, however, could be attributed to the character of the discontinuities: These are, in fact, transition zones in which a gradual, even if relatively rapid, variation of the elastic constants takes place. I n such

ON THE UPPER MANTLE

103

conditions, the comparatively long periodic waves originating from earthquakes are reflected as though there is a true discontinuity, the highfrequency waves obtained from explosionsare not reflected, but only refracted, in correspondence with the character of the discontinuity (transition zone); i.e., the energy of an incident wave does not divide into reflected and refracted waves, but is completely conveyed by the refracted ones. Since C. G. Knott, theoretical studies of reflection and refraction of waves a t discontinuities into an elastic body have been carried out under the hypothesis that “the density or elasticity of the material varies very sharply a t the discontinuities under consideration. It appears, however, that the condition thus assumed could hardly exist in reality, the problem being rather that distribution of material in the immediate vicinity of the discontinuities in the earth would vary more or less gradually, the amplitudes of transmitted (refracted) and reflected waves through these discontinuities differing somewhat with difference in the lengths of the incident waves.” Sezawa and Kanai [40] have written the above reported words introducing their theoretical work about seismic wave transmission in a medium gradually changing its characteristics. Let us assume that po, po, uo and p2, p2, u2 are density, rigidity, and displacement of a point subject to elastic forces in the first and second medium, respectively, with po 3 p2; let us suppose that a n intermediate layer is present in the immediate vicinity of the discontinuity, and that H, pl, pl, u1 are thickness, density, rigidity, and displacement in this layer, in which density and rigidity vary following a certain law. Let us suppose, moreover, that density is the same a t any point and rigidity of the intermediate layer varies following a linear law: p1= A X

(2.1)

in which the x axis is placed as shown in Fig. 10. Let us suppose, finally, a rather sharp variation in the distribution of rigidity between the intermediate and the high rigidity layer. Then we have the following equations of motion in the three media, in case of S waves: a2uo a2uo Po at2 = Pn ax2 P2

where u,, = u

a2a2 a%, at2 = P2 ax2

+ u‘, u’= displacement of the reflected wave in the first layer,

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k--I

1.0

FIG. 10. (a),( b ) . Schematic aspect of problem. Results of calculated cases in ( c ) - ( / I ) , L( = 27~/k)is the length of the incident waves (after Sezawa and Kanai [40]).

In case of P waves, it is general enough to put A The solutions of these equations are

+ 2p instead of p.

+ u’=exp[k( V l t fs)]+ B exp[ik(Vlt

(2.6)

uo = u

(2.7)

u2= E exp[kk”(V,t

f4 3

z)]

106

ON THE UPPER MANTLE

corresponding to the incident waves: (2.8)

u = exp[ik( V l t fs)]

where v3=fi2x;k v , = k " V 3 = p

I n these equations, plus or minus signs are to be used according to the major or minor rigidity of the first layer compared with the second one. If h is the distance from the x-axis origin t o the separation plane between the medium in which primary waves are incident and the intermediate layer (see Fig. lo), we have ~ = h ; u,,= ~ 1 , mAh} -duo = A h - au1 (2.9) Ah ax ax (2.10)

h-H;

x = ( h +H ;

Ul = u2 f

where m is the ratio of the rigidity of the medium with greater elastic constants (poor p2)to the maximum value of the rigidity ( p l )in the intermediate layer. Replacing equations ( 2 . 5 ) ,( 2 . 6 ) ,and (2.7)in these boundary conditions we have : (2.11)

u, = u +u' = cos k( V ' t f x )

(2.12) u '-2

e2 + f 2

(--a

+ b)2 + + d)2 (C

+?]Jm

1

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where M , N , a , b, c, d , e , f , g are some combinations of the Bessel and Neumann functions of the quantities + = 22Jnz) ukh,

4 = 22 f i )

akJh(h -f H )

Using these equations, we can calculate the displacement u' in x = h and the displacement u2 in x = h - H , x = h + H , each of them corresponding to the two different rigidity conditions. Sezawa and Kanai have plotted the graph of the results of their calculations as shown in Fig. 10. The behavior of the transmission through the discontinuity considered can be inferred from this figure. The amplitudes of reflected and refracted waves, in correspondence t o the discontinuity in consideration, depend substantially on the m ratio, i.e., on the elastic constants variation, the incident wave-length, and the thickness of the transition layer. If the incident wavelengths are very large in comparison with the thickness of the intermediate layer, i.e., kH + O , the amplitudes of reflected (u,) and transmitted waves (u,) compared with incident waves (u),will have values as great as those obtained by Knott and others in the completely hypothetical case of a sharp discontinuity. If the incident wavelengths are short, the amplitudes of refracted waves become greater, and the amplitudes of reflected waves smaller. If the rigidity distribution i n the intermediate layer is continuous with the two next layers and the incident wavelengths are small enough, the amplitudes of rejected waves tend to disappear, while the amplitudes of transmitted waves tend to assume asymptoticul values, thus showing that the whole energy of the incident waves is transmitted from the first to the second layer. Sezawa and Kanai proved that incident waves of very short wavelength can lead to asymptotic values of refracted as well as reflected waves. When m = 2, i.e., when a sharp variation in the rigidity distribution takes place, a bit of the energy of the incident wave is always reflected, whatever the wavelength may be. It is then possible to find the maximum value of the amplitudes of refracted and reflected waves. Although, in general, the amplitudes of the waves transmitted through discontinuities are larger and those of reflected waves are smaller together with the decrease of the incident wavelengths, the amplitude variation becomes more or less oscillating in comparison with the incident wavelengths, according to the m value. The higher the m value becomes, the more oscillating is the amplitude variation, which shows that some resonance phenomena are present a t certain wavelengths ; these phenomena are the result of more favorable conditions for multiple reflection inside the intermediate layer.

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107

When the incident wavelength is a few times greater than the thickness of the layer, or a certain fraction of it, the amplitudes of reflected and refracted waves assume values that differ more and more, according to the decrease of the incident wavelengths. Since discontinuities in the earth’s crust are to be regarded as more or less thick transition zones, Sezawa and Kanai’s theory is very important in order to interpret data obtained from explosions. It clearly explains why the oscillations caused by earthquakes, which have always a rather long wavelength, are partly reflected by the intermediate discontinuities placed between the external surface of the earth and the MohoroviEi6 discontinuity, while the elastic energy is almost completely transmitted in case of short wavelength oscillations caused by explosions. I n recording explosions, moreover, we use sets-geophones-that magnify vibrations with periods of some thousandths of a second only, and do not record a t all vibration with periods longer than some hundredths of a second; thus, it is clearly evident how discontinuities that differ somewhat from mathematical surfaces are not recognizable under such circumstances.

2.2.3.4. Dispersion of seismic waves at the highest frequencies. The difference in period, and therefore in frequency, between earthquake and explosion waves may also explain why different values of velocity for the P waves in the granitic layer have been found. It has already been pointed out that highest frequency body waves are subject to anomalous dispersion [41]; i.e., highest velocities are associated with highest frequencies. Therefore highest frequency waves obtained from explosions will travel faster than earthquake waves. I n order t o explain such a phenomenon analytically, Caloi has used Toda’s theory of propagation of seismic waves through elastic media that also present internal friction (duro-elastic media). I n this theory, the P-wave velocity is given by the following equation: (2.14)

j72=-+-1 -

1’

pK,

4p

w2r2

3 p 1 +u2-r2

where p is the density, K , the static compressibility, p the rigidity, -r the relaxation time for tangential strains, and w = 2n/T, where T is the wave period. The S-wave velocity is (2.15)

2-p

W2T2

ys - p l f W z 7 2

Then in both equations velocity depends on w , i.e., on T.

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P. OALOI

Kubotera, starting from Toda’s model, reaches the following relation between k, and the viscoelastic constants: (2.16)

where k,, is the attenuation coefficient dependent on the duro-viscosity of the medium. If we know the values of k,, which a t very short periods are very near to those related to Rayleigh waves [ala], we can find the T value related t o a given period. I n case of explosions, since w is very high, the P-wave velocity becomes v2 ’ex

=-+-1 4P pk,

3p

which may be put in relation with the earthquake’s P-wave velocity: (2.17)

where PIP is the square of the explosion’s S-wave velocity. Then the application of equations (2.16) and (2.17) permits reduction of the velocities obtained from explosions to the Velocities that would be obtained from earthquakes in the same area. Using such a method, in fact, the maximum velocities obtained from small explosions in some Alpine valleys have been reduced from 7 km/sec to about 6.7 km/sec [41]; this value is in accordance with the average value of direct P-wave velocity obtained from earthquakes. This confirms, therefore, that the difference in velocity between explosion and earthquake waves is, partly a t least, caused by anomalous dispersion through the materials forming the granitic layer.

2.2.4. Low-Velocity Channek into the Lithosphere. The difference in Pg-wave velocities between earthquakes and explosions may also be explained by the existence of a low-velocity layer in the granite; explosions, indeed, cause waves to travel in the high-velocity upper strata only, and not the lowvelocity lower strata. Such a hypothesis was suggested by Gutenberg [19] for the first time, after a study of records from near earthquakes carried out in order to solve the problem of the velocity in the granitic layer in southern California. On that occasion, Gutenberg pointed out that the Pg-wave group, whose amplitude was decreasing almost exponentially after 130 km, and whose maximum was shifting from one phase of the group to another, seemed to consist of waves traveling a dispersive medium, and that velocities of about 5.6 km/sec were

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109

most likely to be regarded as group velocities; S waves gave the same impression, and seemed to lead t o slower phases at greater distances. Then such waves were regarded as guided by a low-velocity channel. Waves may be considered as direct ones a t very short epicentral distances only, and their velocity calculated by Gutenberg is about 6.1 km/sec. The constitution of a low-velocity layer is caused by a decrease of the elastic constants, according to a prevalence of the effect of increasing temperature on the opposite effect of increasing pressure. From laboratory data [42] we know that the increase of velocity caused by pressure, from 10 to 50 km in depth, is about 1-2 yoper kilometer. We know very little about the variation of the elastic constants in rocks with the increase of temperature a t a depth of about 10 km, since the temperature variation with depth is not yet well known; it has been observed, however, that the elastic constants of metals decrease rather sharply: At a temperature corresponding to a depth of 20 km, elastic constants of metals are reduced to one-half or three-quarters of the values a t the surface temperature. Therefore, in rocks at a depth of about 10-15 km, the increase of temperature may be predominant, and the increase of pressure subordinate, to cause the elastic constants to vary. Moreover, a decrease of the elastic constants owing t o a phase change of a mineral may occur, as has been observed in the transition from u-quartz to !-quartz [43]. A decrease of velocity owing to the above-mentioned causes may also be found a t any depth down to 100 km; therefore a low-velocity layer may exist in the gabbro layer, too [43,44]. The existence of a lowvelocity layer below the MohoroviEi6 discontinuity has been proved by finding channeled Pa and Sa waves [45]. The main effect of a low-velocity layer is the channeling of the waves that originate in its interior. In fact, the impulses that leave a vibrating point with an angle greater than a definite angle of inclination are bound to reenter rhythmically the low-velocity zone that becomes a channel guide of seismic energy. The more the seismic energy originating from the low-velocity layer travels in directions parallel to the surface boundaries of the layer, the more this energy is conveyed by the channel guide. There is a solid angle with its vertex in the focus, which conveys all seismic rays comprised in it [45]. Channeled waves thus originated may travel long distances without substantial loss of energy. If a wave leaves the focus with an angle of impulse less than the critical angle, it will come to the surface; the critical angle depends on the starting velocity of P or S waves at the source ( V & the velocity a t the surface boundary of the low-velocity channel ( Vwi),and the respective radii (sin i = rJrS v,/vwl)[46]. The shallowcr the source, the greater is the energy that goes toward the surface; this may explain why in relatively shallow shocks damage near the epicenter is sometimes very heavy.

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P. CALOI

Waves of small amplitude, however, reach the surface in the zone in which neither P nor S waves should arrive, because of their channeling; these small-amplitude waves are regarded as confined to the low-velocity channel. Channeled waves leave the channel guide and come to the surface in a way that is not yet very clear; it may be a diffraction phenomenon. In any case, the channeled wavelength is equal to, or greater than, the thickness of material above the low-velocity layer. The presence of a low-velocity channel in the granitic layer would explain the sharp decrease of Pg amplitudes after a certain epicentral distance as well as the different values of their velocities and the difference in time calculations between P and S waves. Indeed S waves, if compared with P waves, seem to originate either a t a different time or from a different point. With regard to the gabbro-basaltic layer, a low-velocity channel could explain why clear P* waves are very seldom recorded. (a) Lg and Rg waves. The hypothesis of a low-velocity channel in the granite layer has been confirmed by finding Lg and Rg waves. Press and Ewing [47] have identified two wave trains with velocities of about 3.5 and 3 kmlsec, periods of 1-6 sec and 8-12 sec, and transversehorizontal and elliptical retrograde motion, respectively; since these two wave trains present all features of slow surface waves of Love and Rayleigh short-period type, and are recorded for continental paths only, they have been named Lg and Rg waves, respectively, where g indicates that in their propagation they are somehow guided by the superficial granitic layer that is present in the continental structure only. I n order t o explain Lg-wave transmission mechanism, Ewing and Press assumed a multiple reflection of SH waves in a superficial layer a t angles of incidence very near t o the normal. I n a careful study of Lg- and Rg-wave propagation through Eurasiatic paths, BBth [48], on the contrary, pointed out that such waves must be related to the existence of a low-velocity channel, as already predicted by Gutenberg. BBth’s studies, afterward confirmed by Gutenberg [44], have shown that Lg waves subdivide into a t least two types: Lgl and Lg2 waves, both showing mainly a transverse-horizontal motion (corresponding to SH) and a small vertical motion (SV); Lgl waves have a velocity of about 3.54-3.58 km/sec, and periods of about 5.8 sec; Lg2 waves have a velocity of 3.37-3.38 km/sec, and average periods of about 6.8 sec; and Rg waves, which present elliptical retrograde motion, and therefore are gcnerally regarded as Rayleigh waves, have a velocity of 3.07 km/sec and periods of about 9.2 sec. The velocity of Lg and Rg waves does not vary at various epiccnt,ral distances, and the velocity of Sa and Pa waves (channeled by the asthenosphere) does not vary either [45]; Rayleigh waves (Ra), moreover, are guided

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111

by the asthenosphere, which confirms the hypothesis that Rg are Rayleigh waves guided by a low-velocity channel in the granitic layer. Another property of Lg and Rg waves, which confirms the existence of a channel guide, is the small loss of energy that affects their propagation on long distances; therefore, Lg waves cannot propagate by multiple reflection on the outer and inner surface limits, as supposed by Press and Ewing. I n fact, if the energy were not to be lost during the multiple reflection of SH waves, the layer should be regarded as limited a t both sides by a fluid or gaseous medium [48]. Moreover, Lg waves cannot be regarded as short-period Love waves as they appear in continental paths only, whereas the abovementioned Love waves should also appear in other structures. In most cases, Lg waves have a clear vertical component which does not appear in Love waves, and carry a lot of energy which could be carried by surface waves only if they had a wavelength longer than Lg. Both Gutenberg and BAth thought that the Lg2 wave, whose velocity is very near to Sg, is a transverse wave channeled by the low-velocity layer. According to Gutenberg’s model of velocity variation with depth, the lithosphere channels, unlike the asthenosphere, have a sharp discontinuity a t their lower boundaries. This explains the absence of channeled P waves which could not be guided because of the rapid loss of energy during the multiple reflections a t the lower boundary. Indeed, it is known that P waves are never completely reflected by the earth’s surface and an inner discontinuity; P waves can be guided only if a gradual variation of velocity takes place, as in the case of the asthenosphere in which Pa waves are present. The prevalence of SH over SV waves is due to the fact that total reflection is more favorable to SH than to SV in case of a n inner discontinuity. Since Lg and Rg waves propagate through continental paths only, they may be used to study the structure of the earth’s crust in some regions. With regard t o the Arctic Ocean [48], large areas of it must have continental structure, since clear Lg waves caused by earthquakes that originated near the northern coast of Siberia have been recorded at Uppsala; the boundary of the Siberian Shield must pass just west of Spitsbergen, since no Lg and Rg waves due to earthquakes from J a n Mayen and Spitsbergen have been recorded at Uppsala. Central areas of the Arctic Ocean, from the North Pole toward the Bering Strait probably have oceanic structure. The Baltic Sea, on the contrary, has quite continental structure, as Lg and Rg waves coming from south Central Europe pass through it. The Mediterranean Sea probably has partly oceanic structure, since no Lg and Rg waves from the southern Mediterranean have been recorded a t Uppsala. Such waves are not always present on records of earthquakes from the Aegean Sea which probably has a very complex structure. A very deformed

112

P. CALOI

continental structure, indeed, seems to obstruct Lg- and Rg-wave transmission. The roots of the mountain ranges, for instance, surely obstruct, or at least interfere with, Lg and Rg transmission. The existence of a low-velocity channel would also explain why microseisms that travel continental paths cover such a great distance [44,49]. Such microseisms are probably caused by channeled waves; they show, indeed, periods and velocities very near to Lg and Rg, and their continental distribution is very different from the oceanic one. No doubt microseisms are a composite wave train, probably containing both Lg and Rg waves [49].

(a) Li waves. BBth [49]found another wave train similar to Lg, with average velocity of about 3.8kmlsec; following his opinion, this wave train should be regarded as guided by the low-velocity channel in the gabbro-basaltic layer, which was already pointed out by Gutenberg [44].The 3.8 km/sec value is in accordance with the velocity of S waves; it should be noted, moreover, that Gutenberg [43]found a velocity of 3.8 km/sec a t the top of the gabbro layer. 2.2.5. Concluding Views. We can conclude, essentially, that the discovery of channeled waves leads to confirm the presence in the earth's crust of two layers at least in the continental areas and of one layer only in theoceanic areas, as already found in the study of body waves and surface waves from near earthquakes. Although regional variations from the standard model occur, discrepant results obtained in some regions from explosion data cannot be regarded as valid. Very high frequencies, anomalous dispersion, and presence of lowvelocity channels fully explain the absence of clear reflected and refracted waves originated by the discontinuities. Explosions, therefore, seem to be of limited use, especially in areas in which the change from one layer t o another may be more gradual than elsewhere. This inadequacy to reveal decp structures is due to the superficiality of the source and t o the characteristic of body waves thus originated. It cannot be denied, however, that in several regions explosions have given results much in accordance with other methods. 2.3. Determination of Structural Features of the Earth's Crust by Studying Surface Waves Methods for determining structural features of the crustal layers by using Love and Rayleigh surface waves are well known: They utilize dispersion that affects such waves during their propagation. Surface waves permit also determinations of averages on long paths. They are particularly useful for

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113

rough determinations of long distances in wide areas of the earth in which seismic stations are lacking, i.e., deserts, oceans, etc.

2.3.1. Use of Love Waves for Atlantic Ocean. The very application of Jeffreys’ theory of Love waves propagation through dispersive media has ensured the explanation of structural features of the crust in the Atlantic Ocean area. Until 1948, it was widely believed that the Atlantic Ocean bottom had elastic characteristics in the intermediate area between the Pacific Ocean and continental areas [50]. I n order to explain such a structure, Rothe [51] assumed that the western area of the Atlantic Ocean would have oceanic (pacific) structure with the gabbro layer only, while east of the Mid-Atlantic Ridge the structure would be continental. Studying Love waves caused by an earthquake coming from the MidAtlantic Ridge, Caloi et al. [52], by examining records obtained a t stations placed on the two coasts of the ocean, have proved for the first time that Love waves have the same velocity, east as well as west of the ridge, a velocity of the same order as that observed for the Pacific Ocean bottom. Thus the ridge does not part two different zones: RothB, on the contrary, following a theory of mountain formation pointed out by the French geologist Glangeaud, maintained that the Mid-Atlantic Ridge was situated at the very boundary zone between continental and oceanic material. Once it had been proved that the Atlantic Ocean bottom consisted of one crustal layer only (with a small amount of sediment above it), the application of Jeffreys’ method led Caloi et al. [53,54] to discover that the MohoroviEid discontinuity is about 17 km deep below the Atlantic as well as the Pacific. 2.3.2. Determination of the Crustal Thickness by Using Rayleigh Waves 2.3.2.1.Croup velocity method. Rayleigh as well as Love waves can be used to determine structural features of wide continental and oceanic areas. The dispersion that affects them depends on differences in elastic properties above and below the MohoroviEid discontinuity; therefore, the group velocity curve depends on the general characteristics of the crust. It is well known that the first exhaustive theoretical treatment of Rayleigh wave dispersion was made by Jeffreys. A group velocity-period curve may be drawn from seismograms and compared with a theoretical curve drawn from a standard model of the crust: In this way variations between theoretical and real structure can be established. Many studies of Rayleigh wave propagation and crustal structure have been carried out by now. Broadly speaking, it can be noted that group velocities in different oceanic

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P. CALOI

areas show an identical structure of the basement anywhere off wide archipelagos. Dispersion curves of Rayleigh waves, in contrast to transversetangential Love waves, are influenced by the effect of the presence of the liquid stratum. This effect explains, among other things, why very low velocities (about 1.2 km/sec) associated with 15-18 sec periods are recorded. These velocities have t o be considered in the determination of the crystal structure, in order to avoid finding a crustal thickness which is much greater than real (Fig. 11 [55,56]).

4.2

3.4

2.6

1.0

1.0

0

10

20

30 40 Period in Sccondr

50 I

FIQ. 11. Comparison of oceanic Rayleigh-wave dispersion for Pacific pat,h with theoretical dispersion curves (a(is the compressional waves and pi the density) (after Press and Ewing [56]).

2.3.2.2.Rayleigh waves on oceanic paths. Surface waves that pass through oceanic bottoms are more rapid than waves of the same period that pass through continental shields: Such a difference has been attributed to the absence of the granitic layer below the oceans. The effect of oceans on Rayleigh waves was more or less carefully studied by Bromwich [57]and Stoneley [58]; however, these authors did not explain seismological data. I n 1931, Sezawa [69] studied t h e problem from a more general point of view in order to clarify the fundamental characteristics of surface waves originating from near earthquakes.

ON THE UPPER MANTLE

115

Extending Lamb's [60] and Pekeris' [61] theories of propagations of explosive sound in shallow water, Press and Ewing [55,56,62,63] studied the propagation of surface waves through a liquid stratum lying above an elastic half-boundless solid medium. The resultant oceanic dispersion was fully explained by propagation through a solid medium showing a compressional wave velocity of 7.8 km/sec, covered with a layer as thick as the water plus incoherent sediments. The observed gradual shortening of the Rayleigh wave period fitted the theory for periods less than about 15 sec and group velocities nearly equal to the velocity of sound in water. The above-mentioned theories and their results are now well known. They are put forth in the references mentioned here and by Ewing et al. [64]. Stoneley, too, came back t o the matter and studied the propagation of Rayleigh waves through a two-layered oceanic bottom [65,66]. On the contrary, a paper by Usami on the same matter seems to have gone nearly unnoticed; in the present author's opinion, this work, besides having priority, solves the problem of the effect of the oceans on Rayleigh wave propagation more completely. I n the above-mentioned theory of Sezawa, we have K: = g2/4c2- K2(V2/c2- l ) , where K = 2n/L, L is the wavelength, V is the Rayleigh wave velocity, c is the water compressional wave velocity, and g = 0.0098 km/sec2. Usami [67] pointed out that Sezawa was in error to put K , h in place of tanh K,h (where h is the depth of the ocean); indeed this substitution is not permitted if h = 5 km and the Rayleigh wave period = 10 sec. Developing the Sezawa theory, Usami reached the following equation which can be used to determine the velocity of propagation V : (2.18) ~.

b2

c v3,/mj +-p b4 J1 (c2/ V 2 ) ~

-

where a , b are the velocities of compressional and distortion waves a t the ocean bottom, p is the density of the ocean bottom, p' is the density of the water, and n is the frequency of Rayleigh waves. Usami solved equation (2.18) numerically, using the following values: p' = 1;

p = 2.7;

b = 3.3 km/sec;

c = 1.43 km/sec;

X = p.

Figures 12, 13, and 14, corresponding to h = 5 , 3 , and 1, respectively, show the results graphically. They clearly show that dispersion curves divide in two groups according to whether V is larger or smaller than c. The second group has a velocity very near to long waves in water, i.e., f i h . This group,

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O 0

o 0

10

2

20

30

40

50

Y L/b

Fm. 12. Effect of ocean upon Rayleigh waves where V is the velocity of Rayleigh waves, b is the velocity of shear wavos, and c is the velocity in water for an ocean 6 km deep (after Usami [67]).

therefore, is t o be regarded as formed by long waves in water more than Rayleigh waves. Velocities of the first group vary between c and b ; the relevant dispersion curve, moreover, shows several branches whose number tends t o infinity as wavelength tends to zero. Water clearly acts to reduce the velocity of Rayleigh waves. In any case, besides the above-mentioned results, Usami’s study led to the following issues:

h=3km

006 -

-----_

-

Asymptote

0 0

1

10

I

I

I

I

20

30

40

50

FIQ.13. Cross section of an ocean 3 km deep (after Usami [67]).

L/b

117

O N THE UPPER MANTLE

(a)Rayleigh waves that propagate through an ocean bottom show normal dispersion, but dispersion becomes anomalous if V is smaller than c and the wavelength is very long. ( 6 )The effect of water is greater on short waves and acts to reduce velocity. (c) In waves having a velocity greater than c, the motion of a water particle a t the free surface is mainly vertical; this phenomenon is a result of the low viscosity of water, which does not permit the transfer of the horizontal motion from the bottom to the surface of the ocean. If the velocity is less than c, however, the motion of a water particle at the free surface is elliptical with vertical major axis, and the amplitudes of both horizontal and vertical

01 0

1

I

I

10

20

30

1-1

40

50 L / b

FIG.14. Cross section of an ocean 1 km deep (after Usami [67]).

displacement diminish with depth as exp( -k(q - h ) J m ) , where r ) is the depth of the free surface. ( d )If h = 5 km, a wave long, 11.6 x 3.3 kmlsec, for instance, has a velocity b , which proves that, among Rayleigh waves that travel the oceanic paths, some are about 10 yofaster than those that travel along continental paths. Thus the greater velocity of Rayleigh waves in the ocean is not due to lack of the granitic layer only, but also to the effect of water on their propagation. Usami also calculated the ratio between radial and vertical maximum displacements of the ground at the ocean bottom in the case of very great distances, in which no azimuthal component has to be considered. I n this case, we have

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The results of the calculations are given in the accompanying tabulation. V/b 1.00 Umrx/Wmax 1.125

0.95 0.762

0.19195 0.681

0.8 0.621

0.6

0.412

0.4334 0.362

0.067 0.333

0.03

We can deduce from the tabulation that the motion of a solid particle a t the ocean bottom is elliptical, counterclockwise when the direction of the propagation is to the right, in which the major axis of the ellipse is horizontal (“prolate”)or vertical (“oblate”) according to whether V is greater or smaller than 0.9% These results of Usami’s theory, which fully solves the main difficulties concerning Rayleigh waves, suggest the opportunity to repeat calculations using data more in conformity with reality, particularly regarding p and b values.

2.3.2.3. Atlantic Ocean. Regarding the Atlantic Ocean bottom, Ewing and Press, in a study of Rayleigh waves dispersion [68], confirm the results reached by Caloi, Marcelli, and Pannocchia: “No significant difference in the nature of the suboceanic basement of the Atlantic and Pacific has been found, since the velocity of shear waves was calculated to 4.45 km/sec for both oceans.” Ewing and Press point out that the average depth of the MohoroviEii: discontinuity in the North Atlantic area is about 10 km, i.e., very inferior to the value of 15-17 km calculated by Caloi et al. The two American authors attributed the latter value to the influence of granitic strata in the neighborhood of the European coasts [64]. Further seismic, magnetic, and gravity surveys, however, seem to contradict the conclusions of Ewing and Press. I n a study of the crustal structure of Iceland, BBth [69] pointed out that in that area the MohoroviEii: discontinuity is 28 km deep. A careful study of the structural features of the Mid-Atlantic Ridge, in a wide area included from about 20” to 45” latitude north and from about 10” to 50” longitude west, has been carried out by Talwani et al. [70],from the Lamont Geological Observatory. The interpretation of a great number of magnetic and gravity observations has led the three American authors to assume that, along three sections more than 600 km wide across the “crest provinces,” the MohoroviEii:discontinuity is about 25 km deep on the average. Talwani et al. [71] reached the same conclusions (see Fig. 15). They believed that the crust under the midoceanic ridges is no thicker than under the oceanic basins. They found, however, that seismic wave velocity is low under the axial zones, indicating an “anomalous mantle.” In order to explain gravity anomalies, moreover, they are compelled to extend the low-velocity zone below the midoceanic ridges (i.e., the earth’s crust) t o a thickness of about 25 km and a latitude of about 1000 km. Since

119

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the so-called “crest provinces” and relative “flank provinces” are largely spread over the ocean bottom, of which they constitute a third part a t least, it seems more correct to assume that in the North Atlantic Basin the MohoroviEi6 discontinuity is placed a t an average depth of 15 km (seeCaloi et al., 52-54) rather than 10 km; indeed, the latter is to be regarded as a minimum rather than an average value. K-eu air

Km

FIG. 15. Gravity anomaly profiles and deduced soction across central portion of the Mid-Atlantic Ridge (after Talwani et al. [70]).

The method is definitely inadequate for determining continental structures, since it can be used with some reliability only in areas where no anomalous structures, which can affect the trend of the curves, are present.

2.3.2.4. Phase velocity method. Local variations in crustal structure may be detect,ed using the phase velocity of Rayleigh waves, following a method developed by Press [72-741. Indeed, the phase velocity also depends on the period, crustal thickness, and elastic constant of the crust and upper niant.le. In any case, the method derives variations of crustal thickness out of variation of phase velocity a t a given period, and does not consider the effect of the elastic constant,s.This is possible, because the heterogeneous substances of which the crust consists have almost equal elastic constants [75,76]. Such a presupposition is rather valid if the average properties of crust and mantle are

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P. CALOI

considered in relation to distances of about 60-100 km, as long as Rayleigh wavelengths. Phase velocity determination a t different periods is obtained, together with direction of propagation, by measuring the difference in arrival times of particular crests and depressions recorded by three seismographs located a t the vertices of a triangle, and as far from the focus as the wavelength value. If the arrival times of a crest or depression a t the three stations are t,, t,,, t,,,, and AI1, A,, are the average differences in arrival times, the phase velocity is given by

-,, -,

C = L,,-, sin A - LIII-Isin@ + a ) A11 - I

Am-I

where L,,,-, and L,,-,are the distances among stations, and A and A + a the angles formed by the segments joining stations I11 and I1 with I and the wave front. Velocity thus calculated is plotted in a velocity-period diagram. The correlation between phase velocity and thickness is obtained by comparing available data with phase velocity curves relative t o a given thickness resulting from experimental or theoretical curves of group velocity. I n the application of this method in California, for instance, Press [72] used a set of phase velocity comparison curves obtained from the group velocity curve in Africa (relative to a thickness of 36 km) and from curves calculated by studying Rayleigh wave propagation in a layered medium. The research was carried out in both coastal and inland stations. Comparison of data obtained with the above-mentioned curves showed a thickening of the crust from the continental boundary zone, in which the crust is about 16-19 km thick, toward the peninsular range zone, in which the crustal thickness is very near the average thickness of the crust in continental areas, and toward the Sierra Nevada Zone, in which the crust is nearly 4 7 4 9 km thick. This value is in accordance with that found by Tsuboi using gravity methods and by Gutenberg [77]. I n some regions of North America, crustal thicknesses have been investigated using the Rayleigh wave phase velocity of the Samoa earthquake of 14 April 1967 [74]. These surveys have shown, first of all, that phase velocities have values clearly lower in the highest regions, i.e., the Rocky Mountains, in comparison with coastal and internal zones; they have average values in the Appalachian area. Such differences in velocity have been correlated with differences in crustal thickness, as it is possible to leave out the effect of casual variations of elastic constants. The average crustal thickness thus calculated are the following ones: peninsular ranges-southwestern desert region, 40 km; basin and range region, 47 km; Rocky Mountains, 47 km; interior plains region, 35-40 km; Appalachian region, 40 km; and Canadian Shield, 36 km.

121

ON THE UPPER MANTLE

km 5

0 -20 40

-60

I(m/xc 40

36

-/-

, I

I

-1

120'

110'

-r-.

4-i. 100'

~

w

b(r

m* w

Fro. 16. Transcontinental North American section of topography and phase velocity of Rayleigti waves with periods of 30 sec (after Ewing and Press [74]).

Figure 16 shows a crustal section throughout the North American continent. The Rayleigh phase velocity method ha.8 recently been used in Europe, too. I n Fennoscandia, for instance, Tryggvason [78] and Luosto [79] have found a value of crustal thickness of about 35 km. It has already been pointed out that phase velocity variations depend also on elastic constant variations; but these variations can be omitted in the application of the method. In the present state of knowledge, indeed, it is not possible to interpret simultaneously velocity variations in terms of both crustal thickness and elastic constant variations. Since the elastic constants are less evident, it is logical to assume that they do not vary. An exact evaluation of the influence of elastic constants on phase velocity variation would become possible only if the thickness of a layer could be regarded as invariable; Press [72], in order to calculate such an influence, carried out research on a model, studying the behavior of flexure waves comparable to natural Rayleigh waves-in Plexiglas and brass plates. His study pointed out that the velocity of these waves undergoes a more or less sharp change both in passing through strata of different composition or thickness and in crossing faults and slopes. Phase velocity anomalies, therefore, may be caused by layer thickness variations and faults, salt domes, porosity variations, high structtures,cliffs, etc. The phase velocity method substantially leads t o calculate thicknesses that

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P. CALOI

cannot be considered very exact. In careful surveys, therefore, body wave methods are preferred.

2.3.3. Higher Modes. A new, interesting method for the study of the earth’s crust and upper mantle is based on the study of the so-called “higher modes” of Love and Rayleigh surface waves. Sezawa and Kani [80,81] first studied the higher mode solutions of the Rayleigh equation theoretically; the studies were carried on by Kanai alonc after Sezawa’s death [82-841. More recently, other authors became interested in this problem, both theoretically and experimentally. Oliver and Ewing [86]have studied long dispersive wave trains corresponding to higher modes of the Rayleigh wave equation in relation to the crustmantle continental system, and then have extended this survey to Love waves. They have tried to explain the formation of the continental surface wave phase Lg with the beginning of the short-period branches of the higher modes. Oliver et al. [86] devoted their attention to the second shear mode of continental Rayleigh waves, and noticed that the comparison bctween the dispersion of observed waves and the theoretical dispcrsion for different crustal mantle models proves the increasing reliability of this method in detecting crustal structure, when data for several modes are available. Using this method i t is possible, among other things, to delimitate crustal stratifications distinguished by very small differences in velocity, which is very difficult, if not impossible, using only body waves generated by earth tremors, especially if obtained artificially. The use of the higher mode surface waves has recently been extended to the study of mantle structure by Kovach and Anderson [87]. Only the fundamental mode had previously been applied to the study of this portion of the earth. With regard to arrivals over oceanic paths a t Pasadena, the two abovementioned authors reached new conclusions and concluded that observational data are more favorable to the Anderson and Tokaoz CIT 11 model (1963) than to the Dorman et al. 8099 model (1960). They confirmed, moreover, the existence of the low-velocity zone beneath the Pacific Ocean; perhaps this zone reaches the dcpth of about 400 km. Channeled waves, such as the Lg, Li, and Sa phases, are rcgarded as higher mode group vclocity dispersion waves; Sa waves would have higher vclocity across continents than across oceans. There is no doubt that useful contributions to the research on crust-upper mantle system features will be possible by the study of higher mode surface waves. Yet the method has a great “rigidity,” and necds excessively refined models that are in sharp contrast to reality: The great differences in rcsults

ON THE UPPER MANTLE

123

using different models confirm this. Regarding Sa waves, the method seems to get unsatisfactory results. It is not t o be forgotten that Pa and Sa waves, as observation proves, have the same origin, and the same excitation and propagation conditions. It cannot be seen, moreover, how the supposed propagation of Sa waves by higher mode can be extended to Pa. One of the merits of the method, on the contrary, is its great power to separate strata that present very small variations of velocity for elastic wave propagation. In this way, too, it has been possible t o invalidate the denial of the existence of crustal stratifications supported by some authors and based on explosion data. In this connection, there can be little doubt that the crust is stratified even in those continental areas where stratification had been denied. In Canada, for example, a survey carried out by studying Rayleigh wave phase and group velocities from Canadian station records [88] proved that the simplest mode: that fits observation data consists of a sedimentary layer 2.5 km thick overlying two layers 36 km thick. An equally good fit might be obtained by a three-layered crust. A study of Brune and Dorman [89] on the structure of the Canadian Shield led to exactly the same result. These authors, too, have found that the model that fits the observational data has a three-layered crust 35.2 km thick.

2.3.4 Rayleigh Equation and Somigliana Waves 2.3.4.1.General. In 1917 Carlo Somigliana, a famous Italian physicist and mathematician, expounded a theory of seismic wave propagation a t the surface of a n elastic, homogeneous, isotropic, indefinite medium [go], which, unlike the original Rayleigh theory, leads to the Rayleigh equation without any previous limitation on velocity. Somigliana’s work passed nearly unnoticed, since very few people were interested in geophysics a t that time, and, moreover, there were some errors and misinterpretations in the treatment of the theory, which distorted its real value. I n 1937 a careful reading of Somigliana’s works led the present author to correct some errors and rectify some conclusions. This revision was completely accepted by Somigliana. Later, the present author devoted his attention to other problems. In the last few years Somigliana’s theory, with the above-mentioned modifications, seemed to the present author to be a very interesting approach toward the detection of the earth’s crustal features. Therefore, it seems useful to give a summary here. 2.3.4.2.Somigliana’s theory. Let us assume that the earth’s surface is plane and refer it to a system of orthogonal axes x , y, z; the x y plane coincides with

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P. CALOI

the earth’s surface and the positive direction of the z axis toward the zenith. It is well known that Rayleigh waves occur in the plane zx. Let us consider a plane wave propagating in unspecified direction through the space; its wave plane is ax

+ yz = k,

k = const,

If d is the distance of the considered plane from the origin of axes, we have (2.20) Moreover, if p is the wave pulsation, v the space wave velocity, and V the velocity of the wave projection on the plane, we have

(2.21) Hence, if e is the wave emergence angle, it follows that

Y

sine =

(2.22)

~

Ja2

+ y2 ’

cose =

u

+ y2

~-

Ja2

Let us now consider two plane waves, one longitudinal, the other transverse, marked with the indices 1 and 2, respectively; vl, v2 are their velocities of propagation, and e, and e2 their angles of emergence a t the surface. From equation (2.21) it follows that

v1 = Vl

COB

el,

21, =

V , cos e,.

I n order that surface velocities V , and V , are alike, it must be

(2.23) We can imagine, therefore, innumerable couples of plane waves, one longitudinal, the other transverse, which present identical surface propagation velocities. If q~ and # arc two general functions of ax yz - p t (where t is time), and u,w are the components on the x and z axis, respectively, we can write the following equation that indicates couples of waves:

+

(2.24)

+, + y1 z --P, t ) w1 = Y d a 1 x + Y z - 23 t ) u1=

a1

1

2

1

1

9 4

+

= y:!+(azx y e z - P z t )

w,= -a, +(a2x

+ y2

-p , t )

The resultants of horizontal and vertical displacements a t the surface

125

ON THE UPPER MANTLE

during the overlap of the two waves can be written u=u1 +u,,

w = w l+wz

The surface propagation of waves that originate from the resultant of a longitudinal and a transverse wave requires that normal and tangential strains are equal to 0 when z = 0

(2.25)

where A, p are Lamd’s constants and 9 = &/ax Once supposed that

(2.26)

C1 = a l x

+ y1

-p1 t ,

+

52 = a2x

is the cubic dilatation.

+ y2z

-p2t,

we have:

(2.27)

N,

= Pal2

+ ( A + 2P)Yl2IP’(51)- 2P.2 Y2 + (722 -

T, = CLP.lY1

v’(51)

*’(52)

.22)*’(52)1

where the apex ’ indicates derivation in C1 and 12,respectively. The new wave that originates a t the surface from the overlap of the waves forming the initial couple needs the unification of the respective representative functions; therefore, y and # are to be combined in the same function. Consequently, once assumed that

(2.28) and referring t o the second equation of (2.21) and (2.26),we have

(2.29) because a t the surface it must be

v1 = v e = v which leads us also to unify the 4‘ arguments. The elimination of T, leads to the condition

(2.30)

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P. CALOI

that can be transformed, using equation (2.22) (2.32)

[A

+ (A + 2p) tan2e,](l - tan2e2)= 4p tan el tan e,

Since (2.33) and, a t the surface, V, = V,

=

V , from equation (2.31) it follows that

(2.34) This is the well-known Rayleigh equation. This theory is very interesting not only because it reaches a formula equal to the Rayleigh equation but mainly because the solutions of equation (2.34) are not conditioned by previous limitations on velocity values: They must satisfy equation (2.32) only. Therefore we can affirm that if equation (2.32) is satisfied two waves, one longitudinal and the other transverse, each with its own period, which interfere in the very point of the surface where they have the same velocity V, cause periodic or nonperiodic surface waves. From this point of view, too, Somigliana’s theory is much more general than Rayleigh’s. Once put x = V 2 / v z ,the Rayleigh equation becomes (2.35)

(;?+ 2)xy - 8(;? + 2)x2 + 8(3 ;? + O ) x

- 16(;?

+ 1) = 0

It can be easily seen that this equation always admits a solution less than unity. It is well known that in Rayleigh’s theory the equation (2.34) is valid only for this solution, as it must be 1’
+ 56%- 32 = 0

The solutions, all real, are XI

=4;

= 3.1547;

xi11 = 0.8453

127

ON THE UPPER MANTLE

Regarding

xrI1,it can be easily seen that it leads to negative values for

Longitudinal and transverse waves that should originate Rayleigh waves a t the surface emerge under imaginary angles. From ,yr it can be deduced that

v = 2v2 and, moreover,

Hence, (2.36)

tanel=&-;

1

tane,=

J3

j--

With regard to signs, equation (2.32) is satisfied by plus and minus, or minus and plus combinations; if there are combinations of equal signs (concordant), they lead to absurdity. This is one of the main errors of Somigliana. Since similar conclusions are valid in the case of three real solutions of equation (2.34) whatever the value of u may be, the very Rayleigh waves do not occur in Somigliana's theory (the wave planes turn into systems of parallel lines). Real solutions greater than unity, on the contrary, have an important meaning: Eventual waves corresponding to equation (2.36) originate from the combination of a longitudinal (transverse) incident wave and a transverse (longitudinal) reJlected wave. The following couples of angles of emergence correspond to equation (2.36) e,

e2 = 7 go",

= f- 30';

or, if i indicates the angle of incidence

i, = f60';

i2 =

30'

Analogously, the following couples of angles el =

i , = j- 77'13'

12'47'

e2 = ? 55'44'

or

i2=

34'16'

correspond t o the solution xII= 3.1547. Consequently, when u = f, if longitudinal (transverse) waves reach the surface under angles of 60" (30') or 77'13' (34'16') and are in relation with

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P. CALOI

transverse (longitudinal) reflected waves with angles of reflection of 30" (60") or 34"16' (77"13'), a new type of waves, which will be called Somiglianu waves, may originate a t the surface. Obviously, Somigliana waves originate only if elastic wave reflection laws a t a discontinuity are fulfilled. Particularly, it must be sin i, sin i,

-=-

v, w1

which really occurs in the above-mentioned cases (corresponding to xr, xII values), as can easily be proved. The author will not explain Somigliana's theory completely here, although it presents other interesting topics. This will be done in another place. It is enough to emphasize the particular importance of the Rayleigh equation in this theory, and the significance of its solutions greater than unity in studying crustal structure. The author will dwell on this subject a bit longer. First of all, it is important t o know in which interval of u values all three solutions of Rayleigh equation are real. Table I V shows Rayleigh solutions corresponding to u values variable between 4 and 0, i.e., between incompressibility and maximum rigidity. TABLEIV. Roots of the Rayleigh equation.

XI XI1

XI11

Vdva Vr1/uz VIIr/va

0.0

0.1

0.2

0.25

0.26

0.26305

0.76393 2.0000 5.2360 0.87403 1.4142 2.2882

0.79764 2.2504 4.9519 0.8931 1.500 2.225

0.82992 2.6873 4.4846 0,9110 1.639 2.117

0.8453 3.1547 4 0.9194 1.7761 2

0.8483 3.3702 3.7813 0.9210 1.8358 1.9445

0.84922 3.5754 3.5754 0.92153 1.89085 1.89085

TABLEV. Anglos of incidence ( f ) and reflection ( - ) for longitudinal (1) and transverse (2) waves, conducive to the formation of Somigliana waves.

Roots

I7

il

i2

~

0.0

XI1

5.2360

=

3.1547

0.25

0.26305

= 2.0000

~ 1 1 1=

XI"

= 4.0000

XI1

=

= 3.5754

90" f38"lO'

r 45" F22"65'

&77"13' f60'

T 30'

i34"16'

f68'51'

731"56'

ON THE UPPER MANTLE

129

If CJ is variable between 0.26305 and 0.5, only the solution less than unity is real; the others are imaginary. Somigliana waves, therefore, may occur only in media in which the Poisson ratio u is less than 0.263, i.e., in sufficiently rigid media. Table V shows the values of the couples of angles i,, i,, to which the Somigliana waves may correspond in the case of u=O.O, u=0.25; and CJ = 0.26305. Figure 17 shows real solutions of the Rayleigh equation when CJ varies between 0 and 4.

0.Bl .. X

0.90

..

0.88

.

086

..

x 6-1

FIG.17. Real roots of Rayleigh equation for u variable from 0 to t.

2.3.4.3. Physical interpretation. Before giving a physical interpretation of the theory developed above, a remark is in order. In the theory we necessarily refer to nondimensional mathematical surfaces, and also the earth's surface is regarded as a mathematical plane. I n reality, this does not make sense: The earth, is in fact, not surrounded by an infinitesimal covering to be regarded as its external surface. Oscillatory phenomena of the earth's surface take place in strata of jnite thickness. On the other hand, such a phenomenon occurs in optics, in the formation

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P. CALOI

of so-called evanescent waves [91], which originate in the second medium when the light reaches the discontinuity surface under a n angle equal to the critical angle of total reflection. The existence of such waves, which are not detectable in ordinary experiments, may be proved by using an apparatus formed by two total reflection prisms, one of which has a slightly convex hypotenuse surface (Fig. 18).

R

R

PIG.18. Diagram showing the forming of evanescent waves in optics.

The experiment shows that around the contact point 0 there is a zone in which light transmission is complete and optic contact is fulfilled, in spite of the air pellicle. The circular zone is surrounded by a ring, reddish for transmission and bluish for reflection: The air peUicle certainly exists here, but it is too thin to ensure total rejection, especially if it is few times greater than the wavelength. This experiment shows that reflection cannot be total if the second medium is too thin: Total rejection of light needs a $finite thickness of the less refracting medium. Somigliana waves also corresponding to Rayleigh equation solutions greater than unity (if they exist) originate from reflection. In our case, the second medium should be the air, which is elastically inert. On the other hand, the earth’s surface, regarded as a mathematical plane, could not transmit surface waves either. It is necessary to replace the nondimensional mathematical surface by a superficial stratum of finite thickness; i.e., crustal layers that present their proper (free) oscillation. In mathematical treatment, the thin external layer is regarded as a surface. Therefore, normal and tangential strains N , and T, are to be regarded as equal to 0 throughout) the layer thickness; the layer is to be considered as a physical body that prescnts free oscillations in contact with an outer medium (the atmosphere) that is elastically inert. This may be done, since during proper oscillations the whole stratum can vibrate like a mathematical surface. Moreover, this is in accordance with observation: Somigliana waves (the present author has called C,,j the Somigliana waves that originate from the incidence of SV waves [ S Z ] ) have periods clearly greater than transverse and longitudinal waves which originate from

131

ON THE UPPER MANTLE

their combination; therefore, their wavelengths are a multiple of the thickness of the layers through which they travel. Another proof of the above-mentioned origin of Somigliana waves is the fact that oscillations caused by incidence of transverse waves (Ci,j waves) can be regarded also as originating from a model similar to the optical model for evanescent waves [93]. 2.3.4.4. Smiglianu waves in the earth's crust. Let us study how Somigliana waves may originate in crustal strata. First consider the case of incident transverse body waves. Surface waves may originate when a = and angles of incidence are of about 30" ( X I I I = 4) and 34'16' (XII = 3.1547).I n practice, as the difference in angles of incidence is very small, suitable conditions for the origin of Somigliana waves occur in the set of values between 30" and 34O.3. When a transverse wave arrives at the basis of a crustal layer under the above-mentioned angles of incidence, surface waves caused by the combination of the incident transverse wave and the corresponding longitudinal reflected wave originate in the overlying layer.

, /

I I

FIQ. 19. Source mechanism of Cg,, waves in the crust. In A , B , G, originate C2.2, Cl.2,

and

C0.2

waves, respectively (exaggerated ratios).

Let us suppose a three-layered crust. In case of transverse uaves, the value of the angle of incidence, to which Somigliana wave (Ci,j) formation corresponds, is reached-under the same conditions-at distances that vary according to its location a t the basis of the first, second, or third layer, respectively; the distance increases from the first to the third case (Fig. 19). Three different periods, of course, correspond to the three above-mentioned

132

P. CALOI

cases; these periods depend on thickness of the layers concerned and their values increase little by little; the maximum value is reached when the required angle of incidence is at the bottom of the earth’s crust. This actually occurs. I n continental regions, indeed, average distances a t which Somigliana waves originate are about 2000-3000 km (C,,,), 40006000 km (Cl,l), 6000-7000 km (Co,l) in the three cases, respectively. Relative periods are of 22, 33, 4 0 4 5 sec on the average (Fig. 20).

D S *E-WGdtzin&T-z!s’) At

-

to!#;

A- b 9 5 ? h

Fro. 20. Examples of C O , and ~ CZ,Zwaves (Kansu earthquake, December 26, 1932).

vary Values of angles of incidence required to originate C,,,, C,,,, and according to different values of a; consequently, relative epicentral distances vary as well. Table V shows that these variations are very small. Furthermore, if the value of u is greater than 0.25, the values of the two real solutions greater than unity become nearer and nearer t o each other (one decreasing, the other increasing) and tend t o the common value 3.5754,corresponding to u = 0.26306.

133

ON THE UPPER MANTLE

I n correspondence with such a value, there is only one couple of incidence and reflection angles which originate Somigliana waves. I n the case of a transverse wave, the required angle of incidence is i = 31'56'; this value may be regarded as the average value of angles i that originate Ci,j waves. If u values vary between 0.26305 and 0.5, the two solutions greater than unity become imaginary. It is evident that in the case of violent earthquakes suitable conditions for the origin of Somigliana waves occur again a t distances twice as great as the preceding ones. Therefore, SS waves will cause new trains of Ci,j waves that the present author has named C,,,, C,,,, and Go,,, respectively. I n their turn, C2.3, C,,,, and C0,3waves will originate a t distances three times as great as the first ones. This is confirmed by observation, Figures 21 and 22 show some examples

FIQ.21. Examples of

C1,1 and C l , 3

waves.

of CiSj waves. At epicentral distances greater than 8000 km, C",, will

originate from PS instead of S. There are sound arguments supporting this

134

P. CALOI

Uccle 25 -Dee: 19.12

A

= 6900 km

FIG.22. Examples of COJ and Co,a waves, which originate a t the bottom of the crust.

fact, too. (Fig. 23).At suchdistances, moreover, examples of three types of C,i,j waves (C0,,, C,,,, and C,,3 associated with granite, intermediate, and gabbro layer, respectively) showing decreasing periods may occur on the same record. The seismograms in Fig. 23 confirm this possibility.

2.3.4.5. Phase velocity of Somigliana waves. The above theory has another consequence: Somigliana wave velocity must be nearly twice as great as the direct waves from which they originate under discontinuity. This really occurs, and has been proved before CCiwaves were properly explained by Somigliana’s theory. Indeed, in 1948,the author tried to explain such waves as elastic “evanescent” waves. The violent earthquake of

Y

FIG.23. Examples of Co.1, (21.2, and c2.3 waves on the same seismogram. (An explanation of these waves-recorded station from the same earthquake-by the so-called “leaking mode” would be impossible.)

at the same

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P. CALOI

3ec

-

2000

'-

1500

--

500

4000

Fro. 24. Travel times of 1946.

a000

C1,1

12000 k,m

and C l , a waves of Turkestan earthquake of November 2,

Turkestan on November 2, 1946 resulted in the recording of remarkable examples of C,,j (particularly Cl,j) waves all over the world. With regard to Cl,l and C,,,, their average velocities were calculated from very clear recordings (see Fig. 24 and [94, 94a, 95]), and it came out that

VcI,l = 7.5 km/sec;

Vcl,a = 7.66 km/sec

Since a very good approximation cannot be reached in such investigations (because Ci,j are long-period waves recorded in the midde of the seismogram, and records have been obtained from seismographs of different construction and characteristic periods), and, moreover, since velocities are undoubtedly affected by different structures and thicknesses of crustal layers in which they originate and propagate, C,,, and C1,2 wave apparent velocities can be regarded as identical. As ordinary transverse wave velocity is about 3.8-3.9 km/sec below the intermediate layer, we can assert that the theory is valid also on this subject. Furthermore, another fact should be pointed out. The angle that originates C,,, waves occurs a t a n epicentral distance of about 4000-5000 km. It follows that Cl,2 (corresponding to SS) should originate at a distance of about 9,OOO-lO,OOOkm. Observation confirms this hypothesis (Fig. 24). Arguments valid for C, are valid also for Co,j and CZj waves.

ON THE UPPER MANTLE

137

2.3.4.6.Somigliana and P L waves. It has already been shown that angles of incidence of longitudinal waves which, in combination with the corresponding transverse wave can originate Somigliana waves, are very high; too high, indeed, t o originate such waves in far earthquakes (i.e., epicentral distances greater than 2000 km). On the contrary, good examples of Somigliana waves caused by the incidence of longitudinal waves occur in records of near earthquakes. On this matter, it is believed that Somville P L waves, which only occur in near earthquakes records when the angle of incidence of longitudinal waves has a value nearly like that required by Somigliana’s theory, are true Somigliana waves corresponding t o incident longitudinal waves that reach the basement of crustal stratifications under great angles of incidence, i.e., 68”51‘ if (T = 0.26305. They may originate, moreover, from low-velocity layers. This explains why in records of earthquakes with focus in the asthenosphere, P and PP waves are followed by long-period oscillations t o epicentral distances of about 9000 km. 2.3.4.7. P L and Ci,j waves. It is useful now to dwell upon Oliver’s interpretation of the PL wave origin. Following Oliver [95-971, P L waves are a normal-dispersion wave train showing periods always greater than 10 sec: they start a t the same time, or immediately after P waves, and sometimes continue to the arrival of the first Rayleigh waves. Comparison between P L and R-dispersion curves would have proved that PL are surface waves. They have been observed at epicentral distances of 25”, but, in Major and Oliver’s opinion, they occur also a t greater distances. Gilbert and Laster [98] carried out experiments on PL wave propagation through a two-dimensional model. R , SV, and PL waves traveling through a single layer overlying a uniform medium have been studied by placing source and receiver on the free surface of the layer. These authors found fairly good agreement between theory and observation. This particular type of propagation has been called the “leaking mode.” Following Oliver [96], the way of propagation of PI, waves would clearly affect the formation of C i , j .He thinks that C,,j waves may be regarded as the result of the combination of SV waves incident on a given surface of discontinuity and PL waves, according to a mechanism shown in Fig. 25.

FIG.25. How PL waves could couple with shear waves (after Oliver [MI).

138

P. CALOI

Before examining Oliver’s hypothesis, it should be recalled that a similar hypothesis, which tried to explain the increasing periods of these waves by dispersion, had already been expounded by Caloi in 1940 [99]. But such a hypothesis was dropped afterward [93]. Dispersion, indeed, is inadequate to explain the range of periods of Ci,j waves (from about 20 to 50 sec).’ Moreover, if these waves originated by combination with PL, they would not have such small amplitudes and periods (or would not be sometimes lacking) on records obtained in oceanic islands. Furthermore, it could not explain why, for instance, Co,i are correlated with PS instead of S waves a t epicentral distances greater than 9000 km; why S do not originate Ci,i a t distances of 9,000-11,OOO km; why Ci,j with the same periods originate again, in the same conditions, a t double (SS)and triple (SSS) distances; why, a t the same seismic station, the combination of PL with S, SS, SSS waves (that have nearly the same periods) originates Ci,i waves of sharply decreasing periods. (Indeed, if the hypothesis shown in Fig. 25 were correct, S, SS, and SSS waves should originate Ci,j of the same period in the same seismic station, which does not happen; on the contrary, we have C,,,, C,,,, and C2,3waves showing sharply decreasing periods, on the average of 50, 33, and 22 sec, respectively.) I n Somigliana’s theory, all these “whys” are clearly explained by combination of transverse waves incident a t the bottom of crustal stratifications under given angles, with corresponding reflected longitudinal waves. Various thicknesses of strata overlying the discontinuity explain differences in period. The author does not deny the effect of dispersion; but, since Co,j, C,,J, and C2,jwave paths are relatively short, such an effect is not important. 2.3.4.8. Applications. The C,,i wave features are utilizable to detect thicknesses of crustal layers in which they originate. As already pointed out, they originate from incidence of SV correlated with reflected longitudinal waves under certain angles. Their period, therefore, depends on the thickness of the overlying layer. A way that ensures a rough estimate of such a thickness may be drawn from the vibrating membrane theory. Let us consider a membrane fixed for r = a . I n this case, the motion is given by [loll:

(2.37)

w = J,(kr)

+ hJ,(ikr) =J,(kr) + Xr, (kr)

where r is the radius of the polar coordinate system (with the center in the 1 In

the case of Love waves, Wilson [loo] calculates an increase of 2.30 x 10-3 sec/km.

139

ON THE UPPER MANTLE

center of the membrane) the membrane is referred to; J,, I , are Bessel functions. From equation (2.37) we have

(2.38)

dw

-= Jo’(kr ) k dr

+ ihJ,’( ikr) = -J , (kr)+ h l , (kr)

where 22

24

(2.39)

I , ( % )= J o ( i z ) = 1 +-++ 22 2 2 * 4 2

(2.40)

I , ( 2 ) = iJ&)

2

=2

...

+22’4+ 22+ + . . 23

25

Since the membrane is fixed for r = a , in this case w and dwldr must be equal to zero. It follows, for r = a,

(2.41) The first solution of equation (2.41) is ka = 3.20, corresponding t o the fundamental symmetric vibration. If T is the corresponding period and p = 2n/T, since [101, p. 3581

p 2 = kq

Eh2 3p(l - u2)

where E is Young’s modulus, h is the semithickness of the membrane, p is the density, and u is Poisson’s ratio, we have

(2.42)

T = 2*a2J3p( 1-- u2)

-

(3 2)2,/E h

hence

(2.43) Under the assumption that the impact of SV waves under crustal layers originates Ci,j waves, the fundamental mode around the impact point is t o be regarded as a circular sector of surface of radius a = VT/4, where V is the Ci,jvelocity of propagation and T is the period. Therefore, from equation (2.43) it follows that

(2.44)

140

P. CALOI

We can put approximately p = 2.8 gm/cm3

E = 2.1012dyne/cm2 u = 0.25

Equation (2.44) becomes d = 0.152 x lo-'

(2.46)

V2T cgs

The study of the Turkestan earthquake of November 2, 1946 [94] ensured the determination of C,,, and C,,, velocity (Fig. 24). This velocity is about 7.6 kmlsec. An exact value of Co,jhas not yet been reached; a few attempts already made seem to lead to the conclusion that COsjwaves also have velocities very near 7.5 km/sec (this is not surprising, since transverse waves have velocities almost equal in the lowest crustal layer and below the MohoroviEi6 discontinuity). Also, the exact velocities of C2,jwaves are not yet known, but they should present values of about 6 km/sec (except in certain continental areas in which tertiary strata ale very thick and transverse wave velocities ere rather low). But we can temporarily assume the following values:

vc,,,= 7.5 km/sec vcl,, = 7.5 km/sec vCz,,= 6 km/sec if T is expressed in seconds and d in kilometers, equation (2.45)now becomes do = 0.855 * T km

d, = 0.855 * T km

(2.46)

d,

= 0.55 *

T km

where do, d , , and d, indicate depths of MohoroviEid, intermediate, and Conrad discontinuities, respectively. Only a few examples of application will be given herein. Records of Ci,j waves are very numerous by now. They are always present (at appropriate epicentral distances) in seismograms given by long-period seismographs, but sometimes they occur also in records given by short-period seismographs. The following average periods for CiSjwaves are given:

Uccle (Brussels)

T sec

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ON THE UPPER MANTLE

Therefore, applying equation (2.46), we have the thicknesses d0

43.5 km

Trieste T sec

di (i = 0, 1,2) km Strasbourg

T sec di (i = 0, 1,2) km Rome

T sec di (i = 0, 1, 2) km Sald

T aec di (i = 0, 1, 2) km oropa T sec di (i = 0, 1, 2) km The Ci,i waves recorded in the continental border zones and oceanic islends are particularly interesting. Let us give a few examples.

Southern California T sec di km

C0.j

C1j

C2,j

34 29

20 17

-

-

Therefore, the MohoroviEi6 discontinuity in southern California, near the sea (Pasadena), is about 30 km deep; this is in accordance with values obtained by Gutenberg through other methods. In Bermuda, Co,jand Cl,j show an average period of about 25 sec; hence, do = 21 km

I n A p k , Samoa, Co,j waves (the only Somigliana waves that occur there)

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P. CALOI

show a period of about 18-20 sec; therefore, do = 15-17 km

this value is in accordance with other values obtained through different methods in the same area. I n Arnboina, the Moluccas, the C,,j period is of about 24 sec; therefore,

do = 20 km I n Reykjavik, Iceland, we have

T sec dikm

co,1

c1,2

35 30

27 23

It is interesting to note that in Iceland B&th [69], through explosion methods, has reached a value of crustal thickness of 28 km. Of course, we can also utilize Somigliana waves caused by SV multiple reflections. At Hwnkayo, Peru, for instance, the Turkestan earthquake of November 2, 1946 gave the following Ci,j waves (epicentral distance= 15,400 km):

T sec di km

c0.2

c1,2

c2.3

48 41

37 32

25 19

All the arguments given above have only the aim of emphasizing the interest that a careful study of Ci,j waves present in order to carry out an accurate determination of crustal structure. I n the author’s opinion, one of the characteristics that make Ci,j waves particularly useful for this purpose is the dependence of their period on the thickness of overlying layers, which ensures the determination of discontinuities from which Somigliana waves originate. Furthermore, the fact that the period is relatively long ensures the cbservation of discontinuities that usually are not found from “body waves” of explosions, these, indeed, are very short-period waves and pass through gradual discontinuities without being reflected. Therefore, in Central Europe, the Canadian Shield, the United States, etc., Ci,j waves clearly show the existence of the “intermediate” discontinuity, which usually is not located by high-frequency waves of explosions. 3. ASTHENOSPHERE AND Pa, Sa WAVES 3.1. General The velocity variation of seismic waves in the earth’s interior is known with fair approximation a t this time. There are still some differences of opinion

ON THE UPPER MANTLE

143

about small zones of the earth, especially the mantle basement and the inner core. Such differences, however, are not so important as to affect the interpretation of distant earthquake records. The gradual improvement of travel-time tables for seismic waves is mainly due to Gutenberg and Jeffreys. Results obtained by these two authors are slightly different with regardto the wide region that ranges from a depth of about 40 km (i.e., immediately below the MohoroviEi6 discontinuity) to about 350 km. The solution of this controversy has led to the discovery of unexpected features of this thick region which has been called the “asthenosphere.”

3.2. The Asthenosphere as Channel Guide of Seismic Energy In seismology a circular zone, with its center in the focus, in which longitudinal and transverse direct waves show reduced amplitudes, is called the “shadow zone.” For crustal earthquakes, one such shadow zone ranges from about 200 t o 1800 kni from the epicenter; in this zone longitudinal and transverse waves show very small amplitudes. I n 1927, in order to explain the existence of this zone, Gutenberg [102,103] assumed that seismic wave velocity slightly decreases a t a depth of 80 km and then increases gradually with depth. I n this case indeed only small amplitude diffracted waves would be recorded in the shadow zone. Further research carried out by Gutenberg during the last decades [lo41 seemed to prove the validity of this hypothesis: I n fact, a decrease in elastic wave velocity which reaches its minimum a t a depth of about 100 km for longitudinal waves and a t a depth a bit greater for transverse waves, should occur below the crust. Jeffreys and his collaborators have continued to ignore the results of Gutenberg’s research, since they were inclined t o ascribe the above-mentioned shadow zone to the presence of a discontinuity nearly 600km deep. Therefore, the valuable tables published by Jeffreys and Bullen in 1940 were calculated leaving Gutenberg’s hypothesis out of consideration. The Table VI gives seismic wave velocity variations in the first 200 km below the MohoroviEid discontinuity, calculated by Gutenberg and Jeffreys. With regard to travel-times of seismic waves throughout the earth, differences are slight, but they are important if we study physical features of only the layers concerned. Indeed, if the results of Gutenberg’s research are valid, this anomalous propagation of seismic energy must be related t o anomalous elastic constants of the medium. Since matter below the crust is regarded as nearly homogeneous, how can

144

P. UALOI

we explain why in a wide zone the velocity does decrease instead of increasing? Hence, we have the great interest in solving the problem in either Gutenberg’s or Jeffreys’ favor. I n 1962, the author tried t o solve it, looking for conclusive proof in favor of one of these two hypotheses. TABLEVI. Seismic wave velocity variations. Jeffreys

Gutenberg

h (km) 50 100 150 200 250

VP

vs

VP

vs

(kmlsec)

(kmlsec)

(kmlsec)

(kmlsec)

8.0 7.85 7.9 8.1 8.3

4.45 4.4 4.35 4.4 4.45

7.75 7.9 8.1 8.3 8.4

4.35 4.46 4.5 4.61 4.7

If Gutenberg’s results were sound, a thick spherical zone, in which the seismic energy could be channeled in order to travel in a spherical channel, should occur below the crust; in other words, propagation parallel t o the boundaries of the layer as well as propagation through different layers should be considered in the earth’s interior, which was a completely new fact unknown up to that time. Of course, this could not occur if Jeffreys’ hypothesis was sound. The author’s study was limited to earthquakes with foci in the asthenosphere (60 t o 300 km deep): It is obvious, indeed, that only earthquakes originating in this zone may originate more or less channeled seismic energy. The seismic wave velocity decreases, which has its maximum a t a depth of about 100 km (Gutenberg), causing conditions favorable to the capture of seismic energy by the asthenosphere; impulses that exceed a given angle of inclination will be bound to reenter the low-velocity zone again and again. Therefore the low-velocity zone becomes a vehicle of seismic energy. Under the same conditions, the greater the energy guided by the asthenosphere, the nearer the focus is to the lowest velocity stratum. Results of the author’s research led to very clear conclusions supporting Gutenberg’s hypothesis. Indeed, in case of earthquakes that originate in the asthenosphere, longitudinal (Pa) and transverse (Sa) wave trains that travel a t velocities of 8.0 and 4.4 kmlsec, respectively, are clearly recorded. Examples are very clear and show maximum amplitudes if focal depths range from 60 to 120 km [106,106]. Thus the existence of a seismic energy channel guide in the earth’s interior was proved for the first time.

ON THE UPPER MANTLE

145

3.3. Dynamic Features of the Asthenosphere Channel

It may seem strange that waves channeled by the asthenosphere can send energy to the outer surface of the earth. However, it is quite normal, as such oscillations have wavelengths ranging from 80 to 200 km. Crustal layers have an average total thickness of 30 b, so that sending energy t o the surface is explainable. Other proofs that confirm the existence of waves channeled by the asthenosphere may be put forward. Such waves must not only show a nearly constant velocity, but they must also arrive a t the surface in a different fashion from normal body waves. Body waves appear to be impulses and often consist of a single impulse only. Waves channeled by the asthenosphere, in contrast, appear as wave trains, since they may pass during their propagation through layers of different thickness in which velocity elastic energy varies slightly. Experience confirms all these arguments (Fig. 26). Something else has to take place. It is well known that, in earthquake records, we generally note longitudinal, transverse, or mixed waves once or many times reflected by the outer surface of the earth. If a shock originates in a low-velocity zone, the nearer the focus to the minimum velocity layer, the greater will be the sector in which seismic waves are channeled. This really occurs in the asthenosphere. Under the above-mentioned conditions, indeed, all rays included in certain impulse directions are bound t o reenter the asthenosphere which thereby captures a more or less high percentage of seismic energy. In fact, a t certain epicentral distances longitudinal and transverse waves-direct or reflected-of only very small amplitude are recorded, or they are very often absent; the corresponding energy is carried by Pa and Sa waves, the recording of which is very clear (Figs. 26 and 27). A further confirmation of the existence of the asthenosphere channel is shown by surface wave propagation when the focus is in the asthenosphere. In these cases, Rayleigh surface waves are always recorded considerably in advance, a fact which has no apparent explanation. The discovery of the seismic energy channel guide below the earth’s crust ensures a clear explanation of this advance; the asthenosphere channels these waves which precede Rayleigh crustal waves, since the medium below the crust allows greater velocity of elastic energy propagation. 3.4. Real Velocities of Pa, Sa Waves Channeled waves were discussed for the first time in 1953, in a communication to the Accademia Nazionale dei Lincei, on the existence of longitudinal and transverse waves guidcd by the asthenosphere [105]. The author called these waves PA and SA. Gutenberg, who shortly afterward confirmed their existence and the author’s interpretation, suggcsted the symbols Pa and Sa, changing into a lower case letter the capital A, initial of “asthenosphere.”

r

FIG.26. Pa and Sa have the aspect of trains of waves. When the focus is in the middle asthenosphere, PP, quite absent. -

_I_-.--11 icm-

, n

6#fitZl"

___ __ m-

1"-

P

p-

cw

W*fm 2

+

----..---I_-

-

pm

---I(irunc

n.-!.u

a-mwmn

nr-tw

5.

.-c-r*r.\*-.uIc-cI--

n

u

L--,-

m

..., SS, ..., are in general

-----.----.-

~.?b---TA-;,P%. -

-

2 )I

-

-

Q

J m l ----------. "

_--_ -

P4 4

. I

FIG.27. When the focus is in the middle asthenosphere, PP, ...,SS,...,are, in general, quite absent.

