Advances In Geophysics, Volume 21

ADVANCES IN

GEOPHYSICS

VOLUME 21

Contributors to This Volume KENNETHP. FREEMAN D. G U B B I N S KUO-NANLIOU T. G . MASTERS RONALDB. SMITH K I I C H ITOMATSU DONALDL . TURCOTTE

Advances in

GEOPHYSICS Edited by

BARRY SALTZMAN Department of Geology and Geophysics Yale University New Haven, Connecticut

VOLUME 21

1979

Academic Press A Subsidiary of Harcourt Brace Jovanovich, Publishers

New York

London Toronto

Sydney San Francisco

COPYRIGHT 0 1979, BY ACADEMIC PRESS,INC. ALL RIGHTS RESERVED. NO PART OF THIS PUBLICATION MAY BE REPRODUCED OR TRANSMITTED IN ANY FORM OR BY ANY MEANS, ELECTRONIC OR MECHANICAL, INCLUDING PHOTOCOPY, RECORDING, OR ANY INFORMATION STORAGE AND RETRIEVAL SYSTEM, WITHOUT PERMISSION IN WRITING FROM THE PUBLISHER.

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

United Kingdoni Editiorr priblished by ACADEMIC PRESS, INC. ( L O N D O N ) LTD. 24/28 Oval Road, London N W l 7 D X

LIBRARY OF CONGRESS CATALOG CARD NUMBER:52-12266 ISBN 0-12-018821-X PRINTED IN THE UNITED STATES OF AMERICA 79 80 81 82

9 8 7 6 5 4 3 2 1

CONTENTS LISTOF CONTRIBUTORS . . . ... .. . . . . . . . . . . . .. . . .. ... .. . .. . . .. ... .. . ..

vii

Driving Mechanisms for the Earth’s Dynamo

D. GUBBINSA N D T. G . MASTERS

....................... I . Introduction. . . . . . . . . . . . . 2. Physical and Chemical P 3. Conservation Laws . . . . ........... 4 . Power Sources for the Magnetic 5. Implications of the Dynamics and Chemistry of the C o r e . . . . . . . . . . . . . ............. ........................ 6. Summary . . . . . . ............................ List of Symbols . . . . . . . . . . . . . . . References . . . . . . ..........................................

i

3 16

21 36 41 43 44

Flexure

DONALDL. TURCOTTE

.................. Introduction. . . . . . . . . . . . . . . . . . . . . . Governing Equations.. . . . Lithospheric Buckling . . . . . . . . . . . . . . . . . . . . . . . . . Flexure with End Loading . . ............................... 5 . Compensation of Loads . . . . . . . . . . . ...................... 6. Axisymmetric Bending . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 . Viscoelastic Flexure . . . . . 8. Stress Relaxation . .......................................... ......................... 10. Conclusions . .......................................... List of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .................. References . . . . . . . . . . . . . . . . . . . I. 2. 3. 4.

51 52

54 55 62 66

70 73 75 82 83 84

The Influence of Mountains on the Atmosphere

RONALDB . SMITH 1 . An Introduction to Mountain Effects.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. The Flow over Hills and the Generation of Mountain Waves . . . . . . . . . 3. The Flow near Mesoscale and Synoptic-Scale Mountains . . . . . . . . . . . . 4. Orographic Control of Precipitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. Planetary-Scale Mountain Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

87 89 142

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

217

V

169 195

vi

CONTENTS

Climatic Effects of Cirrus Clouds

KENNETH P. FREEMAN A N D KUO-NAN LIOU 1. Introduction ..................................................... 2. Review of Previous Work ........................................ 3. Radiative Transfer in the Atmosphere.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Heat and Radiation Budgets of the Atmosphere . . . . . . . . . . . . . . . . . . . . . 5. Effects of Increased Cirrus Cloudiness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

23 I 23 3 237 25 I 272 283 284

Spectral Energetics of the Troposphere and Lower Stratosphere

KIICHITOMATSU I. 2. 3. 4. 5. 6. 7.

.... Introduction.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Governing Equations.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ............ Data Sources and Procedure of Analysis . . . . . . Zonal Mean Climatology of Basic Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . Spatial Energetics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Spectral Energetics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Potential Energy Generation and Kinetic Energy Dissipation Calculated

.......................... Appendix. Tables of Basic Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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

289 292 304 305 313

337 356 362 367 400 40 1

SUBJECT INDEX. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407

LIST OF CONTRIBUTORS Numbers in parentheses indicate the pages on which the authors' contributions begin.

KENNETHP. FREEMAN," Department of Meteorology, University of Utah, Salt Lake City, Utah 84/12 (231) D. GUBBINS, Department o j Geodesy and Geophysics, Cambridge University, Madingley Rise, Cambridge CB3 O E Z , England ( I )

KUO-NANLrou, Department of Meteorology, University of Utah, Salt Lake City, Utah 84/12 (231) T. G. MASTERS, Institute of Geophysics and Planetary Physics, University of California at San Diego, La Jolla, California 92093 (1)

RONALDB . SMITH,Department of Geology and Geophysics, Yale University, N e w Haven, Connecticut 06520 (87) KIICHITOMATSU,Forecast Research Laboratory, The Meteorological Research Institute, Tokyo, Japan (289) DONALDL. TURCOTTE, Department of Geological Sciences, Cornell University, Ithaca, N e w York 14853 (51)

* Present address: Air Force Global Weather Center, Offutt AFB. Nebraska 681 13. vii