147

ON THE UPPER MANTLE

It is well known that in the author's and Gutenberg's interpretation, the proper physical condition for the propagation of such waves is the existence of a wide low-velocity layer beneath the earth's crust usually called the "asthenosphere." After 1926, Gutenberg devoted much of his busy life to the study of this zone. The last study was carried out shortly before his death (on January 5, 1960). I n this paper [107], on page 351, velocity values for longitudinal and transverse waves-at intervals of 10 km each and increasing depth from 40 to 400 km-were reported. According to Gutenberg, the minimum velocity of longitudinal waves (7.8 km/sec; a t other times he had indicated 7.85-7.9 kmlsec) should be found a t a depth of 80-100 km, while the minimum velocity of transverse waves (4.4 km/sec) should be found a t a depth of about 150 km [107]. Real wave velocities are found within this range (7.9-4.4 km/sec) for Pa and Sa waves, respectively. These were, as a matter of fact, the average values for the velocity of these waves, which the author reported in his first work on the subject [105]. Later on, however, others who concerned themselves with these waves calculated higher values for their velocity; this started with Ewing and Press who studied them in 1954 separately from Caloi. There is no need to report here all the results obtained by the various scientists. Only two of the most accurate among such studies shall be mentioned. Magnitsky and Khorosheva [lo81 processed Pa and Sa wave records given by Russian seismic stations, drawn from a study of nine earthquakes, for epicentral distances between 22" and 150". The two Russian scientists found travel-time curves of P a and Sa waves to be straight lines with the equations;

+ 0.2205 A" t(min)= 0.3780 + 0.4180 A" t(min)= 0.9558

Pa Sa

From these equations, we obtain the velocity of 8.3 and 4.4 km/sec, for Pa and Sa, respectively. Guidroz and Baker, in a chapt'er entitled "Channel Waves" [log], gave the following results for P a and Sa travel times:

+0.223 A" t(min)= 0.96 + 0.403 A"

t(mi,l)= 0.85

Pa Sa

from which we derive Pa and Sa velocity of 8.31 and 4.59km/sec, respectively. On the other hand, this is the average value of velocity of these waves obtained also from studying records of a single earthquake. There seems therefore to exist a remarkable difference between the results published by

148

P. CALOI

Gutenberg-by methods connected with the propagation of body wavesfor the minimum velocity in the asthenosphere, and those supplied by the study of channel waves. But are velocities directly drawn from arrival time curves of Pa and Sa waves the actual velocities? It is easy t o prove that they are only apparent velocities. I n fact, if we indicate by r, the radius of the asthenosphere (in sections characterized by the minimum velocity) and by v, the real propagation velocity on the circumference of such a radius, and if t, are longitudinal (transverse) waves guided by the asthenosphere, propagation times obtained at the surface from the relevant travel-time curves, results in [110] t, = to

A +ra- TOa''

(where the epicentral distance is expressed as arc), and therefore

where r, is the average radius of the earth and to may be considered constant (in any case, independent from A). The velocity given by the travel-times curves (straight) is

V , = dA/dt, and therefore (3.3)

va = (Tab())v a

Travel-time curves give V , (apparent velocity); once known, such a value [equation 3.3)] leads us to obtain the real velocity v,. Then, applying equation (3.3) to the V , velocity values obtained by Magnitsky and Khorosheva, we obtain the following value of minimum velocity for longitudinal channel waves a t a depth of 100 km, v,(Pa)

= 8.17 km/sec

and, at a depth of 150 km: v,(€'a)

= 8.1

kmlsec

By calculating transverse waves and real velocity a t a depth of 150 km on the basis of the apparent velocity obtained by the Russian scientists, we obtain v,((Sa) = 4.365 km/sec Applying equation (3.3) to the values calculated by Guidroz and Baker for

ON THE UPPER MANTLE

149

the minimum velocities a t a depth of 150 km, we obtain wu(Pa)= 8.1 km/sec v,(Sa) = 4.4b km/sec The Pa velocity lately given by Gutenberg is slightly loncr than the above reported value. We should point out, however, that in earlier works Gutenberg has given a longitudinal wave velocity average value of 7.9 km/sec. Besides, Lehmann [ l l l ] calculated a longitudinal wave velocity of 8.12 km/sec a t the same depth. There could be no better agreement. As regards Sa waves, then, we may say it is only a matter of coincidence. Let us apply equation (3.3) to another case. The Egyptian earthquake of September 12, 1955, which took place a t 32"24'25"N, 29"52'40"E, a t a focal depth of about 25 km (according to as yet unpublished data obtained by L. Marcelli) has supplied clear records of Pa and Sa waves. The apparent velocities calculated from the travel-time curve have the following values: VJPa) = 8.08 km/sec; VJSa) = 4.54 km/sec. The application of equation (3.3) gives the following real values depth = 100 km

v,(Pa)

= 7.95 km/sec

depth = 150 km

u,(Pa)

= 7.90 km/sec

depth = 150 km

wu(Sa)= 4.43 km/sec

I n conclusion, equation (3.3) allows us t o obtain the real velocity of propagation of channel waves Pa, Sa, which, in the cases under consideration, coincides with longitudinal and transverse wave minimum propagation velocity values a t the same depths obtained by Gutenberg and Lehmann through direct methods. This can be considered further proof of the real existence of channel waves Pa, Sa as well as of the accuracy of the hypothesis on their propagation which the author suggested for the first time in 1953. 3.5. Channeling as a Phenomenon Concerning the Upper Mantle and the Crust

The occurrence of channeling or guiding of seismic energy has not thus far received the attention it deserves. Such a phenomenon plays a role of primary importance in the acquisition of new knowledge on physical and chemical features of the upper mantle. This may be achieved only after a careful study of the mantle itself. This study is much wider than we have so far believed. Under the same conditions, channeling specifically concerns crustal and asthenosphere stratifications, depending upon whether the earthquake

150

P. CALOI

originates in the crust or at depths in the range of a few hundred kilometers. In case of earthquakes of sufficient intensity at intermediate depths, all of the upper section of the mantle, from the asthenosphere up t o the surface crustal layers, becomes a vehicle of channeling. Channeling has major dimensions in earthquakes that originate a t a depth of about 100 km. In this regard the earthquake of July 25, 1960 is an extremely intcrcsting example. Let us study the seismogram reproduced in Fig. 28. It has been recorded by a long-period pendulum (about 90 sec)

FIG.28. A clear example of channeling concerning the asthenosphere and the earth crust: ( I ) dircct and reflected longitudinal waves, (11) channel longitudinal waves, (111) direct and reflected transverse waves, and (IV) channel transverse waves.

operating still experimentally at the new seismic station of l'Aquila, Italy. This earthquake (the data are as follows: tp = 54"N,A = 159"E,H = 11.12.00, h = 100 km (approximately); M = 7 Pasadena, Rome) had a focal depth of about 100 km. Note the long uninterrupted sequence of channeled waves, both longitudinal and transverse, from the asthenosphere and from crustal layers. The seismogram may be divided into four separate sections: (I)short-period direct and reflected longitudinal body waves: (11)longer period longitudinal waves channeled by the asthenosophere and the earth's crust, (111)direct and reflected transverse body waves, and (IV)transverse channel waves, which are to be considered analogs to the channeled longitudinal waves of Section 2 (Love waves, which are recorded among them, excepted).

3.6. Channeling in Case of Deep Earthquakes In case of deep earthquakes (with focus deeper than 350 km from the outer surface, according to Gutenberg's division),is there a possibility of channeling! If so, which zone does it concern? In order to study this possibility, the author investigated the strong deep earthquake of the Japan Sea on October 8, 1960, which took place a t 40"O'N, 129'7'E; H = 05.53.01.1;h = 608 km (approximately)-according to

ON THE UPPER MANTLE

151

the U.S.C.G.S. Its magnitude was evaluated, at Pasadena, between 6.5 and 7.5. Limiting the examination, for the present, t o Sa waves, the author noticed conspicuous examples of transverse channel waves in the about 50 seismograms in his possession, especially among records given by intermediate and long-period seismographs. After resolving their travel times with the method of least squares, the author obtained

VJSa)

= 4.615

km/sec

from which, by means of equation (3.3) v,(Sa) = 4.41 km/sec There are, therefore, channel waves from the asthenosphere. These are among the most evident phases of the seismograms checked; indeed, in many instances, the Sa appears t o be the most remarkable phase in the whole seismogram (Figs. 29 and 30). This proves that upward channeling

FIG.30. Vertical component of Sa wave from deep earthquake is always remarkable.

is not only possible, but it has also the power of conveying a remarkable part of the seismic energy into the asthenosphere. The Sa-vertical component is always very strong (Fig. 30).

152

P. CALOI

I n the earthquake under consideration, no noticeable examples of channeling in the crustal layers have appeared. Finally, the channeling appears to be a general phenomenon, which plays a remarkable role in the propagation of seismic energy. It may concern both the earth crustal stratifications and the asthenosphere. It is most effective in cases of strong earthquakes originating in the asthenosphere, that is, it can affect at the same time the crust channels beside the asthenosphere. In the case of deep focus earthquakes, channeling is especially remarkable only in the asthenosphere.

3.7. Discussion of Pa, Sa Waves Of course, not all geophysicists agree on the physical nature of Pa, Sa waves. Ewing and Press ascribe them t o multiple reflections of longitudinal and transverse waves under the earth's crustal basement. A few believe they are higher mode surface waves. On this subject, BBth [48] had already, in 1954, tried to explain Lg, Rg as higher mode surface waves, but then excluded such a mechanism as not very probable. Bath's opinion on these waves was then reviewed by Oliver and Ewing [85] again, who afterward applied it to Sa waves regarded as higher mode surface waves [112]. BBth pointed out [113] that there are clear records of Pa waves. Indeed, Pa waves cannot be explained as higher mode surface waves because of their high velocity and their particle motion; therefore, they should be channel waves guided by the asthenosphere low-velocity layer. Once this was proved, one cannot see why correspondent transverse waves Sa should not occur; all the more as SH and SV waves are able to propagate along a channel without energy leakage, while P are not able t o do so. BBth, who for many years maintained that Pa, Sa were channel waves, in a work written in collaboration with Arroyo in 1963 [114], believed these waves more likely to propagate by multiple reflections a t grazing incidence under the MohoroviEid discontinuity. Arguments produced by these two scientists are not persuasive. The velocities of these waves, calculated from one record only, on different paths, in case of earthquakes whose focal coordinates are only approximately known, cannot be trusted. The above-mentioned uncertainties can be minimized by the method of least squares, applied to several values obtained in several seismic stations. The search for the apparent angle of emergence has led the two authors to varying values, ranging from 10" to go", but they maintain that Pa and Sa waves have nearly constant angles of emergence, about 51" and 64" for P a and Sa, respectively. The determination of this quantity is always extremely problematical, and becomes particularly hazardous in the case of Pa and Sa, i.e., long-period waves recorded in the middle of the seismogram. Gutenberg thought that the

ON THE UPPER MANTLE

153

longitudinal wave minimum velocity layer is a t a depth ranging from 80 to 100 km; Pa and Sa have wavelengths within this range. Therefore, their recording can be explained also from this point of view, because the crust is very viscous. Moreover, the channel is considerably deeper below the continents than below the oceans. BBth and Arroyo maintain that the best developed Pa and Sa waves are recorded in the case of earthquakes with focal depths less than about 60 km. Apart from the uncertainty of the focal depths of the earthquakes considered (many depths were revised according to recosds at Swedish stations), it is difficult to understand why this should lead to the conclusion that these waves propagate by grazing incidence under the MohoroviEid discontinuity. Furthermore, most of the records considered came from oceanic earthquakes, where channeling is sure to occur a t depths of 3040 km a t most. The author’s studies confirm that channeling is very likely to occur a t depths of about 100 km in continental areas. Moreo-;er, it is very likely to occur in the case of very deep earthquakes [115]. The objections alone could invalidate BMh and Arroyo’s conclusions. But there is something more. BBth and Arroyo found continental velocities of 8.35 km/sec (Pa) and 4.56 km/sec (Sa), and oceanic velocities of 8.01 km/sec (Pa) and 4.45 km/sec (Sa). As already pointed out in Section 3.4,these velocities are apparent, especially as they refer to continental paths of Pa, Sa. It is well known that the crust is more than 50 km thick in wide continental areas: I n mountainous regions of south-central Asia, for instance, Russian scientists have repeatedly found that the MohoroviEib discontinuity is more than 70 km deep. The crust is 50 km thick in the Alpine region on the average, 60 km in the Sierra Nevada, etc. Under the ocean floors, on the contrary, the MohoroviEib discontinuity seems to be not more than 10-15 km deep. Therefore, the asthenosphere channel must be considerably deeper under the continents than under the oceans. The real path of Pa, Sa waves, especially under the continents, is actually shorter than the path calculated a t the surface. Of course, the epicentral distance expressed in degrees does not vary, but, in contrast, the distance expressed as arc does. Therefore, applying equation (3.3) in the case of a minimum velocity layer 100 km deep, we find the following velocities: 8.1 km/sec (Pa) and 4.4km/sec (Sa). Velocities calculated by BBth and Arroyo for oceanic paths, however, are only slightly higher than real, because of the shallow depth of the low-velocity layer that is not more than 3040 km. The difference in depths of the asthenosphere channel, which is considerably deeper under the continents than under the oceans, is not only intuitive but has also been proved. Already in 1958, Kishimoto [116], in a study of some Japanese earthquakes, concluded that the minimum velocity

164

P. CALOI

layer below Japan must be less than 100 km deep. Vemnen et al. [117] concluded that the asthenosphere channel is 46 km deep in Alaska, 80 km in Japan, 95 km in the precontinental region of the Tonga Islands, and 120 km in South America. In the wide oceanic areas, therefore, it is probably 3 0 4 0 km, or even less, deep. Therefore, Pa and Sa wave velocities do not vary in both continental and oceanic regions, but the depth of the asthenosphere channel does. Such a variation in depth causes the apparent increase of Pa, Sa wave velocities with continental paths. Keeping this effect in mind, one can see that the reduced velocities are in full accordance with longitudinal and transverse wave velocities in the low-velocity layer calculated by Gutenberg and Miss Lehmann. In the author’s opinion the results of BIlth and Arroyo’s work furnish further proof in favor of the hypothesis of Pa and Sa propagation as channel waves along the asthenosphere low-velocity layer. Higher mode propagation of Pa, Sa waves, supported by Ewing, Oliver, Anderson, and others is not to be trusted either (see Section 3.6), since such a hypothesis does not explain the shortcomings of PP, . . . , SS, . .., waves in case of Pa, Sa occurrence and almost complete capture of the seismic energy conveyed by P, S waves. Besides, as to Sa waves, Press [118], on the basis of the theoretical curve of displacement versus depth for the second Love mode calculated by Anderson and Toksoz, pointed out that waves “with periods between 10 and 20 seconds (which are the main periods of Sa) have large amplitudes in the low-velocity zone as it could be expected when a wave is trapped or channeled in this zone. This implies efficient excitation of the wave by earthquakes with hypocentres in the low-velocity channel.” This confirms the observations made by Caloi (in 1963).2

3.8. The Problem of Viscosity 3.8.1. Viscosity from Seismic Waves.On several occasions, scientists have tried to explain the existence of the shadow zone relevant to near earthquakes of shallow depth. The author shall limit himself t o mentioning the research into the viscosity of the upper mantle by studying seismic recordings. The results obtained by Sezawa and Kanai [120] are particularly important in this respect. The two Japanese authors used the values of density and velocity in the earth’s interior calculated ‘by Bullen, Jeffreys, and Birch (Figs. 31 and 32). ZRecently, BBth seems to have modified his opinion again. He now admits tho existence of both Pa, Sa, and Pn, Sn waves [ 1191. The manifest upward channeling in case of deep earthquakes seems to have led him to modify his opinion: Such a channeling, indeed, i s a further proof of the existence of Pa, Sa waves, which have periods longer than Pn, Sn waves.

ON THE UPPER MANTLE

166

5.4--

5.0

--

4.6.-

4.2-

3.8--

3.4-

FIQ.31. Density distribution (after Bullen-Sezawa and Kanai [lZO]).

.........

rfkm) . . . . . .

___c

1

Jaw

4w

I

.

5m

.

dllw

32. Distribution of the velocity of body waves (aft,erSezawa and Kanai [lZO]).

156

P. CALOI

As is well-known, if v is the velocity of body waves, and r , I? are the polar coordinates of points of the seismic wave path, it follows that (3.4) where p is the inverse of the apparent velocity, p = dt/dA

(3.5)

where t and A are the time and the epicentral distance deduced from the travel-time curves. Obviously, a t the point of the ray closest to the earth’s center, for every path we have r = r, = vp. Three particular points of every wave front can thus be determined. Equation (3.4)ensures the determination of other points, the slope of the travel-time curve under consideration having been calculated. Figure 33 shows curves of various wave paths thus calculated.

FIG.33. Wave paths in rocky shells: full lines indicate P waves; broken lines, S waves (after Sezawa and Kanai LlZO]). Three-dimensional propagation excepted, attenuation coefficients of longitudinal and transverse waves transmitted through a uniform medium are given by the following equations: exp

[ --2v, . A’A + 2 p

+2p‘ s]

a2

~

e r p [ - - -a2 fs] 2v2 P

where vl, v2 are the velocities of the two kinds of frequency waves, A, p, X, pf are elastic and viscous coefficients, and s is the focal distance. In case of lieterogeneous media, absorption factors can be written as follows

157

ON THE UPPER MANTLE

where s, is the partial path of the wave, and vl, v2, A, p, A', p' are functions of s,. It follows that the two kinds of wave attentuation coefficients multiplied by the focal distances are respectively given by

These relations ensure the determination of the attenuation coefficients of various phases of waves transmitted along the paths shown in Fig. 33. If e-ko is the attenuation factor of waves at any epicentral distance, the coefficient k, is a quantity which can be readily interpreted.

3.8.2. Viscosity in the Crust. Let us suppose A', p' to be constant throughout the earth's crust; k, values relevant t o P, PP, PPP, S, SS, and SSS waves a t various A may be obtained through Fig. 33 and equations (3.7) by substituting vl, v2, A, p for their actual values (Figs. 34 and 35). Starting from a

0

50'

100'

150.

200.

FIG.34. Attenuation coefficient for longitudinal waves (after Sezawa and Kanai [ 1201).

certain value of A, the value of k, increases very slowly, as can be seen from the figures. The reason for this is the fact that the effective attenuation coefficient decreases with depth. The coefficients k, relevant to PP, PPP, SS, and SSS waves are very high, which involves small amplitudes of reflected waves, since the effective attenuation coefficient of waves close to the outer surface is greater than at depth. This conclusion clearly results also from the values for the ScS wave (Fig. 35) transmitted in an almost vertical direction. Indeed it may be noticed that though the path of such waves is longer than 50" their coefficient k, is almost equal to the value it assumes for S waves with path of 20".

168

P. CALOI

I n three-dimensional propagation, the amplitude of waves would diminish according to increasing distance, and k, would increase with increasing A. The study of a deep earthquake (about 0.06Rkm) led to the conclusion that the comparison between the amplitudes of S, SS, and ScS waves recorded a t distances of about 170O.1 and 75O.8, respectively, is valid only if p' 10"' cgs

<

Fro. 35. Attenuation coefficient for transverse waves (after Sezawa and Kanai [120]).

units; this value is much lower than the coefficient of viscosity of surface rocks. Because the value of p' thus obtained is to be regarded as the average value of viscosity a t various depths, it is to be concluded again that viscosity is much lower in the earth's interior than a t the surface. Sezawa and Kanai [121], studying the ratio between Rayleigh and Love wave amplitudes and the variation of the ratio between the periods of such waves at increasing epicentral distances, found that p'/2p = ~ 1 4 ~ '

where c = 5.90 km2/sec2.If p = 2.7, it follows that

p' = 8 x 10' poises (cgs units) Since in continental regions Love and Rayleigh waves (with periods less than 1 min) propagate through strata about 30-60 km thick, we may infer that the viscosity of rocks in strata close to the surface is about 10'' cgs units.3

3.8.3. Viscosity in Upper Mantle Layers. Viscosity in deeper layers may be determined by simultaneously employing P and S wave data supplied by the 3 It is well known that Joffreys calculated that the eart,h's crust. has a viscosity of about 5 x 1020 cgs units. Sezawa and Kanaiattributed thedifference between their value and Jeffrcys' to the fact that they consider only one kind of viscosity, whereas Jeffreys assumed two kinds, namely, elasto-viscosity and duro-viscosity.

159

ON THE UPPER MANTLE

same earthquake. It is to be assumed that the ratio between P and S wave amplitudes is finite also a t the origin. Sezawa and Kanai still used the above-mentioned deep earthquake. If A = 17O.1, the ratio between P and S wave amplitudes is about one-half, while if A = 75O.8, it is about one-third. Let us consider a two-layered crust of 50 km total thickness (20 km the first layer and 30 km the second). In the crust, as seen above, the coefficient of viscosity is 10'' cgs units. Let UB assume that in every layer h = p, A' = p'. As suggested by Sezawa and Kanai, even if the velocities may nowadays seem to be a bit lower than current values, the data occur in the two layers as given by the accompanying tabulation. Density

'v1

212 '

Upper layer (20 km thick)

2.1

5.0 km/sec

3.15 km/scc

Inner layer (30 km thick)

3.0

6.1 km/sec

3.10 km/sec

If density and body wave velocity in the upper mantle vary according to Bullen and Jeffreys' opinion, applying equation (3.7) and using Figs. 34 and 35 we reach the following two relations:

s waves: A ~ - 0 . 4 3 4 .e-2.54P' (3'9)

p waves:

~~-0.310.,-1.22p'

-

-22,

Ae-0.174. e-1.56P' ~~-0.124.,-0.7&3p'

--3

for A= 17O.1 and A = 75O.8, respectively, where A and B are S and P wave amplitudes. The first exponential on every line represents the damping factor in the surface layer, and the second the damping factor in the upper mantle. Solving equation (3.9)we find that the coefficient of viscosity below the earth's crust is p' = 5 x lo9 cgs units, i.e, somewhat higher than the estimated value. Indeed, since the damping action of the medium decreases a t increasing elasticity, the actual value of p' is very likely to be much lower than the calculated value. 3.9. Explumtion of the Shadow Zone, Apart from Channeling The existence of the shadow zone may be explained by both the slight decrease of velocity in the asthenosphere and the effect of the viscosity. I n the first case, following Sezawa and Kanai [122], even if the velocity and depth values are rather far from the current ones, we can assume the following

160

P. OALOI

distribution of longitudinal wave velocity at various depths:

(3.10)

h 0-20 km

6.0 km/sec

20-60 km

6.2 km/sec

Vl

+0.00688~)km/sec (7.488 + 0.006882)km/aec

60-70 km

(7.76

70-83.8 km

The conditions are given in Fig. 36. If vl' is the velocity at the lowest point 0

0

1

2

3

4

. : . : . ; . : ;

5

-

6

7

_ , _ .

8

I

FIG 36. Velocities of body wave6 in the crust and asthenosphere (after Sezawa and Kanai [122]).

FIG.37. Wave path.

of the path z = h (Fig. 37),v1 the velocity a t a generating level z, x and s the straight horizontal distance and the length of the wave path arc, since sin i = vl/v,' = dxlds and therefore tan i = dx/dz = (vl/vl'). The horizontal distance between z1 and z2, [l - (v~~v")~]''~,is (3.11)

ON THE UPPER MANTLE

whence, if v1 = A

161

+ Bz, it follows that

(3.12) Analogously, since dt = dslv, = dzlv, cos i, we have the travel time between the same levels

(3.13) Moreover, since ds = dz/cos i , it follows that (by the same hypothesis of variability of v1 (Fig. 38)

FIQ.38. Wave paths determined by ordinary treatment (after Sezawa and Kanai [122]).

(3.14) I n caw of crustal layers it is B = 0. The wave path may be deduced from equations (3.12) and (3.10).Figure 39

0 50 100 FIQ,39. Wave paths for a velocity drop from 8 to 7.9 km/sec at a depth of 70 km. The region between 6O.63 and 10" for A is a ahadow zone.

shows the longitudinal wave paths affected with the considered decrease, the the angle of refraction of incident waves a t a n epicentral distance of 10" becomes critical. Therefore the zone within the epicentral distances 6O.53and 10" is, in fact, a shuihu zone. Although transmission of diffracted waves is possible in this zone, the energy conveyed by such waves is always very small. Indeed, since the minimum velocity of longitudinal waves in continental areas occurs a t a depth of a t least 100 km, and is less than the value calculated by Sezawa and Kanai, the observed shadow zone is much larger. (In case of transverse waves, the minimum velocity occurs a t a depth of about 150 km.)

162

P. CALOI

In case of absence of the low-velocity layer, the shadow zone may be explained as follows. We suppose that longitudinal wave propagation velocities are given by (3.10)(on the first three lines only), and valid only for z > 50 km. With regard to transverse waves, we have

h

(3.15)

0-20 km 20-50 km >50 km

v2

3.15 km/sec 3.70 km/sec (4.3 0.0019~) km/sec

+

From equations (3.10)and (3.12)we can deduce the wave path as shown in Fig. 37. If the layer 70 to 100 km deep were more viscous than the outer layer, the waves under consideration would be much more damped, since the waves incident at 9O.79(epicentral distance in the shadow zone, Fig. 39)would travel mainly through the more viscous layer. The result would be apparently exactly as the result obtained in the case of a shadow zone. This can be demonstrated. The wave paths a t various epicentral distances having been calculated (Fig. 37), the variation A a t varying viscosities and depths from 70 km downward may be obtained through the formula:

A =A,

(3.16)

e-ko'

where k,, is the absorption coeficient [as shown in equation (3.6)]for longitudinal and transverse waves, and s is the length of the path. If u = 2n/10 sec-l, h = p , x ' = p ' , pl'= 10" cgs (in the crust), equation (3.16) leads t o the results shown in Fig. 40. It clearly follows that slight increases of

'*h

Fro. 40. Amplitude decrease near 10" owing to a viscous layer at 70 km depth (after Sezawe and Kanai [122]).

pz' cause very remarkable decreases of wave amplitude, if A is near 10". Diffraction phenomena are of little importance. Therefore, if a viscosity of about 101o-lO*l poises (i.e., greater than the crust's) occurred a t depths of

ON THE UPPER MANTLE

163

70-100 km, the shadow zone in the case of near earthquakes would be explained independently from the existence of the low-velocity layer. The discovery of Pa and Sa waves, indeed, invalidates this hypothesis, since it cannot explain the channeling of seismic energy by the asthenosphere.

3.10. Shadow Zone and Channeling Zone 3.10.1. The Shadow Zone is Caused by Decrease of Velocity. Once the effect of viscosity as cause of the shadow zone has been excluded, we have to consider the decrease of velocity. As regards longitudinal waves only, the problem has been studied by B%th[123] using the ray theory. The model considered (Fig. 41) was suggested by Caloi [45] and Gutenberg [46].

FIG. 41. Consequences of a low-velocity layer (after Caloi [45] and Gutenberg [as]).

A profile with the velocity value and with linearly varying velocity in each layer was assumed (Fig. 42). The variation in each layer is given by the following relations: Layer A :

w = wo(r/ro)-b

Layer B:

w = vl(r/r,)*

Layer C:

w = w2(r/r2)-*

where b is positive, and ro, r l , r2 have the values shown in Fig. 42. These premises allowed BBth to determine the shadow zones and their boundaries. He reached the following results: (1) the inner radius of the shadow zone is minimum when h = 50 km; (2) the outer radius decreases continuously when foci become deeper and deeper (from h = 0 downward); (3)the maximum width of the shadow zone is reached when h = 50 km (level of maximum velocity); and ( 4 ) the shadow zone disappears when the focus is h = 120 km, i.e., 20 km below the level of minimum velocity. Longitudinal wave travel time and its energies were also calculated. We can include among disregarded effects that may convey seismic energy

164

P. CALOI

inside the shadow zone the reflections on the earth’s outer surface, reflections and refractions on crustal discontinuities, and, chiefly, diffraction, which is probably the main cause of the arrival of seismic energy inside the shadow zone.

l8

7.9

6.0

81

kt

u km/%

FIG.42. Velocity profile for longitudinal waves assumed by BBth [123].

Perhaps the velocity profiles chosen do not fit reality well, and simplifications may be excessive; the values thus obtained, however, are to be regarded as clear information on the effects of the low-velocity layer on the propagation of elastic waves. The existence of such a layer, indeed, cannot be any longer questioned. Besides the indirect observational confirmation supplied by Pa, Sa waves, data from various sources confirm Gutenberg’s conclusion that a low-velocity zone occurs in the mantle at depths of 100-120 km. The width of the shadow zone observed through seismic waves from nuclear explosions is in substantial agreement with Gutenberg’s conclusions drawn from the study of earthquake generated waves. Press and Ewing [124] and Landisman and SIto [125] concluded that G-wave velocity data require the existence of a low-velocity zone. Takeuchi et al. [126] showed that mantle Rayleigh wave dispersion provides additional evidence of the worldwide existence of this zone. Recent calculations and observations on Love and Rayleigh wave dispersion made by Brune and Dorman [89], studying the mantle beneath the Canadian Shield, show that a low-velocity zone must occur in this region and that no difference between SH and SV wave velocities is needed. These conclusions were extended to the suboceanic areas by Sykes and Landisman [127] in a study of physical properties of the crust and mantle beneath oceanic areas, investigated with the aid of Love and Rayleigh waves. More recent research carried out by studying the higher mode surface waves [128,129] has led to the same conclusions.

ON "HE UPPER MANTLE

165

3.10.2.On the Formation of the Low- Velocity Zone. It must now be explained why a low-velocity layer a t a depth from 70 t o 300 km has formed. Gutenberg assumed that the earth's materials become vitreous a t a depth of about 80 km. Three main hypotheses on the physical state of the asthenosphere were considered: the vitreous state of the earth's materials at a depth of about 100 km, asthenosphere holocrystalline, and asthenosphere formed by peridotite in a two-phase mixture. Starting from the relations among longitudinal wave velocities, cubic compressibility, and density of the medium, obtained by Adams and Williamson, Daly [ 1301concluded that the hypothesis of an entirely vitreous peridotite does not explain the high velocities the seismic waves have at a depth of 100 km. Moreover, the high-velocity nearly homogeneous layer from 100 t o 400 km deep excludes a composition of pure vitreous olivine. Meteorites formed by pure olivine are not known, and it is very unlikely that such a thick, pure olivine layer might have formed by gravity differentiation from the relatively thin granitic, intermediate, gabbric, and eclogitic surface layers. These and other considerations lead us to exclude an entirely vitreousperidotitic composition of the asthenosphere. The hypothesis of a holocrystalline asthenosphere fits seismological data better. Using the results of experimental research of Birch, Bancroft, and Bridgmann, Daly concluded that the velocities of propagation in holocrystalline peridotites (especially dunite) a t a depth of 100 km (temperature lOOO"C,pressure 30.000 atm) are very close to seismic velocities. But a holocrystalline asthenosphere, besides contrasting with the recognized weakness of this zone of the earth, is not likely to cause a channel guide for seismic energy. These considerations lead us to assume a multiperidotitic composition of the asthenosphere, following Daly's opinion. It has been proved that peridotitic materials have a rather large interval of crystallization temperature. From laboratory experiments, moreover, it results that these materials, if heated and subsequently cooled, with a drop of several hundred degrees, begin to crystallize; a further cooling leaves as fluid residue only a hydrated solution above the critical temperature. Such an interstitial fluid is believed to have a temperature of about 600°C. The temperature a t a depth of 100 km ranges between 1100 and 15OOOC; therefore, if these deep strata are slowly cooling (as they are likely to do, apart from the effect of radioactive materials), they must include a lot of fluid cells. The recognized weakness of the asthenosphere may be attributed to the probable presence of such interstitial fluids and to their lubricating effect on the largely crystallized peridotite. Propagation velocity values thus obtained may seem not t o be in agreement with those supplied by seismological data; indeed,

166

P. CALOI

seismic wave velocities should be less in a two-phase asthenosphere than in a holocryst.alline asthenosphere. But fluid interstitial cells are probably only a small percentage of the peridotite volume; therefore, velocities would be only somewhat less than in a holocrystalline environment. The observed decrease of dastic wave velocity in the asthenosphere could be explained in this way. It is to be believed that, a t the time of the crystallization of two-thirds of the silicate mantle, fluid cells were wider and formed a large portion of the total volume of the crystallizing peridotitic material. Further crystallization must have progressively reduced the fluid cells. Such fluid cells may have been losing a slight portion of the most volatile elements, which may have slowly accumulated under the basement of the impermeable holocrystalline crust of the earth. The discovery of Pa, Sa waves permits further research into the asthenosphere physical features. I n Magnitsky and Khorosheva's opinion [108], Pa and Sa are cylindrical waves, It is widely believed that the cause of the formation of the low-velocity layer is the prevailing effect of increasing temperature over the effect of increasing pressure. An evaluation of the temperature gradient required t o produce a "low-velocity layer" was made by Valle [131]. He found that in the asthenosphere a low-velocity layer may be produced if the temperature gradient is higher than 14"C/km for P waves and 11"C/km for S waves. With regard to the velocity v , it has a minimum if dv/dh = 0. If T is the temperature, P the pressure, and p the density, it follows that (3.17) Magnitsky and Khorosheva (in accordance with Peierls, Dugdall, Mcdonald, and Clark) think that the dependence of the temperature on the thermal conductivity x may be expressed as follows: (3.18)

x=

A/T

+ BT3

In this hypothesis, x has its minimum in the interval of depth from 50 to 100 km [132]. Let us assume 1.2 x lo-' cal/cm* sec as the average heat flow a t the earth's surface. The two Russian scientists find that the gradient dT/dh may reach the value of 18"C/km a t h = 100 km below the continental crust, and 15 deg/km a t h = 5 0 km below the oceanic crust. These values seem to be enough to explain the formation of a low-velocity layer. Another explanation presumes the vitrification of mantle materials a t various depth. Starting from a formula of Frenkel [133] on the volume increase Y owing to vitrification, Magnitsky and Khorosheva conclude that in the case of dunite and under asthenospheric conditions a velocity decrease of about 6 % is likely to occur, which fits calculated values well.

ON THE UPPER MANTLE

167

Among the various hypotheses on the formation of a low-velocity channel, the partial vitrification of the materials a t depths of 100-200 km seems presently to be the most probable. But the problem is still wide open for further discussion. 4.

THE "20" DISCONTINUITY"

4.1. Discussions of the 20" Discontinuity Gutenberg thought that no clear proof of first-order discontinuities in the mantle existed and that elastic wave velocity varied continuously with depth. Other geophysicists have different opinions. MohoroviEii: thought that discontinuities about 120 and 400 km deep do probably occur. I n 1926, Byerly [134] first emphasized the existence of a sharp variation of the P-wave travel-time curve inclination at an epicentral distance of about 20"; in his opinion, such a variation was about 400 km deep. Gutenberg, in the same year, proved the existence of a decrease in longitudinal and transverse wave amplitudes a t an epicentral distance between 100 and 1600 km (these limits vary in accordance with focal depths and crustal thicknesses), which he explained by the decrease in seismic wave velocity beneath the earth's crust in a spherical zone about 100 km thick. The above-mentioned shadow zone, therefore, was regarded by Gutenberg as a consequence of the low-velocity layer, which he continually studied until the end of his life (1960). Moreover, the 20" discontinuity was regarded by him as a consequence of a low-velocity layer a t the top of the mantle. The reduction of the shadow zone (i.e., the circular zones, with center in the focus, in which P waves are relatively late) a t increasing focal depths, and its disappearance in case of earthquakes deeper than the low-velocity layer, was regarded by him as a confirmation of his opinion. Not all scientists, however, considered the shadow zone as important as Gutenberg did. A few investigators have studied only the point of the P travel-time curve that indicates a sharp increase in the apparent velocity dhldt, and the label "20"discontinuity" has been kept for this anomaly first observed by Byerly, even if further careful investigations showed that change of inclination of the travel-time curve (i.e., the sharp increase of velocity) occurred a t an epicentral distance of about 15". Jeffreys repeatedly interested himself in the 20" discontinuity, and ascribed it to the presence of a first-order discontinuity a t a depth that has been modified many times. I n 1936 [135], he thought that such a discontinuity was 481 & 21 km deep; in 1939, about 410 km, and in 1952 [136], about 500 km deep. In 1958 [137], Jeffreys studied this problem again and concluded that the sharp transition zone was about 200 km deep.

168

P. CALOI

Miss Lehmann also repeatedly studied the anomalous arrival times of longitudinal and transverse waves at the above-mentioned epicentral distances. In a 1934 paper [138],Miss Lehmann, who believed that a discontinuity occurred in the mantle and caused the 20” discontinuity, thought that such a discontinuity was between 250 and 350 km deep. After other studies, Miss Lehmann, in a 1958 contribution [139],concluded that longitudinal waves have a constant velocity of 8.12km/sec a t depths ranging from the MohoroviEi6 discontinuity down to about 220 km; at such a depth they show a sharp increase of velocity to about 8.40km/sec. At greater depth the velocity increases according to Gutenberg. Bullen also thought that a discontinuity occurs a t a depth of about 410 km, interpreted as “the depth of the 20” discontinuity.” I n conclusion, Jeffreys, Lehmann, Bullen, and others thought that the 20” discontinuity is caused by a discontinuity in the upper mantle. Unlike Jeffreys and Bullen, who thought that seiamic wave velocity steadily increases from the MohoroviEi6 discontinuity downward, Miss Lehmann thought that the P waves have constant velocity from the MohoroviEi6discontinuity down to a depth of about 220 km, and did not exclude the existence of a zone of asthenospherical features. This is also confirmed in her most recent studies [la01of the travel times of longitudinal waves caused by Logan and Blanca atomic explosions, and in a study [141]of travel times of P waves caused by nuclear explosions. I n 1962 she confirmed [140] the existence of “a discontinuity surface at about 215 km depth at which the velocity and the velocity gradient increase abruptly, while the velocity varies only slightly or is constant above this depth”; in 1964 she concluded [141]that “low-velocity layers are important features in mantle structure. They do not extend unvaried under continents and they will have to be explored in greater detail.” Therefore, Gutenberg’s and Jeffreys-Bullen’s opinions were antithetical: One excluded the other.

4.2. The Asthenosphere and the 20” Discontinuity An analysis of how antithetical these two opinions were is now in order. I n other words, does the asthenosphere exclude the existence of a discontinuity? On the other hand, does the existence of a discontinuity exclude the asthenosphere “low-velocity layer”? Girlanda and Federico, a t the University of Messina, Italy, have been studying this problem for several years. The two Italian scientists were skillfully taking advantage of the particular position near the southern Tyrrhenian Sea, where intermediate and deep earthquakes very often occur, as compared with European, particularly north European, seismic stations. I n 1963,in a very careful and precise study [142,143],they pointed out that the results strictly depend on the positions of seismic station.

ON THE UPPER MANTLE

169

In the study of a south-Tyrrhenian deep earthquake of January 3, 1960, they found in several ways that P waves arrive earlier at Messina and Reggio Calabria seismic stations. This confirms former conclusions and is to be regarded as a local anomaly. The advance in the Swedish stations, on the contrary, is to be attributed to the presence of the 20" discontinuity. Girlanda and Fedorico came back to the matter and studied [144] the earthquake of December 23, 1959 (14"39'.353; 37"39'.4N; h = 77 f5 km). They pointed out the following factors: (a)The travel-time curve of the first impulses, recorded as far as 30", breaks into two branches, whose most probable equations are, respectively: (4.1)

t

= 5.855

+ 13.982367A-0.02493099A2- 0.0004978825A3 (OO.8

(4.2)

t = 31.650

A < 19O.9)

+ 16.420464A- 0.30528644A2+ 0.004228345A3 (21" < A 5 30")

(b) In the seismograms from Tamanrasset, Uppsala, Skalstugan a second impulse follows thc first one: it is 9' late a t the first station; 7' a t the second, 14'.8 at the third one. The travel time of this second impulse a t the first station (A = 16O.9) fits the second branch extrapolated to 16' of the traveltime curve perfectly well [equation (4.2)], while the travel times of the second impulse recorded a t the other two stations (A = 22O.1; A = 25O.8) fit the first branch (extrapolated to 26") of the travel-time curve [equation (4.1)]. ( c ) The two branches cross a t a distance Als2= 22O.07; suoh a distance is considerably greater than the distance calculated by Jeffreys and Bullen in their travel-time curves for the same depth. (d) The first branch of the travel-time curve, from 0" to 20", does not show any curvature, in spite of the great focal depth. Points (a)and (b) confirm the existence of the 20"discontinuity, since such an existence provokes, a t certain epicentral distances, the presence of two impulses caused by the direct P wave (Pd),and the refracted P wave (Pr), corresponding to a zone of sharp increase of velocity. Point ( c ) shows a remarkable disagreement with Jeffreys-Bullen's traveltime curves, which consider the 20" discontinuity, but leave the effects of the asthenosphere out of consideration. The greater distance, a t which the intersection point of the two branches occurs, may be explained by the presence of the low-velocity layer, since the velocity comes back to its normal increase only below it, and not immediately below the MohoroviEiC:discontinuity. Point (d) is clearly explained by the presence of the asthenosphere, since the earthquake studied by Girlanda and Federico has its focus very near the middle of the asthenosphere channel, where conditions are well suited to the occurrence of seismic energy channeling.

170

P. CALOI

All these facts, therefore, point out that both 20" discontinuity and asthemsphere may exist in the earth's mantle. In the above-mentioned study, the maximum depth reached by the Pd-seismicray emerging at a distance of 26", which is the maximum distance a t which the second impulse has been observed, has been calculated by using a simple method that needs the knowledge of the velocity a t the various depths in the upper mantle; this calculation was made in order to get some quantitative estimates. If the following law of velocity variation in the upper mantle is valid:

I.'= 'V,(r/R,)*

(4.3)

( V , = 7.75 km/sec;

R, = 6337 km;

b = -9/4)

and if we use equation (4.1) of the first branch of the travel-time curve, and leave the effect of the crust out of consideration, the following value of such a depth has been calculated: h=418.8 km. If, on the contrary, a law of velocity variation proposed by Gutenberg is used, the value h = 391.1 km has been calculated. Therefore, it has been concluded that, leaving out of consideration the existence of the asthenosphere, the depth of the discontinuity that causes the 20" discontinuity is not less than 419 kni; if, on the contrary, we consider Gutenberg's velocity curve, such a depth is not less than 391 km. Therefore, if the impulse recorded a t a distance of 25'3 (Skalstugan) were the very last direct impulse corresponding to a seismic ray tangent to the discontinuity at the point of the ray nearest t o the center of the earth, the calculated values would indicate the actual depth of the 20" discontinuity. Since this was not sure, one could conclude that the discontinuity had an average of about 400 km at Zeast. More recently [145], Girlanda and Federico have tried to calculate the depth of the discontinuity without using any law of velocity variation, but using only all the information that can be obtained from the travel-time curves of equations (4.1)and (4.2). It is not possible t o give a short report of Girlanda and Federico's long, careful, and precise calculations here. Suffice it to say that one part of their new and laudable work has been devoted t o the study of some possible quantitative consequences, deriving from the observed disagreement with Jeffreys-Bullen's travel-time curves, and another part to demonstrate that, if the velocity values for 40 to 80 km depth proposed by Gutenberg are considered, equations (4.1) and (4.2) give a probable law of velocity variation that calls for a sharp discontinuity a t 830 km in depth. Precisely, the two authors obtained

hzoQ= 636.0 f 10.65 km

ON THE UPPER MANTLE

171

With regard to longitudinal wave velocity immediately above the discontinuity, they obtained v200+= 8.958 f0.093 km

and immediately below, v200-= 9.575 5 0.223 km

These velocities confirm those obtained by Birch for the same depth (Fig. 2).

4.3. Consequences of the Existence of the 20" Discontinuity The existence of a surface of discontinuity corresponding to the 20"discontinuity has many consequences; among these consequences there is possible formation of surface waves. Their formation had already been predicted by Stoneley in 1936 [146], who had calculated their main characteristics. Since the crustal thickness is about one-tenth of the depth of the 20" discontinuity, the material above the discontinuity may be regarded as a surface layer: therefore. Love as well as Rayleigh-type waves, modified as the moderate-period waves altered by presence of surface layers, are to be expected. Stoneley started from a value of about 480 km in depth, as calculated by Jeffreys in 1936. The jump in vl, velocity of compressional waves, is 9.08-9.81 km/sec, while the jump in v2, velocity of distortional waves, is 5.25-5.66 kmlsec. Stoneley thought that the jump in density at the discontinuity is 3.65-4.11 gm/cm3. In order to simplify the calculations, the outer layer was considered to be of constant density 3.5 gm/cm3,and transverse wave velocity w2 = 4.7 km/sec. The underlying material is supposed to have p'

= 4.11 gm/cm3

and v2' = 5.66 km/sec

The corresponding rigidity values are p = 7.73

x

1O'I

and p'

= 1.32

x 10l2 (dynes/cm2)

Using thcse data, Stoneley calculated the wave velocity v of Love waves of wavelength 2r/f by the well-known formula: tan of h = p'u'/pu

(4.4)

in which u2 = (v'/v:) - 1; uf2= 1 - (v2/vi2), and h is the depth of the 20" discontinuity. Once obtained f h values corresponding to a series of v/v2 values, the group velocity V is calculated from (4.5)

v / v , = 4v2 +f h 4 v l v 2 ) / 4 f h)

by numerical differentiation.

172

P. CALOI

In this way, Stoneley calculated a minimum group velocity of about T = 168’ (period). With regard to the Rayleigh waves, Stoneley used a n approximate formula of Jeffreys, based on the application of the Rayleigh principle. He found a minimum group velocity of 4.0 kmlsec, corresponding to v = 4.55 km/sec and T = 167’. Thereforc such waves have very long periods and are not likely to be recorded by common seismographs, although they may appear as traces on which the usual period waves are superposed. At any rate, their recording should be more likely to occur in the case of deep focus shocks, since the surface waves that occur in normal shocks are generally very negligible during these earthquakes. With regard to records of possible surface waves caused by the 20” discontinuity, the conditions calculated by Stoneley have been realized in 1950 a t some seismic stations, especially with the construction of long-period seismographs by H. Benioff. Ewing and Press [147,148] have made a study of the Rayleigh waves, with periods ranging from 1 to 7 min, recorded on occasion of some violent shocks. Among the results obtained, it is to be noted that the observed group velocity curve shows a minimum value of 3.54 km/sec corresponding t o a period of 225’, which is remarkably near the value predicted by Stoneley. It should be emphasized here that the theoretical study of the dispersion curve, carried out by Ewing and Press, “involves an exact calculation of a layer 516 km thick with =4.48 km/sec over a substratum with &=6.15 km/sec and density 10/9 times greater than that in the layer. As stated in our first paper on the mantle Rayleigh waves [147, pp. 146-1471, we do not intend that our calculation shall imply belief in the reality of layering at approximately 500 km depth. We consider the steplike velocity-depth relation simply as an approximation to the gradual actual increase of velocity with depth [148, pp. 4724731.’’ At any rate, it is important that a two-layered upper mantle fits the observed dispersion only if a discontinuity occurs at about 516 km in depth. After the results obtained by Girlanda and Federico, a two-layered upper mantle is no more a working hypothesis but a physical reality. Regarding eventual recording of surface waves caused by the 20” discontinuity, the problem is not yet solved. Perhaps most violent earthquakes do not give the best examples of such waves, since the free oscillations of the earth prevail. Only very long-period seismographs (more than 4-5 min) may give examples of such waves in the case of intermediate earthquakes. Longperiod pendulums (about 10 min) placed in the “Grotta gigante” in Trieste, Italy, conceived and almost completely constructed by the author in order to study the earth’s tides, very often record “synthetic” earthquakes, in which surface waves of about 3 min periods occur; but such waves have a

4.60 km/sec, which corresponds to v = 4.94 knilsec and

173

ON THE WPER MANTLE

\

FIQ.43. FR

+ RQ = PIP: path of the waves reflected at the 20" discontinuity.

velocity of about 4.4 km/sec and could be G waves channeled by the asthenosphere. They are often preceded by longer period waves (7-9 min) with a velocity of about 5.3 km/sec: These could perhaps be caused by the 20" discontinuity, although their periods seem t o be too long in comparison with the theoretical values. In any case, if the 20" discontinuity really occurs, as seems to be proper to affirm a t the moment, we must wonder if such a discontinuity is sharp enough to affect the body waves, too. Since it is surely proved in the case of refracted waves [149], we could look for occasional reflected waves (Fig. 43). The author has recently studied this problem, and the result seems t o be positive. In distant earthquake seismograms, with very complicated impulses, it is easy to find some oscillations that may be attributed to reflections a t the 20"discontinuity. The author has only studied deep earthquakes (deeper than 600 km), and a t such epicentral distances that cannot cause confusion with reflected waves of different origin (pP, PcP, . . .,). Among the numerous examples, the one shown in Figs. 44-46 seems to be very clear. The wave

P

060116.S

p,P

s,P

At =+16',8 vert cornp

FIG.44. Japan Sea earthquake (4OO.O N; 1 2 9 O . 7 E; H = 05.53.01,l;h = 608 km.) Record of vertical component short period ( 1 S . 8 ) of Quetta (October 8, 1960). Here, pl and 91 are longitudinal and transverse waves reflected at 20" discontinuity (515 km deep).

s

I PIS

I

s,s

AI =+16'.e NS

comp.

Fro. 46. Japan Sea earthquake (see Fig. 43). Record of north-south short period (1.9 sec) comp. of Quetttb (October 8, 1900). Here, PIS, S ~ S ar0 , waves reflected at 20" disoontinuity.

06.07

s

P,S

SIS

At = + 16.8 sec EW comp.

060951

FIG.48. (See Figs. 43 and 44). East-west shortperiod(ls.8)componentof Quetta (October 8, 1900).

groups marked plP, slP, plS, slS correspond to reflections on the 20" discontinuity. The earthquake under consideration occurred on Ootober 7, 1960 at the point with coordinates 4Oo.0N;

12Q0.7E(Japan Sea);

h = 608 km (focal depth).

The epicentral distance of Quetta is 61O.3. From Jeffreys-Bullen's tables, for such a distance and for h=O.OQR, and considering plP, slP, plS, slS wave paths, we find that the differences p1 P - P = 12'.4 81 P

- P = 23'.2

p1S-S=1I8.7 s1

s - s = 228.1

175

ON THE UPPER MANTLE

do fit reflections very well on a discontinuity 515 km deep (i.e., 0.076R, following Jeffreys' symbols). Therefore the travel times observed a t Quetta on the occasion of the above-mentioned earthquake of the Japan Sea lead to a value of 515 km for the 20" discontinuity depth; this value is in complete accord with the value calculated by Girlanda and Federico. 5. ATTENUATION AND MIXED ZONES

5.1. Introduction The attenuation of seismic energy of surface waves is an old problem, which was studied mainly by Angenheister [150,151] and Gutenberg [152] since the first decades of seismic studies. In 1906, Angenheister calculated the absorption coefficient k, per kilometer for several earthquakes. Table V I I shows the values of this coefficient for various paths and periods. TABLEVII. Values of ko for various paths and periods.

20 20 25 30 17 20 20 18 20

Average

3.6 3.7 3.4 3.4 3.0 3.4 3.37 4.6 3.4

0.00038 0.00021 0.00039 0.00039 0.00026 0.00034 0.00037 0.00028 0.00018

0.00031

After Angenheister, seismic wave attenuation and particularly absorption coefficient k, were studied for many years. Gutenberg [1,53] began dividing continental and oceanic paths, Love and Rayleigh waves. It was found that the value of k, depends on the absorption as well as on the wavelength and the duro-viscosity; i t was clearly understood that the longer the wavelength, the less the attenuation effect of the earth's surface layers. No report will be given of the numerous k, determinations made by several authors a t different times, except for that of Kizawa [154], who, studying Rayleigh and W, wave records, recorded a t 23 stations by the Chilean

176

P. C'ALOI

earthquake of January 25, 1939, obtained an average k, valuc of about 0.00029, with periods varying between 33.7 (Rayleigh waves) and 25.7 sec (W, waves). In 1945, Gutenberg indicated a k, value of 0.0002 for Rayleigh waves with a 20-sec period. I n a study of the energy conveyed by Rayleigh waves (1959)) De Noyer [155] obtained k, values of 0.00018 (New Zealand), 0.00029 (New Guinea), and 0.00025 (Turkey). Up to 1954 the attenuation was mainly related to crustal waves. I n that year, Ewing and Press, using records supplied by long-period seismographs, gave the first values of k, for Rayleigh waves propagating through the mantle [1561: k, = 0.000036 for T = 140' ko=0.000022

for T = 2 W

From 1954 on it has been the preference to use the well-known formula, proposed by Knopoff, 1/Q = k,vT/n where Q is a dimensionless constant, v the phase velocity, and T the period. Brune El571 thought it better to use the group velocity than the phase velocity v in this formula. Further information is given by Knopoff et al. [158] on the eubject Especially after the work of Anderson and Archambeau [159], Anderson and Kovach [l60], Knopoff [161], and Kovach and Anderson 11621, the dimensionless quality factor Q has become a new important source of information about composition, state, pressure, and temperature of the earth's interior.

.

5.2. Attenuation and Internal Friction Many theories have been developed to explain k, and Q observed values; but so far none has proved valid. There is no doubt that attenuation is connected with internal friction, although it is likely to depend also on other mechanisms such as scattering. Ragarding the attenuation of Rayleigh waves in the earth's crust, perhaps the theory that, among others, best fits the observation was pointed out by Caloi in 1948 [41a,163,164].In a study of Rayleigh wave propagation through an elastic medium with internal friction Caloi found, among other things, a relation among the absorption coefficient k,, the rigidity p, the internal molecular friction p', and the wave period. If p / p ' = 5 0 sec-l (average observed value), k, values would be as shown in Table VIlI and Fig. 47. The A, values more often observed by many scientists are related to periods of

177

ON THE UPPER MAN'TLE

about 20 sec, and as it is well known that they are 0.0003/km on the average. Indeed, this is the value given by the theory in the case of Rayleigh wave propagation through a solid elastic medium. Moreover, the theory fits the observations very well also in the case of different period Rayleigh waves. 1.99

0.99

0.59

0.59

0.29 0.19

0.09

0.05

a03 aoz 0.01

0.00

0

5

10

15

20

23

3OXC

FIQ.47. Attenuation coefficient ko for Rayleigh waves in a solid elastic medium [41a].

Besides the above-reported examples, Press [166, p. 44181, in very recent work, pointed out that observation shows an absorption coefficient of about 0.00014 for the case of about 40 sec periods; this theory is near the theoretic value of O.ooOo8 calculated by Caloi for a medium with internal friction. It seems very probable, therefore, that the crustal internal friction is adequate to explain Rayleigh wave attenuation.

178

P. CALOI

TABLE VIII. Attenuation in a solid elastic rncdium. p/p' = 60 ~ c - 1 . T(sec)

ko

0 0.05 0.1 0.6 1 2

6 10 20 40 60 120

... 03

12.176 6.4018 0.4626 0.11910 0.03260 0.00741 0.001203 0.000301 0.0000762 0.00003 17 0.0000083

...

0.0

This is not unlikely to occur also in the upper mantle, but in this cam calculations must be done again, as the p/p' ratio is different from that given above.

5.3. Attenuation and Mixed Zones

It is well known that ko is not constant; on the contrary, it varies from place to place, and even in the same place, according t o the combinations we use t o determine it. Changes in the actual value of ko can be caused for many reasons. Among these, the effect of surface wave reflections on vertical or almost vertical discontinuity surfaces may be notable. This occurrence may yield for the absorption factor an apparent value much greater than real; it acts mainly in the case of short wavelength, and, therefore, within the earth's crust. The longer the wavelength, the less its effect on k,. Therefore, a very disturbing effect is expected to occur in the mixed zones. An example is the south Tyrrhenian zone, which is very anomalous (gravity, magnetic, seismic, volcanic anomalies), Fig. 48. This zone is characterized by active volcanism and a great number of intermediate and deep earthquakes, which, as is well known, very often occur in such zones. In the geologic past, this mixed zone was probably much larger, ranging from north-central Italy to Malta, Lampcdusa, and the Pantelleria Islands: This is proved by the great numbcr of extinct volcanoes

ON THE UPPER MANTLE

179

X X

FIG.48. Mantle-crust mix of southern Tirreno (Tyrrhenian Sea): (-) axis of negative gravity anomalies, ( + ) axis of positive gravity anomalies, ( x ) normal earthquakes, deep earthquakes. (a)Situation of some deep focus earthquakes (after Di and (A) Filippo and Peronaci [201); (b) some epicenters of shallow earthquakes and deep earthquakes; ( c ) advances in travel times for P waves of deep earthquakes at Messina (after Girlanda [ 1431).

180

P. CALOI

situated along the above-mentioned boundaries (Fig. 48). At the moment the zone is smaller, and its boundaries are Vesuvius toward the east and Etna toward the south: The active central zone is Stromboli, an uninterruptedly active volcano. Indeed, intermediate earthquakes down to 300 km focal depth are very often recorded in the Lipari Islands zone. Several faults branch out from the center toward the Italian Peninsula and Sicily: One of these faults seems to cross the Strait of Mcssina, leaving on opposite sides Reggio Calabria and Messina. Such a fault seems to be an actual deep fissure of the earth’s crust. Indeed the two seismic stations of Reggio Calabria and Messina-about 3 4 km apart from each other-not only record microseisms of the same period and very different amplitude (much smaller a t Messina than a t Reggio Calabria), but also the surface waves, with periods up to 50 sec, are remarkably more developed a t Reggio Calabria than a t Messina. The crossing of such a vertical discontinuity, therefore, causes a very great dissipation of energy as the wave is traveling westward. Let us consider, for instance, CiSjwave records obtained a t Reggio Calabria and Messina on November 4, 1952 (Kamchatka earthquake) [92]. It is known that, in the case of surface waves of periods unlike the period a t which the group velocity is minimum, we have

where A,, A,, are the epicentral distances of two seismic stations, al, a, the amplitudes of T,, T2period Ci,jwaves observed in the two stations; 6 = A/r,) and k, is the absorption coefficient between the two stations. I n our case it is A1 2: A2 (therefore, 6, N a,), T, 2~ T,; hence,

(5.2)

a2

- = exp[k,(A2 - A1)/2 a1

I

k, = 4.610gl,,- (A, a2

- A,)

Since it results from the observation that a,/% = 2.5 and A, - A,

N-

+4

km, it follows that

k, = 0.45763 As the waves under consideration (ClS2)have a period of about 37 sec, a

k, value of about 0.0003 should correspond to them. The obtained value proves the great absorption effect of the medium lying between Reggio Calabria and Messina. Of course the result also affects the dissipation function l/&.When once supposed that k, = 0.0003 at Reggio Calabria, and since = 7 ktnlsec, T,= 37 sec, it follows that l/&= 0.02493

181

ON THE UPPER MANTLE

Using the same value of v and

T,and ko=0.46763, we have a t Messina

I/& = 37.747 The dissipation function a t Messina is about 1.5 x lo3 greater than at Reggio Calabria. This very great dissipation is to be attributed to the fault of the Strait of Mcssina, which affects the whole thickness of the earth’s crust; indeed, thc different position of the two stations is not enough to explain such a difference. This large subvertical fault that brings into contact rocks of different geologic eras puts the Messina and Reggio Calabria stations ina very interesting position for recording seismic waves. Since such discontinuities, although not so sharp, must often occur also in more homogeneous zones, a consequence of their occurrence is a more or less remarkable increase in the absorption coefficient observed values, as to surfacc waves especially. This is a further confirmation of the author’s theory of propagation through solid elastic media, in which the dissipation is also well explained. The observed ko values, indeed, are to be regarded as highest limits: In reality, they would be much lower in homogeneous media, as demonstrated by theory. In any case, the problem of the attenuation of seismic waves (Rayleigh and Love surface waves, longitudinal and body waves) is strictly connected to the nature of the medium and to the occurrence of horizontal and vertical discontinuities, particularly sharp in mixed zones [1661. Therefore, if we want to solve it, it is necessary to study lateral inhomogeneities with such methods as those suggested by BLth [119].

6. FREEOSCILLATIONS

OF THE

EARTH AND ITS OUTER

SHELL

6.1. Free Oscillations of the Earth During the last decades, the study of the oscillations of the earth, regarded as a vibrating sphere, has become a favorite subject. The interest in the earth’s free oscillations was aroused by the first records of ultra-long-period waves on clinometers, seismographs, and gravimeters caused by very strong shocks from 1950 on. Therefore, this topic is connected with the progress achieved in the construction of instruments capable of recording ultra-longperiod phenomena, which some time ago were completely outside their recording capacity. Along with the progress in the construction of instruments, the electronic calculating machines have made much progress; these machines permit the study of most complex oscillating curves, which some time ago were only partly analyzed by very difficult hand calculations. The mathematical solutions of the problem had already been started during the last century, especially by Lamb [167,168]; it was more recently studied by Love [169] at the beginning of this century.

182

P. CALOI

One of the first applications of such theories, using seismological data, was made by Oddone [170] in a study in 1912. In the case of a solid earth, he obtained a value of 85 min for the whole period of spherical oscillation; such a value is not very far from the value of 94 min calculated by Lord Kelvin (1863) for a fluid sphere of honiogcneous density equal t o the earth’s average density and with the same dimensions; this value was later confirmed by Lamb . This is not the place to report the results of many calculations and new theories pointed out in the last years, especially by Jobert, Pekeris, Takeuchi, etc. Nor is it appropriate to include the theoretical-experimentalresearches of Ewing, Benioff, Alsop, etc., either because we deal only with the outer part of the earth (i.e.,the upper mantle).

6.2. Divtributiou of Earthquake Energy However, we may well inquire if long-period free oscillation may occur in the crust and in the upper mantle. On this subject, it should be noted that the propagation of energy toward the earth’s interior is very different according to the location of the focus (outer upper mantle, 0 to 100 km deep, or deeper than 100 km). The stratification of the earth’s crust and the occurrence of discontinuity surfaces cause the dissipation of a great part of the energy originated by a shallow earthquake by multiple refractions and reflections within the crust; in any case, the energy is bound to scatter into the crustal features. Furthermore, the difference in the records of the two types of earthquakes is very remarkable: Shallow earthquakes give longitudinal and transverse body waves of small amplitude and large surface waves; deeper earthquakes, on the contrary, give body waves of remarkable amplitude, while surface waves are small or completely lacking. There is another argument. In spite of some opposing views, the “channcling” of seismic waves is almost sure a t the present time: It certainly occurs in the asthenosphere and is very likely to occur also in crustal layers. Therefore, a great percentage of the energy of earthquakes originating between the asthenosphere and the external surface of the earth is channeled by the asthenosphere and the crust, and scattered into the outer layers of the earth. 6.3. Distribution of Seismic Energy between Body Waves and Surface Waves

BBth has reached important result8 in a study of the seismic energy distribution between body and surface waves [171], some of these are as follows: 1 . If E is the total energy caused by an earthquake, and EL, the energy of Rayleigh waves, the ratio EIE,, decreases with increasing magnitude according to the forniula:

ON THE UPPER MANTLE

(6.1)

183

log(E/E,R) = 5.34 - 0.56 M,

where M , is calculated from the surface wave amplitudes of distant earthquakes a t the normal depths. 2. The extinction of energy is very remarkable in the case of body waves and affects the total energy by about a factor of 20. The extinction of body waves is not uniform over all the path as in the case of surface waves: It probably is very small after a distance of a few degrees from the source. It increases with decreasing wavelengths, and is greater for transverse than for longitudinal waves. The heterogeneous crust and the irregularities are so important as to cause a remarkable dispersion of the high frequencies. In conclusion, the extinction is mainly due t o scattering within the crust. 3. The extinction is very slight in the mantle, only 10-15 yo out of crustal extinction in the case of normal-depth earthquake body waves over all the mantle path.

6.4. Possibility of Free Oscillations of the Outer Shell of the Earth I n the case of violent earthquakes originating in the earth’s crust or immediately below it as, for instance, the Kamchatka earthquake of November 4, 1952 or the Chilean earthquake of May 22, 1960, the crust is actually torn from the focus toward the surface; therefore, a great part of energy penetrates into the crustal layers. Moreover, the crust is very viscous; therefore, slow free oscillations are very likely to occur in it and in the asthenosphere, as in a spherical shell. The observed “weakness” of the zone 60 to 200 km deep (i.e., the asthenosphere), in which the internal strains are probably disappearing, confirms the independence of such outer oscillations from the earth as a whole. Moreover, they are micro-movements; therefore, such a “weak” transition zone (Gutenberg’s “decoupling zone”) may certainly facilitate the origin and the permanence of slow oscillations in the outer part of the earth. On this matter, Press [172] observed that this zone may be placed where the mantle is effectively decoupled from the crust for tectonic processes and differential movements between crust and mantle. At any rate, the tectonic implications of a zone of reduced strength a t this depth are many. Gutenberg [173, p. 3101, on the same matter, thought that the conditions “at depths of the order of 100 to 200 km are especially favourable to the development of subcrustal currents and that creep and other types of $ow may encounter there a minimum resistance.”

6.5. Periods of Outer Shell of the Earth I n 1962 the author tried to find, in first approximation, the fundamental periods of the earth’s outer part, regarded as a spherical shell [174]. He used

184

P. CALOI

Lamb's theory [168]of the vibrations of a spherical shell of very small thickness in comparison with the radius. Lamb considered two types of vibrations. The first type is that vibration characterized by a completely tangential movement in every point of the shell. The frequency (p/277)is given by the following equation:

k2r?

(6.2)

=

( n - l)(n

+ 2)

where ro is the shell radius, k2 =p2plp, p the density, and p the rigidity of the medium. Therefore, the period is expressed by the following equation:

where v2 is the transverse wave velocity, and the other symbols are obvious. I n the second type of vibration, the movements are partly radial and partly tangential. With regard to the frequency, two values of k2r: correspond to every value of n, as shown by the equation

+ +4) y + n2 + n - 2 ) + 4(n2+ n - 2)y =0

(6.4) k4rr,"- k2r2{(n2 n

+

where y = (1 o)(l - u), u = the Poisson ratio. The two solutions of the equation are one greater and the other smaller than 4y,and the corresponding fundamental modes are completely different. Always the most noteworthy is the mode corresponding t o the smaller solution. If n = 1, k2r? values are 0 and 6.The first solution corresponds to a translatory movement of the shell as a whole parallel to the axis of the zonal harmonic S,. In the other mode, the movement is proportional t o cos 8,where 9 is the colatitude measured from the pole of S,; the tangential motion occurs along the meridian. If n = 2, u = 4, the kr, corresponding values are 1.176and 4.391,respectively, The surface harmonic S, is particularly interesting. The polar diameter of the shell lengthens and shrinks alternately, while a t the same tinie thc equator shrinks and lengthens. In the mode corresponding to the smaller solution, the tangcntial Inotion is toward the poles when the polar diameter lengthens and vice versa. Thc opposite case occurs in the othcr mode. Hence, the reason for the great difference in frequencies. Let us neglect the modes of Lamb'sJirst type, consisting of fully tangential oscillations [if n = 2, and average ro and v2 values, we obtain T = 85 min (approximately)]. With rcgard to the mode of the second type, if a = $, we have y = s a n d equation (6.4)becomes

(6.5) k'r: - k2:r {(n 2 n

+ + 4)1.667+ n2 + n - 2) + 4(n2+ n - 2)1.667= 0

185

ON THE UPPER MANTLE

This equation has been solved in k2r: for n = 2, 3,,4,5, 6, 7, 8, 9, 10, 11,12, 13, 14, 15, 16, 20, 100, 200, . . ., 600. As reported above, the modes corresponding to the smaller solutions are particularly interesting; as shown in Table I X the smallest values of kr, TABLE IX. Roots of equation (6.6) and corresponding free periods.

kro'

n

2 3 4 5 6

7 8 9 10 11 12 13 14 15 16 20 100 200 300 400 500 600

1.176 1.384 1.465 1.504 1.527 1.541 1.550 1.556 1.561 1.564 1.570 1.570 1.570 1.572 1.573 1.576 1.580 1.581 1.581 1.581 1.581 1.581

4.390 6.894 7.473 9.077 10.693 12.314 13.938 15.564 17.391 18.900 20.473 22.080 23.7 11 25.342 26.973 33.450 164.120 327.417 490.716 654.015 817.314 980.613

T

T'

(min)

(min)

131.05 111.4 105.1 102.4 100.9 100.0 99.4 99.0 98.7 98.5 98.1 98.1 98.1 98.0 98.0 97.6 97.5 97.4 97.4 97.4 97.4 97.4

35.1 26.1 20.7 17.1 14.4 12.6 11 9.9 8.8 8.1 7.5 7.0 6.5 6.0 6.7 4.5 0.9 0.4 0.3 0.1 0.1 0.1

tend asymptotically t o 1.581, corresponding t o increasing values of n. I n the application, we may regard the outer shell of the earth as 40 km thick, on the average, with a radius of about 6330 km. If we consider the value v2 = 4.3 km/sec as the average velocity of transverse waves, from the two series of kr, values of n = 2 to 600, we obtain the periods given in column two and three of Table I X and in Fig. 49.

6.6. Observation of Periods of Same Order as Those of Section 6.5 No doubt the values reported in the last two columns of Table I X are to be regarded as approximate values. Of course, the medium below the considered shell, even if i t has almost no strength, affects the free oscillations of the shell. Therefore, the theory must be perfected.

186

P. CALOI

In any case, it is significant that oscillations of about 131 and 110 min have already been observed. Alsop et al. [176], in a work on the free oscillations of the earth observed through the “strain” and pendulum seismograph, made a periodic analysis of the Chilean earthquake records, and noted that: “Additional peaks with periods greater than 60 minutes occur in the strain

80 ’

60 ,

o

l 0

.

: 2

,

. 4

:

: 6

.

.

: 8

.

:

10

:

12

.

14

16

n

FIG.49. Periods of the Lamb second class vibrations for a spherical shell (crust) of the earth. Dashed line represents the free spheroidal vibrations of the earth (after Alsop [1751)*

record spectrum a t 131.5, 109.8, 77.0, and 65.5 minutes, with relative amplitudes of 0.7, 0.3, 0.4, and 0.4. These have not yet been definitely identified.” It is important that such periods can be explained only if we consider the crust and part of the asthenosphere as a vibrating spherical shell. The values of 131m.5,and 109m.5observed by Alsop et al. are, indeed, the fundamental and the further oscillations of the second type (corresponding t o the smaller solutions) of the considered spherical shell (these two oscillations, as shown in Table IX, have values of 131”’.05, and 111m.4).Obviously the highest frequency oscillations have not yet been recorded; indeed, amplitudes of 0.7 and 0.3, i.e., quickly decreasing, correspond to the first two oscillations. Moreover, free oscillations with the above reported periods (about 131 and 110 min), caused in the earth by the Chilean earthquake of May 22, 1960, have also been recorded in the spectrum of gravity records obtained on the

187

ON THE UPPER MANTLE

occasion of that very violent shock. It is enough to mention the records obtained by “Askania” instruments in Kyoto [177] and Brussels which are particularly clear in Fig. 50. Moreover, they clearly occur also in the spectruni of seismograms supplied by the ultra-long-periodinstrument placed at

Period in minutes 200 100

I 0

0

50 40

~Fequoncyin c y c b p a r minute 0.01

0.02

0.04

0.03 (a)

0.05

Periodin minute.s 200 100

50 40

20

30

-1

0

0.01

0.03

0.02

0.04

0.05

(b) FIQ.50. Pwiods of 130m order are present at ( a ) Kyoto and ( b ) Brussels in gravimetrical records of Chilean earthquakes of May 22, 1960 (after Nakagawa et al. [177]).

188

P. CALOI

Berkeley on the occasion of the Alaskan earthquake of March 28, 1964 [178, pp. 514861501. Also, the periods corresponding to the greatest solutions of equation (6.5), reported in the last column of Table IX are rather interesting in comparison with those calculated for the free spheroidal vibration of the earth as a whole. Let us consider, for instance, the values calculated by Alsop [175] for the Gutenberg-Bullen model B (almost equal to those obtained for other models). These are calculated in minutes, and, in the case of the fundamental mode, for the various numbers of order, are given in Table X. TABLEX . Free spheroidal periods calculated by Alsop. Order No.

Periods (in min)

Order No.

Periods (in min)

2 3 4

53.8 35.6 25.8 19.9 16.2 13.7 12.0 10.9 9.9

11 12 13 14 15 16 17 18 19 20

9.2 8.6 8.1 7.7 7.3 7.0 6.7 6.4 6.1 5.9

5 6 7

8 9 10

Table XI compares the values of the periods for the whole earth with those for the outer shell, with numbers of order reduced of one unit. In this comparison, only a slight difference in the values is noticeable (even if those of the outer shell are necessarily approximate). In the case of a contemporaneous occurrence of oscillations of the whole earth and its outer shell, this fact may explain the “doublets” of observed values that differ slightly from each other, a t least for the longest periods. For instance, among the records of the La Coste-Romberg gravimeter in Los Angeles, caused by the Chilean earthquake of May 22, 1960, Ness et al. [179] found the doublet of periods 35.3413537for the spheroidal mode S,.4 Pekeris et al. [ 1801 explained these doublets by thcorctical considerations on the type of reaction of the earth to the violent shocks: The doublets in the spectrum of the earth’s free oscillations were interpreted as multiplets arising from the rotation of the earth (phenomenon similar to the Zeeman effect). Another An analogous comparison may be made of the values observed with a gravimeter in Brussels at the t.imc of tho samc oarthquako where doublets of the order of 2 5 m . 8 9 - 2 ~ 0 7 , 25m.88-25m.09 (minutes) . were fouiicl for oS4.

..

189

ON THE UPPER MANTLE

explanation as reliable as the preceding one could be that the uhole earth and its outer part are simultaneously excited, and that each reacts with its own period. In the author’s opinion, the theoretical-experimental study of the oscillations of the earth’s outer shell will supply several important contribut,ions to the knowledge of physical features of crust and asthenosphere. TABLEXI. Comparison of values for earth and outrr shrll. Order No.

3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

Whole earth periods (in min) 35.6 25.8 19.9 16.2 13.7 12.0 10.8 9.9 9.2 8.6 8.1 7.7 7.3 7.0 6.7

Order NO.

Outer shell periods (in min)

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

7. EARTH’S INTERNAL MOVEMENTSAND

VOLTERRA’S

35.1 26.1 20.7 17.1 14.4 12.6 11.0 9.9 8.8 8.1 7.5 7.0 6.5 6.0 5.7

THEORY

7 . 1 . Polar Movements, Continental Shifts and Subcrustal Currents

It is well known that no agreement exists on the hypothesis of continental drift, Chandler period variations, etc. During the last few years, the important problem of the earth’s rotation stability has been reexamined by using new physical observations (i.e., thermal, magnetic, radioactive, gravity observations) and new hypotheses on the earth’s internal movements. This problem clearly deals with the physical features of the upper mantle, by which it is affected. Therefore, it should be mentioned in the present review. There appears no doubt now that the continents show not only movements relative t o each other but also relative to the poles of rotation. This

190

P. CALOI

does not mean acceptance of Wegener’s hypothesis, which presents some questionable topics, in its mechanism a t least. The main question now is “whether these movements are solely the result of a shift of the individual crustal blocks relative to each other, in addition to a shift of the whole continental crust relative t o the Earth’s axis, or whether shifts of the Earth’s axis relative to the Earth’s interior play a role, too.” (Gutenberg [173].) Regarding this second alternative, the hypothesis of the earth’s internal movements, especially convection currents, is particularly important. If the thermal gradient is in excess in comparison with the adiabatic gradient throughout the earth’s mantle, this becomes gravitationally unstable and convection currents may result. The supporters of this hypothesis do not agree upon the extent of this phenomenon: It might affect the outer shell only, or the whole mantle, or it might consist of intermittent overturns of thermally unstable regions. Vening Meinesz, Heiskanen, Griggs, and others have studied this problem many times. Fekeris and Hales have studied it from a mathematical point of view, examining the possibility of convection currents as a geodynamic process. The main uncertainty in the convection hypothesis is the unknown composition of the earth’s mantle. Convection currents in the earth postulate homogeneity of the material in which such currents are supposed to occur. On this subject, Birch [4,181] demonstrates that the ratio between the bulk modulus (the reciprocal of compressibility) and the density a t about 200 to 900 km in depth is much greater than in any known material: In this connection, such a transition zone forms a big problem, whether it presents changes in phase only, or changes in chemical composition only, or in both (see Section 1). Although the region from 900 km down to the mantle basement seems to be the most favorable to convection currents (although great difficulties are present in this region as well), the opinions on the possibility of convection current occurrence at 200 to 900 km in depth are very different a t the present time. Birch maintained that convection currents are very unlikely to occur in this zone, according t o changes in phase or chemical composition between 200 and 900 km [182]. A careful study on this subject has recently been published by Knopoff [183]. We use to designate as Rayleigh number the following dimensionless quantity (7.1)

where

CY

is the thermal expansion coefficient, g the acceleration of gravity,

/l the vertical gradient of temperature a t the onsct of convection, d the layer

thickness, x the thermal diffusion, and 6 the kinematic viscosity. Knopoff expressed the solution of the linear stability problem in terms of the Rayleigh number. He assigned proper values to the physical parameters of the earth’s

191

ON THE UPPER MANTLE

mantle and found that the Rayleigh numbers range from loGt o lo8,according t o the viscosity. Therefore, they are much greater than the value necessary for marginal instability of a mantle-wide convection; a t any rate, the inhomogeneity of Bullen’s region C is strong enough t o prevent mantle-wide convection from occurring, whether changes in phase or changes in composition take place. Then Knopoff considered the problem on a small scale. He found that the Rayleigh number ranges from lo2 to lo4 in the upper mantle down to 400 km in depth, i.e., in an interval that includes the condition of marginal instability. Therefore, “if the upper mantle can be considered homogeneous on a scale of 1200 km in lateral extent and 400 km in depth, and if the viscosity is high and the strength is low, and the superadiabatic temperature gradients are as high as lo-‘ deg/cm locally, then local convection can take place where the Rayleigh number is of the order of lo4. It is not expected that this will occur uniformly over the earth’s surface.” ([183], p. 109.) In conclusion, convection in the upper mantle may occur (Fig. 51).