This Page Intentionally Left Blank

t

DRIVING MECHANISMS FOR THE EARTH’S DYNAMO

AND

T . G . MASTERS

I . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 . Physical and Chemical Properties of the Earth’s Core . . . . . . . . . . . . . . . . . . 2.1 Evidence from Seismology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Shock-Wave Data and Core Composition . . . . . . . . . . . . . . . . . . . . . . 2.3 The Thermal Regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 . Conservation Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Local Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Global Energy Balance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Global Entropy Balance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Relating the Magnetic Field Strength to the Power Supply . . . . . . . . . . . . . . 4 . Power Sources for the Magnetic Field . . . . . . . . . . . . . . . . . . . . . . . . . 4 . I Radioactive Heating . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Primordial Heat and Freezing of the Inner Core . . . . . . . . . . . . . . . . . . . 4.3 Gravitational Energy Release . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Chemical Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Rotational Energy: the Precessional and Tidal Forcing . . . . . . . . . . . . . . . . 4.6 Other Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 . Implications of the Dynamics and Chemistry of the Core . . . . . . . . . . . . . . . . . 5.1 Results from Dynamo Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Freezing of the Liquid in the Outer Core . . . . . . . . . . . . . . . . . . . . . . 5.3 Boundary Layers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Effects of Stable Density Stratification . . . . . . . . . . . . . . . . . . . . . . . 6. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . List of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

i

3 3 7

12 16 16 18 20 21 21 21 23 24 29 31 35 36 37 38 40 40 41 43 44

1 . INTRODUCTION

Recent work on the origin of the Earth’s magnetic field has focused on developing an understanding of the dynamo problem . Much progress has been made toward answering the question of how electrically conducting I Copyright 0 1979 by Academic Press. Inc . All rights of reproduction in any form reserved . ISBN 0-12-018821-X

2

D. GUBBINS AND T. G. MASTERS

fluids generate and sustain magnetic fields (for a review, see Moffatt, 19781, mainly in the case when the fluid is stirred by thermal convection, because this is the simplest model (for a review, see Busse, 1978). An equally important problem for the geophysicist is to elucidate the mechanism by which the Earth's liquid core is stirred. Fluid motions might be driven by thermal convection, but a bewildering variety of other processes have been put forward in an attempt to explain the presence of the magnetic field. Whatever the mechanism, the generating process must satisfy one fundamental requirement: The energy lost by the electric currents must be replaced. If not, the field would die away in a few thousand years. Not only must we find a power source large enough to replace this lost energy, but the process for restoring magnetic energy must be efficient enough to do so without generating more heat than we believe to be issuing from the core today. The observed surface heat flux amounts to about 4. 1013W, and most of this is believed to originate from radioactivity in the mantle, with at most 25%, or 1013 W, coming from the core. Furthermore the magnetic field is known to have existed for over 3000 million years, with more or less the same strength as it has today; so the power supply must be durable. It is of no use appealing to primordial heat if the requirements of the dynamo would rapidly freeze the whole core. How much energy is required'? That clearly depends on how large and complex the magnetic field is, but it also depends on how the core is stirred. Backus (1975), Hewitt et al. (1973, and Gubbins (1977) have used the laws of thermodynamics to resolve this question, and this new approach has led to considerable new insight into the workings of dynamos. Many early and some contemporary calculations of the power requirements of a dynamo are based on a false representation of the thermodynamics. The energy lost by decay of the electric currents is what must be replaced, and for a dynamo process this is accomplished by the fluid doing work against the magnetic forces. If the convection is driven thermally, then most of the heat is carried directly to the surface by fluid flow without being converted to magnetic energy at all. Bullard and Gellman (1954) and Metchnik et af. (1975) estimate that the fraction of heat converted to magnetic fields is given by the Carnot efficiency AT/ T , but this is not correct (Backus, 1975; Hewitt et al., 1975). Much heat is conducted down the adiabatic gradient in the core and this heat is presumably not available for the dynamo. Many authors have simply added this on to the dynamo requirement for a total estimate, but we shall see (Section 4) that this is not correct either. The main conclusion of recent work (Gubbins, 1977; Loper, 1978a; Gubbins et af., 1979) is to

DRIVING MECHANISMS FOR THE EARTH’S DYNAMO

3

favor stirring of the core fluid by differentiation of its components, probably associated with formation of the inner core. This process stirs the core directly, and most of the gravitational energy must be converted to heat, probably via magnetic fields. The thermodynamic relations quantify these assertions. The attraction of the thermodynamic approach is that it yields information even when details of the system are left unspecified. Its drawback is that without a detailed model, it only yields very crude estimates of the energy requirements. Although the present authors favor the gravitationally powered dynamo, this article is a n attempt at an unbiased but critical review of the subject. We develop a sound theoretical basis from which to discuss and speculate about the various energy sources. A necessary preliminary is a discussion of relevant properties of the core, which is dealt with in Section 2. The basic equations are set out in Section 3, and in Section 4 the following possible power sources are discussed in detail: radioactive heating, primordial heat, gravitational energy release and adiabatic heating, chemical effects such as the heat of reaction between components, and precession and tidal forcing. In a short last subsection some more radical ideas are discussed. The new ideas on the driving mechanism raise some interesting dynamical and chemical questions which are discussed in Section 5 . Finally in the summary, the discussion is placed in the context of the thermal history of the Earth as a whole and its extrapolation back into the past. 2. PHYSICAL AND CHEMICAL PROPERTIES OF

THE

EARTH’S CORE

2.1. Evidence from Seismology

From the point of view of studies of the generation of the magnetic field, the important properties of the core are its homogeneity and adiabaticity, and the compositional difference between the two cores. These properties are amenable to seismological investigation. Short-period data have only been used to determine the compressional velocity in the core, but velocities alone can be used to infer whether or not a region is adiabatic and homogeneous (e.g., Birch, 1952; Bullen, 1963). In such a region, in which the pressure gradient is also hydrostatic, we have

g is insensitive to the density distribution, and there are theoretical

4

D. GUBBINS AND T. G . MASTERS

reasons to expect the bulk modulus to be a smooth function of pressure (Birch, 1952), so that the radial gradient of the seismic parameter, 4, should also be smooth. If d + / d r is not a smooth function of pressure, departures from adiabaticity and homogeneity may be inferred. The compressional velocity distribution near either boundary of the fluid outer core has been the subject of some controversy. Precursors to PKIKP and PKP have been used to infer first-order velocity discontinuities in a region 400 km thick above the inner core boundary (ICB) (e.g., Engdahl, 1968). An alternative explanation in terms of scattering due to inhomogeneities in the lower mantle appears adequate (Haddon and Cleary, 1974). Muller (1975) presents further evidence against core discontinuities by calculating the theoretical amplitudes of the reflections that they would cause, and noting that these are not observed. Interpretation of this region in terms of a complicated phase change between inner and outer cores now seems unnecessary. The velocity near the top of the outer core is difficult to determine accurately from travel time data because of the small number of rays which bottom in this region. The most useful phase for studying this region is SmKS. Choy (1977, 1978) has shown that it is difficult to obtain accurate travel times for SmKS due to phase distortion and suggests that existing travel time data for this phase will be in error. Using full wave theory, he went on to find that the velocity gradient near the top of the core could be consistent with homogeneity. Kind and Muller (1977) have made a study of SKS/SKKS amplitude ratios. They infer a very strange velocity profile for the outer core which might be caused by local solidification of the liquid (Jacobs, 1975). An independent check was made of this profile by investigating the fit of the periods of free oscillations that have high compressional energy in the outer core (Masters, 1978). Conventional models give better fits to the data. It seems safer to presume that there is some unknown factor influencing the data used by these workers than to accept their model at face value. The compressional velocity in the core has been reviewed by Adams and Engdahl (1974), and on present evidence, it seems adequate to model the whole of the outer core as a single layer without anomalous velocity gradients. The compressional velocity is known to about 2% throughout the region; and, though travel time data do admit the possibility of an abnormally low velocity just beneath the mantle (Wiggins et al., 1973), the work of Choy (1977) indicates that this is unlikely. The inner core has long been recognized as a region of high compressional velocity, and it is now generally accepted that it is solid. The main evidence for this comes from the periods of free oscillations which sample the lower core and are much better fitted by models with a solid inner

DRIVING MECHANISMS FOR THE EARTH'S DYNAMO

5

core (Dziewonski and Gilbert, 1971). It also seems likely that an "inner core mode" (i.e., a mode dominated by shear energy in this region) has been observed by Dziewonski and Gilbert (1973) which could not exist if the inner core were fluid. Improved long-period data from Project IDA (Agnew et al., 1977) are presently being analyzed in an attempt to confirm this observation. The average shear velocity in the inner core must be 3.5-3.6 km/sec to be consistent with the free oscillation periods. The compressional velocity in the inner core has been the subject of several studies. An anomalously steep velocity gradient near the JCB has been inferred in studies using geometrical ray theory (Qamar, 1973: Buchbinder, 1971). The theory is of dubious validity for rays near grazing incidence (Richards, 1973), so this feature may be suspect. Muller (1973, 1977), using full wave theory, supports the existence of this steep gradient and a 0.6 k d s e c velocity discontinuity at the ICB. Cormier and Richards ( 1977) have also investigated this discontinuity using full wave theory, but with a different computational technique. They conclude that the amplitudes of PKP are best filled by the simple model PEMA (Dziewonski et al., 1975) which has a 0.83 k d s e c velocity jump at the ICB and a normal velocity gradient at the top of the inner core. There is no obvious reason for this discrepancy. A steep velocity gradient has been interpreted as implying that the inner core is close to its melting point near the ICB because large changes in velocities are seen in ultrasonic studies of low melting point metals near their melting points (Mizutani and Kanamori, 1964). Further evidence on the compressional velocity in the inner core comes from a single observation of PKIIKP by Mass6 et al. (1974). The observation constrains the velocity in the outer part of the inner core to be 1 1 .O k 0.05 kmhec, and the result is apparently insensitive to the velocity model of the outer core. The radius of the inner core has been studied by Engdahl et al. (1974) using differential PKiKP-PcP travel times. They obtained a value in the range 1220-1230 km, the exact value depending on the V p model used, particularly near the ICB. The value obtained from the inversions of free oscillation data may be wrong because, until recently, incorrect FrCchet derivatives were used (Woodhouse, 1976). Some of the free oscillations are very sensitive to inner core radius, and new measurements of the periods of these modes, presently being undertaken, will furnish an accurate value for this parameter. The outer core radius has been the subject of many studies and is now accurately known. The best available estimate is 3485 & 3 km (see Dziewonski and Haddon, 1974, for a review). The density in the outer core has been traditionally computed using the Adams-Williamson relationship with the assumptions of adiabaticity and homogeneity. The addition of direct observational constraints in the