(a)

5 U R FAC E

I I I

I

I I 1

(b)

‘ORE

FIQ.51. Schematic configuration for laminar flow wit,h induced convection in (a)one layer and ( b ) in the upper mantle only (after Knopoff [183]).

Also, Shimazu and Kohno [184] have recently (1964) studied the problem of the convection current occurrence in the mantle, especially with regard to the thermal convection in relation with tectonic processes. It is well known that recent measurements of the heat flow have shown that the upward heat

192

P. CALOI

transfer through the MohoroviEii: discontinuity is greater in oceanic regions than in continental ones, and that a n abnormally high value of heat flow occurs in some areas of mid-oceanic ridges. The transfer of energy from the mantle to the crust may occur as mass transfer (enthalpy), or convection, or differentiation (magma movement): I n the first case, the action of horizontal stresses in tectonic process prevails, while in the second case, vertical differential movements prevail. I n both cases, geological evidence suggests that the tectogenesis is activated in boundary zones between continents and oceans, or in mid-platform basins. On this subject, the two Japanese geophysicists expounded the hypothesis that a differencein physical state between oceanic and continental crust serves as a trigger to generate activation in the upper mantle. The thermal convection is generated by an undulation of the isotherm a t the continent-ocean boundary. It has been demonstrated that differentiation in the upper mantle, due to origin and upward motion of magmas, is thermodynamically equal to convection. I n any case, the mantle convection appears t o be one of the possible mechanisms of generating tectogenesis .

7.2. Volterra’s Theory of the Earth’s Internal Movements 7.2.1.Movement of a System i n Which Stationary Interim1 Movements Occur. Many theories have been proposed to explain the observed changes in geographic latitudes, considering the effects that geological factors, elasticity, and plasticity may have on the earth’s rotation. In 1895, the famous Italian mathematician Vito Volterra studied the problem from another point of view, and examined other causes that may affect the earth’s rotation. Since Volterra’s theory has never been considered (apart from a paper of primarily mathematical character in Astronomische Nachrichten [ 185]), it is reported upon briefly herein, because of the remarkable interest its extension to the earth’s internal movements (convection currents, etc.) may have; 6, ’1, 5 are the main central axes of inertia of a body in which-or at its surface, under the action of internal forces-stationary movements of a part of its material occur, without any change in shape and density. Supposing that the body is homogeneous, let us imagine that under the action of internal forces an inside torus of revolution PQ (Fig. 52) has a uniform movement of rotation around its axis V V ‘ , while the body preserves its rigidity. The center of gravity, the axes of inertia, and the main moments of inertia A , B, C of the body do not change. Following this, given that M , , M , , M , are the components of the couple of momentum due to the stationary movements (whatever they might be) on the axes (,q, 5, that the system has a motion around its center of gravity 0, and p , q, r are the components of the angular velocity of rotation on the

ON THE UPPER MANTLE

193

FIG.52.

axes directions, Volterra [186] reached the subsequent differential equations of the movements of the system considered free round its center of gravity

A dP - + (C - B)qr + M , q - M 2 r = O at (7.2)

Bdq at

+ ( A - C)rp +MI r - M s p = 0

Such equations admit the two integrals

(7.3) where h and k are constant. Furthermore, Volterra obtained [187] the relations in finite terms that connect p , q, r, and t with one another, and improved his theoretical analysis, demonstrating that the nine cosines of direction that a tern of mobile axes g , ~ 5, form with a tern of fixed axes x, y , z are uniform functions of time and that they have polar singularities [188].

7.2.2. Nonstationury Movem'ents. Before continuing the exposition of Volterra's theory, let us consider the following. If a t the instant of beginning the body turns around only one of its main axes of inertia (for instance, then we have

c),

p=q=O,

r$O

194

P. CALOI

and equations (7.2) become (7.4)

dP A -= M , r ; dt

dq B-= dt

dr

c-=o at

- M 1 r ,.

This proves that the derivatives of p and q are not 0 if M , and M , are different from 0, and therefore the main axis of inertia 6 is not a permanent axis of rotation, but is variable. As Volterra wondered at his time, stationary movements certainly occur in the earth’s interior and its surface; these movements do not markedly alter the moments of inertia and the baricenter of the earth, but they may cause values of M , and M , different from zero, and therefore thcy may change the earth’s axis of rotation. With regard to the surface movements, it is enough to mention the large oceanic currents; i.e., the Gulf Stream, North Atlantic, Canary Island, and Brazil, and similar currents in the Pacific, Indian, Arctic, and Antarctic oceans. Such currents, in a certain way, could be regarded as stationary movements of the above-mentioned type. On the other hand, convection movements that occurred in the past, and according to many authors still occur inside the mantle, could be considered much more similar to Volterra’s stationary movements than oceanic currents. An extension of Volterra’s theory to the mantle convection movements is very desirable. The earth’s internal movements, indeed, can be considered only approximately stationary. Although the material distribution is not altered, we cau admit that the earth’s internal movements can sometimes accelerate, sometimes slow down, or that some of these accelerate while others slow down [ 1891. These possibilities are likely to have occurred during the history of the earth as a planet. I n this case, the formulas are to be modified since M I , M,, M , are not constant any more, but variable in time; the position of the axes of inertia in the body’s interior and the main moments of inertia A, B, C, on the contrary, are always constant. In such a hypothesis, equations (7.2) are t o be replaced by the following ones

dP dM A - + (C - B)qr + M , q - M 2 r +L =0 at dt (7.5)

B - + ( A - C)TP+ MI T 0%

at

-M

+ dM2 = 0

~ P

1

195

ON THE UPPER MANTLE

These equations admit the integral (7.6)

+ (Bq + M,12 + (Cr + M,)’

(AP

= k2

On the premise that M , , M,, M , are constant, Volterra indicated the way to study the movement and the various paths (polodies) that the pole of rotation may describe on the ellipsoid of inertia of the body in motion,

7.2.3. Polar Movements Caused by Internal Movements. In connection with Volterra’s theory, the opposite problem, i.e., the determination of internal movements that may cause a given movement of the pole of rotation, is also interesting. Applying his theory to the earth’s actual movements, Volterra [190] assumed that two of the main axes of inertia of the mobile system are equal to each other ( B = A ) , and that the internal movements are not stationary. Thus, the equations of motion would be AdP + ( C - A)qr + M 3 q - M 2 r dt A d9 -

(7.7)

at

+ ( A - C)rp + M , r dr

-

M3p

+-dM1 = O dt

+ dM2 -= 0 at

+ M2P - M , q + dM3 dt = 0

If p , q, are very small, and r variations are also very small, Volterra supposed that r = w E , where w is constant and E is not greater than p and q. From the last equation in (7.7), we have

+

M,

M,O -

1

J: +I*

( M z p - M l 9) at +CE = M,’

+U

where M: is a constant. Therefore, the two first equations become (w + E )

(7.81

A

q=

--if2

-M2(w

On the premise that the terms uq

UP

A’

A’

C-A M ~ EM ~ EC - A A Eq, A EP A ’ A ’

+E))

=OL

196

P. CALOI

may be neglected, we obtain

(7.91

Volterra’s theoretical analytical developments are omitted here. It is enough to recall that starting from equation (7.9) with

C-A M: A u + - =Ap

(7.10)

he showed that the movements of the pole are decomposable in a series of harmonic functions of time. He also managed to express the elements relative to the pole of rotation motion, and those relative to the internal movements that m y Cause the pole shift, by the coefficients of (7.9). I n addition, he obtained the relations between periods and relative constants and reached the following important conclusion: Internal and polar movements have the same periods; apart from two, each of them is peculiar to one of the two motions and i s such that they are unique to each. Exactly, the internal motions may have the period 2n/w,which cannot govern the pole of rotation motion, while the pole of rotation movement may have the period 2nlp, which cannot govern the internal m ~ v e m e n t . ~ We may wonder which are the periods 2nlw and 2nlp in the case of the earth’s motion. Since w represents the earth’s velocity of rotation, 2n/w expresses a sidereal day. From the above-mentioned theorem, it follows that internal movements may have a diurnal period, but this period has no effect on the polar movement that cannot have the same period. It is well known that the ratio (C - A ) / A , calculated from the earth’s precession and nutation, is 11305. Then we obtain from equation (7.10) w

M o

p = - + L 305 A

(7.11)

From which 2n 27r -=-.

p

6

w

305 1 +(305M,O/Aw)

Further information may be found in Volterra [loo, pp. 817-8201,

ON THE UPPER MANTLE

but A

(7.12)

= 305 Cl306;

197

therefore, taking the sidereal day as a unit,

305

2n

-=

p

1

+ 306 (M,O/Cu)

Consequently, we reach this interesting conclusion: 2 r l p is the Eulerian period changed in the ratio 1/(1 306 M,'/Cu). Therefore, if we regarded 2 ~ l p as the Chandler period,6 supposedly constant (430 days) on the average, we would reach the following relation:

+

(7.13) Relation (7.12), therefore, could coincide with the Chandler period if the component of the couple of momentum caused by the internal movements in the direction of the earth's axis were 111053 of that possessed by t h e earth, regarded as a rigid body, for its diurnal movement. Apart from this result, if the component of the couple of momentum of internal movements in the direction of the earth's axis were negligible in comparison with the couple of momentum of the rigid earth, then 2nlp would approximately be equal to the Eulerian period. Therefore, in the preceding hypothesis, the internal movements could not have an appreciable periodical part with a period equal to the Eulerian one. Volterra, moreover, examined which internal movements, under the abovementioned conditions, could be able t o cause the harmonic polar motion with the annual period whose elements were determined by Chandler. On the basis of the elements relative t o the harmonic polar motion with the annual period, reported by Chandler in the Astronomical Journal (1894, No. 329) and leaving out of consideration the M,' ratio, Volterra reached the conclusion that the axis of the couple of momentum of the internal movements able to cause the above-mentioned polar motion must oscillate i n such a way that the projection of its extreme on the equatorial plane describes an ellipse, whose axes he has determined, the major of them being inclined of 45" on Greenwich meridian. i.e., it is in the meridian plane which has 45" W longitude, passing, therefore, i n the middle of the Atlantic Ocean. Volterra, in his very valuable studies, reached these and other conclusions; the above conclusion appears to be particularly interesting if compared with the present knowledge of the Mid-Atlantic Ridge and its probable origin. 7.2.4. Case of Internal Stationary Movements. Volterra also studied the case in which the internal movements are stationary. Then we have the following 6

The Chandler period varies from about 414 t o 440 days [191].

198

P. CALOI

formulas [1921

p

a, + C, cos pt - Czsin pt

= --

P

(7.14) a0

q = - +C1sin pt +C,cos pt

P

+

where C = C , iCz, uo, 3 /, are constants, and p is expressed by equation (7.10) If we put c1 = Cl/w, c2 = C2/w, it follows that

P =MI +c,cos pt -

- czsin pt

AP

(7.15)

q

- = -M2

w

AP

+ c1sin pt + c2 cos pt North

fl'

south FIQ.68.

ON THE UPPER MANTLE

199

Let us suppose that M , is negative, i.e., the internal movements have the component of the couple of niomentum in the direction of the earth's axis, and with sign opposite to the earth's couple of momentum. If we regard a clockwise rotation as positive, and draw from the center of the earth taking as origin two segments O R and OM representing the earth's rotation and the internal motion axis, respectively, and extend them toward the positive direction, the first one will meet the Southern Hemisphere, the second the Northern Hemisphere (Fig. 53). If we now consider, on OR,in the positive direction, a segment OP equal to 1, this will be projected in T on the equatorial plane, and the projections of O r on the axes and 7 will be p / w and q/w respectively. Analogously, in the direction O M , if we regard the segment 0s as equal to OMIAp, the points projected in h on the equatorial plane will be such that the two projections of Oh on the axes 5 and 7 will result equal to M , / A p , M,/Ap. If M and P are the points in which a sphere of unit radius with center in 0 meets the axes OM and OR, respectively, since P is very near the pole of inertia {, we can assume that it approximately describes a circumference whose center is the point H of the sphere, the projection of which on the equatorial plane is h. If H ' , P are the points diametrically opposite t o {, H , P on the considered sphere, the points P', H', {', M will be in the Northern Hemisphere (Fig. 54). We shall give these by the same names a s Volterra: the pole of

t

{I,

Fro. 54.

rotation, center of polar movement, pole of inertia, and center of internal movements, respectively. If h' is the projection of H' on the equatorial plane, we have Oh' = Oh, and, therefore, sin H'C'

= Oh' =

0s sin MI{'

200

P. CALOI

From which (7.16)

sin H'C' -sin M'5'

+

+

,/MI2 MZ2 (C-A)W+MS

-os=-=OM Ap

=&

Supposing that

M = J M , ~+ M Z

we have

M3 = - M

+ M,Z

cos M'l'

From which E=

M (C - A)w - M COB M'5'

P=

(C - A h - M CON M'5' A

(7.17)

I n the case of M 3 positive, let us extend OM in the 0 direction to meet the sphere in the Northern Hemisphere at a point that is still to be regarded as the center of the internal movements denoted by the symbol M'.While in the preceding case the point 5' was in the middle between M' and H', now M' and H' are both on the same side of 5'. If we denote by u and /3 the arcs rM, ('HI, respectively, and if we count these starting from equations (7.16) and (7.17) can be written as follows: [ I ,

(7.18)

sin --/3 -F&, sin a

M

&=

, (C -A)uT M COB tc

p=

(C- A ) u T M cos u A

where we must take the upper sign in the first case; and the lower sign in the second case. Disregarding the earth's plasticity and regarding the internal movements as stationary, Volterra summarized the polar movement laws as follows: 1. The center of the internal movements, the pole of inertia, and the center of the polar movements belong to the same maximum circle of the sphere. 2. If we call u and the distances on the sphere from the pole of inertia and the two centers of internal and polar movements, respectively, we have (7.19)

sin /3 -sin u

-?&=

3=M (C - A ) w F M COB

3. The pole of rotation describes a circumference around the center of the polar movements and has an angular velocity.

ON THE UPPER MANTLE

(7.20)

P=

201

( C - A ) u F Mcosa A

The influcnce of the internal movements on the polar motion is very clearly shown by these laws. If the internal movements did not occur, the pole would describe a circumference around the pole of inertia a t an angular velocity (C - A ) / A . Therefore, the effect of the internal movements is double: ( a ) They alter the center of rotation of the polar motions that is repelled (or attracted) by the center of the internal movements along the maximum circle that joins this point with the pole of inertia. This shift of the center of the polar motions is characterized by the angle B. ( b ) They alter the pole angular velocity of rotation, which varies of the quantity (7 M cos a ) / A . This alteration obviously corresponds to a change i n the Eulerian period. Moreover, it should be remembered that the internal movements make the pole of inertia an impermanent pole of rotation and move this property from point 5‘ to H .

7.2.5. Injuence of Plasticity. Next, Volterra examined the consequences of the earth’s plasticity on the polar movements. He started from the hypothesis that such phenomena occur in a not very long time, such as not t o alter the shape of oceans and continents, while the pole of inertia will shift in comparison with them. The hypothesis that the internal movements are permanent is the same as saying that the center of the internal movements is a fixed point of the sphere ofcenter 0, on which we imagine the earth’s crust is projected, and that the M: is constant. quantity M = JZ1z M ; This being stated, the influence of the plasticity is believed to occur as a constant tendency of the pole of inertia to approach the pole of rotation; the greater the distance between these two points, the greater is such a tendency. In other words, because of the plasticity, the pole of inertia moves, at every moment, along the arc of maximum circle that joins it to the position in which the pole of rotation is at that moment, and the velocity is proportional to the distance between these two points. Volterra called the ratio p between the pole of inertia velocity and such a distance the “coefficient of plasticity.” It is 00 > p > 0. If p = 0, the earth is not plastic a t all; if p = 00, the earth is completely plastic. Because of the internal movements, in a plastic earth, the problem of the earth’s rotation occurs in the following terms: There are four points on the sphere: M‘ (center of the internal movements), 5’ (pole of inertia), H’(center of the pole motions), and P‘ (pole of rotation). These move in accordance wit8hthe following rules:

+

+

202

P . CALOI

1. M' is a fixed point of the sphere. 2 . M', (', H' are on the same maximum circle, and

(7.21)

sin ('HI -sin ('M'

TM (C - A ) w M

-F&;=

COB

('M

3. P' turns around H' at every moment with the angular velocity

(C- A)w F M cos C'M'

(7.22)

P=

A

4. 6' moves at every moment in the ('P' direction with a velocity equal to the product of p by ('P. Then Volterra obtained the equation of the problem. In order to obtain this equation, he no longer regarded the earth as a sphere, but as a plane in stereographic projection. He chose as projection plane the plane tangent to the sphere in M', and as projection center the point M" diametrically opposite to M ' . Volterra's mathematical work will be omitted here. Suffice it to say that if rl, H,, P , are the stereographic projection of (', H', P ,and if x, y are two fixed axes with their origin in M' in the stereographic projection plane, indicating by x, y, xl, yl, x 2 , y2 the coordinates of the points P,, HI, and supposing that

c,,

x = Ce",

y = Ke";

x1 = Clezl,

y1 = KleZ1

(where C , C,, K, K, are constants), Volterra reached the following fourth degree equation in z,

(7.23)

+

+

z4 + 2 p S + (p2 p2)z2F 2 ~ ~ 1 . p2p2e2 ~ ~ 2 =o

and gave its solutions. The two above-mentioned limiting caaes are perhaps more interesting here. If p = 0 (no plasticity), equation (7.23) becomes

(7.24)

z4

+ p2z2 = 0

Two solutions are 0, while the other two are h i p . Therefore, the movement is periodic, with a period of 27rlp, i.e., the Eulerian period varied in the ratio examined in Section 7.2.3. If p =a, we have, dividing (7.23)by p2 and putting l/p= 0,

(7.25)

22

+ p2&2 = 0

This solution has two infinite solutions and the other two are equal to h i p & . The movement is still periodical, with a period of 2 r / p r = 2 n A / M . Following Volterra, we can conclude that, if p = m, the pole of inertia always coincides with the pole of rotation (x=x,, y = y , ) , and that the pole of rotation describes a circumference round the center of the internal movements, with an angular velocity of M I A .

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203

It is clearly understandable that Volterra’s theory extension to the convection movements is likely to be very interesting. The issues reported in Section 7.2.3 are already very important. It does not happen by mere chance that the axis of the couple of momentum of the internal movements, which may cauae the polar movements according to Chandler’s opinion (1894), oscillates in such a way as t o describe-if its extreme is projected on the equatorial plane-an ellipse, whose major axis goes along the middle of the Atlantic Ocean (45”W). On this topic, in more recent studies (1954, 1957) carried out by Melchior [191, p. 2381, which examine the orientation of the major axis during 50 years of observations, it has been found out that this orientation is considerably constant: it ranges from 68”W to 23”E, and is 27’39‘W f.19” on the average. Thus the value Volterra obtained in 1894 fits the average. The annual movement of the pole is believed to be remarkably affected by meteorologic phenomena and mass transfers in the iced Antarctic area. It is still t o be stated which part of such movements is to be attributed to general internal movements, big oceanic currents included.’ ACKNOWLEDGMENT The author expresses his gratitude to the “Consiglio Nazionale dello Ricerche” for gracious cooperation in supplying translations of English text and in drawing several figures. REFERENCES 1. Bullard, E. C., and Griggs, D. E. (1961). The nature of the MohoroviEiO discontinuity. Qeophys. J. 6, 118-123. 2. Bullen, K. E. (1947). “An Introduction to the Theory of Seismology.” Cambridge

Univ. Press, London and New York. 3. Bernal, J. D. (1936). Geophysical discussion. Observatory 59, 268. 4. Birch, F. (1962). Elasticity and constitution of earth’s interior. J . Geophys. Reg. 57, 227-286. 5. Shima, M. (1956). On the variation in bulk modulus density in the mantle. J . Phya. Earth (Tokyo)4, 7-10. 6. Miki, H. (1955). Is the layer C(413-1000 Km) inhomogeneous? J . Phys. Earth (Tokyo), 3, 1-6. 7. Miki, H. (1956). On the earth’s mantle. Mem. CoU. Sci., Univ. Kyoto 27, 363-403. 8. Shimazu, Y. (1955). Chemical structure and physical property of the earth’s mantle inferred from chemical equilibrium condition. J. Earth Sci., Nagoya Univ., 3, 85-90. 7 Incidentally, I want to report. that the mathematician Peano wrote in 1895 a work [ 1931 in which he expounded a theory t o be applied to the polar movement caused by

the Gulf Stream. On the basis of the physical and dynamic data on this current available at that time, Peano found out t.hat “the Gulf Stream imparts to the polar region a velocity of about 3 mm a day, i.e. 1.1 m a year.” He concludes that, because of the atmospheric and sea currents, the polar regions may move a few meters a year in a direction still unknown. Unfortunately, Peano expounded his thoory by his geometric symbols, which are not easily understandable in normal mathematical terms.

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77. Gutenberg, B. (1959). “Physirs of the Earth’s Interior”, 240 pp., Academic Press, New York. 78. Tryggvason, E. (1961). Crustal thickness in Fennoscandia from phase velocities of Rayleigh waves. Ann. Qeofia. (Rome) 14, 267-293. 79. Luosto, U. (1965). Phase velocities of Rayleigh waves in southern Fennoscandia. Qeophysica 9, 167-172. 80. Sezawa, K., and Kanai, K. (1935). Discontinuity in the dispersion curvcs of Rayleigh waves, B i d . Earthquake Rea. Inat., Tokyo Univ. 13, 237-244. 81. Sezawa, K., and Kanai, K. (1935). The Mz seismic waves. Bull. Earthquake Rea. Znat., Tokyo Univ. 13, 471-475. 82. Kanai, K. (1948). On the existence of the Ma waves in actual seismic disturbance. Bull. Earthquake Res. Znst., Tokyo Univ. 26, 57-60. 83. Kanai, K. (1951). On the &-waves. Bull. Earthquake Rea. Znat., Tokyo Univ. 29, 39-48. 84. Kanai, K. (1955). On Sezawa-waves, Part 11. Bull. Earthquake Rea. Znat., Tokyo Univ. 33, 275-381. 85. Oliver, J., and Ewing, M. (1957). Higher modes of continental Rayleigh waves. Bull. Seiamol. SOC.Am. 47, 187-204. 86. Oliver, J., Dorman, J., and Sutton, G. H. (1959). The second shear mode of continontal Rayleigh waves. Bull. Seiamol. SOC.Am. 49, 379-389. 87. Kovarh, R. L., and Anderson, D. L. (1964). Higher mode surface waves and their boaring on the structure of the earth’s mantle. Bull. Seismol. SOC.A m . 54, 161-182. 88. Buchbinder Goetz, G. R. (1963). Crustal structure in Arctic Canada from Rayleigh waves. Tranaacf.Roy. SOC.Canada 1, Series IV, 333-335. 89. Brune, J. N., and Dorman, J. (1963). Seismic waves and earth structure in the Canadian shield. Bull. Seiamol. SOC.Am. 53, 167-210. 90. Somigliana, C. (1917). Sulla propagazione delle onde sismiche. Atti Reale A c d . Naz. Lincei [5a] 26, P a r t I. fasc. 7; Part 11, fasc. 9; 27, Part 111, fasc. 1 (1918). 91. Bruhat, G. (1947). “Optique,” p. 349, Masson, Paris. 92. Caloi, P. (1955), C t , j .Ann. Qeoja. (Rome)8, 293-313. 93. Caloi, P . (1948). Sull’origine delle onde superficiali associate alle onde S,SS, SSS, . . . Ann. Qeofia. (Rome),1, 350-353. 94. Caloi, P., and Peronaci, F. (1948). Onde superficiali associate alle onde S, SS, nel terremoto del Turkestan del 2 Novembre 1946. Ann. Qeojta. (Rome) 1, 520-524. 94a. Caloi, P., and Peronaci, F. (1949). Ancora sulle onde di tipo superficiale associate alle S, SS, . . . Ann. Qeofia. (Rome) 2, 290-294. 95. Oliver, J., and Major, M. (1960). Leaking modes and the PL. Bull. Seiamol. SOC.Am. 50, 165-180. 96. Oliver, J. (1961). On the long period character of shear waves. Bull. Seiamol. SOC. A m . 51, 1-12. 97. Oliver, J. (1964). Propagation of P L waves across the United States. Bull. Seiamol. SOC.Am. 54, 151-160. 98. Gilbert, F., and Laster, 8. J. (1962). Experimental investigation of P L modes in a single layer. Bull. Seismol. SOC.Am. 52, 59-66. 99. Caloi, P. (1940).Sopra alcuni nuovi sistemi di onde sismiche a carattere superficiale oscillanti nel piano principale. Rend. Reule Accad. Italia, Ser. VZI, No. IZ, Parts 1-2, pp. 13-41. 100. Wilson, J. T. (1948). Increase in period of Earthquake surface waves with distance traveled. Bull. Seiamol. SOC.A m . 38, 89-93. 101. Rayleigh, Lord (1937). “The Theory of Sound,” Vol. I, p. 336. Macmillan, New York.

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153. Gutenberg, B., and Richter, C. F. (1936). On seismic waves (third paper). Qerl. Beitr. Qeopfiye. 47, 73-131. 154. Kizawa. T. (1041). On surface waves propagated through the Pacific bottom and the continent. Qeopfiye. Mag. pp. 262-272. 155. De Noyer, J. (1959). Determination of the energy in body and surface waves. (Part 11). Bull. Seiamol. SOC.A m . 49, 1-10. 156. Ewing, M., and Press, F. (1954). An investigation of mantle Rayleigh waves. Bull. Seisml. SOC.Am. 44, 127-147. 157. Brune, J. N, (1962). Attenuation of dispersed wave trains. Bull. Seiamol. SOC.A m . 52, 109-112. 158. Knopoff, L.. et al. (1964). Attenuation of dispersed waves. J . Qeophye. Res. Lettere 69, 1655-1657. 159. Anderson, D. L., and Archambeau, C. B. (1964). The anelasticity of the earth. J . Qeophys. Ree. 69, 2071-2084. 160. Anderson, D. L., and Kovach, R.L. (1964). Attenuation in the mantle and rigidity of the core from multiply reflected core phases. Proc. Natl. A m d . Sci. U.S. 51, 168- 172. 161. Knopoff, L. (1964). Attenution of elastic waves in the earth. Rev. Qeophye. 162. Kovach, R. L., and Anderson, D. L., (1964) Attenuation of shear waves in the Upper and lower Mantle. Bull. Seismol. Soc. A m . 54, 1855-1864. 163. Caloi, P. (1951). Teoria delle Onde di Rayleigh in Mezzi elastici e firmo-elwtici, esposta con le Omografie vettoriali. Arch. Meteorol., aeophya. Biokl. A4, 413-435. 164. Caloi, P. (1960). L'effetto della firmo-viscositil sulla risultante del movimento associato alle onde di Rayleigh. Atti Accad. naz. Lincei. GI. Sci., fie., mat., nat. [8] 29,486-487. 165. Press, F. (1964). Seismic wave attenuation in the crust. J. aeophya. Rea. 69, 44 17-44 18. 166. Cook, K. L. (1962). The problem of the mantle-crust mix: Lateral inhomogeneity in the uppermost part of the earth's mantle. Advances i n Qeophysica 9, 295-360. 167. Lamb, H. (1882). On the vibrations of a n elastic sphere. Proc. London Math. Soe. 13, 189-212. 168. Lamb, H. (1882). On the vibrations of a spherical shell. Proc. London Math. SOC. 14,50-56. 169. Love, A. E. H. (1906). "Elasticity," 2nd ed. Cambridge Univ. Press, London and New York. 170. Oddone, E. (1912). Sui periodi sferoidali propri alla sfera terrestre, rigida per elaaticilti o rigida per gravitazione. Ann. Uff. Centr. Meteor. Qeod. 34, Part I, No. 6a, 1-19. 171. BBth, M. (1958). The energies of seismic body waves and surface wave& I n "Contribution in Geophysics in Honor of Beno Gutenberg," pp. 1-16. Pergamon Press, Oxford. 172. Press, F. (1959). Some implications on mantle and crustal structure from G waves and Love waves. J . Qeophys. Rea. 64,565-568. 173. Gutenberg, B. (1960). Polar wandering, displacements of continents, and subcrustal currcnts. "Pestschrift zum 70 Geburtstag von Ernst Kraus." Akad. Wias. Berlin, K1. IZI, Vol. 1, pp. 306-310. 174. Caloi, P. (1962). Sulle oscillazioni libere della Terra e del suo guscio esterno. Atti Accad. nar. Lincei, C1. Sci., lie., mat., nat. [8] 32, 432-440. 175. Alsop. L. E. (1963). Free spheroidal vibrations of the Earth at very long periods. Part I. Calculation of pcriods for several earth models. Bull. Seismol. Boc. A m . 53, 483-501.

ON THE UPPER MANTLE

M. (1961). Free oscillations of the Earth observed on strain and pendulum seismographs. J . ueophye. Rea. 66, 631-641. Nakagawa, I., Melchior, P., and Takeuchi, H. (1964). Free oscillations of the earth observed by a gravimeter at Brussels. “CinquiAme Symposium International sur les Marbes terrestres.” Obe. Roy. de Belgipue, Sdrie Qkophya. 69, 108-121. Nowroozi, A. A. (1965). Eigenvibrations of the earth after the Alaskan earthquake. J . Ueophya. Rea. 70, 5145-5156. Ness, N. F., Harrison, J. C., and Slichter, L. B. (1961). Observations of the free oscillations of the earth. J. Cfeophye. Res. 66, 621-629. Pekeris, C. L., Alterman, Z., and Jarosch, H. (1961). Rotational multiplets in the spectrum of the Earth. Phys. Rev. 122, 1692-1700. Birch, F. (1951). Remarks upon the structure of the Mantle and its bearing upon the possibility of convection currents. Tramact. Am. Ueophye. U n . 32, 533-535. Birch, F. (1954). The earth’s mantle, elasticity and constitution. Tramact. Am. Qeophye. Un. 35, 79-85. Knopoff, L. (1964). The convection current hypothesis. Rev. Ueophys. 2, 89-122. Shimazu, Y.,and Kohno, Y. (1964). Unsteady mantle convection and tectogenesis. J . Earth Sci., Nagoya Univ. 12, 102-1 15. Volterra, V. (1895). Sulla teoria dei movimenti del polo terrestre. Aat. Nachr. 138, 33. Volterra, V. (1895). Sulla teoria dei moti del polo terrestre. Atti Accad.Sci. Torino: C1. Sca.,js., mat., nat. 30, 301-306. Volterra, V. (1895). Sul moto di un sistema nel quale sussistono moti interni stazionari. Atti Accad. Sci. Torino: C1. Sci., $e., mat. nat. 30, 372-384. Volterra, V. (1895). Un teorema sulla rotazione dei corpi e sua applicazione a1 mot0 di un sistema nel quale sussistono moti interni stazionari. Atti. Accad. Sci. T&no: cd. sci., j e . . mat., nat. 30, 524-541. Griggs, D. E. (1939). A theory of mountain building. Am. J. Sci., 237, 611-650. Volterra, V. (1895). Sui moti periodici del polo terrestre. Atti Accad. Sci. Torino: cl. SCi. fie., mat., W l . 30, 547-561. Melchior, P. (1957). Latitude variation. I n “Prograss in Physics and Chemistry of the Earth,” Vol. 2, 212-243, esp. p. 223. Pergamon Press, Oxford. Volterra, V. (1895). Sulla teoria dei moti del polo nella ipotesi della plasticitti terrestre. Atti. Accad. Sci. Torino: GI. Sci., j s . , mat., nut. 30, 729-743. Peano, G. (1895). Sopra lo spostamento del Polo sulla Terra. Atti. Accad. Sci. T W h : c1. sCi.,fi8., ma*., nat. 30, 2-11.

176. Alsop, L. E., Sutton, G. H.. and Ewing, 177.

178. 179. 180. 181. 182. 183. 184. 185. 186. 187. 188.

189. 190. 191. 192. 193.

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WIND STRESS ALONG THE SEA SURFACE* E.

B. Kraus

Woods Hole Oceanographic Institution, Woods Hole, Massachusetts

Page 213 1. Introduction: The Transfer of Kinetic Energy ............................. 2. The Atmospheric Boundary Layer ....................................... 2 15 2.1. The Region of Constant Stress over a Fluid Intcrface . . . . . . 2.2. The Effect of Density Variation ......................... 2.3. The Role of Bubbles and Spray in Momentum Transfer . . . . 224 3. WavesattheInterface ................................................. 3.1. General Aspects of Wave Processes. .................................. 225 226 3.2. Surface Waves on Deep Water ..................................... 230 3.3. Motion within the Interface ......................................... 3.4. The Generation of Wind Waves. ..................................... 232 3.5. The Effect of Waves on Wind Profiles ................................ 237 241 4. The Transfer of Momentum from Waves to Currents ....................... 4.1. The Problem of Wave Decay ....................................... 241 4.2. Slicks and Three.Dimensiona1 Perturbations of the Boundary Layer. ..... 242 245 5. Conclusionsand Questions ............................................... Appendix ............................................................... 247 ListofSymbols .......................................................... 251 References .............................................................. 253

1. INTRODUCTION: THETRANSFER OF KINETIC ENERGY

Frictional phenomena at the sea-air interface are not identical with those occurring over land. A fluid surface is not rigid and therefore permits transmission of mechanical energy. Because of its extension and relative uniformity, the sea surface also tends to excite a much narrower range of perturbation frequencies in the atmospheric boundary layer than may be the case over land. Kinetic energy can be transferred across a fluid interface by tangential stresses which produce accelerations parallel to the surface and by normal pressure forces which may cause time-dependent deformations of the interface. I n analytical terms these effects are represented by the terms on the right-hand side of the following equation, which describes the approximate kinetic energy balance for either the whole or a mechanically enclosed part of the atmosphere or the oceans, averaged over sufficiently long time.

s s pgwdV-

(1.1)

(1)

* Contribution

8dV=-

(11)

1 s U7dS-

(111)

pvndS (IV)

No. 1647 from the Woods Hole Oceanographic Institution. 213

214

E. B. KRAUS

Terms (I) and (11)represent the volume integrals of potential energy conversion and dissipation. Term (111)-the work done by the stress 7 along the surface S-can differ from zero only if the velocity U is not zero a t the surface. The last term, (IV), represents the work associated with the vector dcformation w , ~of the boundary. I n the atmosphere, over land, both the surface integrals on the right-hand side are zero, except for minor effects due to waving trees or similar phenomena. Over water, these terms differ significantly from zero. In fact, most of the kinetic energy found in the oceans has been derived from the atmosphere through mechanical stresses a t the interface. The wind stress on the ocean surface produces not only quasi-permanent features of the circulation like the Gulf Stream and other current systems but it also has large transient effects both a t the surface and below. For example, Fig. 1 shows a characteristic current meter record obtained in

i

1663

FIQ.1 . Current metor record, July 1964, at 29"11'N, 68"21'W, depth 617 meters. Plotwd in 20 min increments, time shown every six hours, inertial period 23.6 hr. (Supplied by Dr. F. Webster, Woods Hole Oceanographic Institution.)

July 1964 in the Sargasso Sea. The diagram shows the direction and distance of water travel past the meter. The striking oscillations which are the dominant feature of this record have a period of half a pendulum day (12/sin q~ hr).

WIND STRESS ALONG THE SEA SURFACE

215

They are inertial oscillations of the current which were produced by the passage of a storm a t the surface. Inertial oscillations apparently can extend through great depths. They may have energies of the order of l o ’ e r g ~ c r n - ~ . They seem t o develop rapidly with the beginning of a storm, but fade more slowly. The question arises then how energy in such amounts can be transmitted sufficiently fast through the air-sea interface and through the adjacent layers. This is our topic here. 2. THEATMOSPHERIC BOUNDARY LAYER

2.1. The Region of Constant Stress over a Fluid Interface When a real fluid flows along a wall or interface it experiences a tangential stress r which becomes the most significant force in a region close t o the wall. Across this boundary layer, the variations of the tangential stress are small and can be neglected a t least as a first approximation. It is convenient to define a friction velocity u* by the relation r = P U * ~ . Two subregions can often be distinguished within the boundary layer. One is in the immediate vicinity of the wall where the effect of the viscosity is directly felt. In a second region, further away, the motion is still determined by the tangential stress at the wall but the fluid there no longer “knows” how the stress arises and is independent of viscosity. The two regions may, but need not, overlap. Consider the flow of a real fluid over a smooth interface which moves itself with a velocity U,. Within the inner viscous sublayer

Therefore

u-u,=-

U*%

V

The viscous laminar flow will merge into a turbulent regime a t z = S where the Reynolds number

distance

V

reaches a critical value Re. Together equations (2.2) and (2.3) imply that this distance 6 and the velocity there are given by V

s =-u* 1 / j i

U(S)= u, +u*1//R,

216

E. B. KRAUS

The preceding relations are true only for flow over a surface which is aerodynamically smooth. This will be the case if the surface irregularities are so small and gently sloping that they remain embedded and covered everywhere by a viscous sublayer. The surface is hydrodynamically rough if it has large and sharp irregularities, which can cause boundary layer separation and the formation of eddies. It will be seen below that the sea surface behaves in fact like a smooth surface over a wide range of conditions. The tangential stress r cannot be discontinuous across the interface. By definition 7

= pu*2 = pwu*, 2

where pu,, u*, are the densities and friction velocities in the water. It follows: u*,

(2.5)

=JplP,

U*

z 0.03,~,

The friction velocity characterizes the scale of the turbulent velocities. The fact that it is so very much smaller in the water suggests a correspondingly slower rate of mixing and transfer processes a t similar distances from the interface. Reasoning as above, it can be shown that the actual velocity U,(S,) a t the lower boundary of the viscous sublayer in the water will be approximately

(2.6)

U,(6,)

=

U, -u*,JR,

-

= U,

-0 . 0 3 u , D C

The total velocity difference between the upper and lower boundary of the viscous layers is therefore about u , ( R , ) ~ /and ~ 97% of the difference is accounted for by shear in the air. I n the outer layer of the boundary, where viscosity becomes unimportant, it is convenient t o define an eddy viscosity K by the relation

I n hydrostatically neutral conditions in the outer boundary layer, K can depend only on u* and on the distance from the surface. It is therefore likely to have the form:

(2.8)

K

= KU*Z

A tentative physical explanation of the proportionality constant K , known as von KBrman’s constant, will be given below. Integration of equation (2.7) after introduction of equation (2.8) and separation of variables gives the familiar logarithmic law: (2.9)

u 1

- = - In z + const

u*

K

WIND STRESS ALONG THE SEA SURFACE

217

If the integration is determined by the observation of U at some level h, one gets (2.10)

I n the case of smooth flow, there is no reason why the relation (2.7) and therefore the logarithmic law should not be valid down to a region where viscosity becomes important. The integration constant in equation (2.8) can then be determined by a match between equations (2.9) and (2.2) a t the level 6. A viscous sublayer will be maintained if there is a region where the molecular viscosity v overwhelms the eddy viscosity K as formulated by equation (2.8),with K and v assumed equal a t the level 6. From the first equation (2.4) and from (2.8) it follows that this is possible only if (2.11)

1

-=JR, K

From equations (2.1) and (2.7) it follows further that this involves continuity of the velocity gradient across 6. If the deduction is correct, von KBrmBn’s constant may be interpreted physically as the inverse square root of the critical Reynolds number R, which separates turbulent and laminar regimes. The value of K is known from many investigations t o be about 0.4 and this would make R, z 6.25. The validity of this deduction is limited by being based on conditions of smooth flow. Allowing (2.11), a match of the viscous profile (2.2) with the turbulent profile (2.9) gives for z 2 6: (2.12)

I n rough flow such a match is not possible. I n this case there is a region, close to the boundary yet beyond the viscous sublayer, where the scale of the eddying motions depends not only on u* and the distance z from the boundary, but also on the shape and size of the boundary irregularities. I n other words, the flow there still takes “cognizance” of how the stress arises; expression (2.8) which is the basis of the logarithmic law is then not sufficient for the specification of the motion. It is customary to apply equation (2.9) to rough flow, in the form: (2.13)

where z , is the so-called roughness length.

218

E. B. KRAUS

It cannot be implied from equation (2.13) that the mean velocity U vanishes a t the level z = z,, because the logarithmic law is not applicable so close to the boundary. Not only is K, as one approaches a rough surface, no longer a function of only u* and z as indicated by equation (2.8), but the irregularities of the surface will cause the space below their top to be no longer completely filled by air. The concepts of mean velocity and mean stress a t a given level lose their simple physical meaning in such a region. The quantity z, can be interpreted in the first instance as a parameter which specifies a logarithmic profile. This profile does not extend into the immediate vicinity of the boundary. Over a boundary which is very rough, the mean flow will be brought almost (but not quite) t o a standstill some distance away. The profile beyond this distance could then be characterized by equation (2.13) with a largc value of z,. To this extent, z, is a measure of the roughness that is felt below by the current some distance above. It is not possible to use z, directly t o specify the scale of the turbulence a t any particular level,

u,

fM

src-9

FIG.2. Deduced roughness parameters over deep water: A , in Caribbean near Aruba, with offshore winds; and B, in Buzzards Bay, Mass., with onshore winds.

but if observations in the logarithmic flow region do indicate, by extrapolation, a value z, < v / u * K it seems reasonable to assume that this region merges with a viscous sublayer which does not separate from the boundary. The specification of z, depends clearly on the specification of the coordinate

WIND STRESS ALONQ THE SEA SURFACE

219

origin z = 0 and on the surface velocity U,. Over water, if wave observations are available, z = 0 can be specified unequivocally as the mean level of the interface. On the other hand, the mean surface velocity U , is often not well known. Observation of drifting oil lenses suggests it to be usually of order u,, or 3-5 % of the anemometer wind velocity. Similar values were observed by Tomczak [ l ] with drifting plastic envelopes. Lack of specification of U , may easily cause uncertainties in the specification of z, by a factor of 10. Actual determinations of z, over the sea cover a very wide range of values [2, p. 1391. Two sets of determinations obtained a t different locations by profile observations with the same set of sensors each time after careful calibration are shown in Fig. 2. Buzzards Bay is a n enclosed area with often chaotic wave patterns. Near Aruba a fairly regular wave pattern is developed in very steady offshore winds. Surface velocities were not measured. They may have been appreciable off Aruba and if this had been taken into account, it could make the derived values of z, larger by about one magnitude. Even that would leave them much smaller than the observed values in Buzzards Bay. The actual vclocity U a t the 10 meter level rather than u* is used as abscissa in Fig. 2 because this gives some indication of the difference in surface drag over the two sea areas. The drag coefficient a t z = 10 meters (2.14)

was about twice as large over the waters of Buzzards Bay as over those near Aruba. When maritime observations are used to plot z, against u* one finds a positive correlation. This suggests that the number and amplitude of the eddies generated near the interface by boundary layer separation increases with increasing stress. The matter will be raised once more in Section 3.5 after the discussion of waves. (Papers on the nature of the roughness length zo a t a fluid interface were published also by Kitaigorodsky [3] and by Kraus [4] during the months between the writing and the printing of this article.)

2.2. The Effect of Density Va,riation Turbulent kinetic energy is produced not only by shearing instability but also by buoyancy forces. Alternatively, it will be damped if conditions are hydrostatically stable. In the case of hydrostatic instability, the increased turbulence will bring additional momentum from the free stream closer to the boundary. The curvature of the profile there becomes then more pronounced. This has some bearing on wave generation as will be seen later. The opposite holds with stable stratification.

220

E. B. KRAUS

Following Priestley [5] one may allow for the effects of stability by a generalization of the expression (2.8) in the forms (2.15)

where f is a function to be determined and 5 is a stability parameter which depends on the buoyant upward flux of density anomalies p G in the gravitational field. The function f may be said to stretch the z scale in a stably stratified fluid or to shrink it in lapse conditions; in neutral equilibrium f = 0. The eddy viscosity and therefore the boundary layer profile will be similar in all conditions a t equivalent heights: (2.16)

Introduction of equation (2.15) into equation (2.7)and integration, yields the diabatic profile: (2.17)

A stability parameter 4 may be defined in a number of different ways. One is a generalized version of the “flux Richardson number” (2.18)

which represents the ratio of the rates at which potential and mean shear flow energy are transformed into turbulent kinetic energy. A closely related alternative is the “Monin-Obukhov ratio” defined by

(2.19)

If f (5) is expanded into a power series of 5, the substitution 5‘ = z / L permits an evaluation of the integral in equation (2.17). Particularly if conditions are nearly neutral, that is, if 5 is nearly zero, the series can be truncated with its first term: (2.20)

When equation (2.20) is introduced in equation (2.17) one obtains the wellknown log-linear profile: (2.21)

u =1 (In z

u*

K

+c:)+

const

WIND STRESS ALONG THE SEA SURFACE

221

Evaluation of the constant C has been the subject of many investigations and some controversy. Forms for f (4) involving higher powers of 6 have been invoked by some authors, notably Swinbank [6]. Details may be found in the listed references. Less has been published about the causes of the density fluctuations near the air-sea interface and about the role which evaporation plays in destabilizing both the air and the water particularly in the tropics. If the density anomalies p' were a function only of the temperature anomalies T ', one could write: (2.22)

p'w'

=

H -pfIT'w' = - -

(air)

CP

- -pH

(water)

C

where t!? is the coefficient of thermal expansion, and H is the upward flux of sensible heat. In the atmosphere, this would lead to the more familiar forms:

(2.23) I n a humid atmosphere, the density anomaly p' depends not only on T' but also on the specific humidity. The relation (2.22)is therefore incomplete near the sea surface. Approximately: (2.24)

where Q is the evaporation (mass of water per unit area and unit time), r =0.61 is the airlwater molecular weight ratio minus one, and s the salinity in gram of solid material per gram of sea water. I n the tropical atmosphere the contribution of the second term can be of the sameorder or larger than the first; the same applies to summer conditions in higher latitudes as demonstrated in Table I. The fluxes of sensible heat and moisture can have opposite signs. The relation (2.24)leads tlo modified expressions for Rf and L.In the air:

which now incorporates the effect of nonlinear coupling of the three fluxes of momentum, moisture, and heat. Analogous parameters can bc constructed for the sea from the second expression (2.24).

222

E. B. KRAUS

The flux of heat is not continuous across the interface and is generally different in the air and in the sea. This is due to the absorption of infrared radiation which has a delta function behavior a t the interface. Evaporation TABLE I. Contribution in per cent by evaporation to mean monthly upward flux of density defect (mean of noon and midnight observations). Weather station:

J

E (34'N 48"W)

12 12 20 26 26 15 11 15 26 *

K (45"N16"W)

F

M

A

M

J

J

*

62 37 24 18 14 13 * 97 33 19 12 15

*

A

S

O

N

D

* Asterisk indicates downward flux of density defect. as such always causes a n upward flux of density defects both above and below the interface. In the air, however, the resulting destabilization is counteracted and sometimes reversed by the surface cooling produced by evaporation [7].

2.3. The Role of Bubbles and Sprays in Momentum Transfer The air-sea interface is not always a simply connected surface. Spray and bubbles produce a mixture of the two substances, which in a hurricane may become "too thin for swimming and too thick to breathe." I n spray, water is accelerated to wind speeds, while the air bubbles below are forced to move with the current. Spray and bubbles can affect wind and current profiles in two ways. If present in sufficient volume, they can increase hydrostatic stability significantly. This effect is likely to be larger in water than in air because the volume of air entrained below the interface is always much larger than the volume of flying spray above. A second dynamic effect is due to the stress 7, caused by the horizontal acceleration of spray drops (and deceleration of bubbles). The magnitude of 7, increases with the spray load and is therefore a function of height. The total stress 7 is the sum of T, and the Reynolds stress rk T

= rk + r e = pK

au

au

az

a(ln Z)

- + T , = ~ K U *-

+

' R

This total stress T cannot vary with height in the boundary layer, and this causes the logarithmic velocity gradient to be less in the spray zone than in the air above. I n other words, the hydrostatic and dynamic effects of spray both tend to decrease the profile curvature. The effect is the same as that of increased stratification.

WIND STRESS ALONG THE SEA SURFACE

223

The decrease in profile curvature by blowing sand has been studied by Bagnold [8] in a classical treatise. Observational difficulties have presented similar studies for flying spray over the sea, though Monahan [9] has tried to determine r8 from photographic observations of spray droplets. He concluded that re is not more than one per cent of r a t wind speeds below about 10 meters/sec. This is due t o the fact that the drops are relatively small in number and that they are not very efficient in extracting momentum from the air all the time. Big drops often fall back into the sea before they have been accelerated to wind velocity, small droplets may reach that velocity quickly and then will float on without exerting further drag.

WJND SPEED IN M€T€RS PER S€COND

Fro. 3. Spray observations by Monahan [9] as a function of wind speed. (The dashed line is based on visual interpolation.)

There are some indications that the spray stress r8 becomes relatively more important a t higher wind speeds. Figure 3, based on Monahan's relatively few observations, suggests a rapid increase of spray load for winds above 8 meterslsec. No spray observations above 11 meters/sec were available.

224

E. B. KRAUS

The increase in spray concentration a t wind speeds above 8 meters/sec may be associated with the spray producing processes. There are two meclianisms. One is associated with wave breaking. Waves break frequently as a result of interaction between waves of different lengths [ 101. Breaking can occur therefore a t quite low wind speeds, and although it becomes more frequent and vigorous as the wind speed rises, whitecaps can only cover an area of at most 5 yo of the sea surface a t a wind speed of 10 meterslsec [I I]. The whitecaps themselves are bubbles of air which rise after having been entrained by the breaking waves. Drops are ejected vertically into the air when the bubbles burst on reaching the surface. The smaller drops were shown by Blanchard to have ejection speeds up to lo4 cm/sec but their vertical travel is rapidly slowed by friction, and most of them only reach a height of 10-18 cm above the surface. Larger drops have more inertia but are ejected with lower upward speeds and therefore tend to reach about the same height. The highest droplet concentration is found therefore a t the 10-18 cm level where the average vertical speed is least. The acoustic noise produced by bubbles was measured by Wenz [12] who found a continuous gradual increase with wind velocity. The sudden increase in the spray load a t winds above 8 meterslsec indicated by Fig. 3, if true, is probably associated with a second spray producing process-the disruption of wave crests into spindrift. This disruption may be caused by boundary layer separation which can produce relatively large pressure differences in the air on either side of the crest. It depends on the relative velocities of wind and waves-the topic of the following sections.

3. WAVESATTHE INTERFACE Viscous friction along the surface is probably insufficient to account for the energy of the wind-driven ocean currents. It can be shown to be insufficient for the excitation of inertial oscillations. Momentum and energy must therefore be transferred somehow from the wind to the sea by processes which do not involve only viscous shears a t the interface. Stewart [13] estimates that about 20 % of the downward flux of momentum is used to generate waves, but his estimate was based on the growth of the observationally predominant long waves. It will be suggested below that the continued renewal of the shorter, less spectacular waves may well make wave stress the dominant factor in momentum transfer. The waviness of the sea surface is its most striking characteristic. The form, height, and propagation of waves a t any moment are a function of the wind stress in the past; they do influence the instantaneous local wind stress, in turn. Literature dealing with wind waves is voluminous. Besides Kinsman’s [ 141 comprehensive textbook, the author has found a short report

225

WIND STRESS ALONQ THE SEA SURFACE

by Carstens [ 151 particularly instructive and lucid. The present discussion of waves is restricted to those general aspects which bear on a qualitative understanding of vertical momentum transfers.

3.1. General Aspects of Wave Processes The transmission of light, sound, elastic vibration, and other oscillatory processes can all be characterized by some common features. A wave number k and a cyclical frequency w are defined by

k = 2.rrlA

w=

2r/T

where A is the wavelength and T is the wave period. All waves propagate with a velocity -

t

+

c=cn

(3.1)

c=wlk

-D

where n is a unit vector normal to the wave front and c is the phase speed. Waves are also characterized universally by an impulse and energy. Both are proportional to the square of an amplitude a2 in the case of harmonic waves with a given wave number. The energy per unit volume or energy density 6 and the impulse m are related to each other by -t

In the presence of waves with different wave numbers and velocities, the quantities E and 2 become summation functions. In general, it is not possible to generate or observe monochromatic waves, that is, waves of a single wave number or frequency. It is useful, therefore, to deal with impulse or energy densities per unit wave number or frequency interval. Instead of being proportional to the square of the amplitudc a2, the impulse and energy density can then be said to be proportional to a spectrum function, which for two-dimensional waves is defined by -

(3.3)

a2 = t,bka Ak A a ( Ak -+ 0, ha +0)

(Aw-tO)

=+,Aw

The wave number spectrum function is a measure of the mean square amplitude of all waves with wave numbers between k and k A k , coming from directions between a and a Aa. The frequency spectrum function is defined similarly as a measure of the mean square amplitude of all waves in a frequency interval Aw. I n the presence of waves with different wave numbers, the energy and + impulse are advected themselves with a velocity co, the so-called group

+

+

226

E. B. KRAUS

velocity. When phase and group velocity are parallel the latter is given simply by +

cg = awlak

(3.4)

Electromagnetic waves in a vacuum have the same phase end group velocities, but when internal hydrodynamic waves pass through an anisotropic medium, the two become different, both in magnitude and direction. The flow of + energy is represented by the radiation vector r, the Poynting vector of electromagnetic theory. +

+

r = &Cg

(3.5)

The advection of momentum is described by the tensorial product * +

(3.6)

IIYII = 11112 cgll

When the flow of momentum changes, for example, when a wave is absorbed * or reflected, a force P is exerted upon the obstructing medium. Its value per unit volume is given by the convergence of the momentum flow:

(3.7) The tensor y is the “radiation pressure” of electromagnetic theory, though the term “radiation stress” is preferred by Longuet-Higgins and Stewart [lo] who first worked with the concept in fluid mechanics.

3.2. Surface Waves on Deep Water The preceding discussion can be readily applied to surface water waves. The dynamics of the surface is affected by gravity and by the density of water p w . The density of the air can be neglected. In waves of small amplitude, the acceleration g* arising from the combined effects of gravity and the surface tension u ( z 70 dyneslcm) has the form: 9*

=s+-

uk2 Pw

If viscosity played no role, the frequency and phase speed would be given by (3.9)

u2= g,k

(3.10)

The last relation shows immediately that very long waves k -+ 0 and very short waves k + 00 must move faster than waves of intermediate lengths.

227

WIND STRESS ALONG THE SEA SURFACE

The slowest possible surface waves are characterized by the parameters: (3.11)

hmin= 1.7 cm

kmin = 3.7 cm-'

cmin= 23 cm sec-'

-

A plot of c against wave number k is shown in Fig. 4.

X (cm)

10'

10'

100

10

1

0.1

0.01

12C

(m/sec) 10-

8-

6-

4-

2-

0

,

lo-'

1

0.01

0.1

I

I

I

1

10

100

k (cm-')

FIQ.4. Phase speed as a function of wavelength or wave number.

In the presence of a n inviscid wind with velocity U and an equally inviacid current with velocity U,, equation (3.10) assumes the form: (3.12)

( C - U- ) 2 = -9*- - ( UP- U

k

)2 W

Pw

where U- = P U + P W U W Pw

If the inviscid theory were true, relation (3.12) would show that winds only could raise waves if they had a relative velocity larger than the minimum given by (3.13)

U - Uw > m p cmin% 650 cm sec-

In fact, waves do occur a t much lower wind speeds and this led Kelvin to conclude that air viscosity must have a destabilizing effect.

228

E. B. KRAUS

The amplitude u and spectrum function # of surface water waves decreases exponentially with the distance from the boundary:

# = Y exp[ -12kzlI

u2 = A2 exp[ -12ktlI

a = A exp[ -lkzl]

(3.14)

where A is the amplitude and YP the spectrum function at the interface. Figure 5 illustrates the velocity profile and the volume transport along 8

6

4

2

2 $ 0

aJ

E

-2

-4

-6

-8

I

I

I

I

I

I

I

I

I

.2

0

2

4

6

8

10

12

14

meters FIQ.5. Profile of Stokes’ wave ( A = 16 meters, A = 1 meter, c = 5 meterslsec). Distance between adjacent horizontal lines represents volume transport of 1 1113 per meter width per sesond.

verticals through the troughs and crests of a monochromatic wave with small but finite amplitude (Stokes wave), at the boundary of two ideal inviscid fluids at rest. The distance between adjacent horizontal lines represents a volume transport of 1 m 3 per meter width per second. It can be seen immediately that the forward transport under the crest is larger than the back flow under the trough. A mean flow in the direction of propagation is therefore associated with the passage of the waves. The vector of this mean flow is the wave impulse or wave momentum. For waves on deep water it can be expressed in the form [14, p. 2571: -+

(3.15)

-+

m =pwA2k2exp[ --(2kzl]c

WIND STRESS ALONG THE SEA SURFACE

When obtains

229

is integrated over a strip of unit width and infinite depth one

(3.16) Both of the last expressions are applicable only to hypothetical monochromatic waves with a finite amplitude A . I n a wave field with a continuous spectrum, the impulse and energy in the wave number and directional interval dk, do! become

+

-D

Mkff= &pwCkyk, dk d a

(3.18)

Of particular interest is the impulse along the surface z = 0. It is simply -+

-D

mkff(0)= pw Ck2Yka dk da

(3.19)

Energy densities can be derived from the preceding equations with the aid of equations (3.2) and (3.10) - + - +

Eke! = Mka' C = &pw g* y k f f dk du

(3.20) (3.21)

E k a ( 0 ) = pw g*

kyka dk du

The group velocity becomes, by equations (3.4) and (3.9),

(3.22)

=-

2 *

PwS*

-D

The vectors rn and cBare colinear for waves on deep water. This makes the radiation stress isotropic and allows it to be specified by the unit tensor multiplied by a scalar (like pressure in hydrostatics). The value of this scalar for z = 0 can be expressed with the aid of equations (3.2), (3.6), (3.21), and (3.22) as

(3.23)

c

yk,(0) = 1 ~ , ( 0 )=&(p,,gk C

+ 3ok3)ykadk da

If the viscosity v can be neglected, it can be argued on dimensional grounds as discussed in the appendix that the spectrum y k f f as defined here, becomes asymptotically proportional t o k - 3 . When this is introduced into equation (3.23) one finds that the surface radiation stress exerted by the small waves in the wavelength intervals between 0.1 and 1 cm is likely to be about as

230

E. B . KRAUS

large as that exerted by all waves with lengths between 10 and 100 cm. It will be shown below that this may have a n important bearing on the transfer of momentum from the surface to deeper layers in the ocean. Before this conclusion is developed, however, it has to be shown that the wave motion a t the surface is not affected significantly by the viscous stress across the interface.

3.3. Motion within the Interface

A t the boundary of two real fluids there can be no slip. The velocity must therefore be continuous across the sea surface. It might then be asked whether particles a t the waving interface itself will follow the motion of the air or of the water, or neither. If there is t o be no slip, any mean horizontal flow must vary with the distance from the interface. The study of wave disturbances in the presence of a mean shearing flow leads t o the Orr-Sommerfeld equation [l6] which has not been solved analytically except for some special cases. For an approximate assessment of the effect of viscosity on existing waves it should be sufficient to consider the behavior of the waves a t the boundary of two real fluids without mean motion. The effects of viscosity must be largest close to the interface. The orbital motion in the two fluids produces there a reversing shear and therefore a reversing vorticity. Viscous mixing diffuses this vorticity upwards and downwards into both fluids. The air and the water compete in a way for the vorticity produced a t the interface. Success in this competition will be the smaller the greater the mass of fluid which has to be accelerated or decelerated by the diffusion of vorticity from the boundary. This mass depends on the density and on the distance to which the diffusion of vorticity penetrates from the boundary in both directions before the motion reverses. This depth can depend only on the viscosity v and the frequency w . It is therefore likely to be proportional to two lengths 1 and 1, which may be defined, without loss of generality, by (3.24)

lw=J2v,lw

The ratio of the fluid masses affected by viscosity on either side of the interface is therefore given by (3.25)

Following Priestley, the term P ( v ) ~may / ~ be called the “diffusive capacity” of the fluid. At the interface itself, the tangential velocity and the stress must be

231

WIND STRESS ALONG THE SEA SURFACE

continuous (u - uw),= 0

(pv--pwvw;:

(3.26) (7--,)0=

2),=0

With z increasing upward, the equations of horizontal motion are (3.27)

au au -+c-=vat ax

8%

au,

au,

az2

- at+ c ~ = v " -

a2uw h

W

It will be assumed that 1-29

k2

which is equivalent to saying that (3.28)

w 9

2vk2

and that the wave is longer than about 0.1 om. The equations of motion, together with the boundary conditions (3.26) and the boundary conditions a t infinity u = u, = 0, are then satisfied approximately by the real part of the expressions:

(3.29)

z

[ [

A m 2exp -(1 - i ) -

11-.-"I

exp[i(kz - w t ) ]

exp[i(kx-wwt)l

z Awekz

x exp[i(kx - w t ) ] This velocity profile, a t the time of maximum surface amplitude, is shown in Fig. 6. The second equation (3.29) indicates that the wave motion in the water resembles closely that in an ideal nonviscous fluid. At the interface itself z = 0 and (3.30)

u, = u, = Aw 1$-cL exp[i(kz - wt)] z Aw exp[i(kz - wt)] 1-P

That means the interface moves with the water. The horizontal perturbation velocity only becomes zero in the air above the interface a t a distance (3.31)

6, ;=: 1-P

ln2 z (0.38w-"2cm)

232

E. B. KRAUS

The argument does not imply that viscosity must not damp out waves after a large number of oscillations. What it does say is that viscosity has little immediate effect on the pattern of water movements in the interface, 16-

cm 12

-

S-

4-

I 0-

-4

L,

I

-2

I

-1

3 1

2

m/sec

Fro. 6. Near surface vertical velocity profile across crest of wave shown in Fig. 5, with corisiderationof viscosity. The dashed line is the inviscid profile of Fig. 5 .

which is produccd by waves of a t least 0.1 cm length. For longer waves and at greater distances from the interface within the water, the effect of viscosity must be even smaller because of the smaller shear and because Y, < v . 3.4. The Generatiou of Wind Wuves

The momcntuni of wind-driven waves must have been derived from the air above. The stress exerted upon the wind is, therefore, related to the rate a t which wave momentum develops.

WIND STRESS ALONG THE SEA SURFACE

233

When a turbulent wind first begins t o flow over a flat water surface, it will carry-or advect-ddies and associated small pressure fluctuations, which may retain their identity for considerable periods of time. It was shown by Phillips [17] that this leads to a resonant generation of waves a t the water surface. Resonance occurs if (3.32)

Upcosa=c

where U p is the advection velocity of the pressure pattern, and a the angle between U p and the phase velocity c. The power of the air pressure fluctuation is too small for the resonance mechanism t o account for the observed growth rate over the entire spectrum of wind-driven waves. It can produce, however, initial wavy disturbances on a flat water surface. Once formed, these may grow by interaction with the mean air stream. Pressure forces involved in this interaction must be asymmetric relative to the wave crests. There must be greater pressure on the surface behind the crests where the water sinks than ahead of it where it rises, if momentum and energy are t o be transferred. The classical instability theories of Helmholz and of Kelvin [18] involve a symmetric pressure distribution. They prescribe a local pressure minimum above the wave crests where the air moves relatively fast and a maximum over the troughs. This would favor an unstable exponential growth of wave amplitude, if it was not counteracted by the restoring forces of gravity or surface tension. Being symmetric relative to the wave crests, however, the pressure distribution in the Helmholz-Kelvin model cannot transfer momentum from the air t o the waves. In the sheltering hypothesis of Jeffreys [19] and Munk [20] the air is supposed to overtake the perturbation at the surface. This would produce boundary layer separation and a pressure deficit in the “lee” of the wave crest, with a corresponding pressure access on the windward side. Pressure would, therefore, exert a force in the direction of the wave propagation which can transfer momentum t o the water. This mechanism may explain the statistically different steepness of wave slopes in the upwind and downwind directions [all. It can only act on waves which move slower than the air in the boundary layer. A diffcrent approach evolved by Miles [22] and interpreted in physical terms by Lighthill [23] takes the variation of the wind height into account. It is based on the concept of a vortex force. When a vortex-or a fluid particle having vorticity-moves through a fluid, it will be accelerated at right angles to its direction of propagation as shown schematically in Fig. 7. Consider now the pattern of air motion relative to an ideal moving wave as shown in Fig. 8. Figure 8a shows a logarithmic profile which could be --L

234

E. B. KRAUS

FIG.7. Schematic diagram of inertial acceleration experienced by an element moving with velocity W normal to the vorticity 7.

associated with a stress of 1 dynelcm'. The wind overtakes the wave above the critical level z, where U(z,) = c; it lags behind the wave below. The vertical component of the air motion in front of the wave is upward below the critical level and downward above; behind the crest it is downward below and upward above. As the pattern is nearly symmetrical the vortex forces meters r6 -

5-

3-

2-

I

I

I

I

I

-6

-4

-2

0

2

meters (a)

I

I

I

I

I

I

I

I

I

0

4

8

12

16

20

24

28

32

meters (b)

FIQ.8. (a):Logarithmic profile with u , =29 cm sec-1. ( b ) : Schematic (distorted) diagram of air flow relative to a moving wave. Undisturbed wind profile is that shown on left. Wave charaoteristics are the same as for Fig. 6. Critical level shown as dashed line.

235

WIND STRESS ALONG THE SEA SURFACE

are approximately equal and opposite a t the same level in front and behind the wave. This is shown schematically in Fig. 9. meters 5 -

2-

---.

----

)&#-I

7x -c*-

1

0

1

2

1

4

1

Sac-' (a)

3

1 -

0

I

ti

12

16

meters

(b)

FIO.9. ( a )Vorticity distribution associated with profile in Fig. 8. ( b ) Inertial accelerations (vortex force) associated with flow shown in Fig. 8. Critical level shown as dashed line.

Conditions are different in the close vicinity of the critical level z,. The air immediately above slowly overtakes the crest. I n the region of the forward node, it sinks below and is then left slowly behind the advancing crest, until it moves upward again over the rear of the wave. During this slow circulation, air particles gain vorticity by viscosity while in the lower layer; they lose vorticity in the upper layer. The downward air moving in front of the wave has therefore less vorticity than the upward moving air behind it. The resulting horizontal accelerations are different as can be seen again from Fig. 9. The mechanism causes a deceleration of the air a t the critical level. The associated stress has then the character of a delta function. It changes abruptly a t the level z = z, and has different constant values above and below. The deceleration of air behind the wave is associated with a n excess of pressure in the region where the water surface sinks. There is a deficit in pressure in front where it rises. The pressure forces therefore produce a n energy input into the water (pv,&> 0).

236

E. B. KRAUS

The difference between the vortex acceleration in front and behind the waves, and therefore the extraction of momentum and energy from the wind, will be more effective if the amplitude W of the vertical velocity and the change of vorticit,y with height aq/az = a2U/az2across the critical level z, are both large. In formal terms, this is expressed by Miles’ analytically derived formula for the increase in wave momentum (3.33) The associated wave stress on the wind

The amplitude of the vertical velocity W is a function of the wind U ( z )and of COB u.It will be relatively large over high waves. Both Wand the curvature of the wind profile aq/az decrease with height. The efficacy of the process increases, therefore, with increasing wave amplitude or wave energy and with decreasing distance of the critical level z, from the interface. The Miles theory in its present stage is a two-dimensional perturbation theory, which neglects squares of disturbances to the sheared air flow and the dissipative effect on these disturbances of both viscosity and turbulence. It makes no allowance for the fact that the air flow over groups of waves with different energies, directions, and wave numbers is probably very different from the flow produced by a monochromatic wave train. The theory as it stands has been found, however, in spite of these reservations, to give rcasonable agreement with observations by Longuet-Higgins [24] and others. It may also be used tentatively to account for some other phenomena. The air close to the surface is usually slightly unstable, hydrostatically, because of evaporation as discussed in Section 2.2. Hydrostatic instability increases the curvature of the profile, or aq/az close to the surface. This should produce a faster growth of waves. The effect is greatly increased when cold air moves over a warm water surface. Roll [25], Fleagle [26], Deacon and Webb [27], and others, have all reported breaking to be more frequent in this case. Waves have been said also to be higher with larger sea-air temperature differences; but this is a more ambiguous statement [28]. Wave height is determined ultimately by breaking. Gravity waves can never grow to a height a t which their orbital acceleration A u 2 = Akg would exceed g. In other words, Ak < 1 must always be true. Strong wind profile curvature close to the surface, however, does cause the smaller waves in particular to be re-established quickly after having been deutroyed by one process or other. This could have some effect on a mean wave height. I t will be seen that it may also have a bearing on the momentum transport to lower layers in the water.

WIND STRESS ALONG THE SEA SURFACE

237

3.5. The Effect of Waves on Wind Projiles When a wind begins to blow, the surface is a t first relatively smooth. The Miles transfer mechanism from wind t o wave only becomes effective gradually. The wave rnomentum M and the squared velocity amplitude W are both proportional t o the square of the wave amplitude A . Equation (3.33)indicates therefore that wave amplitudes increase exponentially with time. Stewart [13] used data collected by Groen and Dorrestein [29] to show that the associated momentum increase of growing waves can partly balance the downward flux of momentum in the air. Wave development has, therefore, a significant effect upon the drag which the sea exerts on the wind. The growth of waves continues until it is balanced by breaking, dissipation, or other effects, and until waves have developed, which travel with a speed comparable t o the wind above the immediate surface layer. The rate of wave energy and momentum increase must pass through a maximum before that stage is reached. This is shown clearly in Fig. 10 which has been derived from the data given in Stewart’s paper. The question may now be posed as t o whether t,he rate of work done by the wind passes also through a maximum. In other words: Docs the sea surface exert less drag on a wind which has blown for a long time over a long distance? The possibility that this may be the case was mentioned by Neumann [30]. It could have a bearing on the development of atmospheric disturbances over the sea. Two possible mechanisms will be discussed in this section. On the one hand, the wave stress on the wind could become smaller with time because the downward flux of momentum is being tapped a t increasingly high critical levels by the slowly developing, long wave components of the spectrum. This might reduce the amount of energy and momentum available t o the shorter, slower waves with a lower critical level which had developed earlier. It has been argued that this would be tantamount to a reduction of the wind profile curvature. On the other hand, surface roughness may change and possibly decrease with the development of the longer waves. Whatever the mechanism, there are some indications that the work of a wind changes with time or fetch. When a sea develops, the wave components with the greatest spectral energy density are found always a t the low wave number (red) end of the instantaneous spectrum. With time, as longer waves continue to develop, the spectral peak moves t o lower and lower wave numbers. Barnett [31] finds that this is accompanied by a reduction of the spectral energy density in the wave components that made up the peak of the spectrum a t an earlier time. If that were generally true, it would suggest a reduction of the energy and momentum supply to the shorter waves, after longer waves had developed. Tn support of this contention, the author can report qualitative observations based on the handling of a small boat. These indicated that certain short waves had regularly more energy within a limited

I

1

1.0

-

on-

cobt #a4

Q

r

+

0

0

O 0

+

++

0

0

+

-

a -

0.1

1

0

-

2

1

1

.I

+

1

I

I

FIG.10. Effect of wind duration on transfer rates: (0) U = 20 meterslsec, ( + ) U = 15 meterslsec, and (0)U = 10 meters/sec. (a) Change in wave momentum at different times after start.of wind. (b) Change in wave energy at different times after start of wind.

WIND STRESS ALONG THE SEA SURFACE

239

distance downwind from the island of Aruba, N.A., than they had further away where the wind fetch was greater and longer waves had developed. It could be seen from the air that there were also fewer whitecaps further out, although the seas were much larger. If the wind profile, instead of being logarithmic, was less strongly curved, this would appear as a convex curve-or, in the case of observations a t a few levels only, as a kink-on a logarithmic plot. Such kinks have been reported by Kinsman [32], Takeda [33], and others. It is, however, by no means sure that these kinks were really associated with the mechanism suggested above. Reflection of waves and their “bottoming” causes a sea to become noticeably more rough hydrodynamically as a shore is approached. This transient change in roughness could also produce a reduction of curvature, or kinks, in a smooth profile. All the quoted observations appear to have been made close to the shore. There is no obvious reason why &hetransfer of momentum to waves should greatly affect the shape of the balanced turbulent wind profile. This shape is determined by the requirement that the flux of momentum from above into a thin stratum, must be the same as the flux out of the same stratum below. In the case of random turbulence, the eddies become smaller as one approaches the surface. This makes them less efficient transport agents and the vertical gradient of mean velocity therefore becomes steeper close in. Something similar applies, however, to the momentum transfer from the air to waves. The shorter waves, with critical levels close t o the interface, have also less amplitude, and therefore are less effective in removing momentum from the wind. This is compensated by increasing profile curvatures as the surface is approached. The effect of waves on profile curvatures was investigated analytically in a later paper by Miles [34].His analysis is not dissimilar t o that which led to the diabatic profiles discussed in Section 2.2. The transfer of momentum to waves in the atmospheric boundary layer can be allowed for by another generalization of the eddy viscosity coefficient, written as

(3.34) which is equivalent to the expression (2.15). The function f(U/u,) is positive definite, and has therefore a similar effect of stretching the z scale as has a stable stratification. The actual form off as determined by Miles is rather involved. It depends on the existing profile shape and also on the wave spectrum. Using empirical data for the latter, Miles found a reduction of not more that 6 % from the logarithmic profile curvature for a wind of 10 meters/ sec. This deduction is based, admittedly, on not-firmly established assumptions. I n particular, there remains a lingering doubt on the working of the

240

E. B. KRAUS

Miles process in the presence of composite waves with finite amplitude. As far as present knowledge goes, however, it seems rational to conclude that the reduction of profile curvatures by momentum transfer t o the surface waves is small for typical wind speeds. The second factor which may change the stress with fetch or time is the surface roughness. The large variations in roughness which were indicated by Fig. 2, and which were discussed ag,zin in Section 2.3, can probably not be associated with the Miles wave generating process. According t o equations (2.4) and (2.11) the motion a t the edge of the viscous boundary layer must be of order U , U J K . Any surface current U , will advect both the smaller waves and the viscous boundary layer; it does not affect the present argument which deals with their relative velocities. Boundary layer separation is likely to occur, therefore, only over waves which are being overtaken by the air in the viscous sublayer. These wavcs propagate within the limited speed range :

+

Cmin

The roughness length zo is likely to depend on the number and energy of waves which can be found within this interval. It may be assumed to be of the form:

lkm,n k(U*lK)

z~ a

'hz

dk

As shown in the appendix, the spectrum for the shorter waves which may

cause boundary layer separation is likely to be proportional to k - 3 . Therefore, with equations (3.10)and (3.11):

(3.35) A linear dependence of zo on u*'/q was stipulated also by Charnock [35] directly on the basis of a dimensional argument. His and the present dimensional considerations are however only in approximate agreement with observational determinations of zo and u*.These seem to indicate an increase of zo with a higher power than the square of u* if u* is larger than about 20 cm/ sec. The discrepancy does not invalidate the preceding deductions qualitatively. Boundary laycr separation can occur only a t wind speeds which are high enough to give u* a value which makes the right-hand side of equation (3.35) positive and larger than the viscous boundary layer thickness 6 = V / U * K . I t is not possible to relate this process to a single critical wind velocity because of the influence which stability exerts on the relation between the wind a t anemometer height and u*. The matter is affected also by the advection of

WIND STRESS ALONG THE S E A SURFACE

241

small waves by the orbital velocity of much longer waves. The argument may explain, however, in a qualitative way, the rather abrupt increase in spray load, ion conductivity, and other phenomena which were all reported by various authors at wind speeds between 5.5 and 9.0 meterslsec. 4. THETRANSFER O F MOMENTUM FROM WAVES TO CURRENTS 4.1. The Problem of Wave Decay

Viscous skin friction is probably insufficient to energize the observed system of wind-driven currents in the ocean. It is almost certainly insufficient to transfer energy fast enough €or the excitation of inertial oscillations. Momentum and energy transfer from the wind to the sea by wave generation must, therefore, be significant. Stewart [13] estimates that the wave stress accounts for 20 yo of the total stress, but his estimate was based on the observationally predominant long waves. The influence of the short gravity and capillary-gravity waves may well make wave stress the predominant factor in momentum transfer. The questions are then to be asked: How are the alternating wave currents rectified? How is momentum and energy passed on from the waves to the waters below? There is no level at which the current flows as fast as the waves move and no opposite equivalent to the Miles process. It has often been said that the transfer of momentum from waves to current is the result of the destruction of wave motion by breaking, turbulence, and viscosity. There are contradictions in this argument which do not make it altogether satisfactory, Viscosity causes a gradual exponential reduction of wave amplitude a t an approximate rate: y

= 2vk2

If this were t h e only influence, a wave with the length of 16 meters could travel for one year before friction could reduce its height from 100 to 37 cm. This is patently too slow. I n fact, the water is turbulent and it is tempting to replace the molecular viscosity v in equation (4.1) by an eddy viscosity K in the water: (4.2)

y K = 2Kk2

The eddy viscosity K in the top 50 meters of the ocean depends on the wind. A mean value of K z 100 cm2 sec-' seems reasonable, though much higher values were established by some investigators [36]. Introduction of K = 100 om2 sec-' into equation (4.2)yields, however, decay times that are much too short. With this value it would t.ake only 50 min for our 16 mcter long wave to decay from 100 to 37 cm in height, which is again unrealistic.

242

E. B. KRAUS

A value of order 100 om2 sec- for K in the ocean surface region above the thermocline, would be barely enough to account by turbulent mixing for the transient adjustment of currents in the upper stirred layer, for the deepening of this layer by storms, for inertial oscillations, and other dynamic phenomena. On the other hand, it would be too large for the observed persistence of waves and it should cause a faster decrease of wave amplitude with depth than is actually observed. The contradiction led Defant [37] and others to the conclusion that turbulence in the water has little influence on wave motion and that the decrease in wave height is caused mainly by the air resistance encountered by traveling waves. This, however, would leave the problem of momentum transfer from waves to currents quite unresolved. The solution of this contradiction may be found in vertical transfers by relatively large (meso-scale)motion patterns. These may have a n organized, nonrandom aspect. They could affect waves selectively, as may be seen from the following argument. 4.2. Slicks and Three-Dimensional Perturbdon of the Boundary Layer Patches of filmy material can absorb the energy of capillary-gravity and short gravity waves. This is due partly to the work of alternate expansion and contraction of the film. Part of the wave energy may also be lost by a viscous drag of the associated chains of water molecules which are bonded to the molecules of the absorbed surface film [38]. The longer waves have lower frequency and generally less surface curvature. The work of film deformation per unit time is then smaller, and this permits longer waves to pass through the slick patches without significant attenuation. It is the damping of the short waves which often makes the slick visible. Consider now the wave radiation pressure normal t o the sides of a slick patch. There will be pressure on the sides which face the wind, but also some pressure on the sides which are parallel to the wind. As discussed in Section 3.2, the stress in the surface t = 0 exerted by the short capillary-gravity waves which are most likely to be absorbed by the surface may be appreciable. From equation (3.7) and the appendix i t can be estimated that the force per unit exerted by the absorbed waves on the sides of the patch parallel t o the wind is about 20 % of the force exerted on the side facing the wind. As a result, the patch will be squeezed out into a strip along the wind direction. The process is shown schematically in Fig. 11. It can be demonstrated by laboratory expcriments. The skin drag experienced by the wind over the surface active material is smaller than over plain water. The slick streaks, therefore, cannot have been drawn out by the wind, as has been sometimes said; they are shaped probably by the process suggested here. There is always surface-active material on the ocean. Wave radiation

WIND STRESS ALONG THE SEA SURFACE

243

C

FIQ.11. Formation of slick streaks by wave radiation pressure (schematic).

pressure of the short, wind-driven waves will cause this material to become concentrated and to be aligned in more or less parallel streaks. The same alignment was explained by Welander [3Y] as being due to a wind convergence from both sides toward the filmy material with its lower drag, but this effect can be estimated to be probably much smaller than the effects of wave radiation pressure, though it may tend to produce the same result a t a slower rate. Continuing generation of short waves in the slick-free lanes will cause a convergence of mass towards the streak. This must be compensated by a distributed surface divergence between the streaks. The resultant circulation has the form of pairs of longitudinal rolls, as shown schematically in Fig. 12. The force F' exerted by the absorbed waves on the side of the streaks has a cross-wind and a downwind component. The latter causes an acceleration of the streaks and of the water immediately below. This water should move faster and this has, in fact, been established by the observation of dye lines normal to the streaks. As the water below the streaks sinks, it transports momentum into lower regions. Figure 12 also shows, on the right side, the vorticity vectors associated with the mean current shear. The horizontal convergence toward the slick streaks will cause a contraction of the vortex lines associated with the shear of the mean current; in the lanes between the streaks the vortex lines will be stretched. The final motion pattern which results from the stretching and shrinking of vortex lines was studied analytically by Stuart [40]. He started with an initially logarithmic profile and found that the velocity distribution after some time was completely altered by the effect of the longitudinal rolls. I n particular, large systematic velocity differences developed between the

244

E. B. KRAUS

FIG.12. Slick streaks and longitudinal rolls (schematic).

fluid which moved away from the boundary and the fluid that approached it. This is shown by Figs. 13a and b which are reproduced from Stuart’s paper. Somewhat similar deformation of a profile characterized initially by monotonic shear, was also found in numerical computations by Faller and Kaylor r411.

Longitudinal rolls may occur also in the air, particularly over the quasiuniform sea surface. They will tend to cause lateral velocity differences to become almost as large as vertical velocity differences in the disturbed layer. This immediately raises the question of the representativeness of profile observations along a single vertical. The rolls will also cause the mean motion of the perturbed layer to resemble that of a slab with relatively little vertical shear. The Ekman adjustment in the steady case is then produced by angular shears on top and bottom of the layer or by changes in layer thickness. There may be no locally observable turning of the current with depth as is required by the theory for the steady case, and none has apparently been observed so far in the sea surface layer. Conditions are different during transients when a motion pattern is still being established. The wave radiation stress on slick streaks may not provide the major part of the energy input to the longitudinal rolls. Thermal instability and mean shearing instability can produce similar patterns. At most times, all those causes will be effective simultaneously. It is difficult to provide numbers a t this stage, because not much is known either about the selective absorption of waves by slicks, or about the directional spectrum of the shorter waves which are being absorbed. Qualitatively, it can be said, however, that the process should affect the shape of the motion in the longitudinal rolls, and that it should be effective in skimming off the fast-moving surface layer of

WIND STRESS ALONG THE SEA SUBFACE

245

-u FIG.13. Velocity profiles for longitudinal rolls computed by Stuart 1401. Time in notidimensional units. Initial undisturbed profile given by t = 0. (a)Profile for region where fluid moves away from the boundary. ( b ) Profile for region where fluid moves toward the boundary.

the water which can be continuously accelerated again by the re-formation of the short waves. The momentum and energy derived from the surface processes is transported by the rolls directly and effectively to the water below the surface mixed layer. This may provide the answer to the question raised in the introduction.

5. CONCLUSIONS A N D QUESTIONS Some of the mechanical processes which operate across the sea surface have been evaluated and interpreted in physical terms. It would have been difficult to include all phenomena which have a bearing on momentum transfer. An attempt to do so would have made this presentation less readable without adding necessarily to any merit it may have. It will be tried now not only to summarize what has been said, but also to formulate once more

246

E. B. KRAUS

some of the questions that have been left unanswered as well as others that were not even asked. Wind profiles over the sea tend to have a more or less logarithmic shape in the constant stress layer. This description is inadequate, however, in the irnmediate vicinity of the boundary. The von K&rm&nconstant was interpreted in physical terms as the inverse square root of a, critical Reynolds number which separates regions of viscous and turbulent flow regimes. Density differences cause the friction velocity in the air to be about 30 times larger than in the water. Thermal instability and evaporation increase the intensity of the turbulence and the profile curvatures both in the air and in the water. Spray and organized momentum transfers from wind to waves tend to decrease the wind profile curvature. Bubbles tend t o have the same effect on current profiles. The roughness characteristics of the surface itself depend on the number and size of waves which move more slowly than the air immediately above the surface. Physically meaningful processes which can transfer momentum and energy from wind to waves were discovered only in recent years. The advection of momentum by the group velocity of finite wave packets causes a radiation stress which can exert a force on an obstacle. The force exerted by the radiation stress of short capillary-gravity waves along the sides of slick patches can be considerable and causes these patches to be squeezed out into streaks which are approximately aligned with the wind. The decay of waves by breaking and turbulence is too slow to account for observed transient accelerations of currents. It is suggested that momentum is transferred directly from short waves to the current a t some depth by three-dimensional organized perturbations of the boundary. These often have the form of longitudinal rolls. Their presence affects the representativeness of observations obtained from a single fixed vertical. These are some answers; there are many questions that were obviously left unanswered. One might ask specifically whether the conclusions above would be applicable to extreme stresses in a hurricane, for example? How does air move over or around composite waves of finite amplitude, and how does this affect the Miles process? How are waves of different frequencies absorbed by surface active material on the sea? How is the spacing of longitudinal rolls affected by the depth of the stirred layer, and how do the rolls affect this depth in turn? What are the questions that were not asked? The present paper does not probe into the relation between the gradient wind in the free atmosphere and the development of waves and currents below although this is perhaps the most significant and technologically most important aspect of the subject. The time interval necessary for the approach to “fully developed” sea states is

WIND STRESS ALONG THE SEA SURFACE

247

of the same order as the time scale of synoptic perturbations. Transient effects of a n inertial nature are known to be large in the establishment of surface currents, Attempts have been made by Kitaigorodsky [42] and others to predict the development of a sea generated by a stress which does not change with time, but if friction changes as the waves grow, the wind stress could remain constant only if there were a specific, compensating change in the driving force. At present it is not known how significant the frictional changes might be. If they are large, it may not be possible to close the problem of adjustment between air and sea without stipulation of additional constraints. Principles of least work or of extreme entropy production have sometimes been used in similar situations, though their use in finite nonlinear processes cannot be fully rationalized. Closely related to the adjustment problem is the partitioning of energy and entropy between coupled processes. How much energy is dissipated by the wind, how much by waves, how much by current friction? When waves break, a small fraction of the energy is used to produce particles and droplets that can be ejected into the atmosphere with a positive electric charge [ I l l . A polarized electric field represents a higher state of order, the charge separation therefore decreases entropy, though the positive entropy change caused by mechanical dissipation is, of course, much larger. The evolution is analogous to that of biological systems which are characterized also by a local decrease in entropy a t the expense of a greater increase in the surrounding universe. Is there any general law which governs such localized organization and entropy reduction? By raising questions of this type-questions about the time-dependent adjustment between perturbations, or the production rate of entropy in coupled processes and its localized decrease-one could be led by the study of air-sea interactions to the very edge of our knowledge about the physical world. ACKNOWLEDGMENT The author is indebted to Claes Rooth for many discussions and for reading the manuscript of this article. The combined support, under Grant GP-4711, by the At,mospheric Science Section of thc National Science Foundation and by the U.S. Army Electronics Command is acknowledged.

APPENDIX The Dimensional Form of the Gravity Wave Spectrum and Radiation Stress

When a wind blows sufficiently long or far over the ocean, the spectral energy density becomes asymptotically independent of the fetch. The sea

248

E. B. KRAUS

is then called “fully developed.” The energy distribution in the high frequency part of the spectrum is determined in this case primarily by the effect of breaking, which does not permit waves of a given length and frequency to grow beyond a limiting height. This limit depends only on the orbital and gravitational accelerations, regardless of wind velocity. Phillips [43] concluded on the basis of dimensional arguments and observations, that the spectral density in the “equilibrium range” (A.1)

2 c9%-5

I)&,

with the constant C z 7.4x This relation appears to conform reasonably with observational results by Burling [44],Kitaigorodsky and Strelakov [45], Darbyshire as quoted by Burling [46], and others. The expression (A.l) is not applicable to the low frequency, long wave part of the spectrum, which contains most of the energy and which is known to be a function of wind strength. A formula proposed by Neumann [47] covers the whole range but does not agree with Phillips’ dimensionally plausible formula in the equilibrium range. An expression which assumes the form (A.l) a t high frequencies, and can also be fitted to the wind-dependent low frequency components, may be derived if it is considered that there is none or a negligible amount of power in waves that move faster than the wind U a t anemometer levels or a t some other arbitrary greater height. There exists, therefore, for all practical purposes a limiting lowest frequency

Change in the time- and space-dependent density of the frequency spectrum can be associated with the dimensional equation:

where f,,is an unspecified function of its argument in brackets. Term (I) represents the local change; (IT) the change due to advection and dispersion; (III), the mean energy input from the wind through the Miles process; and (IV), t,hc effect of breaking and dissipation assumed to depend only on gravitational and orbital accelerations. The effect of resonance with air turbulence and of nonlinear interactions has not been included in equation (A.3). If there is such a thing as a fully developed sea, it must be characterized by the disappearance of the terms on the left-hand side of (A.3). The right-hand side must then vanish as well.

WIND STRESS ALONG THE SEA SURFACE

as t --f

00,

249

which can be solved for ljlwto give

The unspecified function! is always smaller than one. Its limits aref = 0 for 1 for w + CQ. With

w =wga n d f =

it is possible to write

(A.7)

+w

dw ot

-

~ ~ ~ * edey ( e )

From 1 2 8 > 0 and the limits off it follows that 0 d eY(0) < 1, and that e3f(0)= O for 0 = 1 and 0 = O . The total energy of a fully developed sea can be obtained by integration over all frequencies:

In a similar way, the total resultant wave momentum

The last two results show the total energy and momentum, per unit area, of a fully developed sea to be proportional to the fourth and third power of the wind. Both results might have been derived directly from dimensional analysis. The present derivation provides, however, some additional information. It establishes a consisttent relationship between the total energy and Phillips' relation (A.1);it also permits us to make additional predictions about the form of the spectrum which can be checked against observations. To do this, consider that the function 03f(0) must have at least one maximum and one inflection point between its limits. The location of these is a function of 8 only. It follows immediately that the spectrum has a t least one peak, with a frequency wmaxwhich, like that of the inflection point, is inversely proportional to U :

(A.lO)

250

E. B. KRAUS

Evidence for this relationship is shown in Fig. 14 which is based on data published by Defant [37]. 25

I

1

I

I

I

I X

20

1

-

-

X

X

'5-

-

X

X

3

-

X

loX X

5-

x x m

-

m

m I

I

I

I

I

Fro. 14. Maximum period (croesee)and average period8 (dote) (after Defant [37, p. 641).

The argument can be extended to the two-dimensional wave number spectrum. To do so, the assumption (A.l) is generalized and it is stipulated that waves that propagate into a direction a with the wind cannot move faster than U COB a.There is therefore no energy in waves that propagate into a direction which differs from the wind direction by more than C,ln

a . = arc coa -

(A.ll)

U

and such waves cannot exist. The wave number of the longest waves in the direction a is (A.12)

k O

- U'

9 cos2 a

Proceeding as before, one sets: (A.13) (A.14)

The spectrum becomes then

k = ko15 dk = - (ko/$)d{

WIND STRESS ALONG THE SEA SURFACE

25 1

With f increasing again monotonically from 0 to 1 as 5 decreases from 1 to 0. Combination of equations (A.13), (A.14), ((3.18),and (3.20) yields (A.16)

E,

=

:j

Ekordk = const

9

M,

(A.17)

pw u4 cos4u

= const

pw

U 3 C O S ~a 9

I n reality it is not possible for k to reach very high values because such waves would move again faster than the wind. In fact it can be shown easily that there are two limiting values for k; one expressed approximately by equation (A.12) and a second larger one by (A.18)

k, x

p U2cos2a U

- ko

Existence of this upper limit does not affect the quantities E and M significantly because inequation (A.15)at high wave numbers f([)is of order unity while k - 3 or [ go toward zero. In physical terms this means that the contribution of the capillary waves to the energy density per cm2 of sea surface or to the integrated momentum is small. From equations (3.19), (3.21),and (3.23) one can see, however, that the contribution of these waves to the surface velocity or the surface energy density or the surface radiation stress may not be small. For example, combination of equation (3.23) with equation (A.15) gives (A.19)

s:.’y~,(O)

dk da =

l1

(pwgK2

+ 3 ~ ) f ( 5d)k d a

which becomes large with increasing k,. The last equation has been evaluated for high wave numbers where f ( 5 ) + 1 . The evaluation is simple but cumbersome and is therefore not reproduced here. It lends some support to the qualitative considerations of Section 4. Actual figures have little meaning, however, without quantitative information about spectral wave attenuation by slicks which was not available to the author. LISTOF SYMBOLS A a C c,c C

cp + cm c9

Wave amplitude at surface (L) Wave amplitude at depth z ( L ) Empirical constant Phase speed (LT-1) Specific heat of ma water (cal M-1 K-1) Specific heat of air at constant pressure (cal M-1 K-1) Group velocity (LT-1)

253

kmin

1

L

3 &

a

B

E. B . KRAUY Velocity of slowest possible wave Drag coefficient Wave energy in a prism of unit area and infinite depth ( M T - 2 ) Force exerted by radiation stress (AILT-2) Unspecified function of nondimensional argument Gravitational acceleration ( L T - 2 ) Acccleration due to gravity plus surface tension (LT-2) Flux of sensible hcat (cal L-ZT-1) G.1 Coefficient of eddy viscosity ( L z T-1) Wave number (L-1) Wave number of longest existing wave in direction a Wave number of slowest possible wave Diffusion length for waves Monin-Obukhov length Wave impulse across strip of unit width and infinite depth (AIL-IT-1) Wave impulse per unit area (ML-2T-1) Wave impulse per unit area at z = 0 Unit vector normal to wave front Pressure (ML-IT-2) Evaporation ( M L - 2 T - I ) Reynolds number Critical Reynolds number Flux Richardson number Airlwator molecular weight ratio minus one Radiation vector (Poynting vector) (MT-3) Element of unit nrea of sea surface Salinity Temperature Wave period Time Mean wind and current velocities (generally a function of z ) (LT-1) Mean surface velocity Perturbation velocities in air and water Friction velocity Volume Velocity of boundary normal to itself Vertical velocity Amplitude of vcrtical velocity associated with wavcs ( L T - ' ) Vertical coordinate Critical level Roughness length Angle between wind and wavc speed Coefficient of thermal cxpnntrion (K-1) Radiation stress ( M L - I T + ) Interval Thickness of viscous sublayer (L) Profile inflection height for waves of frequency ( L ) Energy donsity ( M L - I T - 2 ) Energy density at z = 0

WIND STRESS ALONG THE SEA SURFACE

253

Ratio kolk Vorticity associated with shear of mean flow U (T-1) Ratio WO/W Dissipation per unit volume ( M L - I T - 3 ) von K&rm&n’sconstant Wavelength (L) Ratio of diffusive capacities in air and water Kinematic viscosity (L2T-1) Stability parameter Density of air or water ( M L - 3 ) Surface tension ( M T - 2 ) Stress (ML-12’-2) Latitude Frequency spectrum (LZT) Wave number spectrum (Ls) Wave frequency (T-1) Frequency of longest existing wave.

REFERENCES 1. Tomczak, G. (1963). Neuere Untersuchungen mit Treibkorpern zur Bestimmung

2. 3.

4.

5. 6.

des Windeinflusses auf Oberfliichenstromungen im Meer (Zusammenfassung). Ber. Deut. Wetterdienstes 12, No. 91, 57. Roll, H. U. (1965). “Basic Concepts of the Physics of the Marine Atmosphere,” Vol. 7. Academic Press, New York and London. Kitaigorodsky, S. A., and Volkov. Yu. A. (1965). On the roughness parameter of the sea surface and the calculation of momentum flux in the near-water layer of the atmosphere. Zzv. A m d . Sci. USSR Atmospheric and Oceanic Physics Ser. (Engltkh Trand.) 1, No. 9, 973-988. Kraus, E. B. (1966). Aerodynamic roughness over the sea surface. J. Atmospheric S C ~23, . 443-445. Priestley, C. H. B. (1959). “Turbulent Transfer in the Lower Atmosphere,” 130 pp. Univ. of Chicago Press, Chicago, Illinois. Swinbank, W. C. (1964). The exponential wind profile. Quart. J. Roy.Meteorool. Soo.

99, 119-135. 7. Deacon, E. L., Sheppard, P. A., and Webb, E. K. (1956). Wind profiles over the ma and the drag a t the sea surface. AustraZianJ. Phya. 9, 61 1-541. 8. Bagnold, R. A. (1941). “The Physics of Blown Sand and Desert Dunes,” 265 pp.

Morrow, New York. 9. Monahan, E. C. (1966). A field study of sea spray and its relationship to low elevation

wind speed-preliminary results. Thesis submitted to M.I.T. 10. Longuet-Higgins, M. S., and Stewart, R. W. (1964). Radiation stress in water waves.

Deep Sea Res. 11, 529-562. 11. Blanchard, D. C. (1963). The electrification of the atmosphere by particles from bubbles in the sea. “Progress in Occanography,” Vol. 1, pp. 71-202. Pergamon Prcss, Oxford. 12. U‘anz, G. M. (1963). Acoustic ambient noise in the ocean: Spectra and sources. J. AcOUt9t. Soc. Am. 34, 19361956. 13. Stewart, R. W. (1961). The wave drag ofwind over water. J . Fluid Mech. 10,189-194.

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14. Kinsman, B. (1965). “Wind Waves.” Prentice-Hall, Englewood Cliffs, New Jersey. 15. Carstens, T. J. (1964). Stability of shear flow near the interface of two fluids. Tech. Rept. HEL-7-1, University of California, Berkeley, California. 16. Lin, C. C. (1955).“Theory of Hydrodynamic Stability.” Oxford Univ. Press, London and New York. 17. Phillips, 0. M. (1957). On the generation of waves by turbulent winds. J. Fluid Mech. 2. 417-445. 18. Kelvin, Lord (1871). Hydrokinetic solutions and observations. “Math. and Phys. Papers,’’ p. 4. Cambridge Univ. Press, London and New York, 1910. 19. Jeffreys, H. (1925).On the formation of water waves by winds. Proc. Roy. SOC.(London) A107, 189-206. 20. Munk, W. H. (1965). Wind stress on water; an hypothesis. Quart.J. Roy. Meteorol. SOC.81, 320-332. 21. Cox, C. S., and Munk, W. H. (1954). Statistics of the sea surface derived from sun glitter. J . Mar. Res. 13, 198. 22. Miles, J. W. (1967). On the generation of surface waves by shear flows. J. Fluid Mech. 3, 185-204. 23. Lighthill, M. J. (1962). Physical interpretation of the mathematical theory of wave generation by wind. J . Fluid Mech. 14, 385-398. 24. Longuet-Higgins, M. 5. (1962). The directional spectrum of ocean waves, and processes of wave generation. Proc. Roy. SOC.(London),Ser. A 265, 286-315. 25. Roll, H. U. (1952). Uber Grossenunterschiede der Meereswellen bei Warm und Kaltluft. Deut. Hydrog. 2. 5, 111-114. 26. Fleagle, R. G. (1956). Note of the effect of air-sea temperature difference on wave generation. T r a m . Am. Ueophye. Un. 37, 275-277. 27. Deacon, E. L., and Webb, E. K. (1962). Small scale interactions. “The Sea,” Vol. 1, pp. 43-66. Wiley (Interscience), New York. 28. Neumann, G., and Pierson, W. T. (1963). Known and unknown properties of the frequency spectrum of a wind generated sea. Proc. Conf. Ocean Wave Spectra, Prentice-Hall, London, 1961, pp. 9-25. 29. Dorrestein, R., and Groen, P. (1958). Zeegolven, Kon. Ned. Met. Inst. Rept. No. 11. 30. Neumann, G. (1966). Wind stress on water surfaces. Bull. Am. Meteorol. SOC.37, 21 1-217. 31. Barnett, T. P., and Wilkerson, J. (1966). The measurement and interpretation of fetch limited wind wave spectra. (In press). 32. Kinsman, B. (1960). Surface waves a t short fetches and low wind speed-a field study. Chesapeake Bay Inst., Tech. Rept. 19. 33. Takeda, A. (1963). Wind profiles over sea waves. J. Oceanographic SOC. Japan 19, 16-22. 34. Miles, J. W. (1965). A note on the interaction between surface waves and wind profiles. J. Fluid Mech. 22, 823-827 35. Charnock, H. (1955). Wind stress on a watcr surface. Quart. J . Roy. Meteorol. Sop. 81, 639-640. 36. Sverdrup, H. U. (1946). “Oceanography for Meteorologists.” Allen C Unwin, London. 37. Defant, A. (1961). “Physical Oceanography,” Vol. 2. Pergamon Press, Oxford. 38. Garrett, W. D. (1965).The damping of capillary waves a t the air/water interface by naturally occurring surface-active material. U.S. Naval Res. Lab. Rept. 6337. 39. Welsnder, P. (1903). On the generation of wind streaks on the sea surface by action of surface film. Tellus 15. 67.

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40. Stuart, J. T. (1965). The production of intense shear layers by vortex st,retching and convection. Natl. Phys. Lab., Qt. Brit., Rept. 1147. 41. Faller, A. J., and Kaylor, R. E. (1965). A numerical study of the instability of the laminar Ekman boundary layer. Tech. Note BN-410, University of Maryland. 42. Kitaigorodsky, S. A. (1962). Applications of the theory of similarity to the analysis of wind-generated wave motion aa a stochastic process. Bull. Acad. Sci. USSR, Qeophys. Ser. pp. 105-117. 43. Phillips, 0. M. (1958). The equilibrium range in the spectrum of wind generated waves. J. Fluid Mech. 4, 426-434. 44. Burling, R. W. (1955). Wind generation of waves on water. Ph.D. Dissertation, Imperial College, London. 45. Kitaigorodsky, S. A,, and Strelakov, S. S. (1963). On the analysis of spectra of wind-generated wave motion 11. Bull. A d . Sci. USSR, Geophys. Ser. pp. 1240-1250. 46. Burling, R. W (1963). Proc. COT$ Ocean Wave Spectra, Prentice-Hall, London, 1961, pp. 31-32. 47. Neumann, G. (1954). Zur Charakteristik des Seeganges. Arch. Meteorol. Ueophys. Biokl. AT, 352.

Note added i n proof: Attention is drawn to the Eollowing book which appeared recently. 48. Phillips, 0. M. (1966) “The Dynamics of the Upper Ocean.” Cambridge Univ. Press, London and New York.

This Page Intentionally Left Blank

SEVERE CONVECTIVE STORMS Chester W . Newton National Center for Atmospheric Research. Boulder. Colorado Page 1 . Introduction .......................................................... 267 269 2 . General Thunderstorm Structure ........................................ 262 3 . Modes ofConvection ................................................... 262 3.1. Parcel Convection .................................................. 264 3.2. Modified Concepts of Convection .................................... 266 3.3. “Bubble” Convection ............................................... 268 3.4. Continuous Draft Convection ....................................... 3.5. Remarks on the Different Viewpoints ................................. 269 4 . The Severe Thunderstorm Environment and Its Modification . . . . . . . . . . . . . . . . 270 5. Thunderstorms in a Sheared Environment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273 5.1. Dynamical Interaction between Storm and Environment . . . . . . . . . . . . . . . . 274 277 5.2. Organized Convective Circulations ................................... 281 5.3. “Steady State” Severe Storms ...................................... 5.4. Size Sorting and Recirculation of Precipitation Particles . . . . . . . . . . . . . . . . 284 6 . Storm Movement ...................................................... 287 287 6.1. Single-Celled Storms ............................................... 6.2. Influence of Propagation on Movement of Large Storms ................ 287 6.3. Size Discrimination in Relation t o Water Budget ...................... 288 7 . Squall Lines ................................... .................... 291 291 7.1. General Structure ................................................. 7.2. Migration of Storms within Lines and Regeneration Pattern ............ 293 7.3. Over-all Aspects of a Mature Squall-Line System ...................... 294 8 . Severe Weather Manifestations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296 296 8.1. Hail .............................................................. ................................................ 297 8.2. Lightning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297 8.3. Surface Winds 298 8.4. Tornadoes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 300 8.5. HeavyRains ...................................................... 300 8.6. Hazards to Aircraft ................................................ 9 Conclusion . . . ...................................... ................................................. 303 ....................................................... 303

.

1. INTRODUCTION

One of the most majestic sights afforded by nature is a giant cumulo-nimbus cloud. with its boiling tower’sextending high into the atmosphere. its spreading anvil covering hundreds of square miles. and. a t night. its fantastic displays of lightning. At the same time t h e thunderstorm manifests. along with the hurricane. one of the most destructive forces of nature . 267

258

CHESTER W. NEWTON

Strong winds, lightning, and occasional tornadoes do extensive damage to trees and structures and cause many injuries and deaths annually. Flash floods overload the capacities of drainage systems. In some areas, hail causes greater than ten per cent loss of the annual value of crops. Severe turbulence and other phenomena in clouds make it extremely hazardous for aircraft to fly through them. With the increased density of air traffic, and the use of higher speed aircraft which are subject to more severe buffeting, it becomes more and more necessary to have knowledge of the distribution and intensities of convective storms, their movements, and the probable future patterns of their development. Although our present understanding of these storms is in some respects rudimentary, it is vastly advanced over the knowledge of even twenty years ago. At that time, the prevailing viewpoint was that convection was largely a random process that took place when the air mass was potentially unstable and enough solar heating of the ground took place to set off the instability. The existence of large-scale convection in the form of squall lines was recognized but only beginning to be understood. Visual observations were inadequate t o describe even the major aspects of storm distribution. Increased knowledge evolved largely through technological advances. The problem of storm observation was largely solved through the introduction of weather surveillance radar. Augmented networks of radiosondes, and the introduction of radar and radio balloon tracking devices, made it possible t o secure much improved analyses of the thermal structure of the atmosphere, its moisture distribution, and the pressure and wind patterns aloft. It became recognized that successful weather prediction depends on an awareness of the whole three-dimensional structure of the atmosphere. I n the study of convective phenomena, increased attention was given to the ways in which the airmass structure is modified, not by solar heating alone but also by the slow but persistent upward and downward motions accompanying cyclones and anticyclones, and the horizontal advection of heat and moisture. The development of thunderstorms thus came t o be visualized as the endproduct of a grand chain of physical processes, which assume varying degrees of importance. Different types of storms develop under different environmental conditions, ranging from the sporadic formation of short-lived convective cells in a nearly random fashion, to the iiearly steady state, long-lived, and highly organized heat engine characterizing the most sverc? storms. I n this review, an attempt will be made to sketch the main processes pertinent to convective development with emphasis on the structure and mechanics of the storms once they have formed. For discussions of important related aspects not treated here, the reader is referred to reviews published elsewhere [1-31,

SEVERE CONVECTIVE STORMS

259

THUNDERSTORM STRUCTURE 2. GENERAL The general nature of the thunderstorm circulation was described by Suckstorff [4] on the basis of surface observations. Much of our present knowledge of the three-dimensional structure has been derived from field measurements by the U.S. Weather Bureau-Air Force-Navy-NACA Thunderstorm Project in 1946 (Florida) and 1947 (Ohio). These investigations dealt mostly with large cumulus clouds and relatively small thunderstorms of the so-called air-mass type which form as a result of surface heating. Their characteristics are not in all respects similar to those of the larger and more intense storms to be discussed later, but they serve to bring out some of the main aspects of convection. Figure 1 shows a vertical slice through a storm in the “mature” stage [ 5 , 61. I n the preceding “cumulus” stage, the motions are predominantly upward throughout the cloud, as in the left portion of Fig. 1. As indicated by the isotherms, the rising air is warmer than the surroundings, due to the relearre of latent heat of condensation as the air undergoes decompression and cooling during its ascent. The initiation of the downdraft (right side of Fig. 1) was ascribed to the buoyancy being overcome by the weight of accumulated liquid water in some portion of the upper part of the cloud. Chilling by partial evaporation of cloud water, into dry air drawn into the sides of the cloud, results in the downdraft air becoming cooler than the surroundings, and a downward acceleration is imparted to the downdraft air. In the “dissipating” stage, which occurs when the storm has exhausted the supply of warm and moist air in its near environment, and therewith its supply of heat and of vapor for the production of cloud water, the whole cell is occupied by gentle downdrafts. Many thunderstorms contain several cells, as in Fig. 2, in different stages of development. From aircraft measurements [5],the modal width of updrafts was found to be 5000 ft and that of downdrafts 4000 ft. The modal draft speeds (averaged across the draft) were 15-20 ft/sec, increasing somewhat with elevation up to 25,000 f t (the maximum updraft measured was 84ft/sec). The average storm was found to be about as wide as it is high. Radar observations show that the lifetime of an individual convective cell is of the order of half an hour [7, 81. They also show that new cells tend to form most readily in the near neighborhood of pre-existing cells. This is explained as a consequence of the fact that the downdraft air (Fig. 1) must spread out laterally on approaching the ground. Thus, as illustrated by Fig. 3 [9], the region beneath a precipitating storm is occupied by diverging cold air with a depth of several thousand feet. This air, acting as a miniature cold front, converges with the warm air of

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CHESTER W. NEWTON

the surroundings and may lift it sufficiently for condensation to occur. In this manner, a new convective cell is initiated, which in turn goes through the life cycle described earlier. The structure of the storm continually changes as

HORIZONTAL SCALE ORAFT V E T O I SCALE

0 I

o&%lSEC

cn

M

RAIN I

I

SNOW

.

*

ICE CIVSTALS

FIG.1. Circulation in a thunderstorm cell in the mature stage (from General Meteorology, Byers [a], copyright McGraw-Hill,used by permission).

new cells are born and old ones perish, and a multicellular storm as a whole may persist for several hours even though the lifetime of each of its cells is limited. As Battan [ 101 demonstrates, the development of a thunderstorm to great heights is accomplished “in a stepwise fashion as a series of towers

261

SEVERE CONVECTIVE STORMS

0 CELL OUTLINE DOWNDRAFT

I

10

L O

MILES

Q

Fro.2. Plan view of a typical multicellulttr thunderstorm (from Ueneral Meteorology, Byers [6], copyright McGraw-Hill, used by permission).

LOOOFT IEMPERATURf

Fro. 3. Surface wind field bcneath a thunderstorm a t two times (bottom).Each feather on wind symbols (on end from which wind is blowing) represents 5 mph. Solid lines are isotherms ("C); dashed lines, a wind discontinuity near edge of cold air. Upper part show8 temperature profilcs a t surface and 5000 ft, along heavy lines shown a t bottom (after Fujita [9]).

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CHESTER W. NEWTON

forms in quick succession with each tower reaching a greater height than the preceding one.”

3. MODESOF CONVECTION Cumulus clouds and thunderstorms are mechanisms by which heat is transferred from the surface layers, where the heat gained from the sun exceeds that lost by radiation, t o the upper troposphere where there is a net radiative deficit. The heat transferred i s partly sensible and partly latent in water vapor which originates by evaporation from the earth’s surface. Condensation of the vapor, with release of heat to the air, is the essential driving process for the upward motions in convective clouds, and cooling by evaporation is the essential process for downward motions. It is possible to imagine several different modes of convection; the particular mode selected by nature has a strong influence on the intensity of convection, the height to which it penetrates, and the production of precipitation. 3.1. Parcel Convection

The simplest notion of buoyant convection is the “parcel” concept [ l l , 121. An element of air is considered to rise without exchanging heat, water, or momentum with its surroundings. If unsaturated, it cools at the dry adiabatic rate of 9.8”C/km ( A J in Fig. 4); if saturated, the release of heat of condensation

Temperature-

FIG.4. Illust,rating the thermodynamic processes in convection. Heavy line ABCD represents temperature sounding in environment of cloud whose base is at A . For definition of other lines, see text. This diagram is schematic with certain features exaggerated for the purpose of illustration.

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SEVERE CONVECTIVE STORMS

diminishes the cooling rate, and the parcel cools moist-adiabatically (line AECF). If the lapse rate of the surrounding air mass exceeds the rate of cooling of the parcel through part of the layer through which it ascends, the parcel will become warmer than its surroundings and buoyant. Under the assumption that the vertical pressure gradient inside a cloud equals that outside it, the vertical acceleration is given by

dw _

AT dt-’T

Here, w is the vertical velocity, g the acceleration of gravity, and A T the excess of the virtual temperature of the parcel over that of the environment. Integration of equation (3.1)with height gives

):(

A -

=g

E A Tdz = -JpoR P AT d l n p

where A(w2/2)denotes the change of kinetic energy between the level of origin z, and another level z (above or below), R is the gas constant, and p is pressure. As in Fig. 4, the troposphere is in convective situations characterized by a lapse rate which is usually greater in lower than in upper levels. This being the case, AT achieves a maximum positive value somewhere in the troposphere. According to equation (3.2),a parcel gains kinetic energy up t o some level where AT is zero, and loses it above that level. The parcel can penetrate t o a level below which the total “positive area” P equals the “negative area” N. TABLEI. Approximate properties of a convective cloud if the parcel theory were obeyed.”

AT “C

1 2 3 4 5

Maximum Penetration into Maximum AT vertical velocity stratosphere, in stratosphere, “C (at tropopause), meters meterslsec 28 40 49 66 63

1000 1410 1730 2000 2240

-9.8

- 13.8

- 17.0 - 19.6 -22.0

a FT is the mean excess of parcel over environment temperature while in the troposhere. The tropopause, where there is zero buoyancy, is taken to be 10 km above cloud brtse; the mean temperature in the troposphere is - 20°C and in the isothermalstratosphere - 60°C.

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CHESTER W. NEWTON

In light of the assumptions employed, it is clear that equation (3.2) gives an upper limit for the intensity of vertical motion, and for the height to which the convection can penetrate. Petterssen et al. [13] found that the tops of convective clouds over England on the average extended only slightly above level E in Fig. 4, where according to the parcel theory AT is a maximum and the upward acceleration should be greatest. Figure 6 [14] indicates the same feature, with only a few cloud tops surpassing the level of neutral buoyancy.

1 J TEMPERATURE 'C

Ib 2b 3'0

I

I

1

305 ECHOES

4'0 50 60 FREQUENCY

i o do

FIG.6. Frequency distribution of maximum heights reached by radar echoes on 10 days in Texas convective situations. In left part, mean temperature (solid) and dewpoint (dashed) sounding; dash-dotted line shows cloud temperature according to parcel method (after Clark [la]).

As will be indicated, the degree to which the parcel concept is valid is determined to a large extent by the horizontal dimensions of a cloud and by the location within the cloud of the vertical drafts. As a frame of reference, it is useful to note, as in Table I, some characteristics of a cloud which would result if the parcel theory were valid. 3.2. Modified Concepts of Convection Bjerknes [la] and Petterssen [I61 introduced the idea that the amount of air moving upward in cumulus clouds would have to be compensated by an equal downward mass transport between them. The descending air warms by adiabatic compression, thus diminishing the temperature excess of the clouds and their buoyancy. Schmidt [17] was the first to discuss the additional influence of aerodynamic drag in retarding the vertical accelerations of clouds.

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SEVERE CONVECTIVE STORMS

Normand [12] and Stommel[18] initiated the concept of entrainment, which has revolutionized our thinking about convective processes. Outside air is drawn laterally into the rising column of air (Fig. 6a). Since this air is cooler

\

MOTION RELATIVE TO GROWING THEW&

I

I

MOTION RELATIVE

I

To A

a 1

-4 I I

f

I

+ ‘

d (bl

FIG.6. (a) Entrainment into a rising jet, according to Stommel [18]. (b) Circulation in a “t,hermal” or cloud bubble. The black areas indicate successive locations of a fluid element which is deformed as it circulates from the cap into the interior (after Scorer [22], with permission of Cambridge Univ. Press).

than that within the cloud, and is unsaturated, the cloud air is chilled both by mixing of the sensible heats and by partial evaporation of cloud water into the entrained air. Rising parcels then cool a t a rate (dashed line A B in Fig. 4) greater than the moist adiabatic rate dictated by the parcel theory, with a smaller excess temperature. Thus according to equation (3.2) the kinetic energy in the updraft is diminished and the height of penetration of the convection is likewise limited. I n the parcel theory, it was also supposed that descending air within the cloud, having retained its moisture which can be re-evaporated, would also warm a t the moist adiabatic rate BC. Normand [12] observed that entrainment of dry air into a saturated downdraft would further chill the air within it. As a result of entrainment into both the updraft and downdraft, the downdraft air (line BH in Fig. 4)arrives a t the surface with a temperature lower than the air hich initially entered the updraft. I n the precipitation region beneath a thunderstorm or shower cloud, it is consequently colder than in the surroundings (Fig. 3). The process of entrainment thus explains two important aspects of convection: the fact that cumulus clouds do not all attain the heights predicted by the parcel theory. and the coldness of the downdraft which enhances the vigor of its descent. Interactions between the cloud and its environment are also very important for the water budget of the storm. For a storm typical of those measured by

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CHESTER W.NEWTON

the Thunderstorm Project, Braham [19] found that surface rain accounted for only 11 yo of the water vapor flowing into the storm. Of the 60 % of this which condensed, only about one-fifth reached the ground as rain, nearly half was re-evaporated into the downdraft, and the rest was evaporated at cloud edges or remained in suspension as cloud droplets. Computations suggest (see Atlacl et al. El, p. 451 and Fankhauser [20]) that large convective systems such as squall lines are much more efficient as rain producers. 3.3. “Bubble” Convection

Scorer and Ludlam [21] advanced the theory that the active convective elements in a cumulus cloud are discrete “bubbles” of buoyant air, which become evident by the intermittent appearance of semispherical protuberances a t cloud tops. The general form of a laboratory-produced “thermal” or buoyant bubble is illustrated in Fig. 6b. It consists of a ring vortex with upward motion in its buoyant interior and weaker downward motions around its edges. The maximum upward fluid velocity near the axis of the bubble is about twice the rate of advance of the bubble as a whole. As indicated by Fig. 6b, turbulent mixing occurs a t the advancing interface. Mixed air then circulates from the cap, around the lateral edges, and into the interior. The time required for mixed air to enter the core corresponds approximately to the time taken for the bubble t o rise a distance of twice its diameter, so that a large bubble may rise a considerable distance before the core becomes diluted. In the early history of a cloud, it is visualized as consisting of bubbles which penetrate only a limited distance upward into the dry surroundings. The erosion of these predecessor bubbles results in a warming and moistening of the space they occupy. Later bubbles rising from the same cloud base can then pass through a cloudy environment whose properties are not very different from their own, and they are consequently not greatly chilled by mixing until they emerge into the dry air above the previously cloudy space. Thus in a train of bubbles rising through the same channel, each can rise successively higher before suffering erosion, and the cumulus cloud can gradually extend itself to great heights. The behavior of bubbles produced by releasing a mass of buoyant fluid in a water tank has been studied by Scorer [22]. They were found to expand through a conical path, according to the law

(3.3)

r=Qz

where r is bubble radius, z is the height from t h e source, and a is a broadening coefficient defincd in Fig. 6b (this being a measure of the ratc of entrainment). The rate of bubble rise W obeyed a relationship between the drag and the

SEVERE CONVECTIVE STORMS

267

buoyancy force, (3.4)

w =C(gBr)"Z

where g is gravity, B the average buoyancy analogous to ATIT in equation (3.1),and C is a constant. It is significant that both in Scorer's experiments, and in those of Turner [23] wherein release of latent heat was artificially simulated, the linear spread indicated by equation (3.3) was found. These experiments encompassed cases in which deceleration, constant speed, or acceleration of the bubble took place, and Q varied little from about 0.20 t o 0.25. This suggested that the entrainment rate is essentially constant for a wide range of degrees of buoyancy or of static instability in the bubble environments. It is remarkable that Saunders [24] determined, from measurements of the diameters of towers emerging from the tops of cumuli, that these also increased according to equation (3.3)with u z 0.2 which is near the value determined in the laboratory experiments (the smaller value being attributed t o erosion). Furthermore, Saunders found the broadening coefficient insensitive to the stability and relative humidity of the environment in the cloud layer. Equation (3.4) indicates that the vertical velocity achieved increases both with the buoyancy and with the size of the bubble. From measurements of the rate of rise of isolated cumulus towers soon after emergence from the parent cloud, Malkus and Scorer [25] found that this relationship was fairly well obeyed. They also observed that towers, after first emergence from the general cloud mass and exposure t o a dry environment, ceased to rise after ascending about 1.5 diameters. Saunders [24] found a mean value of C in equation (3.4)of 1.5, but with large variations which he attributed to differing buoyancies of individual cloud bubbles. The buoyancy acceleration of convective bubbles is significantly diminished by "form drag" and by incorporation of motionless air from the surroundings. According to Levine [26], (3.5) for a bubble exchanging mass with the environment a t a rate specified by K,; C , is a drag coefficient and the other symbols are defined as in equation (3.4)which is, with certain assumptions, analogous. Properly, B should also include the weight of suspended water; each 4 gm of water per kilogram of air decreases the buoyancy by an amount roughly equivalent to decreasing the excess temperature AT by 1°C. Equation (3.5) shows that as the size of an ascending convective element increases, the influence of drag and entrainment (second term) decreases.

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CHESTER W. NEWTON

Thus the larger the element, the more closely does the vertical acceleration approach that described by the parcel theory, equation (3.1).

3.4. Continuous Draft Convection

At the other extreme from bubble convection is the columnar or continuous draft form, on which viewpoints differing in important respects have been taken. Houghton and Cramer [27] made the assumption that the cloud could be represented by a circular cylinder. Then in steady state an increase of upward motion with height requires, according to mass continuity, a drawing-in of air horizontally through the cylinder walls. Mixing of this air into the convective column diminishes its buoyancy. The amount of entrainment must fit the vertical acceleration and, for a given environmental stability and humidity, is prescribed. An entirely different viewpoint on the mechanism of entrainment was advanced by Morton et al. [28]. According to this, the mixed air of a turbulent sheath is incorporated into a jet a t a rate proportional to the jet velocity. Then if the added mass is also proportional to the surface area, the continuity relationship is

where a is the angular spread of the jet analogous to a in equation (3.3), p is the density of the fluid in the jet, and po that of the outside fluid. Here (.rrr2wp) is the mass flux through a cross section normal to the jet axis. The quantity on the right is the mass incorporated laterally through the walls of a slice of unit thickness, (aw)being the effective radial velocity with which a conical jet extends itself laterally (for a “plume” or jet from a continuous source, experiments indicate that a z 0.1, significantly smaller than the broadening coefficient for discrete “bubbles”). If we write M = .rrr2wp, and po z p, equation (3.6) may be rewritten

(3.7) This equation brings out the important fact that the mass entrainment rate decreases as the diameter of the plume increases (because the surface-tovolume ratio decreases with increasing diameter). Squires and Turner [29] utilized the above hypothesis to construct a n “entraining jet” model. I n this model, the mass entrained is according to equation (3.6) a function of the entrainment constant a, the radius of the draft, and the vertical velocity. The vertical acceleration is specified

SEVERE CONVECTIVE STORMS

269

(a 1 FIQ.7. Radius of updraft (a), excess temperature (b), and vertical velocity (c), from a numerical experiment by Squires and Turner [29]. In case A , the radius at cloud base was 0.5 km; in case B , 1 km, and in case C, 2 km, with updraft rising through saturated environment. Curves B' are for case B with 50% relative humidity in environment. Dotted line in ( c ) shows effect of allowing part of the cloud water to freeze. In (b), heavy dots show altitudes at which maximum updraft velocity resulted.

essentially by the environmental lapse rate and humidity, and by the rate of entrainment. The assumption of a particular mechanism of entrainment eliminates the necessity of assuming a special shape of the updraft. The influence of the updraft radius on its properties is demonstrated by Fig. 7. As the radius a t cloud base is increased, the buoyancy increases, and both the maximum upward velocity and the height of the level a t which it is attained are increased. The liquid-water content also increases with draft radius, proportionately less cloud water being evaporated into the entrained air. Figure 7c shows that low relative humidity in the cloud environment has a marked effect in decreasing the vigor of the updraft and the height to which it extends, in accordance with the findings of Austin [30]. Malkus [31] showed that, in agreement with the above and with equation (3.5),the heights reached by cumulo-nimbus towers increase as their diameters increase. Few measurements exist on the rate of entrainment. Malkus and Williams [ l , p. 601 suggest that (l/M) d M / d z z 1/D, where D is cloud diameter. According t o this, "thunderstorm sized towers ( D up t o 10 km) can shoot through most of the troposphere before experiencing one-to-one dilution. "

3.5. Reniarks on the Different Viewpoints The various concepts of convection differ in fundamental ways. However, they have the common feature that the influence of entrainment diminishes

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CHESTER W. NEWTON

with increasing breadth of the cloud. Bates [32] has suggested that in the large severe storms of the Great Plains, entrainment may be negligible except below cloud base. Thus, as is supported by the considerable degree of penetration of some storm towers into the stratosphere, the unmixed parcel theory may fairly well describe the vertical motions in the core of the draft. It is not necessarily to be expected that the same mode of convection dominates in clouds a t different stages of development, or that the influences of entrainment are the same in all parts of a given cloud. Indications are that the discrete bubble mode characterizes the ordinary cumulus cloud, while many large thunderstorms are dominated by virtually continuous drafts. With regard to the mechanisms operating in the draft form of convection, there are fundamental differences between the viewpoints taken by Houghton and Cramer [27] and by Squires and Turner [29]. I n the former case, the lateral inflow would be greatest below, and zero at, the level of maximum vertical velocity, while in the latter the entrainment would be greatest in the upper middle part of the cloud where vertical velocities are strongest. It is likely that neither of these viewpoints can be excluded. Turbulent entrainment is indicated by the motions of irregularities a t the boundaries of clouds, but systematic inflows and outflows are also shown by wind measurements [6]. The relative extent to which either of these forms contributes t o entrainment has yet t o be investigated. 4. THESEVERETHUNDERSTORM ENVIRONMENT AND

ITS MODIFICATION

It is not the intent of this review t o enter into a discussion of the varied synoptic conditions leading up to the formation of convective storms. Nevertheless, a brief description of one common type of situation is in order, partly because of its intrinsic interest but also because this will set the stage for the discussion which follows. Conditions favorable for the occurrence of severe thunderstorms and tornadoes have been described by Showalter and Fulks [33], Fawbush et al. [34], and others. An idealized weather map near the onset of a severe outbreak is shown in Fig. 8. The significant features are a region of potential instability, with abundant water vapor in lower levels and dry air in the middle troposphere, bands of strong winds in lower and upper levels, veering (i.e., turning in a clockwise manner) with height, and some mechanism which can trigger the release of inetability. I n the situation illustrated, severe storms &re most likely to be initiated near the intersection of the lower and upper level jet streams, and near the west side of the moist tongue. I n discussing the reasons for this, reference will be made to the schematic temperature and dew-point sounding of Fig. 9, which is characteristic of conditions a few hours prior to the onset of severe convection in this region especially in spring.

SEVERE CONVECTIVE STORMS

271

In the lower levels (typically up t o about 6000 f t above sea level) there is humid air which has recently passed over the warm waters of the Gulf of Mexico. The inversion above the moist air is typical of maritime tropical air masses in these latitudes, and is intensified when the air a t the levels concerned has recently traversed the plateau to the west, where it is heated by the surface. Normally the air above the inversion shows a steep lapse of temperature with height through most of the troposphere, with low relative humidity.

FIG.8. Schematic features of a severe weather outbreak. Solid lines are sea-level isobars; dashed lines streamlines of upper tropospheric flow. Shading outlines general area of moist tongue in lower levels; this is in general associated with the region of potential instability.

As observed by Fuiks [35], the inversion plays an important role in contributing to the severity of convective disturbances. So long as the inversion is present, it inhibits the upward penetration of air from the moist layer, in which only stunted cumulus or strato-cumulus can form. The air beneath the inversion can become progressively warmer and more moist through horizontal advection and daytime heating. At the same time, the middle and higher troposphere may become cooler. Thus considering a deep layer, the potential instability can progressively increase without being set off, as long as the inversion is present. Finally, when by some mechanism the inversion is eliminated, explosive overturning can take place. The vigor of this, according to

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CHESTER W. NEWTON

equation (3.2) or (3.5) and Fig. 9, depends on the degree of warming and increase of moisture which has taken place in the lower moist layer, together with the degree of cooling which may have occurred in the upper troposphere. The importance of the inversion for severe weather in the Great Plains area is demonstrated by the mean soundings of Fawbush and Miller [36], which

'\

500 mb

After

lifting

Ground

FIG.9. Schematic distribution of temperature and dew point with height some time prior to (solid lines) and a t the onset of (dashed lines) severe convection. During tho modification between those two states, vertical motions result in temperature changes of parcels shown by triangles, in the manner indicated by the arrows. The temperature of a parcel rising from the moist layer is shown by the line of circles. In. solational heating of lower levels is not taken into account (from Atlaa et al. [ l , p. 341).

show that prior to the occurrence of tornadoes or large hail the inversion is strong, while in the case of small hail no inversion appears and the potential instability is much weaker. Beebe [37] has shown that inversions are not present when and where tornadoes and thunderstorms occur. Beebe and Bates [38] have indicated that one of the principal agents in their removal is organized vertical motion. As indicated in Fig. 9, lifting results in adiabatic cooling of the dry air above the inversion with a simultaneous increase in depth of the moist layer. If, as is likely, the moist air reaches saturation during this process, it will cool a t the lesser moist adiabatic rate being free to penetrate t o high levels once the inversion is removed. The organized vertical motions are, in general, connected with cyclonescale disturbances of the type in Fig. 8, although the upper air perturbation

SEVERE CONVECTIVE STORMS

273

may be of smaller dimensions than that illustrated. Downstream from troughs in the upper troposphere, there is generally horizontal mass divergence in the upper troposphere, coupled with convergence in the lower troposphere (for a general discussion see, e.g., Petterssen [39]). Linked with this, the broadscale upward motions in the middle troposphere commonly have magnitudes of 5 to 10 cmlsec. Acting over a period of 6 t o 12 hr, such vertical motions provide a net lifting of 1 to 2 km, which is sufficient to eliminate even a strong inversion. The upper level divergence is associated with the variation, along the current, of the vorticity (expressed partly by the change from cyclonic or clockwise curvature in the trough t o anticyclonic curvature downstream) and with the strength of the upper winds. For this reason, vertical motions tend to be strongest near the jet stream in upper levels. The low-level jet stream is the region where moist air and heat are advected most rapidly northward, and where as a consequence potential instability is likely first to be generated. These two factors largely account for the tendency for severe convection t o take place first near the intersection of a low-level and an upperlevel jet stream. The onset of convection may be triggered by the gradual upward motions mentioned above, but often other factors come into play. Lifting by the cold front in Fig. 8, as it sweeps into the western edge of the tongue of unstable moist air, may be the decisive mechanism. On other occasions, lifting at the warm front, or over a cold dome residual from earlier thunderstorms may be instrumental. In all cases, diurnal heating or cooling must be taken into account. Surface heating may be sufficient in itself to cause buoyant air to rise to the condensation level and, in the absence of an inhibiting stable layer (Fig. 4),to set off deep convection. Even when dynamical lifting mechanisms are available, solar heating is important because less lifting is needed to set off convection when heat has been added in lower levels. The importance of insolation is attested by the pronounced preference for convective phenomena to occur in rnidafternoon (see Atlas et a2. [ l , p. 1481). In a situation like Fig. 8, squall lines forming near the west side of the tongue of moist unstable air often move eastward through it, finally diminishing when they encounter more stable air on the east side. In some regions, topographic influences arc important either for the formation or dissipation of convective systems [ W , 411. 5. TIIUNDERSTORMS IN A SHEARED EWIRONMENT

As brought out, ill tlie previous section, severe tliunderstorrns tend to occur in localities uhcre there is a strong shear between the winds in the lower and upper troposphere. It is commonly observed that small cumulus clouds, when subjected to appreciable shear, have their tops blown off their bases, and

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CHESTER W. NEWTON

formerly there was a common belief that shear was inimical to the occurrence of thunderstorms. Recent research has disclosed that large storms are, on the contrary, actually invigorated by the presence of shear. I n a sheared environment the storm, moving a t some particular velocity, is in motion relative to the winds a t different levels, and thus is in effect moving through the air mass. Weickmann [42] pointed this out as a distinguishing feature of storms that persist for a long time, by virtue of their being able to sweep up moist air as they migrate through it.

FIG.10. Schematic view of thunderstorm in an environment in which the wind veers with height (see text; Newton [43]).

5.1. Dynamicul Interaction between Storm and Environment Figure 10 [43]shows a simplified view of a storm imbedded in an environment with winds turning with height (say, from south in lower levels and from west near the cloud top). Momentum characteristic of the lower levels is carried up in the updrafts. In the upper part of the cloud, the air tends t o take on the momentum of the upper levels, which in turn is transported earthward in the downdrafts. As a result of vigorous vertical mixing, the vertical shear within the storm is partially annihilated, and the in-cloud air a t both upper and lower levels assumes a mean velocity intermediate between the ambient wind velocities a t upper and lower levels. As a consequence, the motion of the outside air with respect t o the incloud air is as indicated by the broad arrows in Fig. 10. I n relative motion, the lower level air (comprising the moist layer) blows into the right flank of the storm, while the upper level air blows away from that flank. Newton and Newton [44] drew an analogy between the storm column and an obstacle in a wind tunnel. On the relative upwind side of such an obstacle a positive hydrodynamic (nonhydrostatic) pressure is observed, having a value

SEVERE CONVECTIVE STORMS

275

a t the stagnation point, V , being the relative motion. On the downwind side, a negative pressure of about the same magnitude is observed, with negative pressures also on the lateral flanks. According to this analogy, a positive hydrodynamic pressure should be present on the right flank (relative to the mean wind) of the cloud system in lower levels, with negative pressure in upper levels, and vice versa on the left flank. The effect of this induced pressure field is twofold. First, the horizontal pressure-gradient force across the cloud accelerates the in-cloud air along the direction of the relative wind a t a particular level, thus tending to shear the cloud out of the vertical. This tendency is offset by the vertical transfer of momentum. An analysis of the forces involved [44, 451 indicates that with rather modest vertical motions, a storm column may remain erect with relative horizontal motions between cloud and environment in excess of 10 meterslsec. The ability to withstand the shearing forces and the size of the induced pressures depend also on the horizontal dimensions of the storm, according to the proportionality (5.2)

W CC

VR'

CC

(wS)D

where W and S are the vertical motion and the vertical shear, and D is the diameter of the active convective column, ( W S )being a measure ofthe vertical exchange of momentum. The second influence resides in the vertical gradient of the induced pressures. The vertical acceleration is given by (5.3)

where the first term on the right is the buoyancy force in equation (3.1) and the second, where p is the hydrostatic pressure, expresses the influence of the vertical gradient of nonhydrostatic pressure. I n a situation with strong vertical shear, the relative motion between in-cloud and ambient winds can, in the upper and lower levels, be of the order 10 to 20 meterslsec. Corresponding to this, the induced pressure would be around 0.5 to 2 mb. From equation (5.3), the average induced pressure-gradient acceleration through the depth (about 600 mb) of the cloud layer woultl be roughly equivalent to an augmentation (or diminution) of the cloud teinperature by about 1"C, which (Table I) can have a significant influence on tliv intensity of the vertical motions. Probably the most significant influence is in the subcloud layer. In thca outflow layer near the ground (Fig. 3), the relative motion due to momentuni transfer is augmented, as shown elsewhere (see Newton and Newton [441 and Atlas et al. [ l , p. 471). A relatively large vertical decrement of nonliydrostatic pressure can result in the subcloud layer. If the lower layers are it1

r

FIG.11. Aircraft measurements of wind field around a large cumulo-nimbus. In wind symbols, a flag indicates 50 knots, each f d barb, 10 knots speed. Arrows are streamlines;dashed lines, isotachs (courtesy of T. Fujita [46]).

SEVERE CONVECTIVE STORMS

277

all stable, the lifting required to bring the air to saturation will cool it and make it negatively buoyant if only the first term of equation (5.3) is considered. On the right flank of the storm where amlap > 0 , the air can be accelerated upward even when A T < 0. A consequence of the process described above is that new cloud formation is favored on bhe right flank of an existing convective system when the wind veers with height. Thus the existence of strong vertical shear in the environment contributes to a continued regeneration of the storm by new growth. No direct measurements of the nonhydrostatic pressures have been made. An analysis [46] of winds in the neighborhood of a huge cumulo-nimbus (Fig. 11) displays features similar to the flow around a wind-tunnel obstacle, in particular the speeding up of the flow on the lateral flanks of the cloud core and the region of weak winds on its lee side. The general wind speed in the neighborhood of this cloud was about 50 knots. Onthe south side, winds up to 75 knots were observed. The air motions relative t o the moving storm are summarized in Fig. 12.At the middle level, application of Bernoulli’s theorem (with allowance for the movement of the storm) indicates a pressure deficit of up to 1.5 mb near the cloud flank. Figure 11 is of further interest in that it illustrates the enormous extent of the cloud anvil, 200 km long and 65 km wide a t this time which was about two hours after the anvil first started to form. The volume of the plume represents the amount of cloud-filled air which has been pumped into the upper troposphere and eroded from the main storm core [45]. I n effect, the mechanism described above represents a partial conversion of the kinetic energy of the environment winds into kinetic energy of the storm vertical motions via the induced pressure forces. This is probably a factor in the ability of large storms to persist a t night when the moist layer becomes cooler and more lifting is required to regenerate convection.

5.2. Organized Convective Circulations Investigations in recent years have revealed that the air currents in some storms are rather highly organized, and that their configurations have important consequences for the energy processes and other aspects. The general character of the circulation (neglecting superimposed smaller scale updrafts and downdrafts) in a simple convective system (in this case a squall line) is illustrated in Fig. 13 (see Newton [47] and Atlas et al. [l,p. 441). Harrison and Orendorff [48] pointed out the self-propagating nature of the squall lines observed in North America. Once thunderstorms have formed, rain falling from them cools the air beneath, as indicated in Section 3.2. A “pseudo-cold front” characterized by a rapid temperature and humidity change, an abrupt pressure rise, and shifting gusty winds [5, 491 forms a t the

SLANT NUMBERS: WIND SPEED IN KTS

CLOUD BASE (5,000')

-

\

\

\

\

\

GROU ID (I, 0 0 0 ' )

FIG.12. Wind field a t three levels around the storm of Fig. 1 1 . The speeds shown are relative to the movement of the storm (eastward a t 20 knots). At cloud top, arrows indicate relative movements of anvil edge (courtesy of T. Fujita [16]).

a

279

SEVERE CONVECTIVE STORMS

edge of this rain-cooled mass of air. As this front advances, it lifts the moist unstable air ahead of it, causing the development of new thunderstorms which continue the process. The transfer of fast-moving air from the upper levels. together with the divergent winds beneath the storms (Fig. 3), accounts for the rapidity of advance of tlhe leading edge of the cold air in lower levels.

.,............. ..........., ............

..............4 .....

A:::?.

...... :;....;,$::: .......... .;:A?&.......... ....... ...... ....k:

4-

I 200krn.

\\ f;;:. .. -

roo

I

loo hm.

0

Fro. 13. Vertical section through a squall line. Heavy lines indicate squall front (right) and cold front (left); dotted lines, a stable layer with relatively dry air above. Stippling shows approximate cloud distribution; hcavy rain with thunder was observed ncar squall front with light rain toward rear edge of cloud system. Streamlines are relative to moving system (schematic above about 600 mb); each streamline channel carried 4 tons/sec of air per vertical slice 1 meter in width. Redrawn from Newton [47] and Atlas el al. [ 1, p. 441.

One of the most important aspects of the circulation in Fig. 13 is that, contrary to the configuration of small clouds without downdrafts, the updraft leans in an upshear sense (the component of the ambient wind blowing toward the right of the figure increased with height). This has been indicated by Bates [32] and by Browning and Ludlam [50] to be typical of the lower tropospheric portions of severe storms in a sheared environment. An explanation of this arrangement is indicated in Fig. 14. The rapidly advancing squall front of a storm (Fig. 13) overtakes and lifts the air ahead of it in low levels. The updraft is the locus of air particles which rise from the lower layers at different times and lag behind the surface position of the squall front. In t,he case illustrated by Fig. 13, the measured winds and other evidence indicated that dry, potentially cool air entered the circulation from the rear of the storm, in a tongue which extended forward all the way to the squall front. Normand [12] pointed out that since the driving mechanism of a thunderstorm involves not only the updmft but also the vigor of the downdraft, the most effective energy production would be realized if the currents were so disposed that the rain falls into potentially cool and dry air in which

280

CHESTER W. NEWTON

evaporation could readily occur. Evidently the arrangement in Fig. 13 is ideal from this standpoint. A further advantage of the tilted updraft [50] is that condensed water falls out of it; thus, it is in effect invigorated by being partially relieved of the weight of water which would otherwise cut down its buoyancy.

-

cst

FIQ.14. Showing the tilt of an updraft in a moving system, where V represents horizontal speed of air rising from surface layers and c the speed of movement of the squall front (adapted from Bates [32]).

Because measurements of the air motions within clouds are very difficult to obtain, evidence for the tilted updraft is mostly indirect; however, it can be considered conclusive. One piece of evidence is the form of the cloud observed visually or by radar, as in Fig. 15. Serial wind soundings showed that, as in Fig. 13, the moist unstable air feeding the storm entered its forward edge, near A . The highest towers T emerged near the rear edge of the storm column, and extended well above the tropopause, which is only possible km

10

km 100 SO

80

70

60

50

40

30

20

10

0

FIQ.16. Radar profile (at full gain) through a storm west-northwest of Oklahoma City on 21 May 1961. The direction of scan was approximately along the direction of wind shear between lowor and upper levels (from Atlas et al. [ l , p. 611).

SEVERE CONVECTIVE STORMS

281

if a vigorous updraft is present near the tropopause level beneath them. It must be concluded that the updraft slanted across the storm from the general neighborhood of A to T. In rising through the interior of the cloud mass in this manner, the updraft may be sheltered from direct mixing with the dry environment of the cloud. Thus, as illustrated by comparison of curves B and B’ of Fig. 7b, entrainment from the saturated cloudy environment does not so strongly diminish the buoyancy, as would otherwise be the case. Byers and Battan [51] found from radar observations that when vertical shear was present new towers appeared successively on the upwind (or upshear) side of older ones, leaning over a t a rate less than indicated by the winds. On first emerging from the top of a large storm, the horizontal speeds of such towers [50] are very much smaller than that of the ambient air stream; as they drift downwind they accelerate and approach the wind speed. These observations provide direct evidence of the tendency for conservation of horizontal momentum in vertically moving currents.

5.3. “Steady State” Severe Storms Although many thunderstorms are multicellular as in Fig. 2, and are characterized by sporadic development or decay of cells, there has been increasing evidence that some large severe storms maintain nearly steady circulations lasting for several hours. Browning [52] has suggested that these storms, rather than being multicellular, are dominated by a single “supercell” of great size and intensity. During its earlier history, a giant storm studied by Browning and Ludlam [50] was characterized by the successive development of new cells mainly on its right flank, and the sporadic protrusion of towers from its top. When the storm achieved great size, however, its character underwent a pronounced change into an apparenbly steady state configuration. The principal aspects of the radar structure a t this time were similar to those in a storm analyzed by Donaldson [53], illustrated in Fig. 16. This is charaoteristic of many storms producing heavy hail and tornadoes. Such a storm is highly asymmetrical. This is evident not only in its general shape but also in the distribution of weather elements within the storm. The main convective column, in the general neighborhood of A , appears on radar with the highest reflectivity, indicating the presence of heavy rain and generally hail, To the left forward side of this, an extensive region of lesser reflectivity indicates lighter rain at low levels. The anvil or “overhang” streams out in the general direction of the wind shear (from northwest in this case). An important new feature, discussed in detail by Browning and Ludlam [SO], was the presence of a sharply defined nearly vertical “wall” on the forward side of the “hook” on the right rear side of the storm. Ahead of this is

282

CHESTER W. NEWTON

an “echo-free vault,” extending to a much higher elevation than the base of the overhang. This vault is not necessarily void of cloud, but it cannot contain large droplets in appreciable numbers. Browning and Ludlam took this as an indication that large droplets and precipitation are held in suspension above this region, and concluded that the vault is the locale of entry of the main I

i20

naut mi

FRONT V I E W

STREAMERS HEIGHT

SCHEMATIC V I E W S OF S E V E R E STORM x c

CORE

GEARY, OKLA MAY 4. 1961 I800 c

SIDE VIEW

Fro. 16. Schematic plan view, and view toward forward and right sides, of a tornadoproducing thunderstorm (moving in direction of arrow in plan view). The tips of streemera (successive heights indicated in thousands of feet) movod across and toward the storm face in the manner shown (after Donaldson [53]).

updraft which feeds the storm. During the period in which the storms were best developed, both Browning and Ludlam [50] and Donaldson [53] observed that the main convective tower penetrating the stratosphere (u in Fig. 16) remained steady above the “wall” (see also Browning and Donaldson [54]). This was taken as an indication of the steady state character of the circulation. A generalization of the air flow deduced by Browning and Ludlam is shown in Fig. 17. As in Fig. 13, the principal branches of the circulation are the moist low-level air entering the front right side, rising in the updraft and leaving the storm largely in the anvil; and a current originating in middle levels, which enters from the rear of the storm, undercuts the updraft, and leaves (in relative motion) toward the rear in lower levels. As shown by Fig. 18 (which is typical of severe storm environments), the middle tropospheric air has a low wet-bulb potential temperature and is thus susceptible t o strong evaporative cooling.