6

D. GUBBINS AND T. G . MASTERS

form of free oscillation periods has not changed the density models a great deal. The assumption of the Adams-Williamson relationship is compatible with the modal data (Dziewonski er d., 1975; Masters, 1979). Press (1972) uses this assumption in a Monte Carlo study of density models, and his results suggest that the density is known to about 1.5% in the outer core. The density in the inner core has been the subject of much debate, and estimates of the density jump vary from 0.2 to 0.9 gm ~ r n -in~recent on the density jump was Earth models. An upper limit of 1.8 gm given by Bolt and Qamar (1970) who used a small data set of PKiKP/PcP amplitude ratios. Some density jump is required to explain these observations. Buchbinder (1972) estimates the density jump as 0.6-0.7 gm cm-3 following the procedures developed by Bullen (1973, but this introduces many ad hoc assumptions before obtaining a unique answer. Cook (1972) used the relationship K , , = p 4 so that, to first order, jumps in bulk modulus, density, and seismic parameter are related by APIP = A 4 1 4 - AKs,clKs,c Values of the jumps in velocities reviewed above give A 4 / 4 > -0.03 with a preferred value of -0.02 (Cook used a shear velocity of the inner core that was too low). AKs,,/Ks,, is unlikely to be negative and according to Bullen (1975) should be less than 0.05. These estimates give a density jump between 0 and 1.0 gm ~ m - One ~ . recent Earth model, C2 of Anderson and Hart (1976), shows a small density jump at the ICB, but this model also shows a decrease in bulk modulus from outer to inner core. This is at variance with their conclusion that the inner and outer cores are isochemical, because experimental evidence suggests that the bulk modulus will increase on solidification unless there is a significant compositional difference between solid and liquid phases (see later). The free oscillation data are, in fact, only weakly dependent on the density jump, and until recently it was not clear that they required any density contrast between the cores. This problem has been studied by Masters (1979) who concluded that the most likely value of the density jump is 0.7 gm cm-3 with an uncertainty of 0.3-0.4 gm cmP3. The density jump is apparently significantly larger than that found in C2, but many of the sensitive modes have been omitted from the construction of that model. A further property of the core, amenable to observation, is the parameter Q. The determination of Q is usually approached using the decay of free oscillations. Such measurements are fraught with difficulties (Sailor and Dziewonski, 1978; Jobert and Roult, 1976; Deschamps, 1977; Anderson and Hart, 1978). We confine attention here to the results of studies determining Q using spectral ratios of short-period body waves. Of par-

DRIVING MECHANISMS FOR THE EARTH’S DYNAMO

7

ticular interest are studies of Q in the inner core (Sacks, 1972; Doornbos, 1974; Qamar and Eisenberg, 1974). The phases Doornbos (1974) uses to determine spectral ratios are probably sufficiently near normal incidence for full wave theory effects, such as frequency-dependent reflection coefficients, to be unimportant. He finds that Q for compressional waves, Q,,, is about 200 near the ICB, rising to about 600 at a radius of 800 km. This value is surprisingly low and indicates that Q , will be very low indeed, i.e., dissipation will be relatively large. The ratio of Q v p to Qvs for small dissipation is given by (Anderson er al., 1965)

where p* and K* are the imaginary parts of the shear and bulk moduli, respectively. In general, losses in pure compression are much lower than those in shear, i.e., K* -e p * , so Q V , = $ ( V : / V $ ) Q , , = 30

for

Q V , = 200

This is very small and reflects the size of the V s / V p ratio in the inner core. These observations are apparently at variance with the high Q nature of free oscillations that strongly sample shear energy in the inner core (Dziewonski and Gilbert, 1973). Doornbos suggests that the results may be indicative of partial melting near the top of the inner core giving a frequency-dependent Q. In the outer core, Q is infinite, for all practical purposes (Cormier and Richards, 1976). This is usually thought to be qualitatively a t variance with the possibility of a slurry in the outer core (Loper and Roberts, 1978). Gubbins (1978), however, claims that seismic waves are probably not damped at all by the type of slurry likely to be present in the outer core. The seismological constraints on the properties of the core are summarized in Fig. 1. Departures from the Adams- Williamson relationship should be very small in the inner core, and the uncertainty in inner core density is due mainly to the uncertainty in the density jump at the boundary.

2.2. Shock- Wave Data and Core Composition Shock-wave data indicate strongly that the outer core is predominantly iron, with 8- 15% light elements, and, by analogy with meteorites, some nickel (Birch, 1968; Balchan and Cowan, 1966; King and Ahrens, 1973). The main contenders for the dominant light element are sulfur, silicon, and oxygen (see Brett, 1975, and Ringwood, 1977, for reviews). The

8

D. GUBBINS AND T. G. MASTERS

RADIUS FIG. 1. The P- and S-velocities and densities for three recent Earth models (1066A, 1066B, and PEMA). The dotted lines are confidence limits (see text).

choice among these elements is strongly dependent on the model of core formation and so is highly uncertain. It is usually stated that iron has too large a density and too small a bulk sound speed to be compatible with Earth models. The first point is certainly true (Fig. 2), but the second is open to debate (Fig. 3). The bulk sound speed is difficult to estimate along the Hugoniot as this requires a knowledge of the poorly known Griineisen parameter (see later). The velocities shown in Fig. 3 (Altshuler et al., 1960, 1971) correspond to a nonadiabatic temperature distribution, and the actual temperature to which these measurements relate is also poorly known (see Birch, 1972, for a discussion). Estimates of the temperature in the shock front at 1.4 Mbar are very similar to estimates of the temperature at the top of the core, so a direct comparison of densities and velocities should be valid. The velocity of iron appears to be very similar to that of core material at the top. The temperature rises rapidly along the Hugoniot and at 3.3 Mbar (the pressure at the ICB) may be 8000-10,000 K , whereas the temperature of the core is probably only about half this. For a valid

13

c

I,

I

MC e

I

IkB

2

PRESSURE (Mb) FIG.2. Density models of the core compared with shock-wave data (-) for iron: 1066B (- - -), PEMA (-.-,-), C2 (- - - -), corridor of models in the 1066A (-), outer core (= ); MCB = mantle core boundary, ICB = inner core boundary.

PRESSURE (Mb) FIG.3. Shock-wave data recent Earth models (=). between models.