FIG.17. Air flow in a large hailstnrm over wiitheaxt England (mfLer Bmwnhig and L u d h n [l, p 161). A particle moving along the heavy line grows to a hrge hailstone after recirculation in t.he ~ p d m f LLarge . hail falls in a amall region (white area at ground). In this storin no tornadoes occurred, but in othera tornadoes have been obxervpxt in the approximate location shown.

284

CHESTER W.NEWTON

I-

W W b.

B ln a

2

d73

PI-

t I

9 W

I

16

18

20

22

24

FIQ.18. Sounding near storm of Fig. 17. Wet-bulb potential temperature curve (&) indicates temperature which would result if, through having water evaporated into it, the air were brought down t o the 1000-mb level (near the ground) in a saturated state. This temperature is lowest for middle tropospheric air in most convective situations. On this diagram the straight vertical line is a moist adiabat; where this lies to the right of curve 0s (a quantity related to the actual air temperature), a parcel rising from the lower moist layer would be warmer than its environment (after Browning and Ludlam

[W). 5.4. Size Sorting and Recirculation of Precipitation Particlea

As shown in the next section, large severe storms usually move somewhat toward the right of the mean wind direction through the storm layer. Middle tropospheric air (see inset t o Fig. 20) then has an enhanced component into the right flank of the storm. Part of this air is probably entrained into the rear side of the storm, as in Fig. 17,but part also enters ahead of the updraft. Browning [62] has suggested that this air passes through the storm in the manner indicated in Fig. 19. Precipitation falling from the overhang evaporates into it; it descends as a downdraft on the left side of the updraft, and is left behind as a cold layer near the ground as the storm moves forward. Figure 20 shows an interpretation of the form of a radar-detected severe storm, which has evolved from the studies of Browning and Ludlam [50] and of Browning [52]. Since the radar echo is mainly from precipitation-sized particles, it must (as is stressed by Kesder [55]) be interpreted in light of the past histories of the precipitation elements rather than any instantaneous process.

SEVERE CONVECTIVE STORMS

285

According to their sizes, particles fall a t different speeds. All are influenced to some degree by the horizontal winds, but the slower falling particles are displaced more from a vertical fall than are the large ones. This results in a size sorting in the manner indicated by Fig. 20b. Particle 3, for example, is carried farther away from the updraft in which it originated, than are the larger particles 2 and 1 (see also Hitschfeld [45]).

FIG.19. Schematic trajectories, relative to moving storm. of air currents entering the updraft in low levels (L), and the middle part of the storm from the right side (M), after Browning [52]. Circles show fall of precipitation elements as in Fig. 20; the general location of a tornado is shown by t.he dark pendant.

When the wind veers with height, the particles come under the influence of winds from different directions as they fall. Their horizontal paths are therefore as shown in Fig. 20a, curving in a counterclockwise fashion as they fall first through westerly or southerly winds, and later through southeasterly winds near the ground. The extensive area of lighter rain on the left forward side of the main storm column is accounted for by the rain having been carried forward and across the storm face in the manner indicated. Closc to the main storm column, larger particles such as small hail are carried forward only a short distance, and these may fall into the updraft where they are again transported into the upper part of the storm. The size of a precipitation particle growing by coalescence (sweeping up of smaller particles) depends both on the abundance of cloud water and on the time spent by the collecting particle in falling through it. Thus Browning and Ludlam [ 501 have stressed the import.ance of recirculating the precipitation particles in the manner described above in the production of large hailstones. When some of these achieve a large enough size, their fall speeds become so great

286

CHESTER W. NEWTON

0

*

I0 KY

( b)

VERTICAL

SECTION

FIG.20. (a) Plan view of a steady state severe etorm moving with velocity V; (b) vertical section along line A B . I n (a), the stippling represents precipitation of varying intensity which reaches the ground, with hail in the donser area. Dash-dotted and dashed lines indicato the extent of echo in middle and high levels of the storm. Small circlos show precipitation trajectories. Arrows in (a) show motions of small protuberances seen on edges of low-level radar echoes. Tornadoes, if any, tend to occur near flying V. Broad arrows indicate general inflow in updraft and outflow in anvil (after Browning ~521).

that they fall almost vertically through the updraft. Observations show that, as indicated in Fig. 17, large hail falls in a restricted part of the storm. This is identified with the wall in Fig. 16. An analysis by Browning and Donaldson [64],making use of tornado observations by Ward [56], disclosed that the tornadoes were closely associated with the echo-free vault just ahead of t,he

SEVERE CONVECTIVE STORMS

287

wall where the strongest updraft is presumed to occur. Also from Ward's observations, Donaldson 11531 identified the radar wall with the onset of Iiail and rain, and with strong westerly winds indicating the advancing edge of the downdraft.

6. STORM MOVEMENT As mentioned in Section 2, the cold outflow from a storm may t,rigger new cells near its boundary. I n a stagnant environment, the storm as a whole may move in an irregular manner, as the growth and decay of cells proceeds in a sporadic and random fashion. When there is a systematic wind flow, but little vertical shear, the storm will travel more or less with the winds, but will irregularly change direction and speed for the same reason. As will be shown, however, there are some systematic rules of behavior when there is strong vertical shear.

6.1. Single-Celled Storms I n order to circumvent the influences of sporadic growth, the Thunderstorm Project [a] chose to study the movements of small radar echoes which were taken to be single-celledstorms. The movements of these were found to be highly correlated with the mean wind in the 2000-20,000 f t layer, although with strong winds aloft there was a systematic tendency for the storms to move somewhat slower than the winds. Ligda and Mayhew [57] also found a high correlation with the wind a t the 700-mb level. Brooks [58] had observed that while there was a correlation with the winds aloft, there appeared to be some variability in the behaviors of large and small radar echoes, which he suggested might be due to the vertical transport of momentum.

6.2. Injuence of Propagation on Movement of Large Storms From studies of a large number of convective situations utilizing hourly rainfall data, it was found [59, 441 that large storms (which were the only ones revealed by the network) moved on the average about 20" to right of the mean wind in the 850-500-mb layer. In all the situations studied, the wind veered with height, to some degree, as in Fig. 10. This behavior was attributed t o the systematic propagation influences discussed in Section 5.1. The effect of this is illustrated in Fig. 21 [50]. New cell growth on the right flank, accompanied by the decay of old cells on the left flank, result in the movement of the storm as a whole toward the right of the paths of the individual cells. I n the case of the steady state storms discussed in Section 5.3. the propagation takes place [52] in a continuous manner rather than by the discrete formation of cells. A large hailstorm a t Johannesburg [m]moved 60" to left of the mean wind direction. This is probably

288

CHESTER W. NEWTON

DIRECTION OF T R A V E L OF ECHO-MASS

FIQ.21. Showing the influonce of cell propagation on movement of storm (after Browning and Ludlam [50]).

characteristic of large storms in the Southern Hemisphere (the wind in this case turned 90" in a clockwise sense through the storm layer).

6.3. Size Discrimination in Relation to Water Budget In an examination of radar echoes in the Great Plains area, Newton and Fankhauser [61] found that there were systematic differences in movement, depending on storm size. From equations (5.2) and (5.3),the influences of the induced nonhydrostatic pressure field in influencing propagation would be expected to be greater for large- than for small-diameter storms. It is evident from Fig. 21 that the more active the propagation in this manner, the more pronounced will be the movement of the storm as a whole toward the right of the individual cell movement. No way has been discovered in which this mechanism can be related to storm movement in more than a qualitative fashion. An alternative is t o examine the moisture budget of the storm. In a sheared environment, such as illustrated in Fig. 10, a storm is continually ventilated by a fresh supply of moist air fed into its right flank to a degree partly dependent on the storm movement. This dependence is illustrated in Fig. 22. I n Fig. 22a, a simplified wind hodograph is assumed for the purpose of demonstration. Here, V, and V, are the winds a t the base and top of the storm, being the mean wind in the cloud layer and that in the moist layer which is taken to occupy the lower half of the troposphere; is the mean relative velocity of the air in the moist layer, with respect to the moving storm (cf. Fig. 10).

vL

vBL

281)

SEVERE CONVECTIVE STORMS

From Fig. 22b, i t is evident that a cylindrical storm column of diameter D will intercept the water vapor contained in the moist layer a t a rate M I a (V,, D ) . The portion of this, which condenses on entering the storm circulation, is partly disgorged as rain. Assuming some characteristic mean

\

(a)

(b)

FIQ.22. (a) Simplified wind field in storm environment; (b) storm motion relative to wind in moist layer (we text; after Newton and Fankhauser [61]).

rainfall rate over the area of the storm column, water is disposed of through rainout a t a rate M , a 0‘.Thus, unless the migration velocity V,, of the storm relative to the air in tho moist layer is adjusted, a large storm will rain out proportionately more water than a small one, in comparison with the water vapor intercepted. Making the simplest assumption that M , a M , , it follows from the above that a D. From Fig. 22a, it is seen that one way to vary the magnitude of is to adjust the direction of the storm velocity V,. If S is a measure of the vertical shear due t)o the turning of wind with height, as defined in the figure, then [61]

v,,

v,,

Here a is the angular deviation of storm direction from the direction of the

290

CHESTER W. NEWTON

21 May 1961 0

30-db schwa

I

1

1

16

4-

0,

-40.

-30

-20

-10 0 40 a20 *30 DEVIATION FROM MEAN WIND DIRECTION

.40

.SO

.60.

FIG.23. Movements of radar storm echoes to right or left of the wind, in relation to storm diameter. during afternoon of 21 May 1981 near Oklahoma City. Diameters relate to 30 db attentuation on WSR-67 10 cm wavelength radar. Dashed curve, equation (8.1) in text; dotted curve, the same under slightly different assumptions;others are regression lines. Letters B and C indicate same storms at different times (after Newton and Fankhauser [all).

vector mean wind (defined positive toward the right), and Do is the diameter of a storm which moves along the mean wind direction. Figure 23 shows, for one situation, a plot of the observed angle u against storm diameter. Large radar echoes moved as much as 57" to right of the mean wind direction, and small ones as much as 36" to the left. The dashed curve, representing equation (6.1), falls reasonably close to the line of best fit for the data, considering the crudity of the assumptions underlying this equation (among these being that of a circular shape of the storm column). The scatter in Fig. 23 is believed to be largely a result of various sources of error, but it is certainly in part real and must, in any attempts a t shortrange forecasting, be taken into account since it affects the probability that a given storm will move over a particular location [61]. Many storms move in a distinct S-shaped path, as in a n example by Hoecker [62], traveling more nearly with the winds early and late in their lifetimes when they are small, and strongly across the wind direction when they are large. Although the mechanism of movement to the right is accounted for by the considerations of Section 5.1., movement to the left of the winds is not. If no propagation influence were present, and the air making up the storm were drawn in equal measure from all levels of the atmosphere, the storm might

SEVERE CONVECTIVE STORMS

29 1

frocks of 12-db. echoes Storms "a" and "6" 24 May 1962

FIQ.24. Outlines of two storms, at hourly intervals, on 24 May 1982. At right, mean wind velocity and volocity of large storm during two periods (after Newton and Fankhauser [a 11).

be expected to move along the mean wind direction, Movement to the left of the small storms with weak propagation influences is believed to be a consequence of the air within the storm being drawn predominantly from the lower levels (where the wind direction is to the left of the mean wind direction) and to some extent conserving its horizontal momentum. A size discrimination in the speed of movement of storms has been noted. Small- and medium-sized radar echoes (up t o about 10 miles in diameter) move on the average with about the speed of the mean wind through the cloud layer, while very large storms move very much slower. An example is shown in Fig, 24. Storms a and B, which were near neighbors a t their time of formation, moved on diverging paths and, three hours later, were 80 nautical miles apart. Small storm a moved slightly t o left of the mean wind and a t its speed of 35 knots. Large storm B, which was multicellular with new cells forming entirely in its upwind portion, moved on the average 40" to right of the mean wind and a t only 16 knots, or less than half the wind speed. 7. SQUALL LINES

7.1. General Structure Thunderstorms are frequently arrayed in lines, the structures of which [63] may be quite different from one case to another. Sometimes the squall

QQ

FIG.25. Radar rain echoes and cores of heavier precipitation (black, at 24 db attenuation) in squall line of 4 May 1961 at 40 min intervals. I n each case, echoes are located with respect to Oklahoma City at intersection of straight lines (north toward upper left). Letters identify successive locations of individual echoes; asterisks denote echoes not appearing a t previous time. Long-dashed lines connect echoes As and f3 at different times; because the pictures are displaced the lines do not indicate directions of their movements (after Newton and Fankhauser [Sl]).

SEVERE CONVECTIVE STORMS

293

line appears on radar as an unbroken line with a distinct and smooth leading edge, and only weak evidence of cellular composition. More commonly, the structure is irregular, as in Fig. 25, the activity being dominated by a small number of large intense storms, with a larger number of small, relatively weak ones. The tendency for formation in lines is partly understood from the nature of the environment and of the triggering mechanism, as discussed in Section 4. The most clear-cut case is a situation like that in Fig. 8. When a pronounced low-level jet is present, the northward advection in low levels of moist air with different speeds (depending on the distance to right or left of the jet axis where the wind becomes weaker) results in the creation of an elongated tongue of potentially unstable air. The western edge of this is generally quite regular. If a cold front with a nearly parallel orientation sweeps into the moist tongue, thunderstorms may be set off almost simultaneously in a line along its western edge. If the cold front is farther advanced in its northern portion, thunderstorms may be triggered there first, and successively later in regions farther south. Since the storms triggered first will have moved eastward before the ones farther south are initiated, the squall line would assume an orientation clockwise from the western edge of the moist tongue. It is perhaps more common for thunderstorms to start some distance ahead of a cold front (if one is present) in response to the general rising motions downstream from an upper trough (Section 4). I n that case, convection is still likely to be initiated first near the west edge of the moist tongue being overtaken by the trough, or somewhat farther east where the moist air is deepest. The above are only some of several mechanisms which may initiate the convection, in all of which the contribution of diurnal heating at the earth’s surface must be considered.

7.2. Migration of Storms within Lines and Regeneration Pattern According to Boucher and Wexler [41] the modal orientation of squall lines in New England is about 20°, and in Illinois about 40°, counterclockwise from the 700-mb wind direction. This would correspond to a line orientation about north-south or north-northeast-south-southwest in Fig. 8. Individual storms moving more or less with the upper winds consequently have a component of movement toward the left-hand end of the squall line (with respect to its direction of advance), as found by the Thunderstorm Project [5]. The features displayed in Fig. 25 are fairly typical and follow a pattern described by Stout and Hiser [MI. This squall line developed in a situation broadly similar to that in Fig. 8. The migration of individual storms toward the left end (top of page) is evident, for example, in the cases of storms A and E . In general, storms approaching the left end of R squall line gradually

294

CHESTER W. NEWTON

become weaker and eventually perish (in this case, there was a marked decrease of instability and water vapor content northward from central Oklahoma). A t the same time, new storms tend to form mostly near the right-hand end of the line. Thus storm A , which initially occupied the southwest end, was supplanted by other storms by the end of the period shown. In addition, it is noted that storms H and Gr, which developed 50 and 100 miles from storm A , eventually joined the southwest end of the line (the different directions of movement of A , c f , and H being in accord with their different sizes, as discussed in Section 6.3). Thus a regeneration of the righthand end of the line took place both by the development of new storms in its near vicinity and by the incorporation of storms which merged with it from distant locations. The development-and-decay process results in the squall line as a whole being displaced in a nearly parallel fashion similar to Fig. 21. Vigorous young storms characterize the right-hand end, while the left-hand end is more typified by aged storms with relatively weak intensities and extensive stratiform cloud masses. Boucher and Wexler [41] find that the movements of squall lines are correlated with the component of the 700-mb wind normal to the line. In cases wherein lines are oriented nearly parallel to the flow, however, they move with an appreciable component toward the right of the wind direction. This is explained by the behavior of the individual storms in the line, as discussed in Section 6.2.

7.3. Over-all Aspects of a Mature Squall Line System Our present knowledge of the detailed structures of “mesoscale” systems associated with thunderstorms has mainly evolved from analyses by T. Fujita and his collaborators. The reader is referred to a review by Fujita [l, p. 771 for a comprehensive description. Only a single generalization of a thunderstorm system over southeast England by Pedgley [65] will be illustrated here (Fig. 26). Although there are differences from case to case, the general features shown are broadly similar to those which have been observed in mature systems over the United States [66-681. As noted in Section 2, the high pressure area associated with a convective system is due essentially to the dense air produced by cooling through evaporation of rain. Pedgley shows that the maximum pressure likely to be observed is proportional to the square of the height of the cloud base. Fujita [9] had earlier shown that the excess is large when thc rain from clouds with a high base falls through air with low relative humidity, while with low cloud bases and high relative humidity the effect of evaporation in creating a layer of dense air is small. The high prcssurc area is, with a mat,ure system, much more extensive than the area actually occupied by thunderstorms. As a thunderstorm system

205

SEVERE CONVECTIVE STORMS

o

/

60

100

nouticel mllsa

in or00 of l l t t l a or no madium-lava1 clouda

b y turbulance In roin-coolad air

thundaratorm high Crown copyright@ 1963

FIU.26. General featuros of squall lines over southeast England on 28 August 1958; lower figure is vertical section along line A B in upper. Solid lines are isobars at 1 mb intervals; surface winds shown by arrows. Llght and heavy stippling show lighter and heavier rainfall area (after Pedgley [65], by permission of Controller, Her Majesty’s Stationery Office).

moves along it leaves behind it cooled air (see relative streamlines in Fig. 13), which gradually subsides and spreads out over a larger area. Fujita [9] denionstrated that the total mass of the cold air produced in this manner

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CHESTER W. NEWTON

depends on the age of the system, being proportional to the mass of rain reaching the ground, and is equivalent to 9 times the mass of raindrops evaporated inside the downdraft below the cloud base. The cause of the low pressure system that often forms behind mature squall lines (Fig. 26) has not been satisfactorily explained. During the lifetime of a mesosystem the thunderstorm high first builds up; this is followed by generation of the trailing pressure trough when the system has reached maturity; and when the rainfall system decays there is a gradual collapse of both the high and low pressure systems [67]. 8. SEVERE WEATHER MANIFESTATIONS

8.1. Hail Extensive studies carried out in Alberta [69], New England [70, 711, and elsewhere [72, 731 indicate that the probability of hail from a storm increases

60

-

I

OO

20

40

60

80

100

PROBABILITY %

FIQ.27. Probability of occurrence of hail at ground in relation to radar echo height (after D o u g h [ l , p. 1821).

both with its vertical extent and its intensity as shown by radar. The dependence on echo top height is illustrated in Fig. 27 (see Atlas et al. [l, p. 162]).' 1 In

Colorado [73], top heights are similar to those in New England.

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I n the higher probability ranges, the significant feature is the degree of penetration of the storm top into the stratosphere, which (Section 3) is a measure of the intensity of the updraft. In New England [70], the median hailstorm having hailstones 2 in. in diameter or greater penetrated 5000 f t into the stratosphere; the median thunderstorm without hail fell 4000 ft short of reaching the tropopause. Short bursts of hail are associated with storms whose tops rise then decline; protracted hail over a long path occurs with storms whose tops remain steadily above the tropopause [69,70]. I n New England [70], all damaging windstorms and tornadoes were associated with such “hail repeaters.” Analyses from crop damage reports [74, 751 show that some hail swaths, 5-10 miles or more wide, are up to I60 miles or more in length. These are suggestive of steady state storms (Section 5.3) persisting for periods up t o 8 hr or longer.

8.2. Lightning Observations by Kuettner [76] and Weickmann [42] indicate that the greatest lightning activity is closely associated with the area of highest precipitation intensity in a storm. These are borne out by Shackford’s [77] analyses of observations in New England, where on the average heavy rain reached a station 3 to 4 min (0.5 to 1 mile distance) after the arrival of lightning within a radius of 1 mile, with hail on the average 8 min after the onset of lightning. Radar characteristics are similar t o those for hail. Shackford concluded that: “An echo that is merely high is not always associated with violent electrical activity, while one that is both high and intense can be counted upon to produce a great deal of lightning.”

8.3. Surface Winds Byers and Braham [5] state that: “In relatively slow-moving storms, the outflow is almost radial [as in Fig. 31, and as it continues, an area of light wind develops immediately beneath the center of the downdraft area. In most instances, however, the outflow field is asymmetrical, with wind speeds on the downstream side substantially higher than those on the upstream side.” Thunderstorm winds tend to blow directly across the isobars from high toward low pressure [49], although there are exceptions (cf. Fig. 26) due to the movement of the system and possibly to momentum transfer from aloft. The maximum wind strength increases with the temperature drop beneath the storm, which through equation (3.2) is connected with the vigor of the downdraft [78, 791, and with the degree of development of t,he high-pressure cell (Section 7.3). Although there is a certain degree of organization in the wind field connect<ed with squall lines (Fig. 26), outpourings of air from individual storms (Fig. 25)

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cause chaotic wind changes from one portion to another of the active precipitation region. Ward and Arnett [so] found, in the case of the large storm B in Fig. 24 and others like it, that gust fronts emanated in bursts from the rear side of the storm, where new intensifying radar cells formed. This might suggest that each outburst (at intervals of 30-70 min) was associated with the production of a major downdraft. The gust fronts gradually overtook the leading edge of the radar echo, and their influence was felt from 20 to 30 miles south and east of the storms. Brunk [81] found that pressure dips of up t o 16 mb occurred on the rear sides of some squall lines with strong winds proportional to the pressure pulsation. Intense small lows with strong gusty winds and very warm and dry air have been described by Williams [82]. They form on the northwest sides of thunderstorms a t night, and although they are clearly associated with intense local subsidence, the cause of this is a mystery. 8.4. Tornadoes The fact that the probability of tornadoes increases with the severity of thunderstorms is the basis of their forecasting (see Fawbush et al. [34] and Atlas et al. [l, p. 1421). Tornado thunderstorms have higher tops and greater radar reflectivities than others [71], although observations [83] suggest that a t least some tornadoes form beneath developing cumuli on the flanks of, rather than below, the main thunderstorm. Fujita’s [84] summary of several major tornado cases shows that each tornado in a family appeared 5 to 10 miles to the right of the path of its dissipating predecessor, at an average interval of 42 min. This might suggest that the individual tornadoes were connected with successively developing cells in the manner of Fig. 21. Major tornadoes are often surrounded by a weaker “tornado cyclone” [85], 20-30 miles across, which appears [86, 841 to be associated with the “hook” of a storm of the type in Fig. 20. Hoccker’s [87] analysis of the tangential and upward speeds in a tornado is shown in Fig. 28. Tangential speeds up to 76 meterslsec were observed, the total speed being larger (radial speeds could not be measured, but on theoretical considerations an inflow of about 35 meterslsec was probably present 160 meters from the center, with maximum inflow near 50 meters height). The vertical speeds arc representative of a time when the cloud funnel did not reach the ground. Hocoker gives convincing evidence suggesting that upward motions were present only outside the condensation envelope, with probable descending motions inside it. The tangential velocity distribution outside the ring of strongest winds closely fitted a vr-constant distribution a t 300 meters above ground, but the velocity profile was more complicated a t lower levels.

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0

299

50 100 IS0 200 250 300 350 400 450111 DISTANCE FROM CENTER OF VORTEX

Fro. 28. Tangential (rotat,ional) wind component (left), and vertical wind component (right) in a tornado, determined photogrammetrically from movies, by Hoecker [87]. Units are in meterslsec. This is representative of a, stage when the water cloud funnel does not touch the ground.

Laboratory experiments [2] show that the mass flux of a superimposed updraft is the critical factor in the creation of a vortex; this may take the form of a narrow intense draft or a broad, more gentle one. However Ward [88] produced vortices of different types, similar to the variety found in nature, by varying the size and intensity of the simulated updraft. He suggests [80], on the basis of his later laboratory experiments, that the left-hand edge of a tongue of air (where there is cyclonic shear) flowing out from one downdraft region, and moving under a neighboring updraft, might furnish the required conditions for tornado formation. An alternative explanation for the rotation of tornadoes has been suggested by Fulks [89]. This is, that where a cumulo-nimbus column extends into strong winds in the upper troposphere, vortices form on its lee flanks. These are, as described by von KArnirin, a mechanism for shedding the vorticity created in the “skin friction” layer between an obstacle and the ambient flow moving past it (cf. Figs. 11 and 12). For reasons discussed by Fulks, the vortex on the right downwind flank (relative to the upper level flow) is most favorable for tornado development. Tornadoes are observed in the general location of a storm which is predicted by Fulks’ theory. High-level aircraft photographs [go] showed, in storm B of Fig. 24 (at 1637 CST, believed prior to any surface

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tornado reports), a conical hole about 10,OOO f t deep in the cloud top, on the downwind side of the highest tower, which if in centrifugal equilibrium could have contained rotational winds up to 140 meters/sec. This observation suggests that the tornado circulation may extend through the entire depth of a cumulo-nimbus cloud and is consistent with Fulks’ suggestion. The lowering of the condensation funnel of a tornado to the ground indicates that the surface pressure in tornadoes is roughly the same as that a t the cloud base from which the funnel is pendent. This pressure deficit, 100 mb more or less, has not been explained (see discussion by Brooks [91]).

8.5. Heavy Rains Although flash floods may occur locally from a single excessively heavy rainstorm, the occasionally observed extreme accumulations of 20 to 30 cm rainfall are a result of the repeated passages of a number of storms over the area. If a squall line is oriented across the flow, the duration of its passage is likely t o be brief. If, on the other hand, the squall line is oriented nearly along the direction in which individual storms move, several may pass successively over a given location. Huff and Changnon 1921, in a. study of the 10 most severe rainstorms in Illinois during a 10-year period, found that the average orientation of the squall lines concerned was nearly parallel to the average middle tropospheric flow (see also Fankhauser [20] and Newton [93]). In the area of greatest rainfall, an average of five distinct rainfall bursts occurred a t 2.4 hr intervals, the heaviest rain taking place in late evening and early morning. Multiple squall-line passages were involved, and it was observed that these moved with an average speed of 22 mph outside the heavy rain zone, but with a speed of only 9 mph while in it. In these cases, the vertically integrated water vapor content of the air mass averaged 50 yoabove normal. As suggested by the discussion of Fig. 9, high water vapor contents can result from a combination of advection of moist air into a region and an increase of the depth of the moist layer due to the slow but persistent vertical motions associated with synoptic disturbances. Fankhauser [20] found a close agreement between the water precipitated and the convergence of water vapor (which took place entirely below the 700-mb level) into the neighborhood of a squall line. 8.6. Hazards to Aircraft

Hazards in the penetration of tliundarstorms by aircraft include static discharges (damaging to electrical systems), encounters with large hail, or with heavy concentrations of liquid water [94], and stresses duo to turbulence. Results of penetrations of the severe storms of the Great Plains are

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summarized by Van Thullenar and Lee [95]. These suggest that severe turbulence is likely to be present in some part of most growing or mature thunderstorms in this region. Turbulence appears to increase with altitude, up to a t least 5000 f t below the visible cloud top. Appreciable turbulence is also found in the clear air near storms. A U-2 aircraft flying over the large storm of Fig. 24 experienced moderate updrafts and downdrafts at 65,000 ft, a t least 10,000 ft above the highest cloud towers [go]. Van Thullenar [95] generalized that: “Maximum turbulence and the least chance for hail are to be expected on the upwind side of the center of the storm [vice versa for the downwind side].” Lee found that while very high radar reflectivity (exceeding loGmma/meter3)generally indicated severe turbulence, this was not always so. No relationship was found with the spacial gradient of radar reflectivity, but severe turbulence was frequently associated with a rapid increase or decrease in the storm’s radar-indicated intensity with time. The lack of a reliable relationship between turbulence and radar reflectivity has been explained by Kessler et al. [96]. Large quantities of condensed water can accummulate in the upper portion of a storm while its upward currents are vigorous [55]. Since time is required for this water (or hail) to fall out, i t may still be present when the storm passes into a stage wherein the organized currents and the turbulence have degenerated. An example of the vertical motions and lateral gusts in the upper part of a large storm is shown in Fig. 29, taken from Steiner and Rhyne [97]. Superimposed on a broad updraft flanked by downward currents of lesser extent, there were gusts with sharp velocity gradients (correlated with accelerations imposed on the aircraft, top of figure). I n other cases, upward gusts up to 209 ft/sec were measured. Power-spectrum analyses have shown that the horizontal gusts encountered in thunderstorm traverses are equivalent in magnitude to the vertical gusts. In the case of Fig. 29, upward air motions were generally associated with weaker uest winds within the cloud, and downward motions with stronger west winds. Since the west wind increased strongly with height, this correlation is suggestive of the vertical transfer of momentum, although the example shown may be fortuitous. 9. CONCLUSION

Our present knowledge on the nature of severe storms, and convection in general, is greatly advanced over that of twenty years ago. Even so, there are some fundamental gaps in that. knowledge. These cannot be filled until better methods of measurement are devised, and observations are carried out on the grand scale demanded by the complicated and varied nature of convective phenomena.

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VERTICAL 40COMPONENT OF TRUE GUST VELOCITY Wq, FT/S€C 40-

--

. f '

-

-80-170-

LATERAL COMPONENT

40-

GUST VELOCITY 0

-120-

, u

ENTRY 10

1

20

I

2

50

TIME, SEC

I

0

40

30 3

4

5

60 6

7

70

8

DISTANCE, NAUTICAL MILES

Pro. 29. Vertical and lateral true gust components measured at 39,000 ft (about 6000 ft below cloud top) in a storm near Oklahoma City on 17 May 19GO. Derived gust velocity, at top, is rolated to accelerations imposed on aircraft flying through strong true gust gradients. Aircraft heading was 170"; winds at thiR level were from westsouthwest (after Steiner and Rhync [97]). Dashed lines, added by writer, are probable true zero references.

The problems of measurement are great. At present, there is no known method of determining the internal details of chosen parts of storms, except by penetrating them with heavily stressed aircraft which fly a t high speeds. Because of the small scale of convective systems, accurate determination of their details demands the most exact knowledge of the position of the aircraft at all times. One of the fundamental quantities that is difficult to measure accurately is the air temperature. Although several systems have been devised to minimize the influence of aircraft speed on the temperature measured, there are still some uncertainties even in clear air, and these are magnified in the mixed air-and-water environment inside a cloud. Among the uncertain aspects of severe storm structure is the general arrangements of updrafts and downdrafts within the storm. So far, these have only been inferred in an indirect way, mainly from radar data, and observations at the earth's surface. The increasing application of Doppler weather radar will divulge useful information on this aspect, but more direct measurements, such as those in Fig. 29, are needed. Knowledge of the temperatuxe, water vapor, and condensed water-content are required for interpretation in terms of the different possible modes of convection discussed in Section 3.

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A large question that remains to be answered, is that of what typical sizes are achieved by storms under different environmental conditions, and why. As brought out by the above summary, this question enters into all the major aspects of severe storms: the heights they attain, the thermodynamic processes affected by entrainment, the strengths of vertical currents, the production of hail and other phenomena, and the ways in which storms move. Questions Ruch as this, and the question of why some storms are cellular and others in steady state configurations offer great challenges. LISTOF SYMBOLS 9

P C

1 W

z B CO

D K M

R S T

AT V

W a

W

P

V

v -V L VL

VR VU VRL

vs

acceleration of gravity pressure radius of plume or bubble time vertical velocity height buoyancy (Aplp or ATIT) drag coefficient diameter of vertical draft or of radar rain echo entrainment coefficient mass flux of vertical draft gas constant magnitude of vertical shear tcmperature excess of cloud virtual temperature over temperature of environment horizont,al wind speed (magnitude of V ; subscripts listed below) rate of riso of bubble vortex; measure of vertical motion in vertical draft broadening coefficient of plume or of bubble path; angular deviation of storm direction of motion from direction of mean wind in cloud layer (positive toward right) nonhydrostatic (hydrodynamic) pressure air density wind velocity vector mean wind in cloud layer, in undisturbed environment environment wind velocity at storm base vector mean wind in lower level moist layer velocity of ambient wind relative to velocity of cloud system environment wind velocity a t storm top mean velocity of wind in moist layer relative to velocity of storm velocity of storm movement.

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Ludlam, F. H., Malkus, J. S., Newton, C. W., Ogura, Y., Schleusener, R . A., Vonnegut, B., and Williams, R. T. (1983). Severe local storms. Meteorol. Monogr. 5, No. 27, 247 pp.

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2. Aufm Kampe, H. J., and Weickmann, H. K. (1957). Physics of clouds. Meteorol. Monogr. 3, No. 18, 182-225. 3. Weickmann, H. K. (1957). Physics of precipitation. Meteorol. Monogr. 3, No. 19, 226-266. 4. Suckstorff, G. A. (1939). Die Ergebnisse der Untersuchungen an tropischen Gewittern und einigen anderen Erscheinungen. Uerl. Beitr. Ueophy8. 55, 138-185. 5. Byers, H. R., and Braham, R. R., Jr. (1949). “The Thunderstorm,” 287 pp. U.S.

Govt. Printing Office, Washington, D.C. 8 . Byers, H. R. (1959). “General Meteorology,” 540 pp. McGraw-Hill, New York. 7. Workman, E. J., and Reynolds, S. E. (1949). Time of rise and fall of cumulus cloud tops. Bull. Am. Meteorol. SOC.30, 359-361. 8. Battan, L. J. (1953). Duration of convective radar cloud units. Bull. Am. Meteorol. SOC.34, 227-228. 9. Fujita, T. (1959). Precipitation and cold air production in mesoscale thunderstorm systems. J. Meteorol. 16, 45P468. 10. Batten, L. J. (1961). Some properties of convective clouds. Nubila 4, 1-12. 11. Refsdal, A. (1930). Der feuchtlabile Niederschlag. Geofya. Publik. 5, No. 12. 12. Normand, C. W. B. (1948). Energy in the atmosphere. Quart. J. Roy. Meteorol. SOC. 72, 145-167. 13. Petterssen, S., Knighting, E., James, R. W., and Herlofson, N. (1945). Convection in theory and practice. Ueofye. Publik. 16, No. 10. 14. Clark, R. A. (1960). A study of convective precipitation as revealed by radar observation, Texas, 1958-69. J . Meteorol. 17,415-425. 15. Bjerknes, J. (1938). Saturated-adiabatic ascent of air through dry-adiabatically descending environment. Quart. J. Roy. Meborol. SOC.64. 326-330. 16. Petterssen, S. (1939). Contribution to the theory of convection. Uwfye. P d l i k . 12, No. 9, 23 pp. 17. Schmidt, F.H. (1947). Some speculations on the resistance t o the motion of cumulus clouds. Mededeel. en Verhandel. K . Nederl. Meteorol. Inat. B1, No. 8, 54 pp. 18. Stommel, H. (1947). Entraining of air into a cumulus cloud. J. Meteorol. 4,91-94. 19. Braham, R. R., Jr. (1952). The water and energy budgets of the thunderstorm and their relation to thunderstorm development. J. Meteorol. 9, 227-242. 20. Fankhauser, J. C. (1985). A comparison of kinemetically computed precipitation with observed convective rainfall. Natl. Severe Storms Lab. Rep. No. 25, 28 pp.

U.S.Weather Bur., Washington, D.C. 21. Scorer, R. S., and Ludlam, F. H. (1963). Bubble theory of penetrative convection. Quart. J. Roy. Metwrol. SOC.79, 94-103. 22. Scorer, R. S. (1957). Experiments on convection of isolated masees of buoyant fluid. J. F1. Mech. 2, 583-594. 23. Turner, J. S. (1983). Model experiments relating t o thermals with increasing buoyancy. Quart. J. Roy. Meteorol. SOC.89, 82-74. 24. Saunders, P. M. (1961). An observational study of cumulus. J. Meteorol. 18, 451-467. 25. Malkus, J. S., and Scorer, R. S. (1955). The erosion of cumulus towers. J. Meteorol. 12, 43-67. 26. Levine, J. (1959). Spherical vortex theory of bubble-like motion in cumulus clouds. J . Meteorol. 16,853-662. 27. Houghton, H . G.. and Cramer, H. E. (1951). A theory of entrainment in convective currents. J. Meteorol. 8, 95-102. 28. Morton, B. R., Taylor, G. I., and Turner, J. S. (1956). Turbulent gravitational

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convection from maintained and instantaneous sources. Proc. Roy. SOC.(London) A234, 1-23. 29. Squires, P., and Turner, J. S. (1962). An entraining jet model for cumulo-nimbus updraughts. Tellus 14, 422-434. 30. Austin, J. M. (1948). A note on cumulus growth in a nonsaturated environment. J . Meteorol. 5 , 103-107. 31. Malkus, J. S. (1960). Recent developments in studies of penetrative convection and an application to hurricane cumulonimbus towers. I n “Cumulus Dynamics” (C. E. Anderson, ed.), pp. 65-84. Pergamon Press, Oxford. 32. Bates, F. C. (1961). The Great-Plains squall-line thunderstorm-A model. Ph.D. Dissertation, 164 pp. St. Louis University (Univ. Microfilms, Ann Arbor, Michigan). 33. Showalter, A. K., and Fulks, J. R. (1943). Preliminary report on tornadoes, 162 pp. U.S. Weather Bur., Washington, D.C. 34. Fawbush, E. J., Miller, R . C., and Starrett, L. G. (1951). An empirical method of forecasting tornado development. Bull. Am. Meteorol. SOC.32, 1-9. 35. Fulks, J. R . (1951). The instability line. I n “Compendium of Meteorology” (T. F. Malone, ed.), pp. 647-652. Am. Meteorol. Soc., Boston, Massachusetts. 36. Fawbush, E. J., and Miller, R. C. (1953). A method for forecasting hailstone size a t the earth’s surface. Bull. Am. Meteorol. Soc. 34, 236244. 37. Beebe, R. G. (1958). Tornado proximity soundings. Bull. Am. Meteorol. SOC.39, 195-201. 38. Beebe, R . G., and Bates, F. C. (1955). A mechanism for assisting in the release of convective instability. Monthly Weather Rev. 83, 1-10. 39. Petterssen, S. (1956). “Weather Analysis and Forecasting,” 2nd ed., Val. 1, 428 pp. McGraw-Hill, New York. 40. Austin,P.M. (1960).Microstructure of storms as described by quantitative radar data. I n “Physics of Precipitation,” Geophys. Monogr. No. 5 (H. Weickmann, ed.), pp. 86-93. Am. Geophys. Un., Washington, D.C. 41. Boucher, R . J., and Wexler, R . (1961). The motion and predictability of precipitation lines. J . Meteorol. 18, 160-171. 42. Weickmann, H. (1953). Observational data on the formation of precipitation in cumulonimbus clouds. I n “Thunderstorm Electricity” (H. R. Byers, ed.), pp. 66138. Univ. of Chicago Press, Chicago, Illinois. 43. Newton, C. W. (1960). Morphology of thunderstorms and hail storms RS affected by vertical wind shear. I n “Physics of Precipitation,” Geophys. Monogr. No. 5 (H. Weickmann, ed.), pp. 339-346. Am. Geophys. Un., Washington, D.C. 44. Newton, C. W., and Newton, H. R. (1959). Dynamical interactions between large convective clouds and environment with vertical shear. J . Meteorol. 16, 483-496. 45. Hitschfeld, W. (1960). The motion and erosion of convective storms in severe vertical wind shcar. J . Meteorol. 17, 270-282. 46. Fujita, T., and Arnold, J. (1963). Development of a cumulonimbus under the influence of strong vcrtical shear. Proc. 10th Weather Radar Conf., Washington, D.C., pp. 178-186. Am. Meteorol. SOC.,Boston, Massachusetts. 47. Newton, C. W. (1950). Structure and mechanism of the prefrontal squall line. J . Meteorol. 7 , 2 10-222. 48. Harrison, H. T., and Orendorff, W. K. (1941). Pre-frontal squall lines. Meteorol. Circ. No. 16, 12 pp., mimeo. United Air Lines. 49. Tepper, M. (1950). A proposed mechanism of squall lines: the pressure jump line. J . Meteorol. 7 , 21-29. 60. Browning, K. A., and Ludlam, F. H. (1960). Radar analysis of a hailstorm. Tech.

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Note No. 5, Contract A F 61(052)-254, 106 pp. Imp. Coll. Sci. Technol., London. Also (1962) Airflow in convcctive storms. Quart. J. Roy. Meteorol. SOC.88, 117-135. 51. Byers, H. R., and Battan, L. J. (1949). Some effects of vertical wind shear on thunderstorm structure. Bull. A m . Meteorol. SOC.30, 168-175. 52. Browning, K. A. (1964). Airflow and precipitation trajectories within severe local storms which travel to the right of the winds. J . Atmoa. Sci. 21, 634-639. 53. Donaldson, R. J., Jr. (1962). Radar observations of a tornado thunderstorm in vertical section. Natl. Severe Storms Proj. Rep. No. 8 (preprint),21 pp. U.S. Weather Bur., Washington, D.C. 54. Browning, K. A., and Donaldson, R. J., Jr. (1963). Air flow and structure of a tornadic storm. J . Atmoa. Sci. 20, 533-545. 55. Kessler, E., I11 (1961). Kinematic relations between wind and precipitation distributions. J . Meteorol. 18, 610-525. 56. Ward, N. B. (1961). Radar and surface observations of tornadoes of 4 May 1961. Proc. 9th Weather Radar Conf., Kansas Cit,y, Missouri, pp. 175-180. Am. Meteorol. SOC.,Boston, Massachusetts. 57. Ligda, M. C . H., and Mayhew, W. A. (1954). On the relationship between the velocities of small precipitation areas and geostrophic winds. J . Meteorol. 11, 421423. 58. Brooks, H. B. (1946). A summary of some thunderstorm observations. Bull. A m . Meteorol. SOC.27, 557-563. 59. Newton, C. W., and Katz, S. (1958). Movements of large convective storms in relation to winds aloft. Bull. A m . Meteorol. SOC.32, 129-136. 60. Moaeop, S. C., Carte, A. E., and Le Roux, J. J. (1961). Hailstorm a t Johannesburg on 9th November, 1959. NuMla 4, 68-73. 61. Newton, C. W., and Fankhauser, J. C. (1964). On the movements of convcctive storms, with emphasis on size discrimination in relation to water-budget requirements. J. Appl. Meteorol. 3, 661-668. 62. Hoecker, W. H., Jr. (1957). Abilene, Texas, area tornadoes and associated radar echoes of May 27, 1956. Proc. 6th Weather Radar Conf.. Cambridge, Massachusetts (Sec. H.) pp. 18-25. Am. Meteorol. Soc., Boston, Massachusetts, 63. Bigler, S. C., and Inman, R. L. (1958). A preliminary claaeification of radar precipitation echo patterns associated with Midwestern tornadoes. Final Rep., Contract Cwb-9322 (A. and M. Project 169-Ref. 58-llP). Texas A. & M.,College Station, Texas. 64. Stout, G. E., and Hiser, H. W. (1955). Radar scope interpretations of wind, hail, and heavy rain storms between May 27 and June 8, 1954. Bull. Am. Meteorol. SOC. 36, 519-527. 65. Pedgley, D. E. (1962). A meso-synoptic analysis of the thunderstorms on 28 August 1958. Geophya. Mem. 14,No. 106, 74pp. 66. Brunk, I. W. (1953). Squall lines. Bull. A m . Meteorol. SOC.34, 1-9. 67. Fujita, T. (1955). Results of detailed synoptic studies of squall lines. Tellus 7 , 405-436. 68. Williams, D. T. (1953). Pressure wave observations in the Central Midwest, 1952. Monthly Weather Rev. 81, 278-289. 69. Douglas, R. H., and Hitschfeld, W. (1959). Patterns of hail storms in Alberta. Quart. J. Roy. Meteorol. SOC.85, 105-119. 70. Donaldson, R . J., J r . , and Chmela, A. C. (1960). Some behavior patterns ofNew England hailstorms. I n “Physics of Precipitation,” Geophys. Monogr. No. 5 (H. Weickmann, ed.), pp. 354-368. Am. Geophys. Un., Washington, D.C.

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71. Donaldson, R. J.,Jr. (1958). Analysis of severe convective storms observed by radar. J . Meteorol. 15, 44-50. 72. Wilk, K. E. (1961). Radar reflectivity observations of Illinois thunderstorms. Proc. 9th Weather Radar Conf., Kansas City, Missouri, pp. 127-132. Am. Meteorol. SOC.,Boston, Massachusetts. 73. Schleuaener, R. A,, and Grant, L. 0. (1961). Characteristics of hailstorms in the Colorado State University network, 1960-61. Proc. 9th Weather Radar Conf., Kansas City, Missouri, pp. 140-145. Am. Meteorol. SOC., Boston, Massachusetts. 74. Stout, G. E., Blackmer, R. H., and Wilk, K. E. (1960). Hail studies in Illinois relating t o cloud physics. I n “Physics of Precipitation” Geophys Monogr. No. 5 (H. Weickmann, ed.), pp. 369-383. Am. Geophys. Un., Washington, D.C. 75. Frisby, E. M . (1962).Relationship of hail damage patterns to features of the synoptic map in the upper Great Plains of the United States. J. Appl. Meteorol. 1, 348-352. 76. Kuettner, J. (1950). The electrical and meteorological conditions inside thunder clouds. J. Meteorol. 7 , 322-332. 77. Shackford, C. R. (1960). Radar indications of a precipitation-lightning relationship in New England thunderstorms. J. Meteorol. 17, 15-19. 78. Fawbush, E . J., and Miller, R. C. (1954). A basis for forecasting peak wind gusts in non-frontal thunderstorms. Bull. A m . Meteorol. SOC.35, 1P19. 79. Foster, D. S. (1958). Thunderstorm gusts compared with computed downdraft speeds. Monthly Weather Rev. 86, 91-94. 80. Ward, N. B., and Arnott, A. B., Jr. (1963). Some relations between surface wind fields and radar echoes. Conf.Rev. 3rd Conf. Severe Local Storma, 11 pp. Am. Meteorol. SOC.,Urbana, Illinois. 81. Brunk, I. W. (1949). The pressure pulsation of 11 April 1944. J . Meteorol. 6, 181187. 82. Williams, D. T. (1963). The thunderstorm wake of May 4, 1961. Natl. Severe Storms Proj. Rep. No. 18 (preprint), 23 pp. U.S. Weather Bur., Washington, D.C. 83. Bates, F. C. (1963). The mechanics of Great-Plains tornadoes. Conj. Rev. 3rd Conf. Severe Local Storme, 7 pp. Am. Meteorol. SOC.,Urbana, Illinois. 84. Fujita, T. (1960). A detailed analysis of the Fargo tornadoes of June 20, 1957. Res. Pap. No. 42, 67 pp. U.S. Weather Bur., Washington, D.C. 85. Brooks, E . M. (1949). The tornado cyclone. Wealherzuiee 2, 32-33. 86. Fujita, T. (1958). Mesoanalysis of the Illinois tornadoes of April 9, 1953. J. Meteorol. 15, 288-296. 87. Hoecker, W. H., Jr. (1960). Wind speed and flow patterns in the Dallas tornado of April 2, 1957. Monthly Weather Rev. 88, 167-180. 88. Ward, N. B. (1956). Temperature inversion as a factor in formation of tornadoes. Bull. A m . Metewol. Soc. 37, 145-159. 89. Fulks, J. R. (1962). On the mechanics of the tornado. Natl. Severe Storms Proj. Rep. No. 4 (preprint), 33 pp. U.S.Weather Bur., Washington, D.C. 90. Fitzgerald, D. R., and Valovcin, F. R. (1964). High altitude observations of the development of a tornado producing thunderstorm, 19 pp., mimeo. Paper presented at, Conf. o n Phys. and Dynamica of Claude, Chicago, 1964. Air Force Cambridge Research Laboratorics, Bedford, Massachusetts. 91. Brooks, E. M. (1951). Tornadoes and related phenomena. I n “Compendium of Meteorology” (T. F. Malone, ed.), pp. 673-680. Am. Meteorol. Soc., Boston, Massachusetts. 92. Huff, F. A., and Changnon, S. A., J r . (1964). A model 10-inch rainstorm. J . AppZ. Meteorol. 3. 587-599.

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93. Newton, C. W. (1958). Convective precipitation in an occluded cyclone. J . Meteorol. Soe. Japan, 75th Anniv. Vol. pp. 243-255. 94. Roys. C . P. (1963). Penetrations of thunderstorms by an aircraft flying at supersonic speeds. Natl. Severe Storms Proj. Rep. No. 15 (preprint), 19 pp. U.S. Weather Bur., Washington, D.C. 95. Van Thullenar, C. F., Lee, J. T., Arnett, A. B., Jr., and Newton, C. W. (1963). Severe storm detection and circumnavigation. Final Rep., Federal Aviation Agency Contract ARDS-A-176. U.S.Weather Bur., Washington, D.C. 96. Kessler, E., Lee, J. T., and Wilk, K. E. (1966). Associations between aircraft measurements of turbulence and weather radar measurements. Bull. Am. Meteorol. SOC. 46, 443-447. 97. Stainer, R., and Rhyne, R. H., J r . (1962). Some measured characteristics of severe storm turbulence. Natl. Severe Storms Proj. Rep. No. 10 (preprint), 17 pp. U.S. Weather Bur., Washington, D.C.

THE RISE OF OXYGEN IN THE EARTH’S ATMOSPHERE WITH NOTES O N THE MARTIAN ATMOSPHERE*? L. V. Berkner and L. C. Marshall Southwest Center for Advanced Studies, Dallas, Texas

1. Undorlying Premises

..................................................

Page 309

2. Oxygenic Concentration in the Primitive Atmosphcre of the Earth . . . . . . . . . . 310 3. Surface Oxidation in the Primitive Atmosphere .......................... 315 4. Ecology for Photosynthetic Oxygen Production in a Primitive Terrestrial Atmosphere ...................................................... 317 5. The First Critical Level-02 + 0.01 P.A.L. ............................. 319 6. Identification of First Critical Level with Opening of Paleozoic Era ......... 320 7. The Second Critical Level-02 --f 0.1 P.A.L.-the Late Silurian.. ........... 321 8. Oxygenic Levels in the Late Paleozoic and Ensuing Eras .................. 321 323 9. Further Refinement o f t h e Model ....................................... 10. Estimates of the Composition of the Martian Atmosphere and Surface ....... 324 11. Lifeon Mars ........................................................ 328 12. A General Theory of Origin of Planetary Atmospheres .................... 329 References .............................................................. 330

1. UNDERLYING PREMISES

As an initial premise we postulate that upon its formation the earth accumulated without an external primordial atmosphere. This postulate seems established by the very high fractionation ratios of the noble gases following Suess [l], and Brown [2] (e.g., terrestrial neon/silicon abundance compared to its universal ratio), and by the relatively large loss of hydrogen and helium by the inner planets [3]. For example, the terrestrial fractionation ratio of Ne t o N, which are not very dissimilar in atomic weight, with respect to their comparable abundances in the universe, (universally Ne/N- 8.6/6.6), is of the order of one-millionth. This leads to the direct implication that those atmospheric constituents which were not bound chemically in solids of the planetesimals forming the earth were lost from their low gravitational fields prior to their agglomeration into the earth. This postulate leads to the second premise that the atmospheric gases of the earth are of secondary internal origin. Following Rayleigh [4], Rubey

-

* This work was supported in part by the National Science Foundation under Grant No. 768 and No. 4708, and by the U.S. Weat,her Bureau under Grant No. Cwb 10531. t Bascd in part on invited paper prepared for General Assembly, Fifth Western National Meeting of the American Geophysical Union, Dallas, Texas, September 1965. 309

310

L. V. BERKNER AND L. C. MARSHALL

[a, 61, and others, the major and significant juvenile source of atmospheric gases (excepting only oxygen) arises from magmatic differentiation and release through thermal and volcanic processes. Atmospheric and fossil quantities of atmospheric gases and water appear consistent over geologic time with observed volcanic gaseous effluents including H,O, CO,, N,, and H,. This is true also of the solid materials which have formed the crust with its continents and with which the gaseous effluents were presumably associated in juvenile form. For example, from preliminary examination of nitrogenic compounds in the rocks, Rayleigh [4] indicates differentiation of mantle materials in the magmatic processes with a volume of five to fifty times the volume of the crust, and of the volume of juvenile nitrogenofthe atmosphere, together with its fossil counterpart in the sediments. The work of Urey [7] indicates, from the physical chemistry of the earth, that its magma was selectively molten-probably not wholly molten during any geologic periodso that juvenile volcanic effluents were discharged more or less uniformly over the ages. This is in agreement with Hutchinson’s [8] conclusions on the release of nitrogen. Juvenile oxygen appears to be largely absent from the primitive secondary atmosphere. Oxygen is not present in volcanic effluents, and any available oxygen in such effluents would be lost in oxidation processes associated with volcanoes [9]. Partial oxidation of very early rocks is consistent with such low oxygenic levels following MacGregor [lo], Lepp and Goldich [ll], and Rutten [12]. Moreover, oxygen in significant quantities is toxic to the primitive organic materials from which life must have been organized (cf. Abelson [13]). Only in advanced ccllular organisms are developed the enzymes employing vitamins to create defenses against oxygen [14]. The whole evidence concerning the synthesis of living organisms seems to require a reducing and essentially nonoxygenic primitive atmosphere [MI. All free atmospheric oxygen must be derived from H,O either by photodissociation or photosynthesis. Prior to life, and in absence of significant ocean areas, the sources of oxygen were small, and its consumption by surface oxidation through ozone as shown later is large, 10 that equilibrium levels would be low. 2.

CONCENTRATIONS IN THE PRIMITIVE ATMOSPHERE OF THE EARTH

OXYQENIC

In 1959, Urey [lS] suggested that oxygen production in the primitive atmosphere through photodissociation of H,O by ultraviolet radiation would be rcstricted and self-regulated, since the oxygen produced photolytically would shadow the underlying tropospheric water vapor, thereby restricting photodissociation. Berkner and Marshall [17, 181 have calculated this

311

RISE OF OXYGEN IN THE EARTH’S ATMOSPHERE

0

1000

2000 2200 WAVELENGTH (1,

Fro. 1. Solar intensity 1400-3000

2400

2600

2600

A. Composite data from Nawrocki and Papa [19].

restriction on oxygenic concentration in a primitive atmosphere imposed by this “Urey effect.” The distribution of solar radiation in the ultraviolet (uv) bands is now well known, as a consequence of measurements by rockets and satellites, and summarized by Nawrocki and Papa [19] (see Fig. 1). The significant proportion of uv involved in photodissociation of major atmospheric components (99.9%) lies between 1500-2100 -4.This radiation arises from the sun’s photosphere and is relatively stable as contrasted to the strong line radiation from the chromosphere corona at shorter wavelengths. From the study of M-type stars, Wilson [20] suggests that such photospheric radiation will not vary by as much as a factor of 3 over geologic time. Therefore, the sun’s uv radiation between 3000 and 1400A can be assumed as constant over geologic time. Watanabe and his colleagues [21, 221 and Vigroux [23] have measured the absorption of this radiation (Figs. 2 and 3) over the uv band for all significant atmospheric constituents. Of the atmospheric constituents produced as volcanic effluents or related chemical processes into a primitive atmosphere (i.e., in absence of substantial areas supporting living organisms), one should expect H,, H,O, N,, CO,, A, and traces of other elements; 0, O,, and 0, would be present only to the extent allowed by photolytic production from uv radiation acting on H,O.

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313

RISE OF OXYGEN IN THE EARTH'S ATMOSPHERE

With respect to other possible constituents, CO is not stable in an atmosphere containing 0 and 0,. These react a t moderate temperatures and strongly in the presence of uv radiation to produce CO,; NH, is highly soluble and chemically active, being substantially removed in the presence of water. Of the remaining constituents, only H,O, CO,, O,, and 0, will absorb radiation in the dissociative band (1500-2100 A). At low oxygenic partial pressures, 0, is produced in contact with, or very close to, the surface where it is rapidly removed. Therefore, analysis of the photodissociative effects in a primitive atmosphere deals with a three-component system H,O, CO,, and 0,.

-

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FIQ.4. Shielding from photodissociation of atmospheric H20 by

02.

As shown in Fig. 4, H,O vapor is distributed in the present atmosphere according to the Gutnick model [23a], Subsequent discussion suggests that the outer atmosphere may be somewhat drier than this model. Since H,O vapor represents a partial equilibrium of vapor pressure with a wet earth (now 70% ocean areas) and since the earth was substantially wet throughout geologic history with no great temperature differences indicated [16], Fig. 4

314

L. V. BERKNER AND L. C. MARSHAJL

probably represents the distribution of H,O within an order of magnitude through most of geologic history. Prior to withdrawal of CO, from the atmosphere by substantial areas of photosynthetic organisms, CO, may have risen t o higher than present values. However, because of the equilibrium of carbonate ions in water, leading to precipitation and deposition of carbonate minerals from water a t high concentrations, CO, probably never exceeded 10 times the present atmospheric level (P.A.L.)(see Rubey [5, 61 and Holland [24]). Both 0, and CO, are distributed nearly exponentially with altitude. Within these limits, Berkner and Marshall [25] calculate the absorption of dissociative radiation (1500-2100 A) in this three-component system a t each wavelength for each of a variety of atmospheres ranging between 02: 5 x lop3to 5 x P.A.L. and CO,: 1 to 10 P.A.L. with H,O a t 1 P.A.L. 280-

- 240 'y 220 cn 260

I 5110-4

I

I

10-3 2110-3 FRACTION OF PRESENT OXYGENIC CONCENTRATION

FIQ.5. Urey Relf-regulation of oxygen in primitive atmosphere: (-) H2O: I P.A.L.and ( - -) C02: 10 P.A.L.. H2O: 1 P.A.L.

C o t : 1 P.A.L.,

The result gives the quantitative expression of the Urey self-regulation of maximum oxygenic concentration in a primitive atmosphere as illustrated in Fig. 5 . Each point represents the integration of dissociative energy absorbed over the dissociative band by a particular atmospheric constituent. It is apparent immediately that a t low oxygenic levels (i.e., 0, - 5 x lo-' P.A.L.) H,O vapor is exposed to photolytic action, As oxygen is produced

RISE OF OXYGEN IN THE EARTH'S ATMOSPHERE

315

and oxygenic concentration rises, the H,O vapor is shadowed, cutting off the oxygen source. The oxygenic equilibrium will occur a t that oxygenic concentration when the loss of oxygen in oxidative processes, primarily through photodissociation of 0, and loss of 0 and 0,, equals its rate of production. Thus at the Ureyequilibrium limit, the rate of loss equals the rate of production of oxygen. If the rate of loss is increased, the disappearance of oxygen exposes H,O vapor and increases its rate of production to a new and slightly higher equilibrium. If the rate of loss is diminished, H,O is further shadowed and photolytic production is correspondingly diminished. The loss of 0, depends upon its rate of photodissociation for which not more than about 200 ergs om-, sec-' are available, corresponding to a rate of the order of lo'* molecules cm-2 sec-l. Not until this production rate of 0, is significantly exceeded by sufficient areas of photosynthetic organisms can a n oxygenic atmosphere be constructed. From this reasoning, Berkner and Marshall conclude that the oxygenic concentration in the primitive atmosphere of the earth cannot have exceeded about 10-3(i.e.,0.1%) present atmospheric concentration. Hans Suess has properly remarked (see Berkner-Marshall [26]; Barth and Suess [27]) that this can be considered only as a maximum. With H, present in the primitive atmosphere, the summary back reactions

2H2

+ 02+2H20

or

H,

+ 0 + H,O

may represent major sinks for oxygen in the lower troposphere at suitable temperatures, leading to substantially lower concentrations because of very high rates of loss of 0 or 02. As photosynthetic production of 0, is enlarged, it simply substitutes for photolytic oxygen until finally the rate of oxygen production substantially exceeds its rate of loss. Only then can the Urey self-regulation be overridden.

3. SURFACE OXIDATIONIN

THE PRIMITIVE

ATMOSPHERE

The low concentration of oxygen in the early atmosphere, as it rises to the self-regulated limit of
316

L. V. BERKNER A N D L. C. MARSHALL

O3 FOR 02'.10PAL.

6C

5w

\

I

10N (03)/L

FIG.6. Estimated idealized distribution of ozone for various levels of oxygen.

The highly stylized representation of Fig. 6 illustrates the character of ozone distribution, assuming total atmospheric pressure proportional to partial pressure of oxygen. At higher total pressures (most probably due to higher concentrations of nitrogen which is the obvious candidate for the third body in the reaction), the concentrations (but not the altitudes) of ozone would be somewhat increased. A considerable range of atmospheric assumptions relative to atmospheric characteristics controlling ozone production would elaborate, but not substantially modify, the quality of subsequent conclusions. As a result, in the primitive and intermediate atmospheres, atomic oxygen and ozone will be produced in a thin layer in contact with the earth's surface. Since reaction rates of both 0 and O3 are many orders of magnitude times the reaction rates of O,, with respect to surface materials, the loss of 0 and 0, as surface oxidants represents a principal loss of oxygen. Thus, even in the early tenuous oxygenic atmospheres, surface oxidation rates could well have exceeded present values considerably. Berkner and Marshall find the total

RISE OF OXYGEN IN THE

EARTH’S ATMOSPHERE

317

uv energy available over geologic time adequate by this process to account for all fossil oxygen with quantum efficiencies for photodissociation of 0, of less than one per cent. Thus the frequent assumption that crustal oxides in the early and intermediate ages require high concentrations of atmospheric oxygen becomes unnecessary. Wildt [28] has pointed out that this process is now active on the surface of Mars, accounting for the reddish color of its surface (see Section 10). As a further consequence of the production of ozone a t the earth’s surface in early atmospheres, we can now estimate that to build an oxygenic atmosphere, the rate of production of 0, must exceed its rate of photodissociation and consequent rapid loss. Only under this condition can the self-regulated equilibrium of oxygenic levels be upset, since the increase of oxygen concentration can only arise from a small positive differential between the relatively larger processes of production and of removal.

4. ECOLOGY FOR PHOTOSYNTHETIC OXYGEN PRODUCTION IN A PRIMITIVE TERRESTRIAL ATMOSPHERE P.A.L., and Since the oxygenic level of the primitive atmosphere is the oxygenic concentration is now 1 P.A.L., Hutton’s principle of uniformitarianism is inapplicable, for there must have been a considerable variation of oxygenic concentration with time. It is necessary, therefore, to trace in time the oxygen balance and t o search for evidence of the oxygenic concentrations along the geologic column. The only additional source of oxygen capable of overriding the Urey selfregulation is photosynthesis which depends on visible light, unimpeded by oxygen, ozone, water vapor, or carbon dioxide. We therefore inquire into the restrictions on photosynthesis in the presence of the early atmosphere. Figure 7 illustrates the penetration of uv into water, through the atmosphere defined in Fig. 6, to the reduced intensity of 1 erg om-, sec-’ (50A)-’.This defines the early ecology, in approximate consonance with Sagan’s estimates [29, 301, to bottom dwelling organisms in about 10 meters of water. The environment must be protected by sufficient depth of water t o protect from lethal uv, but sufficiently shallow to maximize photosynthesis, with gentle convection to supply nutrients synthesized a t the surface in the presence of uv radiation but no violent convection that would dislodge and circulate primitive benthic organisms (having no advanced forms of control) toward the lethal surface. Thus primitive photosynthetic life must be restricted to shallow lakes or shallow protected seas. The early bioherms described by Hoering and Abelson [31] as supporting living organisms in the era -2.7 billion years appear prime seats for the origin of life. I n particular, pelagic life in the oceans appears forbidden.

318

L. V. BERKNER AND L. C. MARSHALL

Perhaps, as a coincidence, we observe from Fig. 7 that nucleic acids, proteins, and other essential elements for life, whose maximum absorption of available ultraviolet peaks at 2630 and 2750A, respectively, have the most

la00

2000

2200

2400

2600

2800

3000

3200

3400

WAVELENQTH ( 1)

Flo. 7. Path length in liquid watcr in presence of O a and 0 3 for various concentrations of 0 to absorb available uv to “extinction” [ l erg om-2 scc-1 (50 A)-I].

favorable opportunity for nat,ural selection. This appears as a basic characteristic of our solar system since it depends only on the shape of the graph for absorption of 0,and liquid watcr and upon the intensity of solar uv with wavelength. Tn this restricted ecology, Berkner and Marshall make the preliminary estimate that pliotosynthetic organisms a t present densities and effectiveness must cover somewhat more than 1% of present continental areas to upset

319

RISE OF OXYGEN IN THE EARTH’S ATMOSPHERE

the Urey self-regulated equilibrium and begin the construction of an oxygenic atmosphere. 5. THEFLRST CRITICAL LEVEL-O~

+ 0.01 P.A.L.

Figure 8 is a reorganization of the data of Fig. 7. We observe that as oxygen rises to 1 % P.A.L.,the penetration of lethal uv is confined to a thin layer of surface water, opening life t o the oceans. Moreover, a t this oxygenic level, many organisms pass the “Pasteur point,” adapting their metabolism from fermentation t o respiration, thereby increasing available metabolic energies per gram-mole by about 50 times. With the

‘40 3000

a

2900

1

I200

IO-~

10-3

10-2 FRACTION OF PAL

10-1 OF OXYGEN

I

10

FIQ.8. Penetration of uv radiation in liquid water with various combinations of oxygen and ozone atmospheres [intensity at extinction = 1 erg cm-%ec-1(50 A)-]].

appearance of respiration, evolutionary requirements appear for a nervms system to control the process, a circulatory system for oxygen distribution, a digestive system to maximize it, and so on. With the evolutionary opportunities offered by respiration and its consequences, and uith the extensive isolation offered by opening the oceans to pelagic life, one can anticipate an evolutionary explosion toward more complex organisms when 0, +P.A.L.

320

L. V. BERKNER AND L. C. MARSHALL

6. IDENTIFICATION OF FIRST CRITICAL LEVELWITH OPENINGOF PALEOZOIU ERA

We therefore inquire from the geologic record whether such a n evolutionary explosion occurred. There is just one-the opening of the Paleozoic 6 x lo8 years ago. By immediate inference, Berkner and Marshall identify the opening of the Paleozoic with the oxygenic level, 0, lo-, P.A.L. (see Berkner [32]). Following this model, the geologic record should be read exactly as observed with no long interval of advanced evolutionary development interpolated prior to the opening of the Paleozoic. Indeed, such evolutionary development is forbidden until the rise of oxygen permits the beginning of respiration and thus the first evolutionary force toward the organization of multicellular forms. I n a critical study and review of geologic, paleontologic, and geochemical evidence during the Pre-Cambrian, Cloud [33] reaches several conclusions which bear strongly on this model. N

1. There is clear paleontologic evidence of early thallophyta, some probably oxygen producers, as early as 1.9 billion years ago. 2. (Cloud makes the interesting suggestion that): Local and isolated tenters of biological oxygen production in some kind of dependent balance with dissolved ferrous ion as a n oxygen acceptor can account for the different ages and facies of Pre-Cambrian banded iron formation in different parts of the world, this process a t the same time inhibiting oxygen release into an essentially anoxygenic atmosphere. 3. There is no evidence of multicellular organisms prior to the opening of the Paleozoic. The beginning of the Paleozoic should be “defined operationally as the base of the range-zone of Metazoa (or Eumetazoa if we exclude the sponges).” 4. “By the beginning of Paleozoic time, about 0.6 billion years ago, enough oxygen had accumulated to permit evolution and diversification of the Metazoa.”

These conclusions are in general consonance with the Berkner-Marshall model of 1963, with any question of dating hinging on the speed of response of metazoan evolution t o the evolutionary niches opened by respiratory opportunity. Thus Cloud, on geologic reasoning, would leave open the dating of the 1 yo P.A.L. between the limits of 1.2 billion years and the Berkner-Marshall inference of 0.6 billion years. The Berkner-Marshall model P.A.L. the would question the admissibility of this span, since as 0, -+ expansion of even primitive unicellular photosynthetic activity would lead t o a more rapid rise in oxygenic concentrations during this period than can be inferred from the subsequent succession of events. Certainly, further critical examination of this detail of the model will be made.

RISE OF OXYGEN IN THE EARTH'S ATMOSPHERE

321

7. THESECOND CRITICALLEVEL-0, + 0.1 P.A.L.-THE

LATESILURIAN With the evolutionary explosion of the early Paleozoic with pelagic life prolific in the oceans photosynthetic production of oxygen is materially increased. We then observe from Fig. 8 that as oxygen approaches 0.1 P.A.L., the land surface is just protected from lethal uv, opening the evolutionary opportunity ashore. Definite spores of land plants appear first in mid-Silurian, suggesting that a t least in protected locations spore-bearing plants could then first stick their heads above water without lethal sunburn. Immediately, in late Silurian, a number of phyla of plants and insects appear to have evolved ashore simultaneously, with great forests found by early Devonian. At the same time, the advanced respiratory apparatus of the fishes was developed, an apparatus requiring higher oxygenic levels. Almost immediately vertebrate life evolved ashore. By immediate inference, the Berkner-Marshall model identifies the oxygenic level 0, lo-' P.A.L. with the late Silurian evolution of life ashore.

-

8. OXYQENIC LEVELS IN THE LATEPALEOZOIC AND ENSUING ERAS

With the explosion of life ashore, photosynthetic production of 0, is increased some 20 %, further altering the oxygen balance and promoting rapid rise of 0, toward the Carboniferous. From the character of life a t the opening of the Carboniferous, we can infer that the oxygenic concentration had reached or exceeded present values. Because the production of CO, due to decay of organic materials may have lagged in phase behind the rise in production of 0,, some fluctuation in 0, can be anticipated in subsequent geologic periods. More particularly, any induced climatic change that de-adapts specialized photosynthetic producers in substantial numbers, thereby upsetting the oxygen balance, can cause a precipitous drop in 0, concentration due to successive de-adaptation of organisms as oxygenic concentration falls, followed by a slower recovery as new organisms re-adapt. The present quasi-stable oxygenic concentration, involving a rate of 0, production of 7 x 1013 molecules cm-' sec-' over the earth, corresponds to a lifetime of atmospheric oxygen of only some 2000 years until its return through the biological cycle-a time negligibly short compared t o geologic periods (see Rabinowitch [34]). Consequently, fluctuations in oxygenic balance introduced by precipitous drops in oxygenic concentrations followed by slower recovery should be expected in the late Paleozoic and subsequent periods.

322 L. V. BERKNER A N D L. C. MARSHALL

d

RISE OF OXYGEN IN THE EARTH’S ATMOSPHERE

323

Summarizing the best estimates of oxygenic concentration in the several ages, Berkner and Marshall arrive a t the model illustrated in Fig. 9.

9. FURTHER REFINEMENT OF THE MODEL We believe this model represents the most logical synthesis of available information with very considerable self-consistenceover a very wide range of observations. It is, of course, open t o refinement as new data are acquired as a consequence of search stimulated by this model, and their meaning adduced. The present work on paleoatmospheres opens a new vista in the understanding and description of major features of the successive geologic periods, particularly since the opening of the Paleozoic. Specialized geologic features (such as the widespread red-beds of the Devonian or Permian, for example) remain to be related quantitatively to the oxidizing characteristics of the atmosphere, i.e., the concentrations of 0 and 0,, and, through altitude of production and related convective activity, their relative access to the surface iron, sulfur, and other reduced or partially oxidized materials. The oxidizing character of the atmosphere in any geologic period in turn depends not merely on oxygenic levels, but also on inferred concentrations of third bodies such as nitrogen and carbon dioxide, and on the consequent temperature, altitude, and concentration of the ozone producing layer. The conclusions of Hutchinson [8] that nitrogen has increased through geologic time probably uniformly to a first approximation, and of Rubey [6, 61, concerning concentrations of CO,, provide the preliminary basis for commencing such quantitative studies, though further refinement of these estimates is indicated. As the level of production of 0 and 0, rises through the present troposphere, the temperature a t and above the present tropopause will be changed violently, with lapse rates in the troposphere that change convective activity rather radically from one period to another. , Thus, in the primitive atmosphere, with ozone production near the surface, vigorous convection might be induced by convective heat transfer upward. In contrast, during the mid-Paleozoic, for example, this model suggests a maximum of ozone production just above the present troposphere, thus radically modifying the temperature regime with altitude, with consequent radical modification to climate. This study of atmospheric composition and resultant climate in successive geologic periods can be controlled by the requirement that the atmospheric changes between successive geologic periods must lead to a self-consistent succession of atmospheric events. I n particular, more detailed compositional and climatological models of the successive geologic periods raise altogether different and critical paleontological questions that can guide the search for new and related evidence.