(-)

for the seismic parameter (6)for iron compared with in the inner core has been omitted a s it is rather variable

10

D. GUBBINS A N D T . G . MASTERS

comparison both the density and velocity in the shock front must be increased to correct for this temperature difference. At lower pressures, velocities are more temperature dependent than density. In fact I ( l / c ~ + ) ( d 4 / 8 P ) 1~usually ,~ lies between 2 and 3 (Tables Ia and b). The correction to the density is 3-4% (see later) so that we would expect the correction to 4 to be 6-12%. A % increase in the shock-wave value of 4 for iron brings it into agreement with 4 of core material at 3.3 Mbar. It therefore seems likely that whatever impurities there are in the outer core, they reduce the density but have little effect on the bulk sound speed. Discussion of the density at the center of the Earth, and hence inner core composition, has been dominated by Birch’s assertion that the at the pressure of the density of iron is unlikely to exceed 13 gm Earth’s center (Birch, 1961).The direct comparison of Earth models with shock-wave data suggests that the inner core could be pure iron (Fig. 2). At the central pressure the temperature correction to the shock-wave density is about 5%. This is important in any discussion of the relative composition of the inner and outer cores. Nickel may be present in the inner core and would raise the density further. It is possible to see if the density of the inner core is comparable to that of pure iron by the following argument. The density gradient of iron under the pressure and temperature conditions in the core is given by

Assuming the core to be adiabatic and homogeneous, the second term vanishes. We do not know 4Fein the outer core, though it is expected to be similar to 4. Bullen’s K - p hypothesis suggests that the bulk moduli of core material and iron will also be similar, so we write (Ks,c)Fe = tKs,c where t will be roughly independent of radius in the outer core if this region is homogeneous. Substituting into (2.2) gives (2.3)

dpFe/dr

=

-gpF1?/4~

Equation (2.3) can be integrated if t and a boundary condition on +Fe are known. The latter can be supplied by the shock-wave data at 1.35 Mbar, where the temperature difference between shock-wave conditions and those in the Earth will be minimal. If the inner core is pure iron, then r = 1 in this region. An integration of (2.3) using t = 1 in the outer core gave a density of iron at the center of the Earth in excess of 14.0 gm A further experiment using the maximum value of t possible [given

11

DRI VING MECHANISMS FOR THE EARTH'S DYNAMO

TABLEIa. THERMODYNAMIC A N D ULTRASONIC DATA Ultrasonic data at I atmosphere

Substance Olivine" Spinel" Garnet" A1203 Forsteri tea a-Quartza TiO,' MgO" Mercury (li4)b Bismuth (li4)* Salt' a-lrond Sodium (/is)' Tin (/iq)e Indium (li4Y Aluminume CoppelQ SilveP Gold" Sodium'

5

Y

0.25 0.20 0.10 0.58 0.64 -0.9 - 1.3 I .4 4.82 6.05 I .28 -1.4 I .01 2.61 2.39 I .83 2.31 2.70 2.32 2.82

1.3 1.3 1.2 I .3 I .3 0.7 I .3 I .54 2.74 2. I9 I .63 1.7 1.13 2.68 2.43 2.18 2.00 2.42 2.92 1.19

4.88 4.00 5.30 3.40 4.73 7.30 8.10 3.10 4.23 I .89 3.76 6.70

5.13 4.20 5.40 3.98 5.37 6.40 6.80 4.50 9.05 7.94 5.04 5.30

Anderson et a / . (1968). Spetzler el a / . (1975). Spetzler et al. (1972). Guinan and Beshers (l968), Leese and Lord (1968). Data compiled by Stewart (1973).

TABLE Ib. THERMODYNAMIC DATAFOR LIQUIDIRON"

K,C

CPC

1808 1900 2000 2100 2300

7.03 6.95 6.87 6.78 6.61

3.93 3.90 3.87 3.83 3.76

3.4 3.4 3.4 3.4 3.4

x 10-4 x 10-4 x x X

6, = 2.4; I/ay(ay/anP = -0.33. Crawley (1974). Kurz and Lux (1969). Treverton and Margrave (1972).

1.22 I .23 1.25 1.27 I .30

77 1 77 1 77 1 77 1 77 I

2.44 2.43 2.42 2.41 2.39

1086 1057 1026 995 936

12

D . GUBBINS A N D T. G . MASTERS

by the constraint that ( K , , ) , , (outer core) 4 (Ks,c)Fe(inner core)] and a minimum value of the density of iron at 1.35 Mbar of 10.9 gm gave a minimum value of the density of iron at the center of the Earth of 13.5 gm cmP3. The details of the calculations are given in Masters (1978). An Earth model with a pure iron inner core would have a density jump at ~ , can just be accommodated by the the ICB of at least 1 .O gm ~ m - which modal data. The calculations do, however, suggest that the inner core does contain some light elements but not in as large a concentration as in the outer core. Shock-wave data, summarized by Al’tshuler and Sharpidzhanov (1971), also indicate that Poisson’s ratio (up) for metals at high pressures may be high, and a simple extrapolation indicates that u pwill be similar to that inferred for the Earth’s inner core. The high value of this parameter ( u p= 0.45) is often interpreted as meaning that the inner core is near its melting point, but this does not seem to be necessary. In fact, some metals have high Poisson ratios at low temperatures and pressures (Anderson, 1977), e.g., lead has u p= 0.45 at T , / T = 2. 2.3. The Thermal Regime