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L. V. BERKNER AND L. C. MARSHALL

10. ESTIMATES OF THE COMPOSITION OF THE MARTIAN ATMOSPHERE AND SURFACE The success of the present model encourages application of its basic premises to the atmosphere of Mars. The data of the occultation experiment of Kliore et aZ.[35] give accurate estimates of the heights of the ionized layers of the Martian ionosphere. Employing these data to derive scale heights, and guided by visual, spectroscopic, and reflectance data, Johnson [36] concludes that the principal and predominant atmospheric constituent of Mars will be carbon dioxide a t a pressure of about 10 mbars corresponding to 30 gm omp2. The temperature regime derived by Johnson shows that the outer atmosphere of Mars must be very cold ( 85°K) as a consequence of the radiative balance with high concentrations of CO,. The same conclusion is reached by Fjeldbo et al. [37]. This leads to the conclusion that in spite of its lower gravitational field, all gaseous effluents of Mars have been conserved during recent geologic time, and the heavier gases (CO,, etc.) throughout geologic time (with atomic oxygen predominant in the outer atmosphere). Since Mars has relatively little surface water, carbon dioxide has not been significantly removed from thc atmosphere (through precipitation as carbonates or carbohydrates as on the earth), and the present CO, concentrations in the Martian atmosphere appear to represent its total storage over geologic time. In endeavoring to estimate the gross composition of the Martian atmosphere based on the Mariner IV data the following assumptions are implicit: N

1. That the atmosphere of Mars has been produced from internal secondary processes in the same way as the atmosphere of the earth. 2. That the secondary gaseous effluents which have formed the Martian atmosphere are in about the same proportion as the thermal effluents from the earth. 3. That while there are probably primitive organisms with photosynthesis on Mars, they are not sufficiently widespread or abundant to upset the Urey self-regulation of oxygenic pressure. 4. That the Mariner IV results are sufficiently reliable with respect t o derived scale heights, temperatures, and pressures so that the escape of the lighter constituents does not dominate the composition of the Martian atmosphere as has been earlier believed so that the Martian atmosphere represents storage during most of geologic time. We notice on the earth that there has been a deposition of CO, in fossil form of somewhere between 7500 and 30,000 gm om-, (before Rankama and Sahama [38], comprised a t the minimum of precipitated limestone and dolom i t d 6 0 0 , coal and hydrocarbons-800, and CO, in the hydrosphere-25).

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The CO, of the terrestrial atmosphere, however important to photosynthesis, is almost negligible compared to the fossil deposition in the sediments, amounting t o no more than about $ gm cm-, over the surface of the earth. The minimum estimate, 7500 gm ern-,, is perhaps too low because of our lack of knowledge of fossil CO, in the ocean bottoms, which suggests a maximum estimate of about 30,000 gm cm-'. Taking the ratio of CO, on Mars (30 gm cm-*) to the ratio of CO, on earth (7500 to 30,000 gm cni-') we have the CO, on Mars as 0.001 t o 0.004 of the total CO, on earth. This ratio [(C02)~l/(C02)e], represents the proportion of effluents from the solid surface of Mars as compared to the effluents into the atmosphere of the earth. From these basic data the following gross estimates of other elements of the Martian atmosphere can be derived: N2: On earth N, is present t o the extent of 770 gm cmP2in the atmosphere and between 75 and 100 gm cm-2 fossil N, in the sediments (a total represented by about 1000 gm cm-2). Applying the [(C02)b,/(C02)e]ratio we see that N, should fall between 1 and 4 gm cm-2).

0,: The maximum value of oxygen concentration will be subject t o the limitation imposed by Urey self-regulation (as in the primitive atmosphere of the earth). This will lead to a maximum value for oxygen between 0.25 and 1.0 gm cm-, with actual equilibrium value probably more nearly 0.1 gm cm-, for reasons specified in the discussion of the primitive atmosphere of the earth. N I J ~Ammonia : has never been an important constituent of the terrestrial atmosphere partly due to its high solubility in water and partly due to the photodissociation of the residue. Since water is much less abundant on Mars its loss in solution will be low. However, due to the low effluent ratio there appears to be sufficient solar energy to insure essentially complete photodissociation of atmospheric NH, and therefore very low values.

-

AR: Argon in the earth's atmosphere represents about 13 gm Of this total, -0.35 yo is Ar36; 99.65 yois $r4', derived radiogenically from K'O. Therefore, not all atmospheric argon is of direct volcanic origin [3WO] since significant quantities of K40 will have disintegrated in the crust after its ejection from the mantle, and some Ar4' has been released by subsequent erosion. The relative quantities thus released are now under debate though direct volcanic origin strongly dominates. Employing the [(CO,Jl,/(CO2).J ratio, leads t o the estimate of 0.013 t o 0.052 gm cmP2 of Ar in the Martian atmosphere. Since erosion processes are presumably quite different on Mars and on the earth, such atmospheric ratios as Ar/N, when available, will

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provide powerful evidence on the historical process of atmospheric formation on Mars. H,: Since hydrogen does not now escape from the Martian atmosphere (or in any event escapes only very slowly) its accumulation may be significant. Hydrogen will be released from several sources. Among these are: (a) Oxidation of reduced materials in thermal sources through such reactions as 3Fe0 H,O + H, Fe,O,. This is a strong exothermic reaction in terrestrial volcanoes. (b) Photolytic dissociation of H,O with loss of 0, in surface oxides. If photolysis of H,O has led to deposition of 10 meters, complete oxidation of fully reduced materials (see discussion of H,O below), of the order of 100 gm cmPEof H, would be released. This may be enhanced by the photodissociation of small quantities of CH, and NH,. Therefore, since the total Martian atmosphere is limited t o -10 mbars, a considerable escape of H, must have occurred a t sometime during its history. Since there is no way of determining the total effluent of hydrogen from the earth or its historic rates of loss on Mars, we can make only an educated guess as t o its present concentration, but it might be compared to N,. Taking into account the difference of its atomic weight, the mass of its atmospheric column would then be of the order of a few tenths of a gm ern-', although this estimate might be low.

+

+

WATER A N D WATER VAPOR: There are about 2 . 8 3 lo5 ~ gm ern-, of H,O on earth (equivalent to an average of about 2.83 km depth of water over the surface). To this we must add the loss of water over geologic time to produce the atmospheric oxygen and surface oxides (see Section 3). This will raise the estimate of total juvenile H,O produced on earth to - 3 x1O5 gm Employing the [(COE)h,/(COE)e] ratio leads to a n estimate of 300 to 1200 gm om-, produced over the surface of Mars. Of this total, some will be lost by photolytic action of the uv dissociative band on H,O which must be the principal source of the low levels of Martian oxygen, with a subsequent loss of oxygen in surface oxides. If we assume the same proportion of loss as on earth, we obtain an estimate of retained water on Mars of 280-1120 gm cm-'. This estimate also implicitly suggests a laycr of complete surface oxidation as suggested by Wildt [28]. If the oxidized layer on Mars corresponded to that on the earth, this would further suggest an equal layer of completely oxidized sediments of 10 to 100 meters. Since Mars lacks large bodies of water, such oxidation must be due almost entirely to dircct action of 0 and 0, (which must have been produced a t the Martian surface throughout its history in the same manner as in the earth's primitive atmosphere) on iron, sulfur, and other partially oxidized materials. However, since erosive processes involving rain have probably not been involved on Mars, oxidation of surface materials will occur only in thermal activity and because of erosion by wind. Therefore an estimate of 10 meters

-

RISE OF OXYGEN IN THE EARTH’S ATMOSPHERE

327

of completely oxidized surface materials seems most reasonable. This would indicate a lesser removal of 0,, and consequently a lower total photodissociation of H,O and its photolytic loss in the Martian environment. Thus it must be assumed that there has been retained a considerable amount of water on Mars perhaps mostly in the form of dust-covered ice and permafrost in addition to small amounts of exposed frost on the polar caps. The water in the atmosphere of Mars must be below the equilibrium of vapor pressure with a surface with substantial ice a t a temperature in the neighborhood of freezing or below. Based on spectroscopic evidence, together with the knowledge of low atmospheric temperature, the total atmospheric column of H,O probably amounts to no more than 0.6 x to 3 x l o p 4 gm ern-,. This is somewhat lower, though not incomparable to the water vapor in the outer atmosphere of the earth above the tropopause. There would be reason to believe that though the atmospheric mass of H,O vapor is very low, the relative humidity in the Martian troposphere may be quite high and therefore formation of clouds seems entirely possible. In a recent paper by Abelson [15], he estimates that the total accumulation of water vapor in the Martian atmosphere can be attributed to the solar wind. Certainly in absence of a magnetic field, protons will be accumulated in this way. However, if we assume that the Martian atmosphere is predominantly due to a secondary internal source, the water must also be predominantly from internal sources and there must still be relatively large quantities of water stored within Mars’ surface. CRUST OF MARS: The crust of the earth averages about 17.5 km in thickness, and assuming that the solid and gaseous effluent ratios present on the earth are applicable to Mars, wc can calculate an average thickness of the crust of Mars between 20 and 70 meters. The earth’s crust has probably been generated by the average action of 500 volcanoes (the number active now) over geologic time in accordance with the principle of uniformitarianism. This would suggest the generation of the Martian crust and its atmosphere by an average of one-half to two terrestrial-type volcanoes, further suggesting that the Martian crust may be localized and that much of the “Martian mantle” is accessible. The limited volcanic action on Mars may be anticipated by its lower gravitational field and smaller size, leading to significantly reduced internal forces and correspondingly lower rates of differentiation of the mantle. Thus magmatic activity would be correspondingly reduced, and its nature perhaps significantly altered from the volcanic and thermal forms most frequently observed on earth. Nevertheless. such magmatic activity must certainly be present due to internal heat from primordial sources, radioactivity, local faulting, and MacDonald-type expansion and subsequent shrinkage [41]. The surface of Mars will certainly be affected significantly by alternate

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regimes of frost and thaw, and by probable high wind velocities as a consequence of surface differentials in heat absorption. Therefore drifting dust (even in an atmosphere of low dust capacity) must alter surface characteristics noticeably.

A summary of these estimates is given in the accompanying tabulation for the Martian atmosphere, based on the initial assumptions specified. Element

gm cm-a

COa

30 1-4 0.1-1.0

-

Na 0s

>0.1-0.4

Ha Ha0 (vapor) Ha0 (liquid) NH3

Trace

Ar

0.013-0.062

0.6 x 10-4-3 x 10-4 280- 1120

11. LIFEON MARS? I n the light of these estimates, the problem of whether life can and does exist on Mars should be open to further estimate. On the whole, the surface of the planet must be below freezing. Johnson estimates a t the time of the Mariner 1V transit, a surface temperature of 210°K (-43°C) a t latitude 55" south. Nevertheless, visual observation indicates surface temperatures above freezing in the equatorial zone (even though the air temperature may be below freezing), and the temperature regime changes with season as shown by the alternation of the frost on the polar caps, whose disappearance is perhaps due to sublimation. The generally low temperature suggests that even in the nonfreezing zones liquid water will be generally below the 10°C temperature for Martian boiling [El. Thus, local pools of water or saturated soils, perhaps associated with thermal activity or a t the edge of the frceze-thaw belts, should be present, sufficient t o provide uv protection. Likewise, all elements required for synthesis of nutrient organic materials (H,, N,, etc.) appear available. Unquestionably, the Martian ecology, like the primitive ecology of the earth, is intensely severe, accompanied by a greater shortage of protective water, and by somewhat (but not very significantly) lower temperatures. Since we do not know the range of ecologies under which primitive organisms can be synthesized, one cannot state categorically that life does, or does not, exist. Certainly, the low levels of 0, indicate tha.t only the most primitive

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unicellular organisms can have evolved, and these far below a density that can approach a photosynthetic production of 0, a t the Urey self-regulated rate. On the other hand, it seems probable that under the widely varying Martian conditions (within the limits specified), combinations for the synthesis of organisms might frequently exist. Whether these constraints would have conjoined adequately and would be sufficiently constant to insure propagation of living organisms in any locality remains a puzzle. What is evident is that the Martian ecology can provide situations very close to those of the primitive earth. Consequently, its detailed study should provide a wealth of information on the circumstances under which life can or cannot be synthesized and can evolve, through a t least the establishment of definable constraints on those processes. I n the search for life on Mars, we must be keenly conscious of the highly specialized conditions under which it is likely to be found. As in the case of the primitive earth, these would be shallow pools about 10 meters in depth, or equivalently protected waters. Unlike the primitive earth, such favorable localities will be much more rare and more difficult t o find and to identify. On the other hand, the absence of widespread erosion may have better preserved the paleontological record. This suggests broad surveys of the Martian surface designed to identify the more likely sites.

12. A GENERALTHEORY OF ORIGINOF PLANETARY ATMOSPHERES The progress in developing a coherent hypothesis for the organization and history of the atmosphere of the earth, together with the first estimates of the atmosphere of Mars, suggests the steps in development of a basic theory for the constitution, growth, and stability of planetary atmospheres generally. With respect to the earth, only the basic outline has been formed. suggesting search for new forms of evidence. The dctails of constitution and related climatology along the geologic column await further development. I n particular, the only quasi-permanent charactcr of oxygenic levels gives some cause for concern with respect to stability, and there seems some urgency in determining and quantifying all factors influencing this stability. With respect to Mars, we must await detailed measurements by unmanned and manned space probes to teYt our present estimatcs. and to evaluate the feasibility of a general planetary theory of the origin, growth, and stability of their atmospheres. Tlie vacuum of information regarding the atmosphere of Venus warrants its early attention by space probes. It appears established, however, that concentration of oxygen a t or below the 1 yo present atmospheric level is prima ! h i e evidence that aiiy life on a planet must be limited to unicellular forms, and severely limited in its distribution and extent, assuming other necessary ecological factors are favorable.

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Only when oxygenic concentrations exceed respiratory levels are multicellular forms of life indicated. Higher oxygenic levels must be considered as conclusive evidence of widespread and advanced forms of life because of the evolutionary force exerted in selection of more highly organized organisms a t higher oxygenic levels. REFERENCES 1. Suess, H. E. (1949). Die Haufigkeit der Edelgase auf der Erde und im Kosmos. J . Qeol. 57, 600-607. 2. Brown, H. (1949). A table of relative abundances of nuclear species. Rev. Mod. P h p . 21, 625-634. 3. Vinogradov, A. P. (1959). The origin of the biosphere. In “Tho Origin of Life on the Earth, Symposium of International Union of Biochemistry,” Vol. 1, pp. 23-37. Maomillan, Now York. 4. Rayleigh, Lord (1939). Nitrogen, argon, and neon in the Earth’s crust. Part 11: Nitrogen. Proc. Roy. SOC.(London)A170, 459-464. 5. Rubey, W. W. (1951). Geologic history of sea water: An attempt to state the problem. Bull. Qeol. SOC.Am. 62. 1111-1 147. 6. Rubey, W. W. ( 1955).Development of the hydrospherc and atmosphore, with special reference to probable composition of the early atmosphere. Qeol. SOC.Am., Spec. Papere 62, 631-650. 7. Urey, H. C. (1952). “Tho Planets: Their Origin and Development,” 245 pp. Yale Univ. Press, New Haven, Connecticut. 8. Hutchinson, G. E. (1944). Nitrogen in the biogeochernistry of the atmosphere. Am. Scientist, 32, 178-195. 9. Holland, H. D. (1962). Model for tho evolution of the Earth’s atmosphere. “Petrologic Studies: A Volume to Honor A. F. Bucldington,” pp. 447-477. Princeton Univ. Press, Princeton, New Jersey. 10. MacGregor, A. hl. (1940). A Pre-Cambrian algar limestone in Sout,hern Rhodesia. Tram. Qeol. SOC.S . Africa 43, 9-15. 11. Lepp, H., and Goldich, S. S. (1959). Chemistry and origin of iron formations. Bull. Qeol. SOC.Am. 70, 1637. 12. Rutten, M. G. (1962). “Tho Geological Aspects of Origin of Life on Earth.” Elsovier, Amsterdam. 13. Abolson, P. H . (1957). Some aspects of paleobiochemistry. Ann. N . Y. Acad. Sci. 69, 276-285. 14. Gilbert, D. L. (1964). Atmosphere and evolution. “Oxygen in the Animnl Organism,” pp. 641-654. Pergamori Press, Oxford. 15. Abelson, f.H. (1965). Abiogenic synthesis in the Ilartian environment. Proc. Natl. Acad. Sci. 54, 1490-1494. 16. Urey, H. C. (1960). Primitive planetary atmoxphcros and the origin of life. In “Aspects of the Origin of Lift,,” ( M . Florkin, c d ) , pp. 8-14. Pergamon, Oxford; also The atrnosphcres of t,hc planets. I n “Handbuch der Phyaik” (S. Pliiggr, ed.), Vol. 62, pp. 364-418. Springor, Berlin, 1959. 17. Bcrkner, L. V., nncl Marshall. L. C. (l9ti4). Thc history of oxygenic concentrat,ion in the Earth’s atrnosphorc. Disciivsions Faraday SOC.37, 122-141. 18. Berkncr, L. V., and Marshall, L. C. (1965). On the origin and rise of oxygen concentration in the earth’s atmosphere. J . Atmospheric Sci. 22, 225-2ti1.

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19. Nawrocki, P. J., and Papa, R . (1961). “Atmospheric Processes,” AFCRL Rept., Contract AF-19(604)-7405. Geophysics Corp. Am., Bedford, Massachusetts. 20. Wilson, 0. C. (1963). A probable correlation between chromospheric activity and age in main-sequence stars. Astrophys. J. 138, 832-848. 21. Watanabe, K., Zelikoff, M., and Inn, E. C. Y. (1953). Absorption coefficients of several atmospheric gases. Qeophys. Res. Papers ( U . S . ) 21; AFCRC Tech. Rept. No. 53-23. 22. Watanabe. K. (1959). Ultraviolet absorption processes in the upper atmosphere. Advances i n Ueophys. 5, 153-221. 23. Vigroux, E. (1953). Contribution a 1’6tude experimentale de l’absorption de I’ozone (1). Ann. Phys. (Paris)[12] 8, 709-762. 23a. Howard, J. N., and Caring, J. S. (1962). The transmission of the atmosphere in the infrared--a review. Infrared Phya. 2, 155-173. 24. Holland, H . D. (1965). The history of ocean water and its effect on the chemistry of the atmosphere. Proc. Natl. Acad. Sei. U.S. 53, 1173-1 183. 25. Berkner, L. V., and Marshall, L. C. (1966). Limitation on oxygen concentration in a primitive planetary atmosphere. J . Atmospheric Sci. 23, 133-143. 26. Berkner, L. V., and Marshall, L. C. (1965). History of major atmospheric components. Proc. Natl. Acad. Sci. U.S. 53, 1215-1226. 27. Barth, C. A., and Suess, H. E. (1960). The formation of molecular hydrogen through photolysis of water vapor in the presence of oxygen. 2. Phys. 158, 86-95. 28. Wildt, R. (1942). The geochemistry of the atmosphere and the constitution of the terrestrial planets. Rev. Mod. Phya. 14, 151-159. 29. Sagan, C. (1957). Radiation and the origin of the gene. Evolution 11, 40-56. 30. Sagan, C. (1961). Organic matter and the moon. Natl. A d . Sci.-Nat. Rea.Council, Publ. 757. 31. Hoering, T. C., and Abelson, P. H. (1961). Carbon isotope fractionation in formation of amino acids by photosynthetic organisms. Proc. Natl. A d . Sci. U . S . 47, 623-632. 32. Berkner, L. V. (1952). Signposts to future ionospheric research. Ueophya. Rea. Papers, U . S . A . P . No. 12, from Proc. Conf. Ionospheric Phys. 1950, Part B, pp. 13-20. 33. Cloud, P. E., Jr. (1965). Significance of the gunflint (Pre-Cambrian) Microflora. Science 148, 27-35. 34. Rabinowitrh, E. I. (1951). “Photosynthesis and Related Processes.” Wiley (Interscience), New York. 35. Kliore, A,, Cain, D. L., Levy, 0.S., Eshleman, V. R., Fjeldbo, G. and Drake, F. D. (1965). Ocrultntion experiments; results of the first direct measurement of Mars’ atmosphere and ionosphere. Science 149, 1243-1248. 36. Johnson, F. S. (1965). The atmosphere of Mars. Science 150, 1445-1448. 37. Fjeldbo, G., Fjeltlbo, W. C., and Eshleman, V. R. (1966). Models for the atmosphere of Mars based on the Mariner IV occultation experiments. J . Ceophys. Res. 71, 2307-2316. 38. Rankama, K., and Sahama, T. G. (1950). Abundance and general geochemical character. “Geochemistry,” Chapter 23, pp. 574-583. Univ. of Chicago Press, Chicago, Illinois. 39. Faul, H . (1954). “Nuclear Geology,” 414 pp. Wiley, New York. 40. Rankamlt, K. (19G3). “Progress in Isotope Geology,” pp. 350-355. Wiley (Interscicnre), New York. 41. MarDonaltl, G. J. F. (1963). The deep structure of continents. Rev. Ueophys. 1,687665

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THE THEORY OF AVAILABLE POTENTIAL ENERGY AND A VARIATIONAL APPROACH TO ATMOSPHERIC ENERGETICS

.

John A Dutton* Department of Meteorology. The Pennsylvania State University University Park. Pennsylvania and

.

Donald R Johnson Department of Meteorology. The University of Wisconsin. Madison. Wisconsin

..........................................................

Page

1 . Introduction 334 1.1. History of the Concept of Available Potential Energy ................... 335 335 1.2. The Concept of Available Potential Energy ............................ 336 1.3. Generalization of Reference States .................................... 1.4. Present Expressions for the Amount of Available Potential Energy . . . . . . . 338 1.5. Available Potential Energy and General Circulation Theories . . . . . . . . . . . . 339 341 1.6. Purpose of This Article .............................................. 2 . An Exact Theory of the Concept of Availablc Potential Energy . . . . . . . . . . . . . . . 341 342 2.1. The Basis of the Concept ............................................ 346 2.2. Assumptions and Fundamental Relations ............................. 2.3. Properties of the Reference State ..................................... 347 351 2.4. The Amount of Available Potential Energy ............................ 355 2.5. The Rate of Change of A ............................................ 2.6. Avnilahle Potential Energy in Atmospheres with Static Instahilities ....... 358 2.7. Generation and Destruction of Available Potential Energy by Diabatic Processes ......................................................... 361 2.8. Total Diabatic Generation and Frictional Dissipation . . . . . . . . . . . . . . . . . . . 370 3 . Applications to Observational Data ....................................... 373 374 3.1. The Amount of Available Potential Energy ............................ 3.2. The Region of Maximum Contribution to A ........................... 378 3.3. The Structure of the Reference State ................................. 379 380 3.4. Average and Transient Components .................................. 3.5. Comparison of Numerical Results from Exact and Approximate Expressions 381 3.6. The Energy Budget of the Atmosphere ................................ 387 389 4 . Variational Methods in Available Enerm Theory ........................... 4.1. Introduction to Variational Methods .................................. 389 390 4.2. Necessary Conditions for Minimn of the Total Pot.ential Energy 4.3. Comment on Sufficient Conditions .................................... 395 4.4. Variational Approach to the Energy Available for Meridional Flow ....... 396

...........

* Formerly with the Environmental Technical Applications Center. U.S. Air Force (previously. Climatic Center. U.S. Air Force). Washington. D.C. 333

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JOHN A. DUTTON AND DONALD R . JOHNSON

Page 5. Contributions to the Amount of Available Potential Energy and its Relationship

to Other Quantities. . ............................................... 5.1. Contributions fro ytlrostatic Defects. .............................. 5.2. Barotropic Atmospheres and Available Energy. ........................ 5.3. Relationships to Other Quantitics.. .................................. 5.4. The Importance of Variable Static Stability.. ......................... 5.5. Energy Available to Perturbations. .................................. 6. The Dynamics of the General Circulation.. ................................ 6.1. Extrcma of the Total Pot,ential Energy. ............................... 6.2. Least Action Principles and the General Circulation. .................... 6.3. Applications to Rotating Convection Experiments. ..................... 6.4. The Role of Available Potential Energy.. ............................. 6.5. Use of the Least Action Principle. .................................... 7. Cnnolusion . . ......... ....................................... List of Symbols ..........................................................

...................................................

398

398 401 405 409 410 412 412 413 416 424 429 430 431 434

1 . INTRODUCTION

For almost as far back as we can trace man’s curiosity about the world around him, we find evidence of his speculations on the origin of the winds and the weather. The earliest cxplanations naturally attributed these apparently mysterious phenomena to divine intervention. I n the more than twenty centuries of investigations seeking a physical explanation, the answers have progressed from Aristotle’s suggestion that the winds were the breath of the earth to the current studies of the energetics of the general circulation. These recent studies have emphasized the question of how the internal and potential energies of the atmosphere are transformed into kinetic energy and thus maintain the circulations despite the dissipative forces. A recent advance in the study of this particular problem was made by Lorenz’ [l] in his lucid extension to the general circulation of Margules’s [2] concept of the energy available for conversion to kinetic energy. Van Mieghem’ [3, 41 introduced techniques based on variational methods to obtain further generalization. The concept of available potential energy and the many diagnostic studies based upon it have given us new information about the mechanisms which are responsible for energy transformations in the atmosphere; nevertheless, we still know vcry little about why the atmosphere responds with the particular circulation patterns it does to the variable heating which sets i t in motion. 1 Thrsr pc\pew will Iicrrnftrr hc r e f w d to only with thc author8’ names. Other references to thesc authors will inclutle appropriate citation.

THE THEORY OF AVAILABLE POTENTIdL ENERGY

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1.1. History of the Concept of Available Potential Energy In his now classic article, “On the Energy of Storms,” Margules [2] pointed out that only a portion of the internal and potential energies could be converted to kinetic energy and thus originated the concept of available potential energy. He applied the concept to idealized situations representing storms within limited regions bounded by vertical walls. However, he also noted that “sometimes the whole atmosphere” could serve as the closed system. Margules referred to the portion of the internal and potential energies which might be converted as the available kinetic energy. Lorenz chose t o emphasize that although this energy might be converted, it is actually internal or potential energy, and called it the available potential energy, and this term is in general use today. Both investigators defined the available energy in the atmosphere as the difference between the sum of the internal and potential energies of a natural state of an atmosphere and the sum of these energies which would exist after an adiabatic redistribution of the mass to obtain a horizontal, statically stable, density stratification. This sum of the internal and potential energies is commonly referred to as the total potential energy, and this usage has a basis in the fact that the ratio of the internal and potential energies is determined by thermodynamic coefficients provided the atmosphere is everywhere in hydrostatic equilibrium. The total potential cnergy is identical to the total enthalpy of the atmosphere when hydrostatic equilibrium prevails. 1.2. The Concept of Available Potential Energy The concept of available potential energy is founded upon the principle of conservation of mass and the idealization that flows which conserve specific entropy may exist. Under these conditions, the sum of the internal, potential, and kinetic energies is a constant, and therefore a state of the atmosphere which possesses a minimum of total potential energy will likewise have a maximum of kinetic energy. The requirement for reaching a state with a minimum of total potential energy by isentropic readjustment of a natural state of the atmosphere is that the minimum state be horizontally stratified and in hydrostatic equilibrium. During this isentropic readjustment, the continuous isentropic surfaces represent material surfaces with respect to the mass field. Thus the reference state is conceptually attained by imposing a vertical motion field which moves the originally undulating isentropic surfaces into coincidence with the earth’s geopotential surfaces. The attainment of the reference state may not be dynamically possible, however. As warmer air rises and cooler air descends during the readjustment, the release of gravitational potential energy and the work of expansion by the

336

JOHN A. DUTTON AND DONALD R . JOHNSON

internal energy combine to produce kinetic energy. When the horizontal, hydrostatic state is reached, the vertical pressure gradient is balanced by the effects of gravity on the density field and the horizontal pressure gradients vanish. Hence the only forces accelerating the winds are the Coriolis and frictional forces, and thus a t the moment the reference is reached the winds represent a frictionally modified inertial flow. Although the total potential energy of the entire atmosphere is less after the readjustment, this is not necessarily true for the part of the total potential energy associated with a particular portion of an isentropic layer. The readjustment may require the lifting of some parts of an isentropic layer, thus increasing the amount of potential energy and decreasing the amount of internal energy associated with that portion of the layer. On the other hand, portions which descend will reach the reference state with less potential and more internal energy than they possessed in the natural state. Since the changes of internal and potential energy associated with each isentropic layer are generally of opposite sign on portions of the layer which undergo vertical displacement during the readjustment, the change in the total potential energy will be the difference between presumably larger changes in the amounts of internal and potential energy. The value of the concept of available potential energy is that the fraction of the total potential energy that is energetically active in isentropic processes has been separated from the much larger reservoir of unavailable energy. Lorenz’s two examples serve as illustrations. In a hydrostatic atmosphere with horizontal density stratification, the total potential energy is plentiful but none can be converted to kinetic energy. In contrast, even if we make no change in the total potential energy but differentially add or remove heat from a horizontally stratified atmosphere, pressure gradients will be created and these will accelerate the wind. As Lorenz noted, “evidently removal of energy is sometimes as effective as addition of energy in making more energy available.” Uniform heating or cooling of an entire isentropic layer serves primarily to raise or lower the reference atmosphere without directly altering the existing horizontal pressure gradients. The fact that the flat reference state is not realizable on a rotating earth with a n equatorial source and polar sink of thermal energy is not taken into account in the concept of available energy itself. But as pointed out by Van Mieghem, the present applications of the concept to the atmosphere generally involve the tacit assumption that the atmosphere does in fact try to reach this reference state.

1.3. Generalization of Reference States Lorenz notcd that a strictly zonal circumpolar circulation may provide a dynamically stable equilibrium which might serve as a more useful reference

THE THEORY O F AVAILABLE POTENTIAL ENERGY

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state than the one discussed above. Such a distribution certainly corresponds more closely t o the observed circulations than the hydrostatic, horizontal reference state. I n his important contribution, Van Mieghem presented a new method of determining equilibrium states which would serve to define new forins of available potential energy. He applied his technique to both Lorenz’s original definition and then to the circumpolar vortex reference state. Van Mieghem’s approach allows the definition of an available potential energy as the difference of the total potential energies of a natural state of the atmosphere and any reference state which possesses a relative minimum of total potential energy. Some knowledge of the motion field in the reference state is required, however. The difference of energies is then expanded in a temporal Taylor series about the reference state and conditions on the first and second temporal derivatives of the reference total potential energy which are necessary for a minimum are found by inspection. Hence Van Mieghem’s approach permits the introduction of a variety of reference states, provided we possess the insight to determine appropriate conditions on the flow which yield the necessary properties of the time derivatives. This must be regarded as an important extension of Margules’s and Lorenz’s work. Van Mieghem applied this technique to the classical case and found, of course, that isentropic flow to a horizontal, hydrostatic state yielded a minimum. He also showed that the circumpolar vortex could be utilized as a reference state provided it was reached by isentropic flows which, in addition, preserved the absolute zonal angular momentum. The resulting reference state is hydrodynamically stable and serves to define the energy available for meridional motion in the atmosphere. At first, it seems that neither of these reference states is completely satisfactory for application to general circulation problems. Intuition demands a reference state which in some way corresponds to a true equilibrium the atmosphere attempts to reach despite the effects of variable terrain and heat sources. A steady state solution to the equations of motion forced by the average distribution of heating, for example, might be such a reference state if appropriate minimum energy requirements were satisfied. However, when the momentum distribution satisfies the geostrophic and thermal wind relationships, the horizontal density gradients are also determined. Fjrartoft [5] pointed out that under such conditions, the unconstrained production of kinetic energy by the vertical readjustment of isentropic surfaces is impossible. Hence the selection of an optimum reference state is extremely complicated. The search for an optimum reference state would be worthwhile only if available energy in some new form can help us to understand why particular

338

JOHN A . DUTTON AND DONALD R. JOHNSON

modes of atmospheric circulation are chosen. We shall argue in the final section that the concept of available potential energy cannot serve this purpose, and that it is useful only in tracing the energy transformations which do occur while the evolution of atmospheric motion is controlled by other means. In this article we shall therefore be concerned primarily with a rigorous examination of the questions associated with the original definition of available potential energy.

1.4. Presen,t Expressions for the Amount of Available Potential Energy Study and comparison of the analytic expressions for the amount of available potential energy derived by Lorenz and Van Mieghem reveals certain limitations. Lorenz’s basic expression is valid only for natural atmospheres which are already in hydrostatic equilibrium, and thus his available energy represents the portion derived by readjustment of a hydrostatic atmosphere to a horizontal state. Van Mieghem’s expression-a temporal Taylor series-requires that derivatives of the total potential energy of the reference state be known. But these derivatives are not uniquely determined by the natural state, since in part the rates of change a t the instant the reference state is reached depend on how the readjustment is carried out. The result is that Van Mieghem’s final approximate expression for the amount of available potential energy involves quantities determined in the reference state. He has provided no method of determining these quantities from measurements in a natural state of the atmosphere. The rate of change of available energy obviously depends on rates of change in the natural and reference at>mospheres,including those induced by diabatic effects. The attempt to calculate this rate of change from Van Mieghem’s expressions would therefore result in the necessity of evaluating temporal derivatives of reference state quantities which have not been related to quantities in the natural state. I n deriving approximate expressions suitable for computation from observed data, both Lorenz and Van Mieghem introduced restrictive assumptions. These were necessary since the computational formulas were established on thc basis of perturbations of the reference state. To determine the perturbation distance that each unit mass must move vertically in the readjustment, Lorenz introduced a mean static stability factor and Van Mieghem utilized the vertical distribution of potential temperature in the reference state. Lorenz [6] studied the modifications necessary when the lapse rate is nearly adiabatic.

THE THEORY OF AVAILABLE POTENTIAL ENERQY

339

Van Mieghem’s approximate expression, obtained with a perturbation in time from the reference state, and Lorenz’s expression, obtained as a perturbation in space, are nearly equal. I n their approximate expressions the atmosphere is assumed to be near the reference state in either time or space. Van Mieghem found an additional small term involving compressibility effects not contained in Lorenz’s expressions. However, Lorenz’s expression involves the variance of potential temperature on a quasi-horizontal pressure surface while Van Mieghem’s dominant term is the variance of potential temperature on a geopotential surface. Since Van Mieghem’s additional term is proportional to the variance of pressure on a horizontal surface, it scems likely that for typical atmospheric states the two expressions are quite similar. Although these approximate formulations have limitations, it is difficult t o assess analytically a t this time the accuracy that might be gained by exact evaluation. We will show later that application of the exact expression t o observed data produces variations of more than 10 yoin the available energy in comparison with amounts calculated from the same data with one of Lorenz’s expressions.

1.5. Available Poteatial Energy and General Circuhtion Theories The widespread interest and further study of available energy generated by Lorenz’s exposition is obviously motivated by the hope that the concept may lead to improved understanding of the general circulation. Along with Starr [7, 81, Kuo [lo], and others, Lorenz [9] used his approximate results to study the importance of horizontal eddies in a revision of the theory of the general circulation. He separated the physical processes represented by the zonal and higher harmonics of the atmospheric flow by dividing the available energy into two portions. The zonal portion is determined by the strength of the north-south variance of the zonal mean temperature and this component of the available energy is generated principally by the net heating a t low latitudes and net cooling a t high latitudes produced by the earth-sun geometry. The eddy portion of available energy is a measure of the east-west thermal variance. Following Lorenz and others, we might argue that although the zonal available energy is presumably generated by the average radiation field, the long waves in the westerlies distort the pattern and create and maintain the eddy available energy. The direct conversion of available potential energy to kinetic energy occurs in the eddies, as the baroclinic disturbances in the westerlies realize kinetic energy through east-west overturning. The kinetic energy supplied by ageostrophic motion within these disturbances is sufficient to offset dissipation by frictional forces. The kinetic energy of the smaller scale eddies, according to present thinking, is then transformed directly into

340

JOHN A. DUTTON AND DONALD R . JOHNSON

the kinetic energy of the zonal motion and the ultra-long waves, completing the cycle. We can, on the basis of this argument, expect that the total energies and the rates of energy conversion would be quasi-steady, and that the necessary balances between total generation and transformation of available energy and the dissipation of kinetic energy by friction would be maintained. The role of available potential energy in maintenance of the general circulation has been investigated with a variety of observational, diagnostic, and computational techniques. These results are summarized by Oort [ 113 and will not be reviewed here. The studies can be separated into two categories, those based on observational data for fairly extended periods of time and those based on results from numerical integrations based on model atmospheres. On t h e basis of Oort’s summary, the annual net generation of available energy is approximately (2.3 & 1) watts/m2, which is in fair agreement with most estimates of the total frictional dissipation, most notably that of Lettau [12] who found a value of 2 watts/m2 by entropy considerations based on a multiannual heat budget. We shall later cite evidence that this value of both total generation and total dissipation is too small by a factor of about three. It must be noted that most of the diagnostic computations summarized by Oort were based to a large degree on quasi-geostrophic theory and thus must be regarded as tentative in view of observational inadequacies and our lack of understanding of the role of convection in vertical transport of mass, momentum, and energy, the role of the diabatic processcs, and the role of motion a t scales for which the quasi-geostrophic theory is invalid. While the results of numerical experiments are not plagued with sparse data and observational errors, they are also only tentative since such experiments are based on present knowledge of the relative importance of physical processes in the atmosphere and are generally restricted to considering niotion over a spectrum of scales which is probably much smaller than the spectrum of motions which are actually important in atmospheric processes. This restriction results from a compromise between the time and space resolutions necessitated by limited computer capacity. A serious criticisni of the diagnostic studies, however, is that they employed Lorenz’s approximate expression for the generation of available potential energy by diabatic processes. The exact expression for diabatic generation is conceptually quite different from this approximate expression. In fact, the conclusion drawn with the approximate result in these diagnostic studies is exactly the opposite of the correct one in a t least one iniportant case. In particular, we will show that the cyclogenetic areas on the west sides of thc oceans generate, not destroy, available potential energy. Moreover, we shall show that the largest diabatic generation of available energy occurs when low-latitude boundary layer heating is combined with cooling of the upper troposphere in high latitudes.

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341

Another serious criticism, emphasized by Palmen [13], is that energy transformation rates have been calculated in these studies with the vertical velocities derived from the adiabatic, quasi-geostrophic models. Since these velocity fields are smoother than the actual fields and are undoubtedly underestimates, the resulting transformation rates are also too small, leading to underestimation of the rate of dissipation. Palmen [13] estimated that the total dissipation in the winter should be in the range of 5 to 8 watts/m2. We shall cite new observational evidence on the rate of dissipation and compute an estimate of the diabatic generation of zonal available potential energy with the exact expression, both of which agree with Palmen’s estimate. These results will allow us to estimate energy transformation rates and thus construct a new model of the energy cycle of the atmosphere which differs significantly from the one proposed by Oort [ll]. Finally, none of the diagnostic studies to date has shown that the concept of available potential energy yields any precise understanding of why the atmosphere responds as it does to the external forcing. This understanding should be the primary goal of future general circulation research. 1.6. Purpose of This Article

I t is the purpose of this article to: 1. Develop the theory of available potential energy and its rate of change rigorously and exactly, and thereby clarify the roles of convection, hydrostatic departures, static stability, and diabatic processes. 2. Investigate the relationships between the concepts of available potential energy, static stability, and entropy, and investigate contributions to available energy under various circumstances. 3. Demonstrate that the methods of the calculus of variations provide an economical method of obtaining exact results in available energy theory. 4. Propose an application of variational techniques which offers considerable intuitive insight and the possibility of analytic and numerical attacks on the problem of why the atmosphere chooses the modes of circulation it does. 5. Illustrate the role of available potential energy in maintaining the circulation in the context of the proposed theory of the control of the general circulation. 2. AN EXACT THEORY OF

THE

CONCEPTOF AVAILABLE POTENTIAL ENERGY

The concept of available potential energy is based upon the facts that the sum of the kinetic and total potential energies of a mechanically and thermo-

dynamically insulated fluid is a constant, and that under isentropic motion to a particular reference state, the kinetic energy assumes a maximum value.

342

JOHN A. DUTTON AND DONALD R . JOHNSON

The difference between this maximum amount of kinetic energy and the kinetic energy of the original state is called the available potential energy. Although the foundations of the theory have been expounded by Lorenz and Van Mieghem, there has not yet been an exact calculation of the amount of available potential energy in the atmosphere. In addition, there have been a variety of assumptions made, some with and some without justification. In this section, we shall first examine the foundations of the theory, and then, after collecting all of our fairly plausible assumptions about a natural state of the atmosphere in one section, proceed to derive expressions for both the amount of available potential energy and its rate of change in a rigorous and exact manner. In order to develop the entire theory with elementary methods we shall a t first assume that no static instabilities are present in the natural atmosphere. Then we shall drop this requirement and show that only slight modifications of the original method are required to treat the general case. It is worth emphasizing that the results we obtain here are valid for the energy processes associated with, and interacting between, all scales of atmospheric motion. I n addition, the effects of lapses from hydrostatic equilibrium are included and in most equations displayed explicitly.

2.1. The Basis of the Concept The first law of thermodynamics for viscous gases (see List of Symbols) (2.1)

cup dTldt + p V * U = p &/dt

+ F, +V

k VT

+Y

(in which we have reserved pdpldt for external effects such as radiation) combines with the momentum and continuity equations to produce the atmospheric energy equation

for the rates of change of the kinetic energy K and the total potential energy We have applied the usual boundary conditions that the vertical velocity vanishes a t the earth’s surface and that the pressure vanishes a t the top of the atmosphere. For mechanically and thermodynamically insulated dry atmospheres, the right side of equation (2.2) vanishes and the energy of the system is constant. The notion that increases of kinetic energy in the atmosphere result from decreases of the total potential energy is thus valid to the extent that the terms t o the right of equation (2.2) cancel for real atmospheric processes.

n.

2.1.1. Van Mieghem’s Approach. The exposition due t o Van Mieghem of the basis of the available energy concept is summarized in the following. The

THE THEORY OF AVAILABLE POTENTIAL ENERGY

343

temporal rate of change of the total potential energy for a dry, inviscid fluid

with the assumptions of thermodynamic and mechanical insulation and the usual boundary conditions becomes

and is clearly zero when the atmosphere reaches a state specified by the hydrostatic condition

vp

(2.5)

+ p V@ = 0

It can be shown that this condition requires that all thermodynamic variables are constant on geopotential surfaces. The second derivative is

where the subscript T indicates evaluation a t the top of the atmosphere. The hydrostatic condition eliminates the last two terms of equation (2.6). The definition of potential temperature yields

d8= -1 -

8 dt

- -1-dp + 2c- -1 dp

p dt

cp p dt

and combination with the equation of continuity produces a condition for isentropic flow,

Using this condition and the equation of continuity, we may rearrange equation (2.6) in the form

d211 -dt2 - ~ [ ~ [ ~ V . U - W ~ ] ~ Van Mieghem concludes that the isentropic rearrangement of the atmosphere to a state specified by equation (2.5) thus produces a minimum value of the total potential energy, provided that a8/az is positive in this state. The

314

JOHN A. DUTTON AND DONALD R . JOHNSON

rearrangement might conceivably result, however, in a reference state in which the vertical velocity and the divergence both vanish identically; we then have a stationary value of the energy. I n a later paper, Van Mieghem [14] obtained the same result with variational methods.

2.1.2. Lorecnz’s Approach. A quite different approach provides the foundation for Lorenz’s definition of the amount of available potential energy. Upon combining the inertial, Coriolis, and frictional terms leading to vertical accelerations in one function-the hydrostatic defect X-we may write (2.10)

aPP

= -(g - x)p

(This function will be discussed subsequently.) Substitution of equation (2.10)for the density in the expression for the total potentialenergy, equation (2.3), and integration by parts yields (2.11)

Now use of the definition of potential temperature and elimination of density

SO that upon denoting the surface value of potential temperature by 0, and the value a t the top of the atmosphere by 0, ,an integration by parts produces

(2.13)

Lorenz introduced an ingenious convention which avoids the necessity of determining the equation of the earth’s surface in isentropic coordinates. This is accomplished by defining the pressure on an isentrope which intersects the earth and goes “underground” as the pressure a t the surface. Using the convention and defining B,, to be the minimum value of potential teinperature found in the atmosphere we may write

2

For C l J l l V ~ l i i ( ~ i i Cwc ~ ! will use

K =

R/c,

ns an

rxponcnt.

THE THEORY OF AVAILABLE POTENTIAL ENERGY

345

so that equation (2.13) becomes, upon averaging by integrating over the isentropic surfaces and dividing by the area X of the earth's surfacc

It is well-known [15] that if &t) is a function such that + " ( t ) 2 0, then for an integrable function f and a weighting function, w , (2.16) Since + ( t ) fulfills the conditions for 0 5 t putting w = X-' that

< co when $ ( t ) = t1.1

K,

we find upon

(2.17)

or (2.18) with equality obtained only if p ( x , y, 0 ) = p ( e ) . For hydrostatic atmospheres the last term of equation (2.15)vanishes, and we arc assured by the inequality that they will have minimum energy if there is no variation of pressure on the isentropic surfaces. On this basis, Lorenz defined the available energy A h of hydrostatic atmospheres as

We shall show later that this formula agrees with his slightly simpler form. I t will turn out that the same inequality used in this account of Lorenz's approach will be a crucial tool in a rigorous approach to finding a state with minimum total potential energy with thc methods of thc calculus of variations.

2.1.3. Comparison of Approaches. It is obvious that these approaches are quite different, and that both have some advantages. Van Mieghcm's method utilized the first law of thermodynamics and the equation of continuity in direct form and includes nonhydrostatic effects implicitly, but the existence of a minininm depends on the velocity distribution a t the instant the reference state is reached. Lorenz's approach is conceptually simpler, but fails in the form given by him and sunimarized above when nonhydrostatic conditions exist. Neither method yields an exact specification of the reference state for

346

JOHN A. DUTTON AND DONALD R . JOHNSON

nonhydrostatic conditions. Thus the goal of the next few sections is to determine the structure of the reference state explicitly while including the effects of lapses from hydrostatic equilibrium. It is worth observing a t this point that neither of these methods-nor ours -takes account of the dynamics of the atmospheric motion as specified by the equations of motion.

2.2. Assumptions and Fundamental Relations The methods of much of this article are based upon the use of inverse functions, and consequently require that the thermodynamic variables satisfy certain conditions. All of the necessary assumptions and the fundamental relations are collected in this section. We assume, a t first, that any natural state of the atmosphere is such that: 1. Pressure p , density p , and potential temperature 9 are differentiable (and hence continuous) functions of x, y, z, and t . 2. I n addition, ae/az is continuous and positive, and hence 9 is a continuous, one-to-one function of height z. Where an isentrope intersects the earth’s surface and goes “underground” we will define ae/az= co and put azpe = 0. 3. Density and pressure decrease exponentially to zero as z approaches the top of the atmosphere and temperature increases at moat linearly, so that denoting the potential temperature a t the top of the atmosphere by O T , we have = 00 and p(eT) = p ( e T ) = 0. 4. The change in area of geopotential surfaces with height and variations in the force of gravity both may be ignored. The most restrictive assumption is obviously that ae/az > 0 and this will be eliminated later; it is however, justifiable for applications to the free atmosphere in view of the rareness with which superadiabatic lapse rates are observed there. The results based on this plausible model of the atmosphere can justifiably be termed exact. It will be convenient-and revealing-to explicitly take account of nonhydrostatic conditions throughout most of this article with the hydrostatic defect x defined by

eT

(2.20)

x is a well-defined function and its inclusion does not compromise the exactness of mathematical results; for work with observed data, however, presently available measurements necessitate the assumption that the atmosphere is in local hydrostatic equilibrium. The fundamental relation between thermodynamic variables, obtained by

It is clear that for theoretical purposes

THE THEORY OF AVAILABLE POTENTIAL ENERGY

347

eliminating the densit,y between equation (2.20)and the definition of potential t,emperature (2.21)

0 = pcricp(1000)KlpR

is (2.22)

or, upon using assumption 3, (2.23)

p = 1000

[;I;71"" -Xdz

Substitution of this into equation (2.20)then yields (2.24) When these relations are applied to variables in the reference state, x vanishes identically.

2.3. Properties of the Reference State With the preliminaries of t8heprevious section completed, it is possible to state a definition and illustrate methods which are sufficient to determine the structure of the reference state uniquely and explicitly from a natural state of the atmosphere. Definition. The reference state is obtained by readjusting a natural state of the atmosphere to reach a state of hydrostatic equilibrium as specified by equation (2.5), in which al?,/az is continuous and positive. The readjustment process must preserve the distribution of mass with respect to potential temperature. The definition implies that all thermodynamic variables will be constant on geopotential surfaces, that the reference surface potential temperature O,(O) will be the lowest potential temperature 0, of the natural state, and that the potential temperature OT a t the top of the reference atmosphere will be the highest potential temperature found in the natural state. The restrictions on al?,/az imply that &(z) is a continuous, one-to-one function of height z, and hence the existence of a unique inverse function H such that (2.25) is verified.

fW,(z)l

=z

318

JOHN A . DUTTON AND DONALD R. JOHNSON

The mass contained between the isentropic surfaces with values 8, and 8 in the reference state is thus given by (2.26)

~j:~)

pr( z) dz

=

s:

P ~ [ H ( ? ) I H ' (dv ~)

+

where H(8,) = 0. As in equation (2.26), we will henceforth use both q and as variables of integration in place of 8. If M ( 8,, 8) is the mass contained between 8, and 8 in the natural state, the definition requires that (2.27) Putting equation (2.24)in the same form as equation (2.26)and substituting the result into equation (2.27) for pr, we obtain the Stieltjes integral (2.28)

e) =

-

-

This can be integrated immediately and rearranged to give (2.29) which upon differentiation with respcct to 8 yields the diffcrential equation (2.30) whose solution is

Because p,(O)/g is the average mass per unit area of the atmosphere and the mass does not change during the readjustment, p,(O) is determined by the natural state. The height of any isentropic surface in the reference state is therefore determined by the mass and potential temperature distribution in the natural state. Since we know H(8) for 8 , I 8 I 8, we can invert equation (2.31)to obtain (2.32)

e,(z) = ~

- 1 ( ~ )

in the conventional inverse function notation. The variables pr and pr could now be found explicitly with equations (2.23) and (2.24). This proves that the definition is sufficient to specify uniquely the reference

THE THEORY OF AVAILABLE POTENTIAL ENERGY

349

state. I t is clear that the quantity m(O0,0) is invariant under processes which preserve both the mass and the potential temperature of the parcels in motion, and hence the reference state is also invariant under these conditions. Since we have assumed, to begin with, that aelaz is continuous and positive everywhere in the natural state, we are also guaranteed the existence of a unique inverse function h, such that (2.33)

h[@, y, 41 =

and thus we may write (2.34)

in which we have used the assumption that ahla0 vanishes on underground isentropes, which ensures that portions of the integral from 0, to 0, actually make no contribution. Because pand ahla8 are continuous, we may reverse the order of integration to obtain (2.35)

The inner double integral is a surface integral along a pvalued isentrope as (5, y) ranges over the surface of the earth. For simplicity we shall henceforth utilize the overbar to denote the ratio of the value of an integral over an isentropic surface to the area of the earth’s surface. We thus obtain (2.36)

Differentiation of equation (2.27) shows that (2.37)

ahlae =

aalae

and this combined with the hydrostatic equation for the reference state (2.38)

ap,lae = -gpr

aalae

gives the result that

Substitution of this result into equation (2.31)produces

350

JOHN A. DUTTON AND DONALD R . JOHNSON

and so the reference stat,e can be represented explicitly by operations on the variables in the natural state. Differentiation of equation (2.40) now shows that (2.41)

and since ahla0 is positive by assumption everywhere in the natural atmosphere, we verify afortiori the statement in the definition that aOr/az is positive. Furthermore, since a H / a e is a product of functions, all of which are continuous, we conclude that aer/az is also continuous. In summary, we have shown how the structure of the reference state obtained by a readjustment according to the definition is explicitly determined by the structure of the natural state. We have also demonstrated that the isentropic readjustment of a natural atmosphere which meets the conditions of Section 2.2 t o a state of hydrostatic equilibrium as specified by equation (2.5)does in fact lead to a reference atmosphere in which aB,/az is continuous and positive. Use of the hydrostatic defect function x allows an intuitively appealing representation of the reference variables. Conversion of equation (2.20) to isentropic coordinates yields (2.42)

aplae = - (9 -

ahlae

which can be integrated to give (2.43)

and so combination with equation (2.39) produces (2.44)

and hence (2.45)

Lorenz's convention requires that aplai3 vanish on isentropes which have intersected the earth and gone underground. An equivalent statement,

THE THEORY OF AVAILABLE POTENTIAL ENERGY

35 1

according t o equation (2.42), is that ahla0 vanishes. We have adopted this convention, and note that in addition we may choose 0, so that h(&) = 0.

2.4. The Amount of Available Potential Energy The amount of available potential energy in the atmosphere is defined to be the difference between the total potential energy of the natural state and the total potential energy of the reference state obtained from the natural state by a readjustment according to the definition of Section 2.3 with the additional requirement that the readjustment be a frictionless, isentropic process which preserves parcel mass. The additional requirement is sufficient to ensure that the sum of the kinetic and available energy is conserved. The available energy A is therefore given by

A

(2.46)

= IT,

-n,

where the subscript a denotes the natural state of the atmosphere. With the aid of the hydrostatic relation in the reference state, we may write n

(2.47)

R

so that use of equations (2.39) and (2.41) and some combination of factors gives the exact result

It is possible to express the contributions from the differences of internal and potential energy in a form with the same variables in all terms. (This decomposition is made primarily for typographical convenience.) To do so, we put (2.49)

A

's

-

-2

I -

RZ

and then use of the same result as in equation (2.48) and elimination of p in the reference term with the aid of equation (2.21) yields

352

JOHN A. DUTTON AND DONALD R. JOHNSON

Similarly, with equation (2.37), we find that (2.51)

and so changing to the same variables used in equation (2.50), we have

in which we have used ( ) instead of the overbar to indicate averages on isentropic surfaces. Addition of equations (2.50) and (2.52) gives an exact relation for the available potential energy

A=A, + A l .

(2.53)

which involves only the pressure and potential temperature distributions of the natural state. These expressions implicitly include the contributions to the available energy from nonhydrostatic conditions. The role of the hydrostatic defects may be demonstrated exactly by using equations (2.42) and (2.21) to write (2.54)

ah P aP eRpK p g = -(9px)p=-(g-,)(looo)~ae

ap

and thus from equation (2.50) (2.55)

Then with equation (2.51), with equation (2.40), and with an integration of the appropriate modification of equation (2.54), we find

THE THEORY OF AVAILABLE POTENTIAL ENERGY

353

in which we have explicitly used the fact that we may assume h(Bo) is zero since, based upon the convention that aelaz is positive, we may choose 8, to have as small an area above ground as we please. The sum of A , and A , may be ihtegrated by parts and manipulated to produce a slightly simpler expression for A . It is perhaps more revealing, however, to return to the definition (2.46)and derive the result directly. We may write (2.57) and upon using various versions of equation (2.54) along with the condition that h(Bo)= 0, we obtain, after an integration by parts in the potential energy term,

Integration by parts of both this result and equation (2.48) and combination according to equation (2.46) gives the exact expression (2.59)

A=-

-8

g-xav

I

s-xa*

in which we have assumed that the evaluation a t OT vanishes. We shall verify this subsequently for the case in which x vanishes, and thus for 1x1 < g, the result will still be true. In the cases in which x vanishes identically (as we must assume to apply the results to presently available measurements, for example), the expression

354

JOHN A. DUTTON AND DONALD R. JOHNSON

(2.69) reduces directly t o

It can be shown that this result agrees with Lorenz's analytic definition. He wrote (2.61)

But every isentrope with value 6* less than Oo is totally below ground, and therefore by both our and Lorenz's conventions the pressure on these isentropes is given by

Hence the integrand of equation (2.61) is constant from 9 = 0 to 0 = e, and therefore integration of the portion below 0, produces our expression (2.60). The evaluation a t the top of the atmosphere can be assumed to vanish. To show that at 8, (2.63)

we use the definition of potential temperature to find that an equivalent statement is that (2.64)

lim p T = 0 P d

which is true by our assumptions about the behavior of the natural state of the atmosphere a t high altitude. Thus, if natural states of the atmosphere existed which were everywhere in hydrostatic equilibrium,we would have equation (2.60) as an exact expression for the amount of available potential energy. The constant term is not zero, as is shown by the reasoning associated with equation (2.62), since p ( x , y , O0) = p ( x , y, 0) and the difference of the two averages will be positive unless the surface pressure is constant. It would obviously be satisfying to conclude this discussion of the amount of available potential energy by showing directly from equation (2.69) that A is nonnegative. The expression is apparently too complicated for elementary methods [such as the inequality (2.16)] to succeed, however. We must therefore rely on Van Mieghem's proof that the reference state has minimum or stationary total potential energy. In Section 4 we shall show with the methods of the calculus of variations that the flat hydrostatic state is a

THE THEORY O F AVATLABLE POTENTIAL ENERGY

355

necessary requirement to obtain minimum total potential energy under isentropic rearrangement of a natural state of the atmosphere.

2.5. The Rate of Change of A A task of considerable importance, especially for understanding the role of available potential energy in the general circulation, is the derivation of exact expressions for the rate of change of the amount of available potential energy due to both motions of the natural state of the atmosphere and to diabatic processes. We shall find it possible to proceed rigorously from the equation of continuity for isentropic coordinates. This equation can be derived, for example, by extending the usual method (e.g., Thompson [16], t o include nonisentropic processes. The result is

where the subscript indicates differentiations are to be performed with 0 held constant. It is particularly convenient to assume that the externally induced diabatic processes can be represented by the vertical divergence of the net radiation

p dq/dt = - V * Q = -(aQ/&)

(2.66)

and hence with the molecular effects taken into account as in equation (2.1), (2.67)

d8

dt

-

FQ-TQ-W

1

=--

[a&az

-- Da]

C p P ~

(This assumption is made for convenience only and aQ/az can always be replaced by V * Qin subsequent equations to maintain their exact character.) Observing that by the divergence theorem (2.68)

we obtjain our basic result (2.69)

The rate of change of energy in the reference state may be determined from equation (2.48) with the aid of Leibniz's rule since 8, may be a function of time. Interchange of differentiation with respect to 8 and to and an integration

356

JOHN A. DUTTON AND DONALD R . JOHNSON

by parts yiclds (2-70)

Carrying out the differentiations of the second term, T,, and use of the averaged form of equation (2.65) with account taken of equation (2.68) produces

By our conventions, do may be chosen so that it is coincident with the earth's surface on a negligible set E ( z , y) and below ground otherwise. Hence ah/ae is nonzero only on E , and the term may be written

But a t z = 0 on E , U * VB vanishes since the vcrtical velocity must vanish and the horizontal gradient of 8, vanishes. Hence, this second term of equation (2.70) is actually zero. Performing the differentiation in the remaining term of equation (2.70)and using equation (2.69),we obtain

For convenience we put equation (2.43)in the form

and use of the result

allows us to conclude that

The rates of change of thc total potential and kinetic energies may be

THE THEORY OF AVAILABLE POTENTIAL ENERGY

357

written as (2.76)

1'

an

-f = c (U V , p

+ wpx) dV -

and

Using again the definition that A = n , (2.78)

=

j ( U V2p

-nr, we finally have the result

+wpx) d V

It may be more useful to express the energy conversion term in equations (2.77) and (2.78) in isentropic coordinates. Use of the usual coordinate t*ransformationrule [see equations (5.16) and (2.20)] yields V , p = V e p - - VaP eh

(2.79)

az

+ gh) - pxveh

= pVe (cpT

so that (2.80)

-

j(U v , p

+ uypx) d V

=

SJ

+

[U* v e ( g

+ Sh)

X ( W - U ' Veh)]p

ah

- d A dd

ae

These results have been obtained using only the assumptions of Section 2.2, and are thus exact for the natural states of the atmosphere so defined. Therefore, the theory of the role of available potential energy in maintaining the kinetic energy of the atmosphere is summarized in the two exact relations (2.77) and (2.78). Some obvious implications of this theory merit emphasis a t this point. Recalling that aQ/ae < 0 implies local warming, we see that the external diabatic effects, particularly net radiation, generate available potential energy when warming on a given isentrope occurs a t high temperatures relative to cooling at low temperatures. The effects of the frictional addition of heat F , , which is nonnegative, are

358

JOHN A. DUTTON AND DONALD R. JOHNSON

dependent on the stability as well as the temperature distribution. Since the greatest dissipation in the free atmosphere may be expected where the winds and shears are strong in the neighborhood of jets and thus in the region of the steepest slopes of the isentropes, we may assume that T is close to l' where F , is large, and hence that the first factor would be positive and frictional effects in the free atmosphere would tend to reduce the available energy. Since the contribution from heat conduction may be expected t o be positive a t temperature maxima and negative at minima, we also conclude that the molecular conduction in the free atmosphere reduces the available energy, although the effect is, of course, negligible. The effects of condensation are similar to those of radiation, but modified by stability. Condensation of water vapor and the resulting release of heat must occur at above average temperatures and below average stability, as measured by aO/& on a n isentrope, to increase the available energy. The two equations (2.77) and (2.78) emphasize the obvious result that direct conversion between the kinetic and available potential energies requires either a cross-isobar component of the horizontal wind or vertical motion in the presence of a hydrostatic defect. Hence any conversion between the two forms of energy requires a nongeostrophic component of the Aow. These results stress the well-known fact that the geostrophic component of the wind is energetically inactive, and that transformations of energy in the atmosphere depend on the presumably small, nongeostrophic component of the wind. Addition of equations (2.77) and (2.78) shows that the sum of the kinetic and available potential energies is conserved when the diabatic and frictional effects vanish or cancel everywhere. Thermal insulation, in the sense that the total net radiation is zero, is not sufficient for conservation of the sum, however, since, for example, the diabatic term of equation (2.78) depends on the arrangement of the net radiation relative to the temperature field.

2.6. Available Energy i n an Atmosphere with Static Znstabilities The most restrictive assumption used so far in this article has been that a natural atmosphere is everywhere statically stable in the sense that aO/az is everywhere positive. Although this is gcnerally true in the free atmosphere, the equations based upon this assumption cannot be applied to determine rates of change of available energy due t o motion and diabatic processes near the earth's surface where superadiabatic gradients are common. The usual developments of the equations of motion in the hybrid isentropic coordinates depcnd explicitly on the assumption that aO/az is never zero (e.g., Thompson [lG]), which permits interchange of height and potential temperature as independent and dependent variables. When areas of static

THE THEORY OF AVAILABLE POTENTIAL ENERGY

359

neutral stability or instability exist, e(z) is no longer a monotonic function of height and hence h(8)is no longer single valued. To eliminate trhe necessity of this assumption, we utilize any appropriate curvilinear coordinate system,

(2.81)

= {t,

E, 77, e )

for

x = m i , 2,31

in which ( and q specify position on a &valued isentrope. We assume that the Jacobian of the transformation does not vanish anywhere in the atmosphere. Then the system can be inverted locally to yield the parametric equations = Z ( E , rl, e, t ) Y = y ( t , ?l, e, t ) , which, for a particular isentrope 6 at time t , is a locally valid mapping defined

(2.82)

2 =X ( E ,

q , 0, t ) ,

on ([, r ) ) space which gives the Cartesian coordinates of the isentrope. With these coordinates, the total mass associated with parcels whose potential temperatures lie between Bo and t9 is given by

(2.83)

Mceo 0 ) = 9

[:J--

p(5, 7'

e)lJI db

dB

I

where

E denotes ( f , q ) space. Use of

the definition in equation (2.27) that

m(eo, e) = ivqe,, e ) p produces (2.84) In addition to the previous convention that az/ae is zero when an isentrope is below ground, we now add the convention that axla0 = ay/aO = 0 so that J vanishes under these conditions. In order to obtain a continuity equation in the new coordinate system, we introduce tensor methods and the concept of the four velocity; that is

(2.85)

(A

vx = d<

= 0,

1, 2 , 3 )

where

(2.86)

50= t ,

11

=

6,

52

= q,

p= e

so that v') = 1. Similar definitions apply in the original Cartesian space. The tensor continuity equation is

(2.87)

(pd),x = 0

(A

= 0,

1,293)

where we have denoted the covariant derivative with a comma and invoked the customary summation convention. With the aid of the identity

(2.88)

360

JOHN A. DUTTON AND DONALD It. JOHNSON

we obtain, sincc the Jacobian is not zero in the atmosphere, the continuity equation (2.89)

a -(PlJlVX) = 0 8th

(A

= 0, 1, 2,

3)

By definition of a contravariant tensor, the four velocities vh in space can be calculated from the four velocities uh in xh space with the relations (2.90)

Therefore equation (2.88)becomes

which is the required generalization of equation (2.65).Upon averaging over the isentrope the middle two terms vanish by virtue of the divergence theorem, and we have (2.92)

a

Now, when no static instabilities were allowed, we utilized the coordinates in this case the Jacobian is ah/aO.Most of the results of Sections 2.3 through 2.5 hold upon replacing ahla0 with IJI. The function h(0) is not necessarily single valued now, but the theory presented in these sections actually depends only on the amount of mass between two isentropes being well defined. In particular, from equation (2.27) and the hydrostatic equation in the. reference state we obtain ( 2 , IJ,0, t ) and

(2.93)

Examination of equation (2.30) makes it clear that the reference state will be statically stable even though the natural state of the atmosphere may not be, provided that m(0, fl?,) is a monotonically decreasing function of potential temperature-which is certainly plausible for any natural atniosphere. With equation (2.47), equation (2.54) applied to the reference state, and equation (2.93),we find

THE THEORY O F AVAILABLE POTENTIAL ENERGY

36 1

and finally, with the convention that J vanishes when an isentrope goes below ground, the argument associated with equations (2.70) through (2.72) yields

and so the equation for the rate of change of available potential energy is identical to equation (2.78) upon replacing ahla0 with IJI. Thus the contributions to generation or destruction of available energy in all regions, whether statically stable or not, can be determined by considering the product of the rate of diabatic heat addition and the temperature dependent weighting factor of equation (2.78). Since IJJis always positive, the sign of this product determines whether diabatic processes are locally generating or destroying available energy. The magnitude, of course, does depend on J.

2.7. Generation and Destruction of Available Potential Energy by Diabatic Processes The growing interest in available potential energy has brought new emphasis to the important role of differential heating and cooling in the maintenance of kinetic energy. A variety of theoretical and diagnostic studies have been aimed a t providing qualitative understanding of the diabatic processes which are ultimately the source of the atmosphere’s kinetic energy. However, an exact theory of the role of these processes in the generation or destruction of available potential energy has not been utilized, and it appears that the approximations in general use have led to some incorrect conclusions. In this section we shall apply the exact theory developed herein to explore some aspects of the problem. The results are tentative and sometimes speculative, but they make it clear that the exact theory coupled with suitable modeling of both the typical atmospheric structures and associated diabatic processes would provide precise and quantitative understanding of the generation and destruction of available potential energy by diabatic processes. 2.7.1.Comparison of the Exact and Approximate Generation Integrals. Before proceeding with the application of the exact theory to specific atmospheric processes, it will be useful to point out the two fundamental ways in which it differs from the iniplications of Lorenz’s approximate expression. To do so. we will write the generation integral in a more tractable form. and, for convenience. assume hydrostatic equilibrium. Use of eqriation (2.74), the definition of potential temperature, and the results of Section 2.6 allow us to

362

JOHN A. DUTTON AND DONALD R . JOHNSON

write the second term of equation (2.78) as

and to avoid confusion with signs, we will replace the diabatic t e r m with &, which will be positive when the net result of all diabatic processes is local heating. Thus (2.97)

The assumption that IJ1 = ah/aO and use of the hydrostatic assumption allows equation (2.97)to be put in a form which is identical to an expression for the generation derived by Lorenz [6]. In his original paper, however, Lorenz converted an approximate expression in isentropic coordinates for the amount of available potential energy into isobaric coordinates with a linear assumption, and used further approximations t o arrive at its time rate of change. He obtained, using Q for heat addition per unit mass, (2.98)

where the wavy overline denotes an isobaric average and the prime a deviat
THE THEORY OF AVAILABLE POTENTIAL ENERGY

363

quite easily in theoretical investigations. This integral allows us to investigate boundary layer processes. To illustrate how these two expressions yield opposite conclusions in a t least one important case, let us consider the cyclogenetic regions east of the North American and Asian coasts. Lorenz’s approximation predicts destruction of the available energy since the temperature of the cold air mass flowing to sea is presumably colder than the isobaric average temperature while the deviation of the heating function is positive. In contrast, assuming that the isentropes in the surface layer slope over the cold air westward and poleward, the surface pressure will be greater than the isentropic average pressure and equation (2.97) predicts local generation of available energy. We shall consider generation in these cyclogenetic regions in more detail later. There are a variety of diagnostic studies which have used the approximation rather than the exact equation to compute the generation. I n view of the above comments, the numerical results, which in fact vary widely, cannot be accepted without a careful delineation of the situations in which the approximation is adequate.

2.7.2. The Eflciency Factor. The efficiency with which a given distribution of differential heating generates available potential energy depends on the distribution of pressure on the isentropic surfaces and the stability as measured by JJI. For example, an atmosphere with no available energy would have no pressure variations on the isentropes and a sudden application of differential heating would not simultaneously yield available energy. Generation would occur with increasing efficiency if the pressure in the heated regions rose above the isentropic average pressure and if the pressure decreased in the cooled regions. If the pressure variations were in the opposite sense as the result of the differential heating, the available energy represented by these pressure variations would be destroyed by further differential heating. The structure of net heating in the atmosphere is presumably surface and low-altitude heating in the tropical regions and cooling a t almost all altitudes in the polar regions. Assuming we can take [JIto be dominated by ahlad (or that no static instabilities exist), the efficiency factor e (2.99) will clearly be positive near the surface, with highest values being reached in low latitudes. Strongest negative values will generally be obtained near the polar tropopause. Thus the usual structure of the atmosphere, and the resulting distribution of the efficiency factor, implies that maximum generation will occur with low-altitude heating in equatorial regions and highaltitude cooling in polar regions.

364

JOHN A . DUTTON AND DONALD R. JOHNSON

N

LATITUDE

5

(01

FIG. l(a). Average values and standard deviations of the efficiency factor e (mnterl degree) along 75"W for January 1958. The solid lines are isopleths of the morit,hly average values of e, and the dotted lines are isopleths of the standard deviation. The broken lines are monthly mean isentropes. North is on the left.

FIG. l ( b ) . Efficiency factors for April 1988. See FIG. l(a) for additional explanation.

365

THE THEORY OF AVAILABLE POTENTIAL ENERQY

Some monthly averages and standard deviations of the efficiency factor of equation (2.99) for the IGY pole-to-pole cross sections [17] are shown in Fig. 1 . These were computed from equation (2.99),using equation (2.64) to

N

LATITUDE

S

(dl

FIG.l(d). Efficiency factors for October 1958. Seo Fig. I(n) for additional explanation.

366

JOHN A. DUTTON AND DONALD R . JOHNSON

obtain ah/ae as a function of 0, p , and ap/ae. The derivatives ap/ae were calculated with finite differences. Note that the factor 0,by which these values must be multiplied to compute the total generation, has dimensions of energy/(volurne)(time). Lorenz [18] presented a figure showing the distribution of the first factor of equation (2.99) computed from average data.

2.7.3. Contributions of the Diabatic Processes. Determination of the general variations of the sign and magnitude of the diabatic processes allows us to infer their role in generation or destruction of available potential energy. Obviously the available energy which exists in the atmosphere must, by and large, be generated by the latitudinal gradient of heating. However, it is revealing to consider the effect of the various diabatic components in more detail, especially since we know from the previous paragraphs which distributions of differential heating are most efficient. We decompose the radiation field into its solar (R,) and infrared (R,) components, and write (2.100)

We shall consider each term separately.

2.7.3.1. Solar radiation. The divergence of the solar radiation in the lower atmosphere is presumably always slightly negative due to scattering and absorption by clouds, water vapor, and particulate matter. According t o latest radiation budgets, some 20 yo of the solar radiation impinging on the top of the atmosphere is absorbed directly, the rest being reflected or scattered back t o space or absorbed at the earth’s surface. Direct absorption undoubtedly produces available energy on the zonal scale due to greater absorption in low latitudes. Because of seasonal changes, the latitudinal contrast in solar absorption and the resulting generation will be greater in the winter than the summer hemisphere. The total solar radiation impinging on the top of the atmosphere is nearly uniform in the summer [19]. Thus summer zonal energy generation by solar absorption is primarily due to the latitudinal distribution of absorbing constituents. Available energy on the eddy scale is probably also generated by direct absorption. The isentropes which pass through the warm, moist, and cloudy air of a disturbance rise sharply over the cold dry air. Thus we would expect the efficiency factor to be larger where the greater absorption occurs than where only weak absorption is present. 2.7.3.2. Infrared radiation. If we consider the atmosphere as a thermodynamic engine, infrared emission a t high latitudes may be viewed as the condenser associated with the solar absorption a t lower latitudes serving as the boiler, as emphasized by Lettau [12].

THE THEORY OF AVAILABLE POTENTIAL ENERGY

367

An important question is whether, in general, infrared radiation generates or destroys zonal available energy. As Fig. 1 makes clear, the net result will depend strongly on the arrangement of the infrared divergence with both latitude and height. Since the low-latitude atmosphere emits more radiation than the polar atmosphere, we might expect destruction of available energy. However, strong cooling just below the polar and the subpolar tropopause would be associated with large negative efficiency factors and would produce net generation. We shall show subsequently that the atmospheric structure is so arranged that available energy is generated both in receiving and emitting heat, and this implies considerable efficiency in the use of the differential heating and cooling to maintain the reservoir of zonal available energy. Infrared radiation may sometimes generate eddy available energy, since isentropes with positive efficiency factors in the warm sector will have negative factors after rising over the cold air. Therefore combination with virtually no cooling below clouds in the warm regions of disturbances and strong cooling at higher altitudes in the cold regions will generate available energy on an isentropic surface.

2.7.3.3. Frictional dissipation. Determination of the effects of the frictional dissipation and the resulting heating is still speculative. If we assume, as has generally been done, that the greatest portion of the dissipation occurs in the boundary layer, the dissipation would generate available energy due t o the positive sign of the efficiency factor a t the surface. The maximum frictional effects in the free atmosphere would be expected in the neighborhood of the core of the jet streams where the winds and shears are strong. The pressure on the isentropes in the lower part of the jet is generally less than the average pressure and hence the efficiency factor is negative since these isentropes drop into the subtropical troposphere after passing through the jet. The efficiency factor probably changes sign on the isentropes above the jet core as the isentropes assume the typical stratospheric configuration (for example, see Fig. 1). Therefore, since the shears are generally stronger where the efficiency factor is negative, the maximum of frictional heat addition occurs in conjunction with the negative efficiency factor. Hence, dissipation in the jets would probably destroy available energy. The recent work of Holopainen [20] and Kung [21] indicates that considerable frictional dissipation occurs in the free atmosphere. Kung finds that an average dissipation over North America is 1.87 watts/m2 in the boundary layer and 4.51 watts/m2 in the free atmosphere. If these are representative figures, our reasoning above indicates that the presumably dominant dissipation in the jets may result in a small net destruction of available potential energy by frictional processes.

368

JOHN A. DUTTON AND DONALD R . JOHNSON

2.7.3.4. Sensible heat addition. Since the addition of sensible heat by molecular conduction occurs in the very lowest part of the boundary layer, the net generation is probably positive. The average is positive over the continents in the summer, and undoubtedly positive over the oceans in the winter. Sensible heat addition is strong in the tropics and subtropics, and this is an important component of the generation of zonal available energy. Our exact equation also suggests that it is important in generating available energy in the oceanic cyclogenetic areas, as pointed out previously. As the cold, continental air flows over the warm ocean currents, the addition of sensible heat acts to generate both zonal and eddy available energy. Pettersen et al. [22] have found that sensible heat transfer to the atmosphere may exceed 2.5 cal/cm2min (1740 watts/m2)in these regions, implying that strong generation will occur. This result provides a more satisfactory understanding of the energetics of these cyclogenetic areas than the previous conclusion that the available energy had t o be sufficiently ample to provide for both the gain in kinetic energy and the diabatic destruction. Diagnostic studies with the exact equation in these cyclogenetic regions to determine the portion of the generated energy which is immediately converted to kinetic energy would be of interest. 2.7.3.5.Latent heat. Generation of available energy by the release of latent heat is probably important on the zonal scale, and is of prime importance in the energetics of disturbances, most notably cyclones, especially in the tropics. Since the major release of latent heat on the zonal scale occurs most of the year in the lower mid-latitude and tropical troposphere [23], the pressure will be greater than or equal to the isentropic average and hence generation of zonally available energy will be positive and possibly quite large. In a n individual mid-latitude storm, the isentropes rise sharply enough from the warm sector into the stratosphere that we assume that the warm sector pressures are greater than the isentropic averages. Hence the release of latent heat in the warm sectors will generate available energy. Palmen [24] has noted that an intensifying cyclone may act as a source region for both kinetic and total potential energy and established that the release of latent heat can generate large amounts of kinetic energy. Riehl[25] has shown that this process is particularly important in tropical cyclones, and pointed out that they also export large quantities of kinetic energy to surrounding regions. The rapid generation of available energy results as condensation occurs in the ascending currents near the eye wall, where the isentropes begin a rapid downward plunge into the warm air of the eye. With this configuration the efficiency factor should increase rapidly toward the center of the storm.

THE THEORY OF AVAILABLE POTENTIAL ENERGY

369

Palm& [24] estimated that the intense hurricane Hazel, upon becoming an extra-tropical storm, generated 53 watts/m2 of kinetic energy and covered 1/35 of the total area north of 30". From this he concluded that three storms of such unusual intensity would provide sufficient kinetic energy north of 30" to offset a frictional dissipation rate of 5 watts/m2.

2.7.4. Summary of Zonal Ceneration. All of the diabatic components except heating by frictional dissipation generate zonal available potential energy. The latitudinal contrasts in the solar absorption and sensible heat addition increase greatly in the winter, providing a continual generation of available energy. It is undoubtedly true that the strong polar cooling by infrared radiation is an important factor in the generation of available potential energy. Latent heat addition is probably also important on the zonal scale, especially when condensation in summer tropical thunderstorms and tropical cyclones is sufficient to contribute to the zonal average. I n fact, the fall maximum of tropical cyclones with the rapid sensible heat addition from warm ocean waters and the abundant condensation and release of latent heat in the tropical and subtropical troposphere may provide an important component in the generation of increased winter zonal available energy. 2.7.5. Summary of Eddy Generation. It appears that under suitable configurations, all of the diabatic components can serve to generate available energy in a disturbance. It is also apparent that these processes tend to reinforce each other in a developing storm. As an open wave cyclone starts to develop, the release of latent heat in the warm air generates available energy, and in addition, the developing cloud cover accentuates generation due to direct absorption of solar radiation. As pointed out, this cloud cover probably also converts the infrared processes into a local generation mechanism. A t the onset of precipitation, latent heat release probably becomes the dominant generation process. Because of the usual tilt of the low center westward over the cold air, the vertical velocity and temperature fields are in phase throughout the disturbance and the generated available energy is converted into kinetic energy through the rising of warm air and the sinking of cold air. Equivalently, the ageostrophic motion and cross-isobar flow provide the conversion mechanism for the baroclinic release of available energy. Figure 1 shows that latent heat release in the mid-troposphere in the neighborhood of 30" would be associated with large positive efficiency factors. Hencc the disturbance produces enough available potential energy that even though some is immediately converted to kinetic energy to offset local frictional dissipation, it can still export energy to larger scales of circulation.

370

JOHN A . DUTTON AND DONALD R . JOHNSON

However, as the development continues, condensation decreases and begins to occur in the colder regions. Clouds form or are advected over the cold air. Thus generation is decreased or changes sign and presumably the cyclone will dissipate. Sensible heat addition is apparently most important in providing generation of available potential energy when cold air moves over warm oceans, as verified by the existence of the cyclogenetic areas. Although boundary layer friction generates small amounts of zonal available energy, it probably has little effect on eddy generation. The studies of Holopainen [20] and Kung [21] have shown that the cross-isobar flow within the boundary layer and the resultant conversion of available potential energy is just sufficient to balance a boundary layer dissipation rate of about 1.9 watts/m2. They find that the remaining kinetic energy is produced in the free atmosphere and also dissipated there. These results, and our conclusion chat boundary layer sensible heat addition can be an important generating mechanism, suggest that the relation of boundary layer processes to the energetics of the free atmosphere needs additional study. Understanding of the energetics and accurate prediction of cyclogenesis and cyclolysis will therefore depend on adequate modeling of the diabatic processes both in the free atmosphere and the boundary layer.

2.8. Total Diabatic Generation and Frictional Dissipation The long-term averages of the diabatic generation of available potential energy and the frictional dissipation of kinetic energy must be equal, as shown by equations (2.77) and (2.78), since the average of the kinetic and available energies is presumably constant. Therefore we must require that the estimates of generation and frictional dissipation coincide. Obviously, Kung’s [21] careful estimatea of free atmospheric dissipation rates of 4.6 watts/m2 in addition to a boundary layer of 1.9 watts/m2 shed considerable doubt on the previous estimate obtained by diagnostic studies that the total generation of available energy is 2.3 watts/m2. Although the values obtained by Kung [21] over North America and Holopainen [20] over the British Isles are larger than previously expected, it has been known for some time that there was considerable dissipation in the free atmosphere. White and Saltzman [26] found a total dissipation estimate of 5 watts/m2. Therefore, assuming a boundary layer conversion rate of 2 watts/ni2 gives 3 watts/m2 for the free atmosphere. It might be noted that Kung [21] evaluated the average cross-isobar flow while White and Saltzman used the “wa” technique with vertical velocities obtained from a baroclinic model. The discussion of the previous section has shown that considerable diabatic

THE THEORY OF AVAILABLE POTENTIAL ENERGY

37 1

generation can be expected on both the zonal and eddy scales, but the generation by infrared radiation depends strongly on the relationship of the cooling and efficiency factor patterns. To test the theoretical conclusions and obtain preliminary estimates with the exact theory, the generation was computed for a two-layer model using the efficiency factors of Fig. 1 and the zonal heat budget model of Lettau [12]. The primary computation was performed on the 300- and 700-mb surfaces, taken to be representative of the layers 150-500 and 500-1000 mb, respectively. Sensible heat transfers were assumed to occur only in the lowest 100 mb boundary layer with 950 mb used for the computational level. Mean efficiency factors for the representative levels were computed from the summer and winter hemispheres of Fig. 1. The various diabatic components were partitioned into the layers and generation computed from the integral

Each diabatic component &k was represented for the entire meridional cross section with Lettau's [12] estimates q k of the average total heating or cooling for the entire column applied symmetrically to the Northern and Southern Hemisphere. We define pj to be the pressure of the representative level for each layer, AjH to be the standard atmosphere thickness of the j t h layer, and cjk(e)to be a function which, in t h e j t h layer, is the fraction of the heating of the kth diabatic component assigned to the j t h layer, and which is zero outside the j t h layer. Thus (2.102)

Qjk N

qkCjkIhjH

and so approximating the average by using the central latitude of Lettau's 15" sectors gives

and taking the value of e on the representative isobaric surface as the average over the layer and taking Aje to denote the potential temperature difference across the layer, we obtain

The total generation is obtained by summing Gjkover both indices. The results, the values used for ej, (AB/AH)j, and the assumed partitions of the components are given in Table I. The partitions were chosen with the aid of the results of Davis [23].

372

JOHN A. DUTTON AND DONALD R . JOHNSON

TABLEI. Estimates of zonal generation

of available potential energy by diabatic processes.

Layera (mb)

Representative level (mb)

150-500 500- 1000 900-1000

300 700 950

Total

Infrared radiationb (wattlmz) 2.55

- 0.94 0

Diabetic process Solar radiationc (watt/mz) -0.065 0.92 0

1.61

Percentage of total generation

29

Latent heatd (watt/mZ) 0 2.27

0.855

0

0

0 0.83

2.27

0.83

41

15

Sensible heat* (watt/mz)

15

Mean cfficienry factors (mcter/deg) by representative level Latitude 300 mb 700 mb 950 mb

82.5

67.5

- 32.2

- 30.0

- 15.1

- 16.2

-1.5

-1.2

52.5 -21.2 -11.1 - 0.4

37.5

- 12.6 -0.9 0.6

22.5 5.3 11.1 4.2

7.5 16.3 12.6 4.6

Standard atmosphere thickness values used for AH. Partition of Lettau's [12] infrared radiation: 4 to 300 mb, # t o 700 mb. c Partition of Lettau's 1121 absorbed solar radiation: & to 300 mb, # to 700 mb. d All of Lettau's [la] latent heat released assigned to 700 mb. All of Lettau's [ 121 sensible heat transfer assigned to 950 rnb. a

b

This approximate method therefore yields a total zonal generation of 5.6 wattslm'. The result that the net radiation generates zonal available energy as strongly as shown in Table I is due t o the upward bulge of the large negative efficiency factors toward the tropopause just above 300 nib between latitudes 35" and 55". Lowering of this one-third of the radiational cooling to 400 mb would give even stronger generation, as Fig. 1 makes clear. Therefore the total generation is strongly dependent on the contribution from infrared radiation and this in turn obviously requires careful modeling with height. Inclusion of the detailed structure of radiation divergence being revealed by radiometersonde ascents will be necessary to obtain a reliable estimate of the zonal generation. The reasoning of the previous section implies that the diabatic processes will also be effective in generating available energy on the eddy scale. We have already shown how the exact theory will yield strong generation estimates in oceanic cyclogenetic areas, in contrast to the results from the approximate integral that destruction occurs in these areas.

THE THEORY O F AVAILABLE POTENTIAL ENERGY

373

Comparison of our tentative estimates of the zonal generation rate with the calculated values of total dissipation permits us to estimate the eddy generation. Kung’s [21] total dissipation value of about 6.4 watts/m2 and the estimate of 5.6 watts/m2 for zonal generation thus combine to give an eddy generation of about 0.8 watts/m2. Local generation a t this rate is not sufficient to supply the kinetic energy realized during the conversion in a cyclone. Eddy [27] has computed an instantaneous conversion rate of 11 watts/m2 for a n extra-tropical storm, compared with both Palmkn’s [24] estimate of 53 watts/m2 for the conversion occurring after Hazel became an extra-tropical storm and with Kung’s [21] annual average over North America of 8.4 watts/m2. Eddy’s data on the conversion occurring over North America while the storm intensified on January 21 and 22, 1959 yield a mean rate of 9 watts/m2 or a total conversion for the 48-hr period of 432 watt-hr/m2.Thus both his and Kung’s data imply a conversion in 48 hr about equal to the seasonal change of 420 watt-hr/m2in the total kinetic energy between 20” and 70”N reported by Spar [28]. These results make it clear that the diabatic generation of available potential energy must be considerably larger than the currently accepted 2.3 watts/m2. Therefore, we expect that when diagnostic studies are carried out with the exact theory and adequate data on radiation divergence, latent heat release, and sensible heat transfers, the estimates of total generation will in fact correspond with Kung’s [21] estimate of total dissipation of about 6.4 watts/m2. The rapid conversion rates which have now been demonstrated in both individual storms by Palmen [24] and Eddy [27] and on the average by Kung [21], along with our estimate of zonal generation, show that the atmospheric energy transformations are more intense than generally realized. 3. APPLICATIONS TO OBSERVATIONAL DATA Numerical estimates of the amount of available potential energy are obviously of interest if the concept has any relevance to atmospheric processes. However, the data presently available impose a severe restriction on the accuracy with which we can study atmospheric energetics. To some degrec, even our intuitive sense of which scales of motion and which physical processes are important is colored by the present observational data. Although the final results cannot be more accurate than the observational data. except for what can be gained by appropriate statistical methods, errors of estimation have been frequently compounded by trcating the limited data with approximate methods, some of which were not even designed for the task at hand. As we shall show here, the computation of the amount of available potential energy requires care. The exact results

374

JOHN A. DUTTON AND DONALD R . JOHNSON

of the previous section will be applied to an unusual set of observational data to obtain estimates of the amount of available information about the structure of the reference state. All of our computations utilize the daily isentropic cross sections for 1958 along 75"W from the north to south poles prepared by the US. Weather Bureau [171 from the International Geophysical Year data. Isentropes were analyzed on the cross sections a t 10" intervals between potential temperatures of 240" and 400°, with a 50" jump to 450". The pressures on each of these isentropes were read a t every 10"of latitude from 240" t o 450" for every day of 1958. This is, of course, only a two-dimensional slice of the atmosphere, but these cross sections provide unusually complete meridional representations, and the data permit immediate application of the exact results. The same data were used to compute the available potential energy with two versions of Lorenz's approximate expression in isobaric coordinates. The exact results thus allow us to determine the accuracy of the isobaric approximation, and hence make suitable adjustment of present estimates of the total hemispheric available potential energy. Combination of these estimates with the results cited and developed in Section 2 then makes it possible to make new estimates of the amounts of energy and the transformation rates involved in the atmospheric energy cycle.

3.1. The Amount of Available Potential Energy Since the hydrostatic assumption has been utilized in the original reduction of the data on the meridional cross sections, we may use the expression originally given by Lorenz and rederived here for the amount of available potential energy. For the contribution t o the total available potential energy from the portion of the atmosphere below a &valued isentrope we thus obtain from equation (2.60)

It is worth noting that this "contribution to the total" is not identical to the amount of available energy below 8, since this latter quantity would involve both the "zp" term and the "Opl +K" term which arise in the integrations by parts used to obtain equation (2.60). The expression (3.1) was evaluated with finite sums approximating both the horizontal and vertical integrals involved. The usual cosine weighting was applied in computing all latitudinal averages and sums, and the surface pressure was taken uniformly t o be 1000 mb. The computations were performed independently for the Northern Hemisphere (0" to 90"N), the Southern Hemisphere (0" to 9O"S),and for the entire cross section (90"N to

THE THEORY OF AVAILABLE POTENTIAL ENERGY

375

90"s).In addition to the evaluation of the available energy, the total potential energy and the total entropy were also computed using formulas expressed in isentropic coordinates. The contribution to the total potential energy is the sum of the two positive terms of equation (3.1),that is

The entropy is given by

so that in accord with the convention used in the other integrals we have (3.4)

With average pressures at 450" of about 70 mb, these computations to 450" represent about 93 yo of the total mass along the meridional cross sections. The results for the Northern and Southern Hemispheres are shown in Fig. 2. The annual cycle of high available energy and low values of both total potential energy and entropy in the winter changing to low available energy and high total potential energy and entropy in the summer is readily apparent in both hemispheres. Monthly average values of these quantities will be presented later. There are two features of Fig. 2 which merit special mention. First, the sharp drops in the entropy are all associated with conditions when the 240" isentrope has nonnegligible area above 1000 mb. I n these cases, the mass between 240"and 250"is proportional to about 990-70 mb, compared with the usual case of about 1000-70 mb, or a drop of about 1 %, which is about the same magnitude as the sharp decreases in entropy. Hence a plot of entropy per unit mass would not have the sharp excursions present in Fig. 2. Second, the Northern Hemisphere atmosphere shows a transition to summer conditions which is more gradual than the transition to winter. The latter change appears in the available energy curve as two fairly sharp jumps occurring about October 1 and November 15. Note that the first plateau of winter values is interrupted in all three curves by a definite return, for a week-long period, to summer conditions. Examination of the synoptic analyses confirms an interpretation of this period as Indian summer. The Southern Hemisphere seasonal variation is similar, although the first jump to winter conditions about April 1 is less distinct.

Y c; 7.000 BOO0 5.000 4.000

4

0

m

3.000

2

?

2.000

5.940

Entropy

S.9 I 5

5.9 I5

S.B90

5.890

5.865

5.865

' I 2.600

-

'

-

2.600

FIG.2. Daily values of the available potential energy, total potential energy, and entropy computed for the Northern and Southern Hemispheres along 75"W for 1958. The Northern Hemisphere results are shown with the dark line, those for the Southern Hemisphere with the light line. The units for available energy are 109 ergslcmz; for total potential energy, 1012 ergslcmz; for entropy, 1010 ergslcmzdeg.

td

THE THEORY OF AVAILABLE POTENTIAL ENERGY

377

The large variations of the available energy over periods of a few days are associated with the advection of baroclinic zones through the cross sections, and are strongly correlated with intensification of the jet stream and the development of steeper slopes of the isentropic surfaces below the jet regions.

AVAIL ABLE POTENTIAL

f a

is'

i0-'

FREQUENCY

FIG.3. The variance spectra of the daily values of available potential energy, total energy, and entropy for the Northern Hemisphere cross-section data along 75'W for 1958. The spectra are given in relative units of variance per frequency. Selected periods in days are indicated by arrows.

The cyclic behavior of the available energy, total potential energy, and the entropy is revealed by the variance spectra shown in Fig. 3. The spectra of all three quantities are remarkably similar. The major interruption of the general decrease of variance with frequency is the broad peak spanning

378

JOHN A. DUTTON AND DONALD R . JOHNSON

periods near 40 days. Although the statistical reliability of these spectra is low since long periods are included (90 lags on 365 days), the identification of these peaks a t 40 days with the atmospheric index cycle seems reasonable. The slope of the available energy spectrum is close t o -2, which implies, to the extent that A is a measure of pressure variance on isentropes, that the amplitude of the pressure fluctuation increases more or less linearly with the period of the disturbance.

3.2. The Region of Maximum Contribution to A The exact formula and the results of the computations performed on the meridional cross sections make it possible to illustrate which features of the atmospheric structure provide the bulk of the available energy. TABLE11. Cumulative contributions to the monthly average available potential energy as a function of potential temperature for the Northern Hemisphere croo8s section along

75OW. January

July

240 250 260 270 280

997 990 976 949 901

0.026 0.034 0.063 0.152 0.390

1000 1000 1000 1000 994

0.00 0.00 0.00 0.00 0.03

290 300 310 320 330

833 714 565 440 344

0.91 1.72 2.61 3.36 3.85

958 859 669 607 383

0.055 0.26 0.68 0.89 1.12

340 350 360 370 380

253 188

4.07 4.12 4.14 4.18 4.21

273 183 160 133 121

1.23 1.27 1.30 1.35 1.40

390 400 450

107

4.25 4.27 4.34

110 100 70

1.44 1.47 1.57

156

134 119

97 68

Table I1 gives the monthly averages values of A(@and @ ( O ) for January and July in the Northern Hcmisphere. For the winter case, the layer between isentropes 2 N " and 340" contains nearly 60 yo of the total mass, but yields

THE THEORY OF AVAILABLE POTENTIAL ENERGY

379

88 yoof the total available energy. These isentropes, as Fig. l a illustrates, rise sharply from the equatorial troposphere t o just below or into the polar stratosphere. Thus the strongly baroclinic zone below the subpolar jet contributes the majority of the available energy. The summer situation is not as striking, since the isentropic layer between 290" and 340" contains nearly 70 yo of the mass, but yields only 75 % of the available energy. Some 20 yo of the total is contributed by the layer 350" to 450" and in fact, the 3.4 x 10' ergs/cm2 in this layer is even larger than the 2.2 x lo8 ergs/cm2 of available energy in the same layer of the January average. Thus the available energy is more uniformly distributed in the vertical in the summer than in the winter. The results on generation of available energy in Section 2 demonstrated that equatorial heating and polar cooling in this same isentropic layer would produce a considerable amount of available energy. The probable importance of this layer in energy conversion processes has undoubtedly been obscured by the use of isobaric coordinates.

3.3. The Structure of the Reference State It is intuitively appealing to consider that the reference atmosphere has a lower center of mass than the natural state, and that the available energy is thus proportional, under hydrostatic conditions, t o the total potential energy released in a vertical contraction of the atmosphere. But we find, somewhat to our surprise, that there are layers of the atmosphere which expand during the flow to the reference state. An integration by parts of equation (2.41),assuming hydrostatic conditions and that H(8,) = 0, yields (3.5)

and from equation (2.54) the equivalent formula for the natural state is (3.6) These two heights were computed from the meridional cross-section data, and the January and July averages of the daily values for the Northern Hemisphere are shown in Fig. 4. The plot of H ( 0 ) illustrates that the summer and winter reference atmospheres are quite similar, except a t low altitude where the lowest valued isentropes are below ground in the summer. More revealing is the graph of h - H , also shown in Fig. 4. The strongest vertical contraction occurs in the same baroclinic layer, 290" t o 340°, which

380

JOHN A. DUTTON AND DONALD R . JOHNSON

ti-H

i\

January

250

300

350

400

4

Potential tapemtun

FIQ.4. The structure of the reference state as illustrated by the monthly average heights of the isentropes. The figure on the left compares the January and July monthly average heights H ( 8 ) of the isentropes in the reference atmosphere. The figure on the right illustrates the difference between the monthly average of the mean height L(6)of an isentrope in the natural state and the average of the reference state height If(@.

contains the majority of the available potential energy. There is a slight expansion in the winter near the boundary of the stratosphere, and a more pronounced expansion in the upper stratosphere. The July average shows that a region of minimum contraction occurs in the same layer, but the sign is never quite reversed in the portion of the atmosphere considered here.

3.4. Average and Transient Components Time averages of the estimates computed from the meridional cross sections will presumably be more valid estimates of global or hemispheric values than the daily values. In addition to monthly averages of the various quantities, the monthly mean cross sections were computed and the available energy, total potential energy, and entropy of these average states were calculated. The results are given in Table 111. (We note in passing that, in view of the spectral results, the choice of a monthly average was perhaps unfortunate.) The transient contribution to the available energy is defined as (3.7)

A,=A-A,

THE THEORY OF AVAILABLE POTENTIAL ENERGY

38 1

where A, is the available energy of the time-averaged state. The monthly average of A, is on the order of 10 % of the total available energy. Since by definition

where the square bracket denotes a monthly average, the variance of the transient component (3.9) equals the variance of the available energy (3.10)

0 2 = [ ( A-

There are several features of Table 111 which suggest that averages of the results computed from the cross sections may be representative of larger portions of the atmosphere. First, the January and July figures of 2.51 x 10" and 2.58 x 10l2 ergs/cm2 for the total potential energy of the Northern Hemisphere compare well with the values of 2.6 x 1OI2 and 2.65 x 10" computed by Spar [28] with data from a larger area. Second, although the summer values of available energy in the Southern Hemisphere are larger than those of the Northern Hemisphere, the winter values are almost identical and the progress of the annual cycle is very similar. Finally, we shall show subsequently that the available energy of the monthly average state gives estimates of the zonal available potential energy which compare well with those computed from much larger data samples.

3.5. Comparison of Numerical Resultsfrom Exact and Approximate Expressions All of the numerical work done previously has, to our knowledge, used the approximate expressions developed by Lorenz to illustrate certain features of the behavior of available potential energy. Since these expressions were given in isobaric coordinates, they were adopted immediately for computations with observational data. Lorenz's first step was t o use a binomial expansion of p 1+ K to obtain

We computed daily values of A for all of 1958 with this expression and found them to be generally 5 % lower than the exact value, with a few extreme deviations approaching 10 yo.With further manipulations Lorenz obtained

TABLE111. Monthly average values of the available potential energy A , the total potential energy i,and the entropy 3. Month

A AT^ At* OAC n it* (109 ergslcmz) (109 ergslcmz) (108 ergs/cmz) (108 ergs/cme) (1012 ergslcmz) (108 ergslcmz)

W

00

E3

Sd

(10'0 ergs/

cme dep) Northern Hemisphere January February March April May June July August September October November December Average Standard deviation

4.34 4.46 4.25 3.98 2.72 2.23 1.57 1.64 2.30 3.14 4.15 4.78 3.30

3.85 3.96 3.98 3.69 2.47 1.95 1.41 1.49 2.08 2.81 3.76 4.44 2.96

1.16

1.06

January February March April May June JdY August September October November December Average Standard deviation

2.76 2.73 2.75 3.63 4.18 4.79 4.41 4.84 4.44 3.76 3.15 3.00 3.70

2.57 2.49 2.52 3.29 3.78 4.36 3.94 4.38 4.05 3.48 2.87 2.68 3.37

0.82

0.74

4.94 5.05 2.73 2.93 2.52 2.76 1.56 1.52 2.22 3.32 3.88 3.41

7.83 6.30 4.50 3.24 4.99 3.91 3.20 3.74 4.52 6.42 7.87 5.54

2.51 2.51 2.53 2.54 2.56 2.57 2.58 2.58 2.57 2.55 2.53 2.51

6.29 6.36 3.48 4.07 3.45 3.40 1.87 1.48 3.10 3.94 4.97 4.10

5.88 5.88 5.89 5.90 5.91 5.91 5.91 5.91 5.91 5.90 5.89 5.89

2.56 2.56 2.56 2.54 2.53 2.52 2.52 2.51 2.52 2.53 2.54 2.55

2.00 3.32 2.55 4.67 5.35 4.80 6.39 5.61 4.13 2.97 3.27 4.03

5.91 5.91 5.91 5.90 5.90 5.89 5.88 5.88 5.89 5.90 6.90 5.91

4

z ?

+-2

U

Southern Hemisphere 1.87 2.36 2.30 3.27 4.06 4.35 4.71 4.57 3.84 2.73 2.75 3.15

3.84 4.72 4.14 6.40 4.55 6.54 5.29 6.82 7.26 4.31 5.52 4.35

cd 4 0

1

January February March April May June July August September October November December Average Standard deviation

3.86 3.91 3.70 3.95 3.65 3.89 3.46 3.80 3.74 3.60 3.83 4.15

3.46 3.48 3.41 3.58 3.24 3.48 3.08 3.43 3.38 3.24 3.42 3.76

3.79

3.41

0.21

0.20

3.99 4.30 2.88 3.68 4.08 4.08 3.85 3.60 3.56 3.57 3.99 3.91

Pole to Pole 5.64 4.80 3.75 3.86 2.64 4.57 2.97 4.18 4.77 4.23 6.18 4.18

2.53 2.53 2.54 2.54 2.54 2.54 2.54 2.54 2.54 2.54 2.53 2.53

4.06 4.96 3.13 4.23 4.39 4.50 3.94 4.39 3.70 3.84 4.20 4.58

5.89 5.89 5.90 5.90 5.90 5.90 5.90 5.90 5.90 5.90 5.90 5.89

The subscript T indicates values computed from the monthly mean cross section. b The subscript t indicates monthly averages of the transient component, defined to be the difference between the daily values and the values computed from the monthly mean crow section. c The quantity U A is the standard deviation of both the daily values of the available energy and the daily values of the transient component (see text). Note that [S]= ST and that [St]= 0. Z ,

E d

a

2 0 4

%

k* !s W Y

8W 2

E W

2s

z

12.00

1

'

1

'

1

+---I

12m

10.06

I OD0

2.00

O.OO

12.00

1

0.00

lroboric Approximation lP,

(t~r,)

12.00

10.00

10.00

8.00

am

6.00

6JnJ

4.00

4m

2.00

2.00

0.00

Qoo

FIG.5. Comparison of daily values of the available potential energy for the Northern Hemisphere cross sections along 75OW for 1958 as computed with the exact formula in isentropic coordinates and two versions of the isobaric approximation. The exact curves are the dark lines, the approximate values are shown by the light lines. The lower curve was obtained with the lapse rate of the standard atmosphere used for v ( p ) . The upper curve was computed with y ( p ) determined daily in 50 mb increments from the cross-section data. The units are 109 ergslcmz.

THE THEORY OF AVAILABLE POTENTIAL ENERGY

385

the expression in isobaric coordinates which has been in general use (3.12)

We have retained the isentropic boundary term; this (small) term has been ignored in all computations of which we know. The digitized p ( 8 ) data from the cross sections was interpolated to yield O(p),and hence T ( p ) ,data for isobaric surfaces a t 50 mb increments between 1000 and 100 mb. The upper surface is close enough t o the 450" isentrope (which has an average pressure of about 70 mb) that available energies computed by these two methods may be compared. The estimates with the approximate formula were modified to apply to the same amount of mass as the exact results by multiplying them by the factor [f(240) - P(450)]/900. The comparison of the daily values is shown in Fig. 5 . The lower curve gives the values computed using F(p)= ys, where ys is the standard atmosphere lapse rate. The results with this choice of lapse rate are clearly unsatisfactory. The upper curves in Fig. 5 compare the exact results and the values computed with f ( p ) obtained daily and directly from the cross-section data. These approximate values follow the exact values, but seriously overestimate the available energy in the winter. We shall argue later that the neglect of horizontal variations of static stability is a major defect of the approximation, and this defect would be expected to be more serious with the strong stability variations of the winter season. I n order to obtain a calibration of the isobaric approximation which might be used to determine the reliability of the average hemispheric values previously computed, the monthly average mean values for the Northern and Southern Hemispheres computed by exact and approximate methods uere plotted on the regression diagram shown in Fig. 6. Both the use of the standard atmosphere lapse ratc and the average lapse rate as a function of height in the stability factor (y (,-7) lead to overestimation of the available energy. One of the most successful applications of the approximate formula to determine the amount of available potential energy in the atmosphere is the work of Krueger et al. [29]. They used a fixed lapse rate of Fi.l"C/km in their computations with five years of data. The curve for this lapse rate, derived from the curve for the standard atmosphere lapse rate by multiplication by the appropriate factor. shov s that this approximation agrees fairly well with exact results in the neighborhood of average values of 3.0 x 10" ergs/&, but underestimates in the summer and overevtimatrs in the winter.

386

JOHN A. DUTTON AND DONALD R . JOHNSON

Exact Formula

FIG.6 . Regression curve of the exact monthly averages compared with those calculated with the isobaric approximation. The curves through the data points were drawn subjectively. The dotted line illustrates a one-to-onecomparison. The solid line illustrates the comparison between exact values and those computed with the lapse rate a variable function of pressure. The upper dashed line compares the results using the lapse rate of the standard atmosphere, and the lower dashed curve is obtained from this by appropriate adjustment for a lapse rate of 6.1°K/km. The unit8 are 109 ergslcrnz. Computations over most of the mass of the atmosphere for 10 days allowed Krueger et al. [29] to adjust their five-year average estimates of the available energy in the 86O-MN)mb layer to apply to the whole atmosphere. They obtained 2.72 x lo9 and 7.3 x 10'eergs/cm2 for the annual average of the zonal and eddy available energy, respectively. With the calibration curve we thus obtain 2.8 x lo9ergs/cm2 for the zonal available energy and 3.35 x lo9

THE THEORY O F AVAILABLE POTENTIAL ENERGY

387

for the total, compared with the average 3.3 x lo9 computed with the exact formula on the meridional cross sections alone. The average of the available energies computed from the monthly average Northern Hemisphere cross sections is 2.95 x lo9 ergs/cm2and by comparison, is probably a good estimate of the zonal available energy. The ratio of 27 yo of eddy to zonal available energy obtained by Krueger et al. [29] would thus imply an associated eddy component of 8 x lo8 ergs/cm2. We must conclude that, although Krueger et al. [29] were apparently successful in determining annual average values due to cancellation of seasonal errors, almost all daily and even seasonal average values of the available energy computed to date with the approximate formula are suspect, and that modification to include horizontal static stability variation is undoubtedly necessary to obtain an expression which will give reliable results in isobaric coordinates. Figure 6 makes it clear that the isobaric approximation in its present form cannot be multiplied by a constant factor to be made accurate over the annual range of daily values of available potential energy.

3.6. The Energy Budget of the Atmosphere The new estimates of the average available energy of the Northern Hemisphere and the material of Section 2 on generation and dissipation rates combine to yield a different picture of the atmospheric energy cycle than the one currently accepted. I n this section we will present estimates of the rates involved in the cycle based, as far as possible, on the results obtained from direct computations with relatively reliable observational data such as wind and temperature fields. We accept Kung’s [21] estimate of the total dissipation of 6.4 watts/m2 as representative, and we shall nse the tentative estimate of 5.6 watts/m2 of diabatic generation of zonal available energy obtained in Section 2. Since the generation must equal the dissipation for average conditions, we estimate the eddy generation a t 0.8 watts/m2. Because the average energy budget of the atmosphere must balance, the specification of just a few of the transformation rates, given the total dissipation and zonal generation, is sufficient to uniquely determine the others. Thus it is advantageous to choose to estimate the transformations which can be measured directly with reliable observational data and which have no, or as little as possible, dependence on vertical motion fields. The transformations between available energy and kinetic energy on the zonal scale and between eddy and zonal kinetic energy are sufficient to complete the cycle and to meet the above requirement. Figure 7 shows the estimates of the energy budget derived in this manner.

388

JOHN A. DUTTON AND DONALD R. JOHNSON

Gz 3 I It 1.0)

5.6 AZ

T

29.5

0.25” 0.1 ( i O . 2 )

’

0.55

KZ

8(+3)

DZ

0.5(_+0.2)b

Dz -T

Gz-T 3.0 Ii I.Of”

0.4 I2 0.2)

5.35

0.3‘2’

T

-0.81t1.0l

D.0,

+ DE = GZ+GE

FIG.7. Estimates of the amounts of energy and the transformation rates involved in the atmospheric energy cycle. The unit of amounts of energy in the boxes is 108 ergs/cmz; the unit of energy transformation rates along the arrows is watts/mz. The figure compares our estimates, derived as explained in the text, with those of Oort [ll]. Our estimates are shown on the exterior of the arrows, Oort’s on the interior side. His estimates of kinetic energy were retained. Footnotes: (1) The conversion A z to K z is estimated with results Oort obtained from statistics reported by V. P. Starr and R. M. White for the observed winds a t seven levels. (2) Average of results compiled by Oort from computations on observed winds reported by V. P. Starr. (3) Estimated by Oort with results from the isobaric formula which requires w fields and the stability factor of the expression for A. (4) The data used by Oort for this estimate was either the result of computations using adiabatic, quasi-geostrophic w fields or from numerical general circulation models.

The two conversion rates we specified are estimates based only on direct computations with observational data as compiled by Oort [ll] and are within the limits he placed on these transformation rates. The other two conversion rates and the partition of the dissipation into zonal and eddy components follow from the total dissipation, and the entire model is based on direct observational results and balance requirements. The partition of the generation into zonal and eddy components affects only the rate of transformation between zonal and eddy available energy. The biggest difference in conversion rates between this model and Oort’s is the magnitude of the two conversions we inferred by balance requirements. Our estimates follow from the independently derived but consistent estimates of zonal generation and dissipation; the figures given by Oort for dissipation, generation, and these two transformation rates are summaries of studies based

THE THEORY OF AVAILABLE POTENTIAL ENERGY

389

on vertical velocities derived with adiabatic quasi-geostrophic models, or with the isobaric approxiniation for the rate of generation. In addition, some of the difficulty with the A , to A , transformation undoubtedly arises from the same static stability factor which occurs in the isobaric approximation for the amount of available energy. To summarize our conclusions: I . Various versions of the isobaric approximation have led to varying estimates of the average amount of available potential energy, most of which are too large, and the importance of the eddies as reservoirs of available energy compared to the zonal circulation has been frequently overestimated. 2. The generation of available energy and the dissipation of kinetic energy have both been generally underestimated by approximate methods. 3. As shown by our model in Fig. 7 , atmospheric energy transformations involving eddy available energy are more intense than presently accepted values. Energy conversions computed with the large-scale, smooth vertical velocity patterns obtained from present operational numerical methods undoubtedly are underestimates of the true conversion rates. 4. The gain in accuracy by studying available energy in isentropic coordinates may well be more important than the convenience of utilizing isobaric coordinates. I n the study of atmospheric energetics, as in any field, care must be taken in rushing to computers with approximations designed to aid in theoretical exposition. 4. VARIATIONAL METHODSIN AVAILABLE ENERGYTHEORY

The concept of available energy is founded upon comparison of natural states of the atmosphere with a reference state which possesses a minimum energy property. As illustrated in Section 2, the specification of this reference state originally evolved from observations that certain conditions ensure stationary or minimum values of the total potential energy. Now we turn to the methods of the calculus of variations to find necessary conditions for minimum or stationary values of the total potential energy. It will become clear that the ease with which we accomplish this objective depends upon the basic methods of Section 2. 4.1. Introduction to Variational Methods

The determination of distributions of the thermodynamic variables which minimize the total potential energy under the requirement of isentropic flow is an isoperimetric variational problem. Stated generally, in such a problem we wish to find minima of the functional

390

JOHN A. DUTTON AND DONALD R . JOHNSON

where

and

y w

=

(ayiaz,, aYiax2,ay/ax3)

for all functions y which satisfy certain accessory conditions or constraints. In the case of available potential energy, all of the functions y must be derivable from each other by isentropic, mass preserving rearrangements. The most straightforward of the isoperimetric problems involves integral constraints; that is, the functions y must satisfy a condition of the form (4.2)

s

g(x, y, y') dx = const

It can then be shown that the minimum of (4.3)

for some Lagrange multiplier p to be determined later with equation (4.2) is the desired solution. The classical necessary condition for weak relative extrema (and hence, also necessary for strong relative extrema) of a functional such as equation (4.3) is that the functions yi satisfy the Euler equations

where the integrand is to be differentiated by formally considering yi, and i3yi/axk as independent.

4.2. Necessary Conditions for Minima of the Total Potential Energy Our basic problem is to minimize (4.5)

s

II = (c,T + p ) p d V

subject to the condition that all admissable density and temperature distributions are isentropic readjustments of a natural state of the atmosphere. This becomes a tractable variational problem with the aid of the methods of Section 2. That is, we shall minimize (4.6)

THE THEORY OF AVAILABLE POTENTIAL ENERGY

39 1

as a function of p and h, subject to the condition that (4.7) where F(0) is determined by the natural state of the atmosphere and c1 = [ R / ( l O 0 0 ) ~ ) c ~It I ~ is ~ . clear from equation (2.37) that the constraint (4.7) requires preservation of the distribution of potential temperature with respect to mass. The constraint (4.7) is a mixed differential and integral condition and the classical isoperimetric results are not immediately applicable. We shall follow Akhiezer [30] in developing results appropriate for this problem. Assuming that a minimum stationary distribution pe exists, we put (4.8) where 6(x,y, 0,)

= p,(x, Y, = 6(x,y,

e) + E ~ ( z ,Y, e)

0,) = 0. Now we define

(4.9)

h = h(z, Y, 0, E )

with h to be determined from the constraint (4.7).For maximum generality, we put (4.10)

ahla8 = ~ ( xy, ,e)

where = F(0)for every 8. Hence,

or

(4.11) Now, (4.12) so that [ a h / a ~ ] , is = ~zero a t 8,, and its other zeros depend on the arbitrary function 6. Writing the integrand of the energy integral as f(x,y, 8, p, h, ah/%) and the constraint as k = p ah/aO - A(x, y, 0 ) we define

392

JOHN A. DUTTON AND DONALD R . JOHNSON

By the definitions (4.8) and (4.9), when E is zero, thus requiring that

n ( ~is)a minimum or a t least stationary

[ d I l ( ~ ) / d ~ ] e= = o0 Furthermore,

for every 8 and that

s

E.

kdA=O

Thus for any bounded function, p

= p ( x , y,

O ) , we know

(4.14)

Writing out this derivative, using subscripts to denote partial differentiation with respect to 0, p, and h, we find that (4.15)

Integration by parts with respect to 0 of the terms with a2h/aEa0produces, in the Euler derivative notation of (4.4),

But [ah/&], 0 vanishes for 0 = O0,and by the boundary condition that p(0,) is zero the other factor of the last term vanishes a t OT. Therefore, since 6 is arbitrary and [ah/a~],= 0 is not identically zero by equation (4.12) we have the Euler equations for our problem: ~

(4.17)

[f + @I,.

=0

(4.18)

[f f p k l h e

=

which must bc solved subject to the boundary conditions that ~ ( 0 , )= p(0,) = 0 and that O(r,y, 0 ) 2 O,,with equality attained a t least once. The function p is then to be determined from the constraint. Applying the Euler operation to the integrand of equations (4.6) and (4.10)

THE THEORY OF AVAILABLE POTENTIAL ENERGY

393

we find that

and (4.20) where we have dropped the subscript e. Elimination of p between the equations immediately shows that gp ah/ae

(4.21)

+ appe =o

which is, of course, the hydrostatic equation. Solving equation (4.19) for p and substitution of the result in equation (4.20) yields the relations

and

e ap/ae -

(4.23)

= gh

Differentiation of equation (4.23) and combination of the result and equation (4.23)with equation (4.22) now gives (4.24) and therefore (4.25)

ah

p-

ae

1 aZp

= --

g

ae2

[

ap

-1 K

ae C, --

culR

=

n(z,y, e)

or, upon rearrangement,

(4.26) which yields

where a is an arbitrary function of integration. However, the boundary condition that p(0,) is zero implies by virtue of equations (2.21)and (4.24)that a must in fact be identically zero.

394

JOHN A. DUTTON AND DONALD R . JOHNSON

Integration of equation (4.27) shows that

The condition on A was that k = F ( 0 ) ;but we shall show that to achieve a minimum we must have A = F ( e ) ,where P ( 0 )is the function derived from the natural state. From equations (4.24) and (4.27) (with a = 0), we obtain (4.29)

= C,-CC/CP

[j:‘ gA dp,]c.bp

and therefore (4.30)

p

=f’qA

dp,

0

Utilizing equations (4.23) and (4.25), we have the result (4.31)

which, upon combination with equation (4.27) becomes, (4.32)

By virtue of the hydrostatic relation (4.21) and equation (4.30)we may write

The function ~ ( t=) t l + r is convex and g,”(t) = (1 -t K ) K t- ‘ vlCp > 0 for 0 5 t < 00 so the classical inequality for convex functions (Hardy et a2. [MI, p. 161) applies and shows that (4.34)

and (4.35)

“.m+~ I M,.

y,

e)il+~

Thus the minimum or stationary value is had for A

3

F(B)andp(f&)= @((Iu)

THE THEORY OF AVAILABLE POTENTIAL ENERQY

395

As a result, equation (4.27), with CL = 0, implies that ap/atl is a function of 0 only, and hence by equation (4.24), the density distribution of the minimum energy atmosphere is barotropic. Furthermore, equation (4.32) shows that the height of an isentrope is a function of 0 only, and therefore the reference atmosphere is horizontally stratified. Returning to the convention of denoting the values of the minimum or stationary state with the subscript r and identifying S(0)= p ah/a0, as determined by the natural state, we find

which is identical to equation (2.48). To summarize: Assuming that a minimum or stationary value of the total potential energy can be reached by isentropic flow, we have demonstrated that the necessary condition for such a minimum is that the atmosphere be hydrostatic, barotropic, and horizontal-the classical “flat” reference state.

4.3. Comment on Suficient Conditions In the preceding analysis we determined necessary conditions for the minimum total potential energy, but we do not know whether these conditions are also sufficient. The most easily applied sufficient conditions of the calculus of variations depend on calculating second derivatives of the integrand. For example, if the integrand is f(x, y ( x ) ,z(x),y‘(x),~ ‘ ( x ) then ) , the strengthened condition of Legendre is that

(4.37)

T2fuy

+ c2

+ 2TCf#,Z* +

C2fZ’Z‘

>0

for v2 = 1. Referring back to equation (4.6), we see immediately that a n inequality of this type will not be satisfied for our case since the only derivative present is ahlatl, which occurs as a factor multiplying the remainder of the integrand. In Van Mieghem’s [14] study of available energy with variational techniques, he shows that the second variation is positive. But as pointed out by Bliss [31], the positiveness of the second variation must be coupled with the strengthened Legendre condition in order to prove sufficiency. Van Mieghem’s integrand is also the total potential energy, and hence will apparently not satisfy the required Legendre condition. We conclude that, as is generally true in the calculus of variations, the sufficient conditions for minimum total potential energy will be a more recondite problem than the necessary ones. The crux of the issue is that a sufficient condition surely must include a statement, as does equation (2.9),

396

JOHN A . DUTTON AND DONALD R . JOHNSON

about the motion. It may even be enough t o specify the existence of some kinetic energy in the flat, hydrostatic state.

4.4. Variational Approach to Energy Available for Meridional Flow Lorenz suggested that it might be desirable t o define the available energy so that it becomes zero when the conditions of stable hydrodynamic equilibrium are realized. Hedid not, however, carry out the necessary derivations on the grounds that it was preferable not to introduce the momentum distribution of the natural state of the atmosphere, as would be required. Van Mieghem acted upon Lorenz's suggestion, and by extending some results of Fjortoft [32] concerning incompressible zonal motion, derived approximate equations for the energy available for meridional flow. He showed that the sum of the internal, potential, and zonal kinetic energies of a natural state of the atmosphere reaches a minimum value when an isentropic readjustment which conserves the parcel angular momentum leads to a mass and momentum distribution specified by (4.38)

which is the classical equation of a permanent circular vortex in hydrodynamic equilibrium. We shall now examine this type of available energy with variational techniques similar to those of the preceding section. For this problem it is necessary to take specific account of the spherical geometry of the earth. Thus we seek to minimize, for 5 = h(0) a, the energy

+

(4.39)

..+n=J[

p(u

+ s25 cos + c1 ; c

(pe)cp/cu

subject to the isentropic constraint

(4.40) and the condition that the zonal angular momentum be preserved

(4.41)

ac COS' p(u + Q l cos 4)C3-

ao

4 d+ dX = Fa(0)

The verification of the applicability of the Lagrange multiplier technique proceeds essentially as before. We put

THE THEORY OF AVAILABLE POTENTIAL ENERGY

(4.45)

(1

x3

-p- = A n ,

ao

3

z

397

A,cos$+dh=F,(O)

and thus find that (4.47)

which permits us to solve equation (4.46) for u.We conclude that [a5/ae],=, is zero a t 0" and that its other zeros depend on 6 as before, and that [au/aE],=, is also not in general zero since this would require a particular relationship depending on 6 between A, and Az. The remainder of the derivation proceeds as before, although we must add the boundary condition that the momentum vanishes a t the top of the atmosphere as well as the density and pressure. We let m = (u !2[ cos 4) and E = pm2/2 c,,p/R p@,, and find the Euler equations

+

+

(4.48) (4.49)

m

E

+

+ p2 5 cosd = 0

+ P + p1p + p2 pm 5

COB$

=0

and (4.50)

[25E

+ pmQ cos $P + pGM + 2p15p + p2(3pmC2 + pa c0s24c3)1atpe a - - [52E + p,pC2 + p p 4 3 $1 = 0 ae COB

(305

Elimination of p1 and p2 in equation (4.50)with the aid of the other two equations yields (4.51)

Combination of equations (4.48) and (4.49)to eliminate p2 gives (4.52)

398

JOHN A. DUTTON AND DONALD R . JOHNSON

We assume, possibly as a special case of a more general solution, that p1 = p l ( @ and p2 = p2(0).Thus, equation (4.48) may be rearranged and

differentiated to show that (4.53)

where the subscripts indicate differentiation to be performed with 0 constant and hence, using the obvious result (4.54)

we obtain (4.55)

ap 840

+ p--

GM ac iL 340

-

-

- - 5 tan4)

prn2

a40

5

=

o

Conversion of Van Mieghem’s defining equation (4.38) to vertical and 4 components gives equations (4.61) and (4.55). Therefore we have shown that his reference state can be obtained, possibly as a special case, from a variational formulation. I t does not seem worthwhile t o embark on the tedious computations necessary to solve the Euler equations in the fullest possible generality. 6.

CONTRIBUTIONS TO THE AND

AMOUNTOF AVAILABLE POTENTIAL ENERGY

Its RELATIONSHIP TO OTHER

QUANTITIES

In seeking a better understanding of the role of available potential energy in atmospheric processes, it is useful to study both the contributions t o the total amount under various circumstances and the relation of the concept to other atmospheric quantities. First, we shall attempt to determine how specific characteristics of a natural state of the atmosphere contribute to the available energy. Next, we shall explore the relation between available potential energy, a gross static stability function, and entropy by expressing these quantities as somewhat novel, although formal, representations of each other. Finally, we shall show that perturbation form of the amount of available energy has appeared in several contexts. 5.1. Contributions from Hydrostatic Defects It is tempting to conjecture that addition of hydrostatic defects to a natural state of the atmosphere will increase the amount of available energy. However, as we shall show, this is not necessarily true.

THE THEORY OF AVAILABLE POTENTIAL ENERQY

399

We consider a column of small enough horizontal area u that horizontal variations of the atmospheric variables may be ignored. The total potential energy in this column is then

We shall find the extrema of this energy subject to the constraint that parcel mass and potential temperature are conserved

The Euler equations of the problem

(5.3)

are identical with those of Section 4.2 for the extrema of the total potential energy when we restrict p to be a function of potential temperature only. Thus, as in equation (4.21), a necessary condition for an extreme value of the total potential energy in a vertical column is that it be locally hydrostatic. The results of Section 4.2 show how to calculate the variables in the new hydrostatic state, to be denoted with a subscript h. Thus from equations (4.30) and (4.2) we find that (5.4) or upon using the fact that (5.5)

we obtain (5.6)

and similarly from equation (4.32)

If we are willing t o assume that x and p were differentiable along isentropic

400

JOHN A . DUTTON AND DONALD R . JOHNSON

surfaces in the natural state, then equations (5.6) and ( 5 . 7 )show that ph and ahh/aO will be differentiable, and hence continuous, if every column ie readjusted to extremize the energy in that column by eliminating the hydrostatic defects. The essential question, however, is whether this presumed extremum value is a maximum or a minimum. Conversion of equation (5.6) back to Cartesian coordinates yields

and we know that (5.9)

The available energy in the column due to the hydrostatic defects may now be written (5.10)

which, with equations (5.8)and (5.9) and an integration by parts yields (5.11)

and hence (5.12)

The fact that x is arbitrary (if we ignore the Coriolis and viscous terms in the Cartesian vertical equation of motion we have x = -dw/dt) shows that A, may be either positive or negative. This niakes it clear that the hydrostatic balance and flatness of the reference state must act in combination to give the necessary conditions for a minimum total potential energy. Furthermore, since we may write exactly that (5.13)

we have the result that (5.14)

A

=-

R

cp

s

( p -p,)dV

R -- A , cv

Therefore hydrostatic defects have a more complicated effect than merely

THE THEORY OF AVAILABLE POTENTIAL ENERGY

40 1

contributing to A as an additive factor, and obviously also affect the integral of the difference of pressures. It is of interest to put equation (5.11) in another form. Use of the definition of y, [(2.20)]and another integration by parts yields (5.16) where I, and P, are the internal and potential energies of the column, respectively. When the column is in hydrostatic balance, as is well known, the ratio of internal to potential energy is c J R . Hence A , will be positive or negative depending on variations of this ratio in the natural state of the atmosphere.

5.2. Barotropic Atmospheres and Available Energy Considerable theoretical and numerical use is made of the concept of barotropy, and it is accordingly of interest to investigate the available potential energy in barotropic atmospheres. For simplicity, we shall consider only barotropic atmospheres which are also in hydrostatic equilibrium everywhere.

5.2.1. The Structure of Hydrostatic, Barotropic Atmospheres. Concise, and to our knowledge new, representations of barotropic atmospheres are possible with the methods of Section 2. We shall use the two familiar equations for transforming derivatives (5.16)

V2

0

1

+ Veh

( )

- v2e

(

0

(a/az) ( 1 = Vs

0

(

1

and (5.17)

v2

( m e )(

= Ve

(

in which the small circle indicates any appropriate operation. By definition of a barotropic atmosphere, we know that at any point on an isentrope above ground (5.18)

Vsp =0

and applying this to equation (2.23) with the aid of equation (5.16) produces

Application of equation (5.17) to h(8)yields

(5.20)

fah/aO)V, 8 = - Ve h

so that the right side of equation (5.19), upon changing to integration with

402

JOHN A. DUTTON AND DONALD R . JOHNSON

respect to potential temperature, becomes (5.21)

Differentiation then yields

Ve ahlae = o

(5.22)

and the general solution to this equation is obviously

h(x, Y,0) = F ( e )

(5.23)

+

Y)

where F and G are arbitrary differentiable functions. Taking into account the boundary condition that the height and its derivative with respect to potential temperature must vanish for surfaces which intersect the ground, we may write the solution aa

h(8)= max

(6.24)

and

-

(6.25)

ae

if

F+G>O

F+GtO

It will be convenient in the sequel to use the mathematical notion of the support of h(x, y, which we define t o be the closure of the set of points (x,y) such that h(x, y, 0 ) >0. I n other words, the support includes both the points a t which the &valued isentrope intersects the ground and the points where i t is above ground. We let C(0) be the characteristic function of the support, so that C ( e )= 1 on the support and C(0)= 0 otherwise. Hence,

e),

aqae = c(e)arlae.

[Note. The result (6.22) is most easily obtained with the definition (5.18) and the corresponding statement about the isentropic gradient of the density. Differentiation of equation (5.18) with respect to potential temperature and use of the hydrostatic equation then gives equation (5.22). A virtue of the above method is that equations (5.19) and (5.20) combine to yield an equivalent differential equation for Cartesian coordinates (5.26)

aejaz

v, aelaz - a2e/az2 vze = o

which is less tractable than equation (5.21). However, it is easily shown that particular solutions of the form (6.27)

e = e[z - g(z,Y)I

THE THEORY OF AVAILABLE POTENTIAL ENERQY

403

satisfy the equation. The derivation of this equation from (5.22) requires a cumbersome transformation of i32h/i3e2.]

5.2.2. Construction of Barotropic States from Natural States. The arbitrary function G(x,y) of equation (6.24) allows us to isentropically rearrange a natural state of the atmosphere into an unlimited number of barotropic states, all to be denoted with the subscript B. In order to retain 0, as the lowest potential temperature above ground, we require that

with G = 0 on the support of 8, which, with equation (5.24), yields h(8,) = 0. We now replace F ( 0 ) with H B ( e ) , require H B to be nonnegative and thus the support of an isentrope is the set on which f ? B ( e ) G(s, y) 2 0. We shall require that averages involving aH,/aB include contributions only from the support of an isentrope. The hydrostatic equation in the barotropic state and the condition that the rearrangement preserve parcel mass and potential temperature vields the relations

+

(5.29)

so that (5.30)

The retention of the averaging operator here is essential since, by equation (5.25), ah,/at? vanishes off the support of an isentrope. Thus, on isentropes which do not intersect the ground in the barotropic state, we have

On isentropes which do intersect the ground, the situation is more complex. Since we must have (5.32)

the pressure a t a point on an isentrope below the ground is the pressure of the isentrope which is intersecting a t that point and p B is not constant off the support of an isentrope. Thus. off the support of an intersecting isentrope Oi,

401

JOHN A . DUTTON AND DONALD R . JOHNSON

we have ps(Oi) < p,(ei)and since f j B = p , , on the support it must be true that PE(ei) > P r ( O i ) .

So there are variations in pressure on isentropes which intersect the ground, and hence, by equation (2.60),sloping barotropic atmospheres possess available energy. In particular, denoting by 8* the isentrope with the largest value of tl which intersects the ground, we obtain

This expression also serves to define the available energy of a n arbitrary hydrostatic, barotropic atmosphere. When the barotropic state is derived by isentropic readjustment of a natural state, the available energy will be less than that of the natural state provided we have some condition on p,. A sufficient one, for example, would be that

[P,(z, Y, e)il+ K

I

y, e)il+ K

and this is ultimately a condition on G(x, y). Turning again toward the explicit construction of specific barotropic states by isentropic readjustment of a n arbitrary state, we combine equation (2.23) in isentropic coordinates with equation (6.29) to obtain

The quantity under the averaging operator in the right-hand expression is constant on the support of 0 and vanishes elsewhere, so that denoting the support by E(B),we obtain

Once G is specified, the integral over the support can in principle be calculated as a function of HE and the vertical structure of the barotropic state is then determined. When ell the isentropes are flat, or for those that do not intersect the ground, then E(B)= C and equation (6.36) is identical to a differentiated version of equation (2.28); in these cases we therefore obtain (6.36a)

HB(0)= H,(e)

(e > e*)

and hence for the nonintersecting isentropes in sloping barotropic states

(5.36b)

h , ( ~Y, 0) =

w e ) + ~ ( zY), I w e )

upon taking account of equation (6.28).

(e > o*)

THE THEORY O F AVAILABLE POTENTIAL ENERUY

405

For isentropes which intersect the ground, and this is in part determined by G, equation (5.35)leads to complicated expressions even for simple functions G. For example, if G = - ax, then (5.37)

;I,,, =; p"" 7 HBl dA

Idy

ax =

dy

and the addition of this factor of H , in equation (5.35)leads to a non-linear differential equation. The preceding results make it clear that in principle it is possible to isentropically readjust a natural state of the atmosphere to obtain a variety of barotropic states. However, there is nothing unique about the total potential energy of these states and, apparently, we can by choicc of the function G, construct barotropic states with total potential energy less, equal to, or greater than that of the natural state. These conclusions emphasize that although hydrostatic defects and baroclinicity in the natural atmosphere are important for the existence of available potential energy, their exact role is difficult to specify quantitatively.

5.3. Relationship to Other Quantities The concept of available potential energy is obviously best considered as a part of atmospheric energetics. but it is useful to compare it t o other atmospheric parameters in attempting t o understand how the role it plays in energetics is related to other concepts. Lorenz [33]pointed out that the amount of available energy is related in some manner to the static stabilities of the two states and introduced a gross or global static stability function which increased during the flow to the reference state. However, his gross static stability (a pressure-weighted integral of aO/ap) was not directly related to available potential energy, and thus did not give quantitative clarification of the precise sense in which the static stability increases. It is also appealing to search for a relationship between available potential energy and atmospheric entropy due to the intuitive similarity of the two concepts.

5.3.1. Integral Representations. In this section we shall show that rather unusual formal relations between available potential energy, an appropriate static stability function, and entropy can be derived. We consider the common static stability factor (5.38) the specific entropy

= (g/e)

ae/az

406

JOHN A. DUTTON AND DONALD R. JOHNSON

(5.39)

a(e) = cP In

(ep,)

in which we have set the usual additive constant equal to - cp In 0"; and the total entropy of the atmosphere

s

S = cp p In (ep,) d V

(5.40)

Adopting the convention that the natural state is either in hydrostatic equilibrium or has been isentropically readjusted to such a state, we will put (5.41)

[see equation (5.14)] and since

aqae = g/eU

(5.42)

we obtain (5.43)

in which p , = and = g/R. Thus, Ah can be represented as a Stieltjes integration of the difference of two pressure-weighted inverse static stability factors. This provides a precise quantification of the statement that the static stabilities must be less in the natural than in the reference state. Naturally, the total potential energy itself can be expressed in this manner, and, for example, (5.44)

For convenience we put

(5.45)

rh(e)

= (Ph/uhn) -

and then define (5.46)

Differentiation of this expression yields (5.47)

or in more compact form,

THE THEORY OF AVAILABLE POTENTIAL ENERGY

407

(5.48) Since aslae = c p p

(5.49)

we also have

(5.50)

a A h / a e = (cpYarh)/e

so that aA,./atI has the same sign and zeros as the factor Rearrangement and integration yields

f

(5.51) The definition (5.40) and the use of the hydrostatic variables allows us to define the total entropy below a given isentrope as

(5.52)

Because equation (5.50)shows that aAh/i%and F h have a bounded ratio, we may integrate a rearrangement of equation (5.47) to obtain (5.53)

and thus find from equation (5.52) that

Integration by parts produces (5.55)

The initial term on the right can be shown to vanish a t the top of the atmosphere with the argument associated with equation (2.60),although in this case we must consider

(5.56)

lim p In 8 Ilim p e = lim P+O

P-rO

P+O

T’+ r u / c

=0

408

JOHN A. DUTTON AND DONALD R . JOHNSON

Therefore the entropy of the entire atmosphere is given by (5.57)

Substitution of equation (5.50) into equation (5.57) produces, of course, the same result as can be obtained from a n integration by parts of equation (5.52). This family of formal results provides a compact statement of the relationships between the concepts of available potential energy, gross static stability, and entropy.

5.3.2. Further Entropy Results. The difference i n the average entropy contained between any two isentropes in the two states of the atmosphere can be calculated with equation (5.40)and is given by

which, according to equation (2.37), vanishes identically even with hydrostatic defects present in the natural state. Thus, as expected from the preservation of parcel mass and potential temperature during the flow to the reference state, the total entropy of the two states is identical. Since the entropy contained between isentropes is the same in both states, the distribution of entropy with respect to geometrical coordinates will be changed if the isentropes change position during the readjustment. Section 3 suggests that this will almost always be the cam. It is interesting to note that if we define (5.59) we obtain

(5.61) We may conclude that although the isentropic average of the ratio of pressure t o the stability factor changes during the flow to the reference state, the

THE THEORY OF AVAILABLE POTENTIAL ENERGY

409

average of the ratio of density to the stability factor does not. This latter conclusion is, however, only a restatement of equation (2.28). Expression of the well-known relation for the rate of change of the entropy in the notation of Section 2 yields (5.62)

dS - = - j - [ - -1D aQ Q]dV dt T a%

which upon change to isentropic coordinates becomes

(5.63) Comparison with the expression (2.78)for the rate of change of the available energy in the case in which the net diabatic effect is zero shows that the generation of available potential energy and the decrease of entropy on a given isentropic surface are similar processes. The additional weighting factor in the available potential energy expressions introduces a dependence on altitude which is appropriate to energetics but not to entropy processes. Although the generation of available potential energy is thus related to the processes which decrease the entropy of the natural state, we know that the entropy of both the natural and reference states decreases by an identical amount.

5.4. The Importunce of Variable Static Stability The role of variable static stability in available energy theory has largely been neglected in both theoretical and diagnostic studies. Lorcnz [6] has rioted the problem of determining the amount of available energy contributed from regions in which the lapse rate approaches the dry adiabatic rate, but his approximate expressions in general use ignore variations in static stability. To our knowledge, all diagnostic studies of available energy have been based on model atmospheres with isobarically constant stability factors, Most authors who have used Lorenz’s approximate equations (e.g., Oort [l l]), assigned a negligible importance to variations in static stability. But Gates [34] found large variations even in the monthly mean static stability computed across 100 nib layers. Later in considering baroclinic instability of numerical models, Gates [35] showed that the wavelength of maximum baroclinic instability shifts markedly to shorter wavelengths with decreasing static stability, and that the energy transformations in a variable static stability numerical model indicate a significantly different evolution of the disturbances. The results of this section illustrate analytically the importance of static

410

JOHN A. DUTTON AND DONALD R . JOHNSON

stability variations in available potential energy theory. Due to our use of isentropic coordinates, the mass in the isentropic layer bounded by surfaces with potential temperatures 0 and 0 d9 depends on the static stability. Furthermore, when we neglect the hydrostatic defect for clarity, equations (2.55) and (2.56)combine upon integrating (2.56)by parts, to yield an equation equivalent to (5.43)

+

(5.64)

The observation that (5.65)

makes it clear that the available potential energy depends on the difference of two pressure-weighted inverse static stability factors. It is obvious that in regions where the lapse rate approaches the dry adiabatic rate the contributions to the first term will become an order or two of magnitude larger than they will be in regions of higher static stability. Such conditions occur not only in the boundary layer, but are usually quite apparent in any vertical cross section through the upper troposphere, especially in regions of strong subsidence. We conclude that, rather than being of negligible importance, variations of static stability are an important factor in determining the amount of available potential energy. I n the exact expression for the available energy of a hydrostatic atmosphere [equation (2.60)], the importance of static stability is masked since the available potential energy is determined in this form solely by the pressure distributions. Hence diagnostic studies conducted in isentropic coordinates with equation (2.60) will automatically include the effects of variable static stability; future ones utilizing isobaric coordinates should, in our opinion, be designed to include static stability variations. 5.5. Energy Available to Perturbations One of the most important applications of the concept of available energy has been its introduction as a perturbation form of the total potential energy in a variety of theoretical studies and numerical integrations of the equations of motion. Following Charney and Drazin [36], Van Mieghem [37] showed that the equations of motion of a perturbation superimposed upon a zonal current lead t o a conservation theorem for a quadratic form with dimensions of energy density. Denoting the basic flow with subscript zero and the perturbation by primes, this energy form may be written

THE THEORY O F AVAILABLE POTENTIAL ENERGY

41 1

(5.66)

Van Micghem [37] referred to the terms in square brackets as the available energy of the perturbation flow. This perturbation available potential energy is identical to the integrand of Van Mieghem’s [3] expression for the total available potential energy of the atmosphere, provided of course, we take the zero order state to be the reference atmosphere and the primed quantities to refer to the difference of the natural variables and the reference variables. After Lorenz’s [I] paper and a few months before Van Mieghem’s [3] paper appeared, Eckart and Ferris [38] published a review of the various systems of perturbation equations. They pointed out the difficulty of defining a suitable form of perturbation energy from the usual energy expression and suggested the use of a quadratic integral of the equations of motion which they called the “external energy.” Their expression involved entropy as a basic variable, and in our notation is

Since Eckart and Ferris used both (5.68)

+ R(a‘/a,)

S’ = cu(T’/T0)

(for an ideal gas) and s‘ = s - so, we may write (5.69)

s’ = s - so = cp In (O/Oo) = c, In

(T/ To)+ R In (./ao)

Use of only the first term of the expansion of ln(1 +z) for 1x1 < 1 on the grounds that products of perturbations may be ignored yields (5.70)

SI

= cP

w/e0

which shows that their expression and Van Mieghem’s are identical. I n a 1960 book, Eckart [39] again introduced the external energy without reference to available potential energy and called particular attention to the “unfamiliar” potential temperature term which he then named the thermobaric energy. This term is Lorenz’s 1955 basic approximate expression (except for the difference in coordinate systems) for available potential energy and Van Mieghem’s 1956 dominant term. These perturbation expressions for the amount of available potential energy have considerable advantage when the reference or basic state is known. However, when we start only with the natural state of the atmosphere, no method other than that of Sections 2 or 4 is now available for calculating the reference variables from the natural ones.

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JOHN A. DUTTON AND DONALD R . JOHNSON

6. THE DYNAMICS O F THE GENERALCIRCULATION

Theoretical study of the general circulation-or perhaps more precisely, circulations of planetary scale-is generally aimed a t one or both of the two questions pointed out by Lorenz [40]. 1. Why does the atmosphere rcspond with the observed patterns or modcs of circulation to the external forces driving it? 2. How, once the mode of response is determined, is the kinetic energy maintained despite frictional dissipation? In this section we propose that the classical least action principle can be applied to yield both intuitive understanding and, hopefully, mathematical analysis of the control of the general circulation. The role of available potential energy in maintaining the kinetic energy can then be examined as a possible answer to the second question. 6.1. Extrema of the Total Potential Energy The theory of available potential energy is based entirely on finding a condition which insures that the total potential energy is a minimum. Both Van Mieghem’s original and variational approaches and the variational methods of Section 4 arrived a t the requirement that under isentropic flow a minimum or stationary value is obtained for horizontal, hydrostatic stratification. At the other extreme, we know that the total potential energy will be a maximum when there is no kinetic energy a t all. This leads to the question of whether the atmosphere, in its natural motion, tries to approach either of these two extreme states. Observational data indicate that there is always an ample supply of available potential energy and that the majority of it is not converted to kinetic energy. So we can conclude that the atmosphere does not attempt to reach the reference state of minimum total potential energy. This conclusion does not, however, contradict the results of Section 2, which show that the kinetic energy which is realized can be considered to be derived from the available potential energy. But on the other hand, the kinetic energy is such a small portion of the total atmospheric energy that i t does indeed seem that the atmosphere very nearly attains the state of maximum total potential energy. The kinetic energy that does exist is manifestly a response to differential heating and obviously provides an important transportation mechanism which is necessary for balancing the atmosphere’s heat budget. These observations illustrate that available potential energy is basically a kinematic concept-in the sense that its theory takes no account of the

THE THEORY OF AVAILABLE POTENTIAL ENERGY

413

actual forces at work in the atmosphere. Thus, to be of maximum use in answering the question of how the general circulation is maintained, the concept of available energy should be utilized within a theory which provides the proper dynamical restrictions.

6.2. Least Action Principles and the General Circulation The classical least or stationary action principle has proved both reliable and advantageous in a wide variety of applications. A virtue of this powerful concept is that it allows the introduction of variational techniques which may avoid the necessity of solving systems of differential equations to determine the evolution of a mechanical system, or the flow of a continuous medium. Although it was known by Lagrange that the material equations (or as commonly called, the Lagrangian equations) of motion of a fluid could be derived from a variational principle, i t is only recently that the Eulerian equations of inviscid, isentropic motion have been obtained in this manner. This recent development is summarized by Serrin [41].Van Mieghem [I41 subsequently, by a different method, derived the inviscid, isentropic equations of atmospheric motion. A crucial constraint in obtaining the Eulerian equations, proposed but not published by C. C. Lin (according to Serrin), is that the fluid particles retain their identity. Van Mieghem’s passage to Lagrangian coordinates in the course of his development apparently is equivalent to this requirement in an Eulerian coordinate system. To our knowledge, the viscous equations of motion have not yet been produced from a macroscopic variational principle. It would be revealing to be able to derive them from some tluch principle with the equation of continuity and the thermodynamic energy equation (including external and viscous effects) postulated as constraints. It certainly seems likely that the successful principle will be a least action one. The action is defined to be an integral of the difference between the local kinetic and total potential energy densities. and the least action principle3 states that the natural motions will minimize this integral. Since, as we have already observed, the kinetic energy in the atmosphere is such a small fraction of the total potential energy, the motions do indced seem to be governed by the least or stationary action principle. Once the action is formulated, constraint,s generally need to be applied, and the resulting solutions are determined by the constraints and the boundary conditions. For the atmosphere, we would attempt to find the minimum 3 More accwrately, Hamilton’s principle. We choose, however, to criptive term, “least action.”

UHU

tho more des-

414

JOHN A DUTTON AND DONALD R . JOHNSON

or stationary value of

(6.1)

&=/I 0

subject to appropriate side conditions. The classical approach to using the least action principle is t o operate on the Eulerian equations in such a manner that they lead t o a tensor conservation equation for energy and momentum. The original form of this tensor is usually modified to achieve symmetry properties which also ensure conservation of angular momentum. The equations of irreversible fluid motion can be written in this compact conservation form if we allow Greek subscripts t o range from 0 to 3, implying t , rl,x2, z3,and Roman subscripts to imply xl, x2, xg. Then, using the usual convention of summing double indices, the equations of irreversible motion and the energy conservation theorem in Cartesian coordinates may be expressed as

upon choosing the components of T,, to be

where

(6.4) and the X , are body forces (for the atmosphere, Xv = - g&,,). This form of the equations does strongly suggest that it should be possible to derive them as a variational principle. The energy-momentum tensor formulation also implies, as noted by Serrin [41], that it will undoubtedly be necessary to postulate the equation of continuity and the first law of thermodynamics as constraints in order to derive the Eulerian equations. The introduction of Lagrange multipliers into the Lagrangian of the problem and the fact that we are attempting to derive Eulerian, rather than Lagrangian or material, equations of motion raises several difficulties in applying the classical approach which we have been unable to overcome so far.