An adiabatic and homogeneous core can satisfy much of the seismological data. There are also good theoretical reasons to expect departures from this state to be small in the fluid outer core (Masters, 1979). In such a region, the temperature gradient is given by (2.4)

where y = a K , , / p C , = a+/C,. The temperature distribution is given by integrating (2.4). This requires a knowledge of the ratio d C , or y as a function of pressure and temperature, together with a boundary condition on the temperature. y is also used in the reduction of shock-wave data. Expressions for y have been derived by Slater (1939), Irvine and Stacey (1973, and others using classical continuum mechanics under the assumption that y is independent of temperature. Lattice dynamics have also been used, mainly in the harmonic approximation in which u is again independent of temperature (see, e.g., Palciauskias, 1975). However, at the temperatures in the interior of planets the harmonic approximation may break down (Mulargia, 1977; Fazio et a1 ., 1978) and y can no longer be assumed to be independent of temperature. This means that there is no correspondence between the thermodynamic y and the “ y ” in the Mie-Griineisen equation of state (Knopoff and Shapiro, 1964). The an-

DRIVING MECHANISMS FOR THE EARTH'S DYNAMO

13

harmonic contribution to y depends strongly on the assumed interatomic potential, and Fazio et ul. (1978) showed that this is not sufficiently well known to set useful limits on y in the Earth's core. Even if the temperature independence of y is a valid assumption, the method of Irvine and Stacey requires zero-temperature extrapolations of Ks,c and d K s , c / d P from the values in the deep Earth, which introduce further inaccuracies. A further point to note is that all the preceding studies are valid for the solid state only and so application to the fluid outer core is probably inappropriate. (The usual justification is that, at high pressures, the fluid exhibits a solid-like structure-albeit transiently.) Another difficulty with this approach is that the electronic contribution to y must be estimated separately and this involves further theoretical problems. Knopoff and Shapiro (1964) have investigated the effect of several different theoretical and empirical determinations in the reduction of shock-wave data to isothermal equations of state (all assume y to be independent of temperature). The different relationships produce widely disparate results, and the authors could only conclude that the assumption of a constant y is invalid: y must be a decreasing function of pressure or volume. The temperature independence of y can be directly related to the variation of C , in an adiabatic and homogeneous region using the relationship

C , can be considered as the s u m of several contributors:

c, = c,,,,+ c,, + C,? + ... C , , and C,, are the harmonic and anharmonic contributors from vibrational effects, and Copis the electronic contribution. In the core there are other possible contributions [e.g., magnetic ordering and Schottkey anomaly effects (Jamieson p r a / . , 1978)], but these are thought to be small. The temperatures in the core are well above any estimates of the characteristic or Debye temperature of core material (Stacey, 1977). In this high temperature limit C , , takes on the classical Dulong-Petit value which can be estimated from the mean atomic weight of the core and is -500 J kg-' K-l. The anharmonic contribution may be important but is probably smaller than the electronic contribution (Jamieson et a / . , 1978), which is given by C o p =ARGT A is in general a function of volume. C , is by no means constant along

14

D. GUBBINS AND T. G . MASTERS

an adiabat, and because of the large factor l / a T y in (2.5) this can lead to a strong temperature dependence of y . Most estimates of the electronic contribution to C , give a total C , of about 700 J Kg-' K-' which is very similar to C , in the core because

c p= C,(1 + a T y ) and a T y e 1 . The variation of C , in the core is given by the relationship (Birch, 1952)

The factor in square brackets in (2.6) is of the order of several units and is positive (Birch, 1952) so that C , is a slowly decreasing function of pressure. In fact, the decrease in C , is unlikely to exceed 3% across the core because a T y e 1, so it may be regarded as constant for our purposes (700 J kg-' K-l). The variation of y with pressure in an adiabatic region is given by (2.7)

(2) =-c Y

5.C

(1 + Y -

5)

where

Equation (2.7) can be integrated if 4 can be estimated and a boundary condition placed on y . It turns out that the distribution of y which results is much more sensitive to the boundary condition on y than on any reasonable choice for 4. A crude upper limit can be placed on 5 by noting = p/+(a+//ap),. The quantity that (d+/dT), is always negative so that tmax [p/+(dp/d+),] can be estimated in the core if the core is assumed to be homogeneous and adiabatic. An indication of the size of (d+/a T), can be obtained from the reduction of shock-wave data (Fig. 3) assuming that and +Fe are similar under the same temperature conditions. Low-pressure ultrasonic data are also useful indicators (Table I), and these suggest that velocities are generally more temperature dependent than density, i.e., I I/+(/a+/dT), I >a so that 4 < p / + ( a + / d ~ )-~ 1 . Furthermore 4 is usually positive at low pressures, and if 5 ? 0, (a+//aT), is constrained to lie between -3000 and -800. The shock-wave estimate gives (a+/dT), = -2000 which is consistent with the ultrasonic data. Models of the outer core give rather variable distributions of p/+(d~#~/dp), which should be a slowly decreasing function of pressure (Anderson et a / ., 1968) because

+

DRIVING MECHANISMS FOR THE EARTH’S DYNAMO

15

pl+(~3+ldp)~ is just ( d K , , , / d p ) , - I , but the calculations are insensitive to the particular Earth model used. The boundary condition on y is very difficult to estimate. Some recent estimates of y in the core are shown in Table 11. Most have yMcein the range 1 .O- 1.5, and this range has been used to do representative temperature calculations. The actual temperature distribution is much more sensitive to the boundary condition chosen for the temperature when integrating (2.4). It is conventional to fix the temperature distribution in the core at the ICB using estimates of the melting temperature of iron [-SO00 K according to Birch ( 1972) and Liu (19731. This takes no account of the effect that the light elements in the outer core have on the melting temperature. A rough calculation using ideal solution theory indicates that the melting point depression will be several hundred degrees. An alternative approach is to fix the temperature at the top. Here the temperature cannot exceed the solidus of mantle material (Tozer, 1972) which is estimated to be 3600 K by Kennedy and Higgins (1972) [this is thought to be an upper limit by these authors; Stacey (1977) prefers a value of 3200 K]. The temperature at the top of the lower mantle is reasonably well known, and if the lower mantle is adiabatic and homogeneous, the temperature can be extrapolated to the base of the mantle and is in the range 2800-3100 K (Masters, 1978). Phase changes in the lower mantle will tend to increase this. There is some evidence for a thermal boundary layer between mantle and core from seismology. The elastic structure at the base of the mantle is consistent with such an interpretation (Jones, 1977). These arguments lead to a choice of temperature at the core mantle boundary of 3300 K which should not be in error by more than a few hundred degrees. Figure 4 shows the effect of the uncertainties in y and t, and the calculation marked (*) has been used in the calculations of the various integrals required in the energetics problem. The choice of boundary condition on T leads to a temperature of about 4100 & 300 K at the ICB, consistent with the estimates of the melting temperature of iron after adjustment for a depression due to impurities. The actual temperature in the core is probably uncertain by 10 to 15%. TABLE11. ESTIMATES OF y Al’tshuler et a / . (1971 Anderson, 0. L. (1977) Jarnieson et a / . ( 1978) McQueen et a / . ( 1970) Mulargia and Boschi ( 1977) Mulargia and Boschi (1979) Stacey (1977)

IN THE CORE

YMCS = y = y = y =

1.45

1.5

1.2-2.0 1.1-3.1 YMCB iz 1.0 yMrB == 1.45-1.58 yvrs = 1.42

16

D. GUBBINS AND

T.

G. MASTERS

T(K1

.

4400

,,

4000

3600

I MCB

RADIUS

1

FIG.4. Adiabatic temperatures for the outer core. The value T o is fixed at 3300 K. Curves b , and b, are calculated with Y M C B = 1.2 and 5 = ( p / + ) ( d + / / a p ) , - 1 and -yMcs = 0 and 5 = 0, respectively. Curve c 1 is calculated with YMCB = 1.5 and $. = ( p / + ) ( d + / / a p ) , - I , and c 2 with -yMcB = 1.0 and 6 = 0. Curves c 1 and c q give an idea of the uncertainty in the temperature distribution for a fixed temperature boundary condition at the MCB. The curve marked (*) has been used as a representative adiabatic temperature distribution in the calculation of the energetics of the core.

3. CONSERVATION LAWS

The liquid of the outer core is modeled by a two-component liquid, representing a heavy constituent (iron) and a smaller proportion of a light element (e.g., sulfur). Modifications to allow for extra components are straightforward, and the phase changes are treated in Section 5 . The aim is to derive mathematical expressions for the conservation of energy and entropy of the whole core and to cast these expressions into a physically meaningful and useful form. Background is given in Malvern (1969) and Landau and Lifshitz (1959). Symbols are defined in a List of Symbols at the end of the chapter. 3.1. Local Equations

Conservation of mass gives (3.1)

ap/at

+ v-vp + pv-v = o

DRIVING MECHANISMS FOR THE EARTH’S DYNAMO

17

and when applied to the light component alone it gives

+ pv.Vc + V-i = 0

p(ac/at)

(3.2)

The equation of motion in an inertial frame is p(av/at) = - p v 2 - V p

(3.3)

+ V T ’ + J x B + pV+ + F

where the gravitational potential satisfies (3.4)

V2+ = 4 r G p

and F is some arbitrary applied body force. Maxwell’s equations, in the “nonrelativistic’’ approximation valid when v G c (speed of light), gives aB/at

(3.5)

=

V x (V x B)

+ 7V2B

V-B = 0

(3.6) Ohm’s law is

J

(3.7)

u(E + v

=

X B)

Use will be made of the thermodynamic relation

+ (p d p / p 2 ) + p dc

de = T ds

(3.8)

Conservation of energy gives at

ExB +- v . 7 ’ + q + H + PV*V++ v * F P0

Now expand the left-hand side of (3.9) and substitute for dv/df and aB/ a t from (3.4) and (3.6). Then using (3.8) for e and the continuity equations (3.1) and (3.2) gives an equation for the local changes in entropy: (3.10)

p ~Ds- = - V . q

+ H + J-2 + V v . 7 ’ + pV.i

Dt

U

Further details on the derivation of this equation will be found in Landau and Lifshitz (1959). The Onsager reciprocal relations give (3.11)

q - pi

=

-kVT

(3.12)

i

=

-a,Vp

(Landau and Lifshitz, 1959).

+ (PT/a,)i -

PVT

18

D. GUBBINS A N D T. G . MASTERS

Global equations are found by integrating (3.9) and (3.10) over the volume of the core. A very accurate account is given by Backus (1975) who deals with a single-component system only. The energy equation (3.9) differs from that of Hewitt et al. (1975) which is only valid for gravitational potential independent of time. Another difference between this treatment and that of Backus is that we allow the mantle to have finite electrical conductivity in order to retain the electromagnetic core mantle coupling responsible for driving the precessional dynamo.