THE THEORY OF AVAILABLE POTENTIAL ENERGY

415

However, the full equations of motion are derivable from the principle that, on the microscopic scale, the thermodynamic probability will be a maximum. This principle yields the Boltzmann distribution of molecular velocities, which, in combination with the Maxwell-Boltzmann collision equation, produces the irreversible phenomenological equations of fluid mechanics. Since a maximum of the thermodynamic probability is likewise a maximum of the entropy, the equations are derivable from a principle which maximizes the entropy on the microscopic scale. The derivation of the Boltzmann distribution from a maximum entropy principle requires the use of conservation of energy and mass as constraints. The Herivel-Lin derivation of the reversible equations (as summarized by Serrin [41]) similarly employed an action functional constrained by the equation of continuity and by an isentropic requirement. These two derivations suggest that the concepts of least action and maximum entropy are either closely related or may act in conjunction to produce a Eulerian variation principle for the irreversible flow of the atmosphere. When the constraints are properly formulated and the full equations of motion can be derived from the least action principle, knowledge and understanding of the control of the general circulation should advance rapidly. For example, it may be possible to apply the direct methods of the calculus of variations to the principle to obtain numerical results while avoiding the familiar difficulties of working with nonlinear partial differential equations. However, if the introduction of the constraint that fluid particles conserve their identity is necessary, as it was in the derivation of the inviscid equations, the usefulness of direct methods would be seriously compromised. The success of the least action principle in the widest possible variety of physical problems and the observed fact that the kinetic energy is a negligible fraction of the total energy certainly suggests that we can tentatively adopt it as an aid t o intuitive understanding of atmospheric processes. In contrast, we observe that the concept of available potential energy is based upon maximizing the difference K - n, subject to isentropic motion, so that in fact the concept of available energy is derivable from a maximum action principle. Conversion of any substantial fraction of the available energy would therefore violate the least action principle, and as pointed out earlier the theory of available energy does not depend on the dynamics embodied in the equation of motion. We conclude that the concept of available energy cannot by itself serve t o determine why particular modes of circulation are chosen. This is the fundamental reason why the inadequacy of the reference state is not a serious problem. The concept of available energy can apparently only be used in tracing energy conversions. not predicting under what conditions they will occur. Thus although, in principle, it is possible to use the methods of Section 4 to produce an intuitively appealing reference state in

416

JOHN A. DUTTON AND DONALD R . JOHNSON

n,

which we maximize K , subject to the constraint of isentropic motion and the constraint that the horizontal kinetic energy K , , be geostrophic, such a reference state will suffer from the same defect as the flat reference state. I n particular, there is no assurance that the atmosphere tries to reach such a state of maximum geostrophic kinetic energy, and like the flat reference state, such a geostrophic reference is derivable from a maximum action principle. [Note added in proof. Professor R. L. Pfeffer, Florida State University, has investigated the consequences of just such a definition of available energy utilizing the constraint that the winds in the reference state be geostrophic. Starting with a somewhat restricted model, he finds that the minimum amount of total potential energy in such a state is larger than the total potential energy of the flat reference state, and hence the available potential energy defined with such a reference state is less than the available energy considered here. Since the Coriolis parameter is introduced in the geostrophic constraint, the amount of available energy in this referencc state depends on the earth’s rate of rotation. However, as shown in Section 5 , the statement trliat sloping barotropic atmospheres have no available energy is incorrect. The equations (2.77) and (2.78) already contain the essential feature of this new model: conversions of total potential energy to kinetic energy will cease when a totally geostrophic wind field is attained, and will resume only when the motion field becomes nongeostrophic, possibly after diabatic processes have generated additional available energy. A nondivergent geostrophic state, as shown by Van Mieghem’s result (2.9), will represent a stationary value of the total potential energy. A zonal geostrophic current with no meridional variation would represent such a stationary value. But since no northward heat transport is possible in such a flow, it cannot be an actual equilibrium which the atmosphere is attempting to reach. An account of Pfeffer’s work is now available in project report form and is to be published in the Proceedings of the International Symposium on. the Dynamics of Large-Scale Processes in the Atmosphere, Moscow, 1065 (in press).]

6.3. Application to Rotating Convection Experiments A logical way t o test the applicability of the least action concept is with observations of flows generated by differential heating in the well-known rotating thermal convection experiments of Fultz et al. [42], Fultz [43], and Hide [44]. In these experiments, a rotating annulus filled with fluid is differentially heated and different flow types develop under different rates of rotation and

THE THEORY OF AVAILABLE POTENTIAL ENERGY

417

heating. For conditions similar to those of the atmosphere, the flows are nrrestingly similar to the observed planetary circulations. One advantage of working with these experiments is that the equations can be simplified somewhat. First, we may consider the equation of state of the fluid to be

where p, and To are suitable initial values. Secondly, it is convenient to replace the vertical coordinate with h(r, f?)(, where h is the height of the upper free surface and the nondimensional variable 6 ranges over (0, 1). Thus, the action functional becomes

6.3.1. N o Heating. As a very simple application of the least action principle, let us consider a rotating annulus of inner radius ro and outer radius r l , which contains a fluid and is so arranged that the fluid is locally thermally insulated on all sides. We assume steady state conditions, and then with the nondivergent character of the motion takcn into account, the first law of thermodynamics is

or in integrated form

f

S f

c , T U * ~ d u = FQdV+

(6.8)

kVT.qdu

But the normal velocity must vanish on both the rigid and free surfaces, and there can be, by our assumptions, no heat conduction on any surface; thus the integral of the dissipative forces must also vanish. The expansion of the dissipation tensor in cylindrical coordinates and consideration of the boundary conditions show that the relative velocities must vanish, as is expected intuitively. Returning to equation (6.7), these results imply that the temperature is a linear function, and therefore by the requirement that there be no conduction at the boundary, the tempcrature, and hence the density, must be constant throughout the fluid. On the assumption that the container is being rotated, independently of the moment of inertia, a t a constant rate, the variational problem reduces to finding the equation of the free surface, subject to the requirement from

418

JOHN A . DUTTON AND DONALD R . JOHNSON

the equation of continuity that mass must be conserved. I n particular, dropping the constant terms and performing an integration in the vertical, we obtain

subject to the constraint that (6.10)

The Euler equation with the Lagrange multiplier pt is (6.11)

and substituting in equation (6.10) and integrating we find that (6.12)

p. - euT = gh, - R2(r?

+ r:)/4

and so finally (6.13)

which is, of course, the same answer obtained by conventional means (c.g., Lamb [46]).

6.3.2. Flow Regimes in Rotating Experiments. The results of the rotating experiments have shown that the fluid responds with different modes of motion to different external conditions. The symmetric Hadley regime occurs, with sufficient rate of rotation, a t either relatively low or high differential heating rates, with the less regular Rossby regime similar to atmospheric motion occurring in between. Figure 8 is adapted from Fultz’s [43] review of experimental work. An important feature is that the transition into the Rossby regime with increasing heating is marked (for water) by a n almost linear curve with a slope of minus one-a line of constant radial temperature gradient. As shown by Powlis and Hide [46], however, this linear shape is not present for fluids of other viscosities a t the same rates of rotation. Considerable simplification of the least action functional is possible under the conditions of the experiments summarized by Fig. 8. These experiments were performed with a rotating annulus cooled on the inner wall and heated on the outer wall, with the average temperature of the working fluid reported to be essentially constant. Pultz et al. [42] observed that the deviations due

419

THE THEORY OF AVAILABLE POTENTIAL ENERGY

to heating of the free surface from the equilibrium paraboloid (6.12) are on the order of microns and so we may ignore them. For simplicity we will use the overbar t o denote averages defined by (6.14)

I0'

I 0-

.: a

' b r T 40°

16 .brT

10"

:

I 0

ArT = lo

10-

FIG.8 . Flow regimes (for water) in a rotating annulus (rim heating, center cooling) after Fultz [43].The thermal Rossby number R * o , is proportional to the ratio of the radial temperature gradient to the square of the rotation rate, and Q*-1 is proportional to the square of the rotation rate. The Rossby regime is enclosed by the heavy line and dominant wave numbers are shown. The external temperature difference is shown by the slanting lines.

Thus for (6.15)

h = hr

+ h'

with h' defined by comparison with equation (6.12) in the obvious manner, we find that

-

(6.16)

h'=O

420

JOHN A . DUTTON AND DONALD R . JOHNSON

The average temperature is

-

Th/h,= T,,

(6.17)

and thus upon expanding the left side with T = To (6.16), we find that

+ T‘ and using cquation

c

T‘h= (T,,- To)h,

(6.18)

Upon multiplying out the various factors of the functional (6.6) and using equation (6.18),we obtain 1

(6.19) d = 27r(r,2 - r;2)po(- Ioc,,[Ta- eT0(T,- To)]h,dt

+ [{( 1 - E T ’ )(ui $- Rrsi1)2 + 2

C,E(

0

!7”)2 - (1 - ET’)ghf

Because (for water) E is of tho order 2.4 x deg-’, for simplicity, the term ET)may be ignored against unity in the first term of the second integral. It would be difficult to design an experimental apparatus in which the average temperature was maintained a t To despite the varying rates of differential heating. However, Fultz et al. [42] reported that their experimental procedures usually resulted in fairly constant mean temperatures for external gradients in the range we are considering here. We shall therefore assume that mean temperature effects are negligible in equation (6.19) and put T , = To. However, such effects would be important in other types of experiments-in particular, if the fluid were heated differentially, rather than being both heated and cooled. Upon assuming that T , = To, and eliminating the constant terms, the variable part of the functional becomes

1 1

(6.20) d’= 27r(r,2 - r t ) p o

((ui+ ~ T s i 1 ) 2

+c , E ( T ’ + ) ~eT’gh4

0

The dominant terms are the variance of temperature and the absolute kinetic energy. When the differential heating is initially imposed, there is presumably no relative kinetic energy. But with increaseddiffcrential hcating, the development of heat transport mechanisms requires kinetic energy but apparently gives less total action than allowing the variance of the temperature to increase. From Fig. 8 and the above simplified functional, we may conclude that in the symmetric regime a t low heating rates, the direct circulation yields sufficient transport to keep the variance of temperature adequately small

THE THEORY OF AVAILABLE POTENTIAL ENERQY

421

without too great a gain in kinetic energy. But as the radial temperature gradient is increased, a new compromise must be found. Almost independently of rotation rate, as shown by Pig. 8, the transition into the Rossby regime occurs (for water) at a fixed radial temperature gradient. Apparently, the quasi-horizontal transport of the RosRby regime represents a means of accomplishing the reduction of temperature variance with less total kinetic energy than would be required by a sufficient intensification of the Hadley regime. At even higher heating rates, the flow changes into the so-called fast Hadley regime, where apparently the least action principle dictates the necessity of rapid, efficient, and direct heat transport to decrease the temperature variance even though considerable kinetic energy is required. It appears that the least action principle offers an intuitive understanding of the hysteresis observed in the change of transition point for increasing and decreasing temperature gradients. The transition with increasing gradient is probably determined by the increasing variance of temperature and the least action necessity of controlling this even a t the expense of additional kinetic energy. With decreasing gradient, the flows change to the higher wave number at lower external temperature gradients than the gradient a t which the transition occurred when the gradient was being increased. Since the flow at a given wave number and gradient is presumably controlling the temperature variance adequately, a decrease in the external gradient requires no change until the transition to a new regime with presumably less variance and kinetic energy is possible. It is also appealing to consider the vacillation phenomena first reported by Hide [47], similar to the atmospheric index cycle, as a temporal least action response. Fultz et al. [42] clearly demonstrate the existence of the cycle with both periodic variations in the heat transported to the cold source and the strength of the westerlies. The heat transport and the average zonal winds are in opposite phase, with the major variation in the zonal wind being due to appearance of a westerly jet near the inner core at times of minimum transport. The interesting feature is a decrease, after several cycles, of the amplitude of the oscillations of the speed of the zonal wind. Undoubtedly this signifies a decrease in the total kinetic energy as well. Hence, by vacillating. the flow apparently accomplishes the necessary average heat transport with less kinetic energy than a steady state response would require. One of the most significant results of the rotating convection experiments for application to atmospheric phenomena is that the flow in the Rossby regime changes to lower wave numbers as the differential heating is increased. Atmospheric flow appears to follow the same pattern, with longer waves becoming more important in the winter. The specific energy spectra shown in Fig. 9 illustrate this tendency for more of the energy of the meridional

422

JOHN A. DUTTON AND DONALD R . JOHNSON

component to be a t lower wave numbers in the winter than in the summer; we have chosen the meridional component to emphasize the wave structure. If the least action principle is valid, we can conclude that the long waves should be more efficient transport mechanisms since they contain the most kinetic energy. We shall return to the observational evidence that this is indeed true in the next section.

I 2 3 4 5 6 7 B 9 1 0 1 1 1 2

Wavr Numbrr

FIG. 9. Mean winter and summer specific kinetic energy spectra. Data for 300 mb from noriton and Kahn [48] and data for BOO mb from Saltzman and Fleisher [4!3].

Although the changes in wave number exhibit a striking similarity in response in the atmosphere and the experiments, part of the atmosphere’s behRvior is undoubtedly governed by the location of the major orographic features and the distribution of heating over the eontincnts and oceans. Experimental determination of the effect of cquivalent barriers and heat distributions in rotating experiments would be of considerable interest, and Fultz et al. [42] report that this work has been initiated. Certain additional features of the transition curves for the experimental flows with constant average temperature can apparently be explained qualitatively by the least action concept. To do so, we obtain a non-dimensional version of the functional equation (6.20),which may be written

THE THEORY OF AVAILABLE POTENTIAL ENERGY

423

There are two important points t o be considered. First, as the rotation increases, temperature variance becomes less important due to the division by i2'. Second, increasing rotation amplifies the antisymmetrical function (r2 - (r,' + r $ ) / 2 ) thus giving less weighting to temperature variations and kinetic energy near the core.

FIG.10. Flow regimes (for water) in a rotat,ing annulus (rim heating and centcr cooling) adapt.ed from a graph of Fowlis and Hide [46]. The Rossby regime is enclosed by the heavy line and the dot,tecllines show, from left to right, the location of wave number six and the transition from steady to irregular waves. The quantity T(r1) - T(r0) is the external temperature difference and R is the angular velocity of the annulus.

From this we would expect that the transitions to smaller wave numbers would occur at higher variances of temperature with increasing rotation rate. Figure 10, which has been replotted against new axes from a graph of Fowlis

424

JOHN A. DUTTON AND DONALD R. JOHNSON

and Hide [46], shows that this does actually occur, if we can assume that internal temperature variance is proportional t o the imposed external temperature difference and that mean temperature effects are not important. The decrease in weighting near the core also suggests that if the flows are under least action control, then the relative kinetic energy should either be concentrated near the poled or symmetrical about (r: rI2)/2. Fultz et al. [42] show that the strongest zonal velocities do indeed occur near the core in the Hadley regime. The Rossby regime generally exhibits a zonal velocity profile with either a maximum near ( r 2 rI2)/2 or two maxima in a profile more or less symmetrical about this point. The same reasoning may be applied t o the cases in which there is rim cooling and core heating. An important change in the functional occurs since the term with the first power of T' is now in opposite phase with the weighting function. Thus we would expect, a t higher rotation rates, that the transitions will occur a t even higher values of the temperature variance than in the center-cooling experiments. On the assumption that the temperature variance is proportional t o the external temperature gradient, a transition curve of Fultz [43] bears out this conclusion for the Hadley t o Rossby transition. Furthermore, the flows which develop for this heating arrangement take advantage of the possibility of further reducing the action by being predominantly easterly.

+

+

6.4. The Role of Available Potential Energy The fact that the total kinetic energy of atmospheric motion generally increases with decreasing wave number has been established by a large number of observational studies. If, as we are postulating, the planetary circulations are controlled by a least action principle, these longest waves must be of prime importance in fulfilling the constraints since they contribute a major share of the total action. This conclusion receives support from the rotating convection experiments, which show that increased differential heating in the Rossby regime causes transitions to flows with a longer dominant wavelength. Solutions obtained from a principle requiring minimization of the action integral in time would presumably show least action control of the transient features of the general circulation and the observed tendency for vacillation of the major patterns. The exact results of Section 2 show that the atmospheric kinetic energy is derived from the reservoir of available energy created by the diabatic processes. But rather than available energy being converted to kinetic energy spontaneously by virtue of its existence, both observations and the least action principle imply that its conversion is controlled and restricted.

THE THEORY OF AVAILABLE POTENTIAL ENERGY

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Recent observational results indicate how available energy is released to maintain the kinetic energy of the planetary circulation. Computation of energy transfer and conversions in the Northern Hemisphere for the year October 1962 to September 1963, by Murakami and Tomatsu [50] are particularly interesting in view of the least action concept. They find that an average conversion of available energy to kinetic energy occurs at all wave numbers in both summer and winter, with the highest rate of conversion a t wave numbers two and three in the winter, and with a weak maximum a t wave number six in the summer. This differs from the results of Wiin-Nielsen [51] and Saltzman and Fleisher [52, 531, who reported conversion rates with sharp changes from one wave number to another and with relative maxima a t wave numbers two or three and absolute maxima at wave numbers six or seven. Saltzman and Fleisher found a relatively sharp drop from wave numbers two t o three and then a sharp increase continuing to wave number six. Murakami and Tomatsu [50] used data from the 1000 and 500-mb surfaces, while the other results mentioned were based on data from 850 and 500 mb. They point out that their analysis probably includes more emphasis on the important surface diabatic effects. I n view of the results of Section 2.7, the inclusion of these effects apparently has important and explainable consequences. Furthermore, Kung [21] shows that on the average four times more kinetic energy is generated below 850 mb than between 850 and 500 mb. Hence, Murakami and Tomatsu presumably included the major region of conversion while the earlier studies did not. These new results, indicating important conversions of available energy a t wave number three, certainly agree with our conclusions that the ultra-long waves are a major component of the least action circulation, especially in the winter. This conclusion is borne out by earlier work of slightly different nature. Perhaps most significant is the finding of Wiin-Nielsen et al. [54], that the maximum northward transport of sensible heat in the Northern Hemisphere in January 1962 occurred in middle latitude at low altitudes and that 50 yo of tho transport was accomplished by the ultra-long waves (numbers one through four) while waves with numbers five through eight accounted for only 25 yo of the transport. Approximately the same percentages held for the fractions of the total northward transport of angular momentum accomplished by these waves. The dominance of wave number three in the results of Murakami and Tomatsu [50] may be due to the use of a quasi-geostrophic model, but is to be expected partly from the distribution of the earth’s major mountain ranges and atmospheric heat sourcetj. The anchoring of this wave t o these features is clearly illustrated in the mean monthly maps of 500 mb wind

426

JOHN A . DUTTON AND DONALD R . JOHNSON

characteristics (8 years of data) of Lahey et al. [55]. The isotach patterns of the meridional component of the geostrophic wind show that large regions with northerly components form over or in the lee of the Rockies, the Alps, and the Himalayas and the Mongolian plateau in September and persist through March. These three regions are separated by three equivalent regions with southerly components with about equal speeds. In the summer months, as also indicated by the spectral data, this pattern is less distinct. The dynamics of this situation are illustrated by a figure of Murakami and Tomatsu [50] showing isopleths of the conversion of available potential energy to kinetic energy on a time vs wave number grid. The ridge of the maximum rate of conversion goes from wave number two in October to wave number 3.6 in December and then reverses sharply to reach a highest rate of conversion in January on a rise spanning both wave numbers two and three. The ridge line then goes slowly back to 3.5 by March. In April maximum rates of conversion occur at both wave numbers three and six. This split situation persists, with the maximum a t low wave number moving toward the one a t higher wavc number and finally in July becoming the only one a t wave number five. A new maximum will apparently develop a t wave number three again in September, although unfortunately the October 1962 and September 1963 ends of the graph are quite different. This qualitative agreement of the calculated rates of energy conversion reported by Murakami and Tomatsu [BO] with the situation shown on the average monthly charts indicates that their results may be representative of the average situation despite strong blocking in January and February of 1963. Effects a t wave number three are important in other results of Murakami and Tomatsu [60]. The available potential energy associated with wave number three is transferred to the available potential energy of both longer and shorter waves, while the kinetic energy a t wave three grows a t the expense of the kinetic energy of both the zonally averaged motion and all shorter waves. Therefore, the longest waves apparently are necessary for efficient heat transport as required by the least action principle, but part of the kinetic energy at wave number three is derived from kinetic energy realized a t other wave numbers. An important result not emphasized by Murakami and Tomatsu is that the conversion of available potential energy a t each wave number t o the kinetic energy of that wave number exceeds exchange of kinetic energy with all other waves by a t least four times, and exceeds the exchange of kinetic energy with the zonal current by about 40 times. Kinetic energy a t each wave number is primarily lost by transfer t o the atmosphere above 600 mb, with the transfer at wave numbers two and three being strongly dominant. It thus

THE THEORY OF AVAILABLE POTENTIAL ENERGY

427

appears that, a t least in a quasi-geostrophic model, each wave is to a large extent self-maintaining below 500 mb. However, additional details of the transfer of energy between each of the waves would be of interest. The technique of studying energy transfers with Fourier series around the latitude circles is disadvantageous in one respect. I t is known that asymmetry of the waves is essential for accomplishing the transfer of momentum. Therefore, the physical feature is an asymmetric wave while the equivalent statistical feature is a dominant wave and smaller waves a t higher harmonics. Thus some of the energy associated with wave numbers greater than three, for example, presumably belongs to an actual, asymmetric wave whose form is dominantly a lower harmonic. These results and observations combine to imply that the ultra-long waves are the dominant and controlling feature of the general circulation, as suggested by the least action principle and the rotating experiments. Of course, wave number three may be forced by orography and the distribution of heat sources (for a summary, see Smagorinsky [56]). However, the fact that ultra-long waves dominate the flow in the rotating experiments under conditions similar to those of the atmosphere suggests that the geographical features may determine the position and some of the asymmetry of this wave rather than its number. It therefore appears that the present emphasis on the question of the maintenance of the zonally averaged circulation is misplaced. Most importantly, such a mean circulation is a statistical and not a physical concept. Moreover, the results quoted indicate that the planetary circulation and its dynamics are regulated by the ultra-long waves. The predictions of a least action principle applied to a temporally varying circulation would obviously be of considerable interest. It is intuitively appealing to consider that the transient features of the circulation represent efficient means of satisfying the constraints and accomplishing the necessary transports with less total kinetic energy than would result from an equivalent intensification of the planetary circulation. It is also tempting to speculate that much of the current confusion in baroclinic instability theory could be cleared up with a suitable least action approach to the problem. If the entire circulation of the atmosphere is in fact controlled by a least action principle, then the intensification of a cyclone or anticyclone must be considered as one of many possible evolutions of the circulation, and its contribution to the satisfaction of the constraints must be weighed against the contribution to the total action functional. In particular, present theories of baroclinic instability do not take into account the possibility that the basic circulation may undergo a readjustment which relieves the potential or developing instability a t shorter wavelengths.

428

JOHN A. DUTTON APJD DONALD R . JOHNSON

6.4.1. Smaller Scnle Motion. If the least action concept is valid for the atmosphere, it should apply to all scales of motion. However, quite different responses may be dictated for large- and small-scale motion. An important part of the large-scale response is controlled by the dominance of the earth’s rotation in the absolute kinetic energy. For example, the existence of equatorial easterlies where the earth’s linear velocities are largest tends to reduce the total action. With smaller scale motions, however, the zonal averages of the velocities are probably quite small and hence the functional, upon expansion of the absolute kinetic energy, would involve the absolute kinetic energy of large-scale motion and the relative kinetic energy of small-scale motion. The generally negative correlation between density and temperature for small-scale motion is the analog of the temperature variance of the rotating convection experiments. The perpistence of strong correlations a t smaller scales would give a contribution to the action functional which would be integrated over long periods of time. We may speculate that it is more efficient in the least action sense to annihilate the small-scale portion of this correlation by an immediate response involving flows which last for only short periods of time. This would imply that a t small scales the available energy generated by diabatic processes is immediately converted into kinetic energy, in contrast to the large scales a t which a considerable reservoir of available energy always exists. The small-scale response would therefore be controlled by small-scale diabatic effects, the distances between external heat sources and sinks, and the existence of internal heat sources and sinks created by interactions with either the larger or the turbulent scales of motion. On this reasoning, we would expect the very early stages of cyclone formation induced by small-scale differential heating to be thermally direct circulations, perhaps a local form of Hadley flow. As the potentially cyclonic circulation intensifies due to continued heating and with the aid of the positive feedback mechanisms in Section 2.7.5, it will constitute an increasingly important fraction of the total action. Then increased interaction with the larger scales and a transformation toward the quasi-geostrophic regime and the typical baroclinic reRponse would be expected. As the disturbance becomes identifiable as a cyclone, it will aid in accomplishing the required poleward transport of heat. The even smaller scale motion of the boundary layer can also be viewed in the least action context, with the primary transport now being in the vertical. The general structure of the turbulent motion produced by convection and by boundary layer shear provides for the fairly uniform conversion of the kinetic energy into thermal energy, thus decreasing the total action.

THE THEORY OF AVAILABLE POTENTIAL ENERGY

429

6.5. Use of the Least Action Principle Development of a variational principle for irreversible motion would provide both an increased intuitive understanding and may allow the analytical problems to be formulated in a more tractable manner than the equivalent problem of solving a system of partial differential equations. The direct methods of the calculus of variations offer considerable hope that a variational principle can be used to make explicit predictions about the structure of a planetary flow which can be tested against observations. The Ritz method is one of the most widely used. For the atmosphere or for rotating convection problems, we would choose a complete polynomial expansion for each of the m variables. The first n terms of these expansions would then be substituted into the constraints, which would produce a series of relations which the coefficients of the polynomials must satisfy. The same expansions substituted into the action integral would also yield a finite sum of products and sums of the m x n coefficients. We thus have a numerical problem of finding values for the m x n coefficients which minimize the action and satisfy the constraints. As we increase the number n of coefficients, the a.ction will obviously not increase. There are, of course, problems of convergence to be taken into account, and the success of the method will depend in part on the choice of the polynomials. It would appear that this technique is especially suited to problems in the Rossby regime since the observational data indicate that the longest waves contain the majority of the kinetic energy and play the greatest role in satisfying the constraints. Perhaps the most difficult part of the problem for both the rotating convection experiments and for the atmosphere is proper formulation of the heating term. As a first approximation in the rotating experiments, we might assume that the temperatures of the container walls are fixed, and represent the temperature with a polynomial constrained to give the proper temperatures at the boundaries. For the atmosphere, it may be advantageous to assume that the solar heating is released a t the ground as an upward stream of infrared radiation. It would then be necessary to develop a suitably simplified version of the equation of radiative transfer to complete the specification of aQ/az. Introduction of surface variations in orography and heating corresponding to the characteristics of continents and oceans would be of considerable interest in determining the relative effects on the planetary circulations. Since most of the frictional dissipation occurs a t very high wave numbers, introduction of eddy coefficients is undoubtedly necessary, and might in fact provide a means of determining values for such coefficients appropriate to the reaction of motion at the turbulent scales with the planetary circulation.

430

JOHN A . DUTTON AND DONALD R . JOHNSON

7. CONCLUSION The variety of approaches and methods used to investigate the general circulation have led to many conclusions about its energetics, its dynamics, and its control. Some of these seem contradictory, and some seem to be of restricted value and application-a situation reminiscent of the disputes of the six blind men of Indostan who went to see, but could only touch, the elephant. All were partly right, and yet all were also wrong. The results of the previous section indicate that the least action concept offers a possibly fruitful approach to a variety of problems associated with atmospheric motion of all scales. It provides a dynamic control on the evolution of the motion which is presumably equivalent to the equations of motion, and seems to yield considerable intuitive understanding. It may be the gateway to future analytic and numerical success, or it may be still another voice in the blind men’s argument about the elephant none of them has seen. Studies of energy transformations in general and those involving available potential energy in particular, can never yield information about the control of the circulation unless they are coupled with a dynamic theory. By themselves, such studies are of necessity diagnostic and kinematic, and can yield only a detailed understanding of the mechanisms by which energy is conserved. The discovery and application of variational and integral techniques in atmospheric research is just beginning, but their success in other fields is sufficient guarantee that their exploitation will dispel many mysteries. It will be an exciting revolution as atmospheric science joins the era of the integral.

ACRNOWLEDQMENTS

The authors are indebted t o both Professor J. Van Mieghem, Royal Belgian Meteorological Institute, co-editor of Advances in Ueophysica, whose suggestions and comments led to simplification and clarification of the material of Section 2, and t o Professor Edward N. Lorenz, Massachusetts Institute of Technology, for several helpful diseusrrions and for reviewing the manuscript. Discussions with Professor Raymond Hide, Massachusetts Institute of Technology, provided additional understanding of rotating convection experiments. Almost all the authors’ colleagues a t both the Universit,y of Wisconsin and The Pennsylvania State University have assisted with suggestions and questions, and by listening patiently to lengthy ticcounts of new devolopments. Professors H. H. Lettau and Lyle H. Horn, University of Wisconsin, were partirularly encouraging and helpful. The research was partially supported by a U.S. Weather Bureau grant, WBG-52, to the University of Wisconsin, and by the National Centnr for Atmospheric Research, which provided one of us ( D R J ) with both a stimulating summer environment and an opportunity for profitable reflection. The numerical work was done at the Computation

43 1

THE THEORY OF AVAILABLE POTENTIAL ENERGY

Center, The Pennsylvania State University, with computer time and facilities provided by the University. The final computer programs were prepared and the processing of the data organized by Mr. Dennis Deaven, The Pennsylvania State University. Initial programming and test efforts were carried out by Messrs. James Polston and Gary J. Thompson, Environmental Technical Applications Center, U.S. Air Force, and by Mrs. Barbara Stroh, of The Pennsylvania State University. The prodigious task of preparing punch cards from the IGY cross sections was undertaken by Mr. Lawrence Dillehay, University of Wisconsin. The assistance of Mrs. Linda White in typing several drafts of the manuscript, correcting crrors, and meeting deadlines made completion of the article a more pleasant task than it might have been. Finally, we should like to acknowledge the considerable assistance arid encouragement received from Dr. H. E. Landsberg, Environmental Science Services Administration, coeditor of Advances i n Ueophysica. The promptly rendered decisions and the patient understanding of the unavoidable difficulties made working with him a n enjoyable experience. LIST OF SYMBOLS

Physical Variable8 and Conatants (All thermodynamic quantities i n mechanical unila) Pressure Absolute temperature Density Specific volume (in Section 4, an arbitrary function of integration) Potential temperature: 0 = T(1000/p)R/~p Variables of integration implying potential temperature Specific heat at constant volume Specific heat a t constant pressure Gas constant: R = cv - cv RlCP Speed of sound: C2 = c p p/cvp The hydrostatic defect: x = ap/paz + g Wind velocity vector 3 component of U y component of U z component of U Potential energy of unit mass: #. = Q M ( l / a - l / r ) where Q is Newton’s constant and M is the mass of the earth - tR2R2 Gravitational potential energy of unit mass (geopotential: # = Acceleration of gravity: gk = -V@ Rate of energy gain or loss per unit. volume from external sources (for example, radiation) Divergence of external heat flux vector Q : V Q = - p dq/dt Sum of internal diabatic terms: D p = P p Tp + 0 Molecular condition of heat (including conduction from boundary surface):

ma

+

Tp=V*kVT

JOHN A. DUTTON AND DONALD R . JOHNSON

Coefficient of heat conduction Rate of heat addition per unit volume due to condensation or evaporation Frictional addition of heat per unit volume Frictional dissipation of kinetic energy F Q ]dV Boundary frictional effects: $FB q du = j [ F u Sum of all diabatic processes: = - V . Q T Q F Q 'if Solar radiation Infrared radiation Total mass contained between 81 and 82. M(el, Lapse rate Dry adiabatic lapse rate: yd = g/c, Autoconvective lapse rate: y n = g / R Stability factor: u = ga8/8& or a small area Efficiency factor for generation of available potential energy [defined in equation (2.99)] Specific entropy A difference of stability measures [defined in equation (5.46)] Arbitrary function used in variational proofs Lagrange multiplier A small quantity used in variational proofs; in Section 6, volumetric expansion coefficient Isobaric vertical velocity: w = dp/dt [ R/( 1000)~]cp'% Angular velocity of earth or annulus Mean radius of the earth Distance perpendicular to earth's axis Mean height of fluid in annulus Inner radius of annulus Outer radius of annulus

8

+

+

+

+

wc

Integral Quantitiee Total kinetic energy ( R the same per unit ar?) Internal plus gravitational potential energy (rI the same per unit area) Available potential energy: A = J& - ITr ( Athe same per unit area) Contribution t o A from internal energy Contribution t o A from potential energy Generation of available potential energy by diabatic processes [defined in equation (2.96)] Generation at j t h level by kth diabatic process Functionals depending on y Absolute zonal kinetic energy Total potential energy of column of unit area Available potential energy in a unit column due to hydrostatic defects Available energy due to hydrostatic defects alone Total entropy of the atmosphere (8 the m e per unit area) Action functional [defined in equation (S.l)]

433

THE THEORY OF AVAILABLE POTENTIAL ENERGY

Coordinates and Transformations t

Time

X h

Cartesian or spherical coordinates: z or h increases toward the east; y or increases toward the north, z or r increases upward along the zenith

Y4 z r

4

Arca of the earth’s surface Height of a &valued isentrope in reference atmosphere Height of a 8-valued isentrope in natural atmosphere h(@ a Height of an isentrope in a hydrostatic, barotropic atmosphere: hB = HB(@ Q(z,y ) where H is a function of 8 only and Q is a function of x and y only zs = z Cartesian or spherical coordinates: 2’ = t , zl = z or h, 2 2 = y or or r Four velocity: uh = dxA/dt(h = 0 , 1, 2, 3) Generalized isentropic coordinates: 5 0 = t , 6 1 = 5, 5 2 = 77, 5 3 = 8 Four velocity: vA = d t A / d t (h = 0, 1, 2, 3) Jacobian of the transformation: J = a@, y, z)/a(t, 7,8) Denotes (6, 7)space

+

+

4,

Mathematical Operators and Symbols

J dV

Integral over the entire atmosphere or annulus Integral over the earth’s surface Integral over the entire bounding surface

9 V

d

a [f IV (7 (

1’

(7 8ik

In

Exterior unit normal vector on bounding surface Hamiltonian operator (del or nabla); may be restricted by subscript Total derivative Partial derivative Euler derivative: [ j I Y = af/ay - apzt (aj/a(%/azt)) Average, generally over an isentrope, see equation (2.36);in Section 6.3, as defined by equation (6.14) Deviation from average or unperturbed state Isobaric average Kronecker delta: = 1 if i = k, = 0 otherwise Natural logarithm

Subscripts a r e

T h

B 0

e 2

Denotes natural atmosphere (except ya, CP,) Denotes reference atmosphere Denotes extremum Implies evaluation a t the top of the atmosphere Denotes hydrostatic conditions Denotes hydrostatic and barotropic conditions (except FB) Denot,es initial or unperturbed conditions ( 8 0 the lowest potential temperature in the atmosphere except in Section 5.6 where 00 is unperturbed potential temperature) Denotes differentiations to be performed with 8 held constant Implies horizontal; differentiations to be performed in the horizontal only

434

JOHN A . DUTTON AND DONALD R . JOHNSON

REFERENCES 1. Lorenz, E. N. (1965). Available potential energy and the maintenance of the general circulation. T e l l w 7 , 157-187. 2. Margules, M. (1903). “Ober die Energie der Stiirme. Jahrb. Zentralanat. Meteorol. 40, 1-28; translation by Abbe, C. (1910). Smithaonian Inst. M b c . Coll. 51, 533-695. 3. Van Mieghem, J. (1958). The energy available in the atmosphere for conversion into kinetic energy. Beitr. Phya. Atmoaphare 29, 129-142. 4. Van Mieghem, J. (1957). Energies potentielle e t interne convertibles en dnergies cindtique dans l’atmosphbre. Beitr. Phya. Atmoaphdre 30, 5-17. 5. Fjertoft, R. (1980). On the control of kinetic energy of the atmosphere by external heat sources and surface friction. Quart. J . Roy. Meteorol. SOC.86, 437-453. 8. Lorenz, E. N. (1955). Generation of available potential energy and the intensity of the general circulation. I n “Large Scale Synoptic Processes” (J. Bjerknes, Proj. Director), Dept. Meteorol., Final Report, 1957, Contract AF19(804)-1288. Univ. of California, Los Angeles, California. 7. Starr, V. P. (1948). An essay on the general circulation of the earth’s atmosphere. J . Meteorol. 5 , 39-43. 8. Starr, V. P. (1958). Modern developments in the study of the general circulation of the atmosphere. J . Qeophya. Rea. 61, 334-340. 9. Lorenz, E. N. (1958). A study of the general circulation and a possible theory suggested by it. Sci. Proc. Intern. Aaaoc. Meteorol. Rome, 1954, pp. 457-482. Butterworth, London and Washington, D.C. 10. Kuo, H. L. (1958). Energy releasing processes and instability of thermally driven motions in a rotating fluid. J . Meteorol. 13, 82-101. 11. Oort, A. H. (1984). On estimates of the atmospheric energy cycle. Monthly Weather Rev. 92, 483-493. 12. Lettau, H. (1954). A study of the mass, momentum, and energy budget of the atmosphere. Arch. Meteorol., Qeophya. Biokl. A7, 135-157. 13. PalmBn, E. (1980). On generation and frictional dissipation of kinetic energy in the atmosphere. Soc. Sci. Fennico, Commentationea, Phya.-Math. 24, No. 11, 1-15. 14. Van Mieghem, J. (1980). Le principe variationnel de la mdcanique de l’atmosphbre. Beitr. Phya. Atmoephiire 32, 136-152. 15. Hardy, a. H., Littlewood, J. E., and P d y a , G. (1934). “Inequalities,” 314 pp. Cambridge Univ. Press, London and New York. 18. Thompson, P. D. (1981). “Numerical Weather Analysis and Prediction,” 170 pp. Macmillan, New York. 17. U.S. Weather Bureau, (1982). “Daily Aerological Cross Sections Pole to Pole along Meridian 75OW for the ICY Period.” U.S. Dept. of Commerce, Washington, D.C. 18. Lorenz, E. N. (1980). Generation of available potential energy and the intensity of the general circulation. I n “Dynamics of Climate” (R. L. Pfeffer, ed.), pp. 88-92. Pergamon Press, Oxford. 19. List, R. J. (1968). “Smithsonian Meteorological Tables,” 527 pp. Smithsonian Inst., Washington, D.C. 20. Holopainen, E . 0. (1983). On the dissipation of kinetic energy in the atmosphere. T e l l w 15, 26-32. 21. Kung, E. C. (1986). Kinetic energy generation and dissipation in the large scale atmospheric circulation. Monthly Weather Rev. 94, 87-82. 22. Petterssen, S., Bradbury, D. L., and Pcdersen, K. (1982). Tho Norwegian cyclone models in relation to heat and cold sources. aeojys. Publik. 24, No. 9, 243-280.

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23. Davis, P. A. (1963). An analysis of the atmospheric heat budget. J. Atmoa. Sci. 20, 5-22. 24. Palm&, E. (1959). On the maintenance of the kinetic energy in the atmosphere. I n “The Atmosphere and Sea in Motion (The Rossby Memorial Volume)” (B. Bolin, ed.), pp. 212-224. Rockefeller Inst. Pross, New York, and Oxford Univ. Press, London and New York. 25. Riehl, H. (1959). On production of kinetic energy from condensation heating. I n “The Atmosphere and Sea in Motion (The ltossby Memorial Volume)” (B. Bolin, ad.), pp. 387-399. Rockefeller Inst. Press, New York, and Oxford Univ. Press, London and New York. 26. White, R . M.,and Saltzman, B. (1956). On conversion between potential and kinetic energy in the atmosphere. TeZlw 8, 357-363. 27. Eddy, A. (1965). Kinetic energy production in a mid-latitude storm. J. Appl. Meteorol. 5, 569-575. 28. Spar, J. (1949). Energy changes in the mean atmosphere. J . Meteorol. 6, 411-415. 29. Krueger, A. F., Winston, J. S., and Haines, D. A. (1965). Computation of atmospheric energy and its transformation for the Northern Hemisphere for a recent fiveyear period. Monthly Weather Rev. 93,227-238. 30. Akhiezer, N. I. (1962). “The Calculus of Variations,” 247 pp. Ginn (Blaisdell), Boston, Massachusetts. 31. Bliss, G. A. (1946). “Lectures on the Calculus of Variations,” 292 pp. Univ. of Chicago Press, Chicago, Illinois. 32. Fjertoft, R. (1950). Application of integral theorems in deriving criteria for stability of laminar flows and for the baroclinic circular vortex. Qeojys. Publik. 17. No. 6, 1-52. 33. Lorenz, E. N. (1960). Energy and numerical weather prediction. Tellw, 12, 364-373. 34. Gates, W. L. (1961). Static stability measures in the atmosphere. J . Meteorol. 18, 526-533. 35. Gates, W. L. (1961). The stability properties and energy transformations of the twolayer model of variable static stability. Tellus 13, 460-471. 36. Charney, J. C., and Drazin, P. G. (1961). Propagation of planetary scale disturbances from the lower into the upper atmosphere. J . Geophys. Rea. 66, 83-109. 37. Van Mieghem, J. (1963). New aspects of the general circulation of the stratosphere and mesosphere. Proc. Intern. Symp. Stratospheric Meaospheric Circulation, Berlin, 1962, pp. 5-62. (Meteorol. Abhandl., Freie Univ., Berlin, 36.) 38. Eckart, C. and Ferris, H. G. (1956). Equations of motion of the oceans and atmosphere. Rev. Mod. Phys. 28, 48-52. 39. Eckart, C. (1960). “Hydrodynamics of Oceans and Atmospheres,” 290 pp. Pergamon Press, Oxford. 40. Lorenz, E. N. (1966). Energetics of atmospheric circulation. I n “The International Dictionary of Geophysics.” Pergamon Press, Oxford (in press). 41. Serrin, J. (1959). Mathematical principlcs of classical fluid mechanics. I n “Handbuch der Physik” (8.Fliiggc, ed.), Vol. 8, Part I, pp. 125-263. Springer, Berlin. 42. Fultz, D., Long, R. R., Owen, G. V., Bohan, W., Kaylor, R., and Weil, J. (1959). Studies of thermal convection in a rotating cylinder with some implications for large scale atmospheric motion. Meteorol. Monogr. 4, No. 21, 1-104. 43. Fultz, D. (1961). Developments in controlled experiments on larger scale geophysical problems. Advances i n Qeophys. 7, 1-89. 44. Hide, R. (1958). An experimental study of thermal convection in a rotating fluid. (London) A250, 441-478. Transact. Roy. SOC.

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45. Lamb, H. (1932). “Hydrodynamics,” 8th ed., 738 pp. Cambridge Univ. Press, London and New York, and Macmillan, New York. (Reprinted, 1945, by Dover,

New York.)

R. (1985). Thermal convection in a rotating annulus of liquid; effect of viscosity on the transition between axisymmetric and nonaxisymmetric flow regimes. J . Atmos. sci. 22, 541-558. Hide, R. (1953). Some experiments on thermal convection in a rotating liquid. Quart. J. Roy.Meteorol. SOC.79, 161. Benton. a. S., and Kahn, A. B. (1959). Spectra of large scale flow at 300 millibars. J. Meteorol. 15, 404-410. Saltzman, B., and Fleisher, A. (1982). Spectral statistics of the wind at 500 mb. J . Atm08. Sci. 19. 195-204. Murakami, T., and Tomatsu, K. (1985). Energy cycle in the lower atmosphere. J. Meteorol. SOC.Japan, Ser. I I 43, 73-89. Wiin-Nielsen, A. (1959). A study of energy conversion and meridional circulation for the large scale motion in the atmosphere. Monthly Weather Rev. 87, 319-332. Saltzman, B. and Fleisher, A. (1960). The modes of release of available potential energy in the atmosphere. J. Qeophys. Rea. 65, 1215-1222. Saltzmann, B. and Fleisher, A. (1961). Further statistics on the modes of release of available potential energy. J. Qeophys. Rea. 66, 2271-2273. Wiin-Nielsen, A., Brown, J. A., and Drake, M. (1983). On atmospheric energy conversions between the zonal flow and the eddies. Tellus 15, 281-279. Lahey, J. F., Bryson, R. A., Wahl, E. W.,Horn, L. H., and Henderson, V. D. (1958). “Atlas of 500 mb Wind Characteristics for the Northern Hemisphere.” Univ. of Wisconsin Press, Madison, Wisconsin. Smagorinsky, J. (1960). A synopsis of research on quasi-stationary perturbations of the mean zonal circulation caused by topography and heating. In “Dynamics of Climate” (€4. L. Pfeffer, ed.), pp. 44-49. Pergamon Press, Oxford.

48. Fowlis, W. W., and Hide,

47. 48. 49. 50.

51. 52. 63. 54. 55.

68.

AUTHOR INDEX Numbers in parent,heses are reference numbers and indicate that an author’s work is referred to although his name is not cited in the text. Numbers in italic indicate the page on which the complete reference is listed. Abelson, P. H., 310 (15), 317, 327, 328 (15). 330, 331 Adams, L. H., 98 (35), 101 (35), 102 (35), 109 (42), 205 Akhiezer, N. I., 391, 435 Alsop, L. E., 186, 188, 210, 211 Alterman, Z., 188 (180), 211 Anderson, D. L., 122, 164 (128, 129), 176. 207,209, 210 Angenheister, G., 175, 209 Archambeau, C. B., 176, 210 Amett, A. B., Jr.,298, 299 (80),301 (95), 307, 308 Arnold, J., 276 (46), 277 (46), 278 (46), 305 Arroyo, A. L., 152, 208 Atlas, D., 268 ( l ) ,266, 269 (l),272, 273, 275, 277, 279, 280, 294 (l),296 (1), 298, 303 Aufm Kampe, H. J., 258 (2), 299 (2), 304 Austin, J. M., 269, 305 Austin, P. M., 273 (40), 305

Bagnold, R. A., 223, 253 Baker, R. G., 147, 208 Barnett, T. P., 237, 254 Barth, C. A., 316, 331 Bates, F. C., 270, 272, 279, 280, 298 (83), 305, 307 BBth, M.,110, 111 (48), 112 (49), 118, 142, 152, 154, 163, 164, 181, 182, 205, 206, 208,210 Battan, L. J.. 259 ( 8 ) , 260, 281, 304, 306 Beebe, R. G., 272, 305 Benton, G. S., 422, 436 Berkner, L. V . , 310, 314, 315, 320, 330, 331 Bernal, J. D., 82, 203 Bigler, S . G., 291 (63), 306 Birch, F., 82, 119 (75), 190, 203, 206, 211 Bjerknes, J., 264, 304

Blackmer, R. H., 297 (74), 307 Blanchard, D. C., 224 (1l), 247 ( 1I ) , 253 Bliss, G. A,, 395, 435 Bohan, W., 416 (42). 418 (42), 420 (42), 421 (42), 422 (42). 424 (42), 435 Bontchkovskii, V. F., 67 (23), 68, 77 Booker, D. R., 258 ( l ) , 266 ( l ) , 269 (1). 272 ( l ) ,273 ( l ) , 275 ( l ) ,277 ( l ) ,279 (1). 280 ( l ) , 294 (1). 296 ( l ) , 298 ( l ) , 303 Bouasse, H., 93, 204 Boucher, R. J., 273 (41), 293, 294, 305 Boussinesq, J., 28, 76 Bradbury, D. L., 368 (22), 434 Braham, R. R., Jr., 259 (5),266, 270 (5), 277 (5), 287 (5), 293 (5), 297, 304 Brookamp, B., 98, 204 Bromwich, T . J. A., 114, 206 Brooks, E. M., 298 (85), 300, 307 Brooks, H. B., 287, 306 Brouet, J., 49 (14), 62 (14), 72 (14), 76 Brown, H., 309, 330 Brown, J. A., 425 (54), 436 Browning, K. A., 279, 280 (50),281 (50), 282, 284, 285, 286, 287 (50, 52), 288, 305, 306 Bruhat, G., 130 (91), 207 Brune, J. N., 123, 164, 176, 207, 210 Brunk, I. W., 294 (66), 298, 306, 307 Bryson, R. A,, 426 (55), 436 Buchbinder Goetz, G . R., 123 ( 8 8 ) , 207 Bullard, E. C., 81, 203 Bullen, K. E., 82, 203 Burling, R. W., 248, 255 Byerly, P., 167, 209 Byers, H. R., 258 (l),259 (6, 6), 260, 261, 266 ( l ) ,269 ( l ) ,270 (5), 272 (1). 273 ( l ) , 275 (l),277 (1, 5), 279 (l), 280 ( l ) , 281, 287 (5), 293 (5), 294 (l), 296 (11, 297, 298 ( l ) , 303, 304, 306

Cain, D. L., 324 (35), 331

437

438

AUTHOR INDEX

Caloi, P., 87 (15), 88 (16, IS), 89 (17, 18), 91, 92 (21), 95 (17), 99, 107 (41), 108 (41,41a), 109(46),110 (45), 113,119, 130 (92), 131 (93), 136 (94, 94a), 138, 140 (94), 144 (106, loo), 145 (105), 147 (105), 148 (110), 163 (115), 163, 176, 177 (41a), 180 (92), 183 (174), 204,205, 206, 207,208,210 Caputo, M., 40, 42 ( l l b ) , 76 Carstens, T. J., 225, 254 Carte, A. E., 287 (60), 306 Changnon, 6. A., Jr., 300, 307 Charney, J. G., 410, 435 Charnock, H., 240, 254 Chmela, A. C . , 296 (70), 297 (70).306 Clark, R. A., 264 (14), 304 Clarkson, H. N., 52 (16). 76 Cloud, P. E., Jr., 320, 331 Cook, K . L., 181 (lee), 210 Cox, C. S . , 233 (21), 254 Crarner, H. E . , 268, 270, 304

Daly, R. A., 165, 209 Davis, P. A., 368 (23), 371, 435 Deacon, E. L., 222 (7), 236, 253, 254 Decae, A., 76, 77 Defant, A., 242, 250, 254 De Noyer, J.. 176. 210 De Panfilis, M . , 89 (18). 91 (le), 204 Di Filippo, D., 89 (18). 91 (18), 92, 179, 204 Donaldson, R. J., Jr., 281, 282, 286, 287, 296 (70, 71). 297 (70), 298 (71), 306, 307 Dorrnan. J., 122 (86), 123, 164, 207 Dorrestein, R..237, 254 Douglas, R. H., 258 ( l ) , 266 ( l ) , 269 ( l ) , 272 ( l ) , 273 ( I ) , 275 (1). 277 (1). 279 (l), 280 ( l ) , 294 ( l ) , 296 (1, 69). 297 (69), 298 ( l ) , 303, 306 Drake, F. D., 324 (35). 331 Drake, M., 425 (54), 436 Drazin, P. G., 410, 435

Eckart, C., 411, 435 Eddy, A., 373, 435 Eshlernen, V . R., 324 (35. 37), 331 Ewing,M., 101, 110, 114 (55, 56), 115, 118 (64, 71), 119 (74, 76), 120 (74), 121, 122,

162, 164, 172, 176, 186 (176), 205, 206, 208,209, 210,211

Faller, A. J., 244, 255 Fenkhauser, J. C., 266, 288, 289, 290, 291, 292, 300, 304, 306 Faul, H.. 326 (40), 331 Fawbush, E. J., 270, 272, 297 (78), 298, 305, 307 Federico, B., 168 (142, 143), 169, 170, 179 (143), 209 Ferris, H. G., 411, 435 Fitzgerald, D. R., 299 (go), 301 (go), 307 Fjeldbo, G., 324 (36), 331 Fjeldbo. W. C . , 324 (37). 331 Fjertoft, R., 337, 396, 434, 435 Fleagle, R. 0.. 236, 254 Fleisher, A., 422, 426, 436 Fortsch, O., 96 (24), 96 (24), 98 (24), 204 Foster, D. S . , 297 (79). 307 Fowlis, W. W., 418, 423, 424, 436 Frenkel, J. I., 166, 209 Frisby, E. M.,297 (76), 307 Fuchs, K., 96,97 (29,30), 98 (30), 204 Fujita, T., 258 (1). 259 (9), 261, 266 (I), 269 (l), 272 ( l ) , 273 ( l ) , 275 (l), 276, 277 (1. 46), 278, 279 ( l ) , 280 (l), 294 (67), 295. 296 (1, 67), 298 (1, 86). 303, 304, 305, 306, 307 Fulks, J. R., 270, 271, 299, 305, 307, Fultz, D., 416, 418,419, 420,421, 422, 424, 435

Galperin, E. I., 98, 99 (32), 100 (32), 205 Garnburcev, G. A., 99 (31), 205 Cane, P. C., 98 (39). 102 (39), 205 Garing, J. S . , 313 (23a), 361 Garrett, W. D., 242 (38), 254 Gast, P. W., 86, 204 Gates, W. L., 409. 435 Gilbert, D. L., 310 (14), 330 Gilbert, F., 137, 207 Girlanda, A., 168 (143), 169, 170, 179, 209 Goldich, S. S . , 310, 330 Grant, L. O., 296 (73). 307 Griggs, D. E., 81, 194 (189), 203, 211 Groen, P., 237, 254 Guidroz, R. R., 147, 208

AUTHOR INDEX

Gutenberg, B., 91, 92, 108, 109 (43, 44, 46), 110, 112 (44), 113 (50), 120, 143, 147, 163, 175, 183, 190, 204, 205, 207, 208, 209, 210

Haines, D. A., 385 (29), 386 (29), 387 (29), 435 Hales, A. L., 98 (39), 102 (39), 205 Hamilton, G. R., 101 (33), 205 Hardy, G. H., 345 (15), 394, 434 Harrison, H. T., 277, 305 Harrison, J. C., 188 (179), 211 Hart, P. J., 102 (36), 205 Heezen, B. C . , 118 (70). 119 (70), 206 Henderson, V. D., 426 (55), 436 Herglotz, G., 7, 76 Herlofson, N., 264 (13), 304 Hersey, F. B., 101 (33), 205 Hide, R.,416, 418, 421, 423, 424, 435, 436 Hiser, H. W., 293, 306 Hitschfeld, W., 275 (45). 277 (45), 285, 296 (69), 297 (69), 305, 306 Hochstrasser, U., 115 (66), 206 Hodgson, J. H., 98, 102, 205 Hoecker, W. H., Jr., 290, 298, 299, 306, 307 Hoering. T. C., 317, 331 Holland. H. D., 310 (9), 314, 330, 331 Holopainen, E. O . , 367, 370, 434 Horn, L. H., 426 (55), 436 Houghton, H. G . , 268, 270, 304 House, D. C., 258 ( l ) , 266 ( l ) , 269 (1). 272 ( l ) , 273 (l),275 ( l ) , 277 ( l ) , 279 (1). 280 ( l ) , 294 (l),296 (l), 298 (l),303 Howard, J. N.,313 (23a), 331 Huff, F. A., 300, 307 Hutchinson, G. E., 310, 323, 330

Inman, R. L., 291 (63). 306 Inn, E. C. Y., 311 (21), 312 (21), 331

James, R. W., 264 (13), 304 Jardetzky, W. S., 115 (64). 118 (64), 206 Jarosch, H., 188 (180), 211 Jeffreys, H., 23, 76, 91, 167, 209, 233, 254 Jobort, G., 74, 77 Johnson, F. S., 324, 331

439

Kahn, A. B., 422, 436 Kamitsuki, A., 173 (149), 209 Kanai, K., 103, 104, 122, 154, 165, 156, 157, 158, 159, 160, 161, 162, 205, 207, 208 Katz, S., 287 (59), 306 Kaylor, R., 244, 255, 416 (42), 418 (42), 420 (42), 421 (42), 422 (42), 424 (42), 435 Kelvin, Lord, 233, 254 Kessler, E., 111, 284, 301, 306, 308 Khorosheva, V. V., 147, 166, 208 Kinsman, B., 254, 228, 239, 254 Kishimoto, Y., 153, 173 (149), 208, 209 Kitaigorodsky, S. A., 219, 247, 248, 253, 255 Kizawa, T., 175, 210 Kliore, A., 324, 331 Knighting, E., 264 (13), 304 Knopoff, L., 176, 190, 191 (183), 210, 211 Kobayashi, N., 164 (126), 208 Kohno, Y . , 191, 211 Kosminskaya, I. P., 98, 99 (32), 100 (32), 205 Kovach, R. L., 122, 164 (129), 176, 207, 209,210 Kraus, E. B., 219, 253 Krueger, A. F., 385, 386, 387, 435 Kuettner, J., 297, 307 Kung, E. C., 367, 370, 373, 387, 425, 434 Kuo, H. L., 339, 434

La Coste, L. J. B., 52 (16), 76 Lahey, J. F., 426, 436 Lamb, H., 116, 181, 184, 206, 210, 418, 436 Landisman, M.,164, 208, 209 Laster, S. J., 137, 207 Lee, J. T., 301 (96), 308 Lehmann, I., 149, 168, 208. 209 Leontiev, G . Y . , 67 (22). 77 Le Pichon, X . , 118 (71), 206 Lepp, H., 310, 330 Le Roux, J. J., 287 (60), 306 Lettau, H., 68, 77, 340, 366, 371, 372, 434 Levine, J., 267, 304 Levy, G. S . , 324 (35), 331 Ligda, M. G . H., 287, 306 Lighthill, M. J., 233, 254

440

AUTHOR INDEX

Lin, C. C., 230 (16), 254 List, R. J., 366 (19), 434 Littlewood, J. E., 346 (M), 394 (16), 434 Long, R. R., 410 (42), 418 (42), 420 (42), 421 (42), 422 (42), 424 (42), 435 Longuet-Higgins, M. S., 224 (lo), 226,230, 253,254 Lorenz, E. N., 334, 338. 339, 362, 306, 406, 409, 411, 412, 434, 435 Love, A. E.H., 7. 76, 181, 210 Lubimova, H. A., 166 (132), 209 Ludlam, F. H., 268 ( l ) , 266 ( l ) , 269 ( l ) , 272 ( l ) , 273 ( l ) , 276 ( l ) , 277 ( l ) , 279 (1). 280 (1, 60), 281 (60), 282, 284, 285, 287 (60), 288, 294 ( l ) , 296 ( l ) , 298 ( l ) , 303, 304, 305 Luosto, U., 121, 207

MacDonald, G. J. F., 86, 204, 327, 331 MacQregor, A. M., 310, 330 Magnitsky, V. A., 147, 166, 208 Major, M., 136 (95), 137 (95), 207 Malkus, J. S . , 268 ( l ) , 266 ( l ) , 267, 269 ( l ) , 272 ( l ) , 273 ( l ) , 276 ( l ) , 277 ( l ) , 279 ( l ) , 280 ( l ) , 294 (l), 296 ( l ) , 298 (1). 303, 304, 305 Marcelli, L., 89 (l8), 91 (18), 113 (52, 63, 64), 119 (62, 63, 64), 204, 205, 206 Margules, M., 334, 336, 434 Marshall, L. C.. 310, 314, 316, 330, 331 Martin, H., 101 (34), 205 Mayhew, W. A., 287, 306 Melchior, P., 23 (6,28), 24, 43 (6),49 (12, 14), 64 (18), 57 (18, 19), 00 (6),62 (14), 60 (21). 72 (14), 76, 77, 187 (177). 197 (191), 203, 211 Menzel, H., 97 (30), 98 (30), 204 Meyer, R. P., 94, 204 Miki, H., 82, 84, 86, 203 Miles, J. W., 233, 239, 254 Miller, R. C . , 270 (34), 272, 297 (78), 298 (34), 305, 307 Molodensky, M. S . , 23, 76 Monahan, E. C . , 223, 253 Morton, B. R., 268, 304 Mossop, 8. C . , 287 (60), 306 Miiller, St., 96 (29), 97 (29), 98 (29), 204 Munk, W . H.,233 (21), 254 Murakami, T., 426, 426, 436

Nakagawa, I., 187 (177), 211 Nawrocki, P. J., 311, 331 Ness, N. F . , 188, 211 Neumann, G., 236 (28), 237, 248, 254, 255 Newton, C . W., 268 (l), 260 ( l ) , 209 ( l ) , 272 (I), 273 (l),274, 276 (1, 44), 277 ( l ) , 279 ( l ) , 280 ( l ) , 287 (44, 59), 288, 289, 290, 291, 292, 294 ( l ) , 296 (l),298 ( l ) , 300, 301 (96), 303, 305, 306, 308 Newton, H. R., 274, 276 (44). 287 (44), 305 Nishimura, E., 173 (149), 209 Nishimura, G . , 32, 76 Nishitake, T., 86, 204 Normand, C. W. B., 262 (12), 206, 279, 304 Nowroosi, A. A., 188 (178), 211 Nurmia, M., 164 (117), 208

Oddone, E., 182, 210 Ogura, Y., 268 (11, 266 ( l ) , 269 ( l ) , 272 (1). 273 ( l ) ,276 ( l ) , 277 ( l ) , 279 ( l ) , 280 ( l ) , 294 ( l ) , 296 ( l ) , 298 ( l ) , 303 Oliver, J., 122, 136 (96), 137, 162, 207, 208 Oort, A. H., 340, 341. 388, 409, 434 Orendorff, W. K., 277, 305 Owen, G. V . , 416 (42), 418 (42), 420 (42), 421 (42), 422 (42). 424 (42), 435

PelmBn, E., 341, 308, 309, 373, 434, 435 Pannocchia, G., 113 (62, 63), 119 (63. 64). 205, 206 Papa, R., 311, 331 Piquet, P., 67 (19). 77 Peano, G., 203, 211 Pedersen, K.. 368 (22), 434 Pedgley, D. E., 294, 296, 306 Pekeris, C. L., 116, 188, 206, 211 Peronaci, F., 91, 92, 136 (94, 94a), 140 (94), 179, 204, 207 Peterschmitt, E., 96, 96 (29), 97 (29, 30), 98 (29), 204 Petterssen, S., 264, 273, 304. 305, 368, 434 Phillips, 0. M., 233, 248, 254, 255 Pierson, W. T., 236 (28), 254 Pblya, G., 346 (M), 394 (16), 434 Porkka, M. T., 164 (117), 208

AUTHOR INDEX

Press, F., 101 (33). 110, 114 (55, 56), 115 (64), 118 (64), 119 (76), 120 (74), 121, 154, 164 (126), 172, 176, 177, 183, 205, 206, 208, 209, 210 Priestley, C. H. B., 220, 253

Rabinowitch, E.I., 321, 331 Rankama, K., 324, 325 (38, 40), 331 Rayleigh, Lord, 138 (101), 139 (lo]), 207, 309, 310, 330 Refsdal, A., 262 ( l l ) , 304 Reich, H., 95, 96, 98, 204 Reynolds, S. E., 259 (7), 304 Rhyne, R. H., Jr., 301, 302, 308 Richter, C. F., 113 (50), 175 (153), 205, 21 0 Riehl, H., 368, 435 Ringwood, A. E., 86, 204 Roll, H. U . , 219 (2), 236, 253, 254 Rothd, J. P., 95, 96 (29), 97 (29), 98 (29), I1 3, 204, 205 Roys, G. P., 300 (94), 308 Rubey, W. W., 309, 310 (5, 6), 314, 323, 330 Rutten, M. G., 310, 330

Sagan, C., 317, 331 Sahama, T. G., 324, 325 (38), 331 Saltzman, B., 370, 422, 425, 435, 436 Slto, Y . , 164, 208 Saunders, P. M., 267, 304 Schleusener, R. A., 258 ( l ) , 266 (l), 269 ( l ) , 296 (73). 272 (l), 273 (1). 275 ( l ) , 277 ( l ) , 279 ( I ) , 280 ( l ) ,294 (l),296 ( l ) , 298 (l),303, 307 Schmidt, F. H., 264, 304 Schulze, G. A,, 95 (24), 96 (24), 98 (24), 204 Schulze, R., 51 (15), 76 Scorer, R. S . , 265, 266, 267, 304 Serrin, J., 413, 414, 415, 435 Sezawa, K., 103, 104, 114, 122, 154, 155, 156, 157, 158, 159, 160, 161, 162, 205, 206, 207, 208 Shackford, C. R., 297, 307 Sheppard, P. A., 222 (7), 253 Shima, M., 82, 203 Shimazu, Y., 85, 86, 191, 203, 204, 211

441

Shimozuru, D., 86, 204 Showalter, A. K., 270, 305 Slichter, L. B., 40, 76, 188 (179), 211 Smagorinsky, J., 427, 436 Somigliana, C., 123, 207 Spadea, M. C., 89 (18), 91 (18), 204 Spar, J., 373, 381, 435 Squires, P., 268, 269, 270, 305 Starr, V. P., 339, 434 Starrett, L. G., 270 (34), 298 (34), 305 Stein, A., 96 (29), 97 (29), 98 (29), 204 Steiner, R., 301, 302, 308 Steinhart, J. S . , 94, 204 Steinhauser, F., 33 (9), 76 Stewart, R. W., 224, 226, 237, 241, 253 Stommel, H., 265, 304 Stoneley, R., 114, 115 (65, 66), 171, 206, 209 Stout, G. E., 293,297 (74), 306, 307 Strelakov, S. S . , 248, 255 Strobach, K., 96 (29). 97 (29), 98 (29), 204 Stuart, J. T., 243, 245, 255 Suckstorff, G. A., 259, 304 Suess, H. E., 309, 315, 330, 331 Sutton, G . H., 122 (88), 186 (176), 207, 21 1 Sverdrup, H. U., 241 (36), 254 Swinbank, W. C . , 221, 253 Sykes, L., 164, 209

Takeda, A., 239, 254 Takeuchi, H., 164, 187 (177). 208, 211 Talwani, M., 118, 119, 206 Tatel, H. E., 98, 101, 102 (36), 205 Taylor, G. I., 268 (28), 304 Tepper, M., 277 (49), 297 (49), 305 Teraxawa, K., 30, 76 Thompson, P. D., 355, 358, 434 Toksoz, M. N., 164 (128), 209 Tolstoy, I., 115 (63), 206 Tomatsu, K., 425,426, 436 Tomczak, G., 219, 253 Tryggvason, E., 121, 207 Tulina, Yu. V . , 99 (31), 205 Turner, J. S . , 267, 268 (28), 269, 270, 304, 305 Tuve, M. A., 98 (35), 101 (35), 102 (35), 206

442

AUTHOR INDEX

Urey, H. C., 310, 313 (16). J30 Usami, T., 115, 116, 117, 206

Vallo, P. E., 166, 209 Valovcin, F. R., 299 (go), 301 (go), 307 Van Mieghem, J., 334, 344, 396, 410, 411, 413, 434, 435 Van Thullenar, C. F., 301, 308 Veitsman, P. S., 99 (31), 205 Verbaandert, J.,43, 46, 49 (12), 76 Vesanen, E., 154, 208 Vicente, R. O., 23, 76 Vigroux, E., 311, 312, 331 Vinogradov, A. P., 309 (3), 330 Volkov, Yu. A., 219 (3). 253 Volterra, V., 192 (186), 193 (ME), 196, 196, 198 (192), 211 Vonnegut, B., 268 ( l ) , 266 (l), 269 ( l ) , 272 (l),273 (1). 276 (l),277 (1). 279 ( l ) , 280 ( l ) , 294 (1). 296 ( l ) , 298, (1) 303

Weickmann, H. K., 268 (2, 3), 274, 297, 299 (2). 304, 305 Weil, J., 416 (42), 418 (42), 420 (42), 421 (42), 422 (42). 424 (42), 435 Welander, P., 243, 254 Wenz, G. M., 224, 253 Wexler, R., 273 (41). 293, 294, 305 White, R. M., 370, 435 Wiin-Nielsen, A , , 426, 436 Wildt, Et., 317, 326. 332 Wilk, I<. E., 296 (72), 297 (74). 301 (96), 307, 308

Wilkerson, J.. 237, 254 Williams, D. T., 294 (68). 298, 306, 307 Williams, R. T., 268 ( l ) , 266 ( l ) , 269, 272 ( l ) , 273 ( l ) , 276 ( l ) , 277 ( l ) , 279 ( l ) , 280 ( l ) , 294 ( l ) , 296 ( l ) , 298 ( l ) ,303 Willmore, P. L., 96, 98, 102, 204, 205 Wilson, J. T., 138, 207 Wilson, 0. C., 311, 331 Winston, J. S., 386 (29), 386 (29), 387 (29), 435

Wahl, E. W., 426 (65), 436 Ward, N. B., 286, 298, 299 (EO), 306,

Workman, E. J., 269 (7), 304 Worzel, J. L., 101 (33), 118 (70), 119 (70), 205, 206

30 7

Watanabe, K., 311, 312, 331 Webb, E. K., 222 (7), 236, 253, 254 Weichert, E., 98, 204

Zadro, M., 74. 77 Zelikoff, M., 311 (21), 312 (21), 331

SUBJECT INDEX PL waves and, 137 principal data, 91, 98 Rg waves and, 110 Somigliana waves and, 123-142 surface-wave study, 112-142 thickness of, by Rayleigh waves, 113122 viscosity of, 167-158 Crust deformation, 1-77 atmospheric effects, 67-74 deformation tensor components, &lo deviations of vertical, 11-13 direct observations, 60-62 distribution of, 25-26 of elastic semi-infinite body, 28-36 elastic-sphere deformation, 36-43 elastic type, 7-8 by external potential, 6-14 hydrological effects, 62-67 Love numbers, 7-8 luni-solar potential form, 14-23 measurement units, 4 measuring instruments, 43-60 particle-accelerator stability and, 74-76 from semidiurnal sectional forces, 26 Terazawa’s problem and, 30-36 types, 2

A Aircraft, storm hazards to, 300-301 Asthenosphere, Pa,Sa waves and, 142-167 Atmosphere( s), earth’s 309-329 planetary, theory of origin, 329-330 Atmospheric energetics and available potential energy, 333-436

B Body waves, crust study by, 87-112 C Clinometers, 43-50 Convective storms, 257-308, 8ee also Thunderstorms aircraft hazards of, 300-301 hail in, 296-297 heavy rains in, 300 lightning in, 297 modes of convection, 262-269 “bubble” type, 266-268 continuous draft type, 268-269 modified concepts of, 264-266 parcel type, 262-264 movement of, 287-291 propagation effects, 287 single-celled, 287 size discrimination and water budget, 288-291 organized convective circulations of, 277 size sorting. 284-287 squall lines in, 291-296 squall lines in, 291-296 “steady state” severe type, 281-283 storm-environment interaction, 274-277 surface winds of, 297-298 tornadoes, 298-300 Crust, 87-142 asthenosphere and, 142-167 body waves study, 87-112 C,,, waves and, 137 explosion study of, 93-108 higher mode surface wave study, 122-123 Lg waves and, 110-112 Li waves and, 112 Pg waves and, 108-1 10

E Earth, free oscillations of, 181-189 internal movements of, 189-203 Volterra’s theory, 192-203 Earth’s atmosphere, energy budget of, 387-380 oxygen rise in, 309-329 Earth tides, analysis and prediction of, 23-26 Explosions, crust study by, 93-108

G Gravimeters, 60-64. 68-60 calibration, 64-67 data interpretation, 64-67

H Hail, 296-297

443

444

SUBJECT INDEX

L Lightning, 297 Love numbers, 7-8

M Mantle, upper, aee Upper mantle Mars, atmospheric composition and surface of, 324-328

life on, 328-329 Milligal, definition, 4

latent heat and, 368 least action principle and, 429 to perturbations, 410-411 role of, 424-428 solar radiation and, 366 of static instabilities, 368-361 theory of, 341-373 Lorenz's approach, 344-346 variational methods of, 389-398 variable static stability and, 409-41 1 zonal generation and, 369

0

R

Oxygen rise in earth's atmosphere, 309-

Rain, in convective storms, 300 Rayleigh waves, crustal thickness from,

329

first critical level, 319-320 paleozoic era and, 320 in late paleozoic and ensuing eras, 321-

113-122

S

323

in primitive times, 310-316 ecology of, 317-319 surface oxidation in, 3 1 6 3 1 7 second critical level, 321

P Pa& waves, asthenosphere and, 42-167 Particle accelerator, stability and crust deformation, 74-76 Pendulums calibration, 67 horizontal, 43-47, 68 installation, 48-60 Verbaandert-Melchior type, 47-48 Planets, atmospheric origin on, 329 Potential cnergy, available, atmospheric energetics and, 333-436 amount of, 338-339, 361-364, 374-378 average and transient components, 380381

concept of, 334-336 contributions to, 398-411 barotropic atmospheres and, 401-406 from hydrostatic defects, 398-400 dynamics of. 412-429 eddy generation and, 369-370 efficiency factor, 363-366 frictional dissipation and, 367, 370-373 general circulation theories and, 339-341 generation and destruction of, 361-370 infrared radiation and, 366-367

Sea-surface wind stress, 213-266 atmospheric boundary layer, 216-224 effect of density variation, 219 kinetic energy of, 213-216 motion within interface, 230-232 role of bubbles and sprays, 222-224 slicks and, 242-246 surface waves on deep water, 226-230 waves, effect on wind profiles, 237-242 at interface. 224-226 wind-wave generation, 232-236 Shadow zone channeling zone and, 163-187 explanation of, 169-163 Somigliana's theory of seismic wave propagation, 123-129 Squalls, 291-296, 8ee aleo Convective storms Storms, convective, aee Convective storms

T Terazawa's problem, 30-36 Thunderstorms, aee alao Convective storms environment of and its modification, 270-273

in sheared environment, 273-274 structure, of, 269-262 Tides, crust deformations and, 20 Tornadoes, 298-300 20" discontinuity, 167-176

445

SUBJECT INDEX

V

U Upper mantle, 79-211, see

Crust attenuation and mixed zones of, 175 composition, 80 earth’s internal movements and, 189-203 free oscillation of earth and, 181-189 problems of, 80 20’ discontinuity of, 167-175 viscosity of, 154-159 a180

Verbaandert-Melchiorpendulum, 47-48 Volterra’s theory of earth’s internal movements, 192-103

W Wind stress along sea surface, ~ e eSeasurface wind stress

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