3.2. Global Energy Balance Forming the scalar product of the induction equation (3.5) with u and integrating over the whole core gives (3.13)

-dI - d V B2=

-1v.J

xBdV-IcdV-f----.& E x B

dt 2Po PO Backus (1975) integrates this equation over all space, when the surface integral must vanish because of the behavior of E and B at infinity. The other two integrals on the right-hand side may be replaced by integrals over the core alone if the mantle is an electrical insulator because then J = 0 there. Allowing for mantle conductivity leads to an equivalent formulation of (3.13):

d (3.14) dt

1%

I

B2 -2p0 dV=-

v.JxBdV-IFdv -]&v.JxBdV-

where ‘8‘ denotes all of space and A is the region occupied by the mantle. The extra terms in (3.13) and (3.14) are due to electromagnetic forces acting across the core mantle boundary and constitute a driving force for the precessional dynamo. Equation (3.14) makes explicit the dependence of this driving on the electrical conductivity of the mantle. These terms are discussed further in Section 4.5. Integrating the energy equation (3.9) over the core and making use of (3.14), (3.4), and (3.3) gives (3.15)

$ q.dS

=

dI

- - p e dV + dt

-

f

pv.dS

+

I

I

H dV +

v-F dV -

I

pv.VJI dV

D R I V I N G MECHANISMS FOR THE EARTH'S DYNAMO

19

Equation (3.15) has a straightforward physical interpretation. The heat flowing out through the core mantle boundary, Q , is equal to the sum of the sources of power within the core. These are the rate of loss of internal energy R =

I

HdV

the rate of radioactive heat generation

GI =

/

pv.V$ dV

the rate of loss of gravitational energy

TI

J

v . F dV

the rate of working by applied body forces

P

=

= -

f

pvdS

the rate of working on the surface by pressure forces the rate of loss of kinetic energy the rate of loss of magnetic energy

s,

=

)!

the rate of working of viscous shear stresses on the boundary

V T ' d S

the rate of working of magnetic forces on the boundary minus the ohmic heating in the mantle and (3.15) becomes (3.16)

Q

=

R

+ X + (GI + P ) + ( K + M + T i + Sv + S m )

The core will contain all of the energy sources on the right-hand side of (3.16), to a greater or lesser degree. We will simplify the equations by studying individual mechanisms in isolation. The last five terms are grouped together because they constitute the driving for the precessional and tidal dynamos. Radioactive heating requires only the R term, and cooling, Z. Gravitational energy necessarily involves Z, P, and G . Each separate mechanism will be discussed in the next section, where it will

20

D. GUBBINS AND T. G . MASTERS

be shown that further algebraic manipulation is required before these expressions can provide useful numerical estimates.

3.3. Global Entropy Bulance Dividing Eq. (3.10) by T and integrating over the core gives an expression listing all the contributions to the change of entropy:

-1 d dt

ps d V =

-17 1 dV+

+

+dV

1

-Jd2 v CTT

+

V V :7’

dV+

E d V T

This equation includes Backus’s (1975) equation (4) as a special case because it is valid for a two-component system. Using (3.11) and (3.12) for q and i this equation may be written

d l p s d V = dt

(3.17)

--+

To

+

1

1

FdV

(‘a2

k - d V + . / --zdTV

+

1

5 d V

+

1

Y

d

V

where it has been assumed that there is no flux of material across the boundary i-dS = 0. More details of the derivation of (3.17) are given by Gubbins (1977). Again the physical interpretation of this equation is straightforward. Entropy changes come from

--Q

heat flowing out across the boundary

To

conduction of heat diffusion of light material radioactive heat sources

E,, =

1

S d V vV:T’

E , = ] y d V

ohmic heating viscous heating

DRI VING MECHANISMS FOR THE EARTH'S DYNAMO

21

k , a, v , and CT are all positive, and so the four dissipation terms E k , E , , E o , and E , are also positive and constitute the entropy gains that must be compensated for by the other source terms in (3.17).

3.4. Relating the Magnetic Field Strength t o the Power Supply

Many authors have tacitly assumed that the ohmic heat losses, JJz/ u d V , were a good estimate of the energy that had to be supplied. A

striking feature of the energy conservation equation (3.16) is that it contains no reference to the ohmic heating. This is because this electrical heating arises from other sources listed in (3.16) and is deposited inside the core again, so that it does not represent a loss of energy by the whole system. Ohmic heating cannot feature in the global energy balance. In order to relate the ohmic heat losses to the total heat flowing out through the core mantle boundary (and hence to the overall power requirements), we must use (3.16) in conjunction with the entropy equation (3.17), which does contain the ohmic term Eel. Substituting for Q from (3.16) into (3.17) gives (3.18)

Eo + E ,

+ Ek + E,, = E , +

PS

dt

dV

+Q-R ~

TO

where ER=,H($-i)

dV

Equation (3.18) represents a balance between the entropy gains of dissipative processes to the losses of the energy source terms on the righthand side. Given any simple model for the evolution of the core we can estimate these source terms and compare them with the requirements of the expected magnetic fields. This idea forms the basis for the calculations of the next section.

4.

POWER SOURCES FOR T H E

MAGNETICFIELD

4 . 1 . Radioactive Heating This case was treated by Backus (1975) and Hewitt et a f . (1975). The core is assumed to be composed of a single-component liquid and is in steady state. Entropy changes due to the splitting of the atoms is ne-

22

D. GUBBINS AND T . G . MASTERS

glected. Taking the mantle to be an insulator, (3.18) becomes (4.1)

E R =

Eo

+Ek

Taking the temperature gradient to be our estimate for the adiabatic gradient (see Section 2) gives E k = lo8 W K-l. An evenly distributed heat source ( R ) of 1013 W with T o = 3300 K could therefore provide for E o = 1.5 X lo8 W K-’, corresponding to a magnetic field of about 150 G. The driving term, E R , contains the small factor (l/To) - ( l / T ) in the integrand. Backus (1975) and Hewitt et al. (1975) show that the optimum thermodynamic “efficiency,” or the ratio of ohmic heating to heat supThis differs from the corresponding Carnot plied, is (T,,, - Tmin)/Tmin. because the ohmic heating reengine efficiency of (T,a,y - Tmin)/Tmin mains inside the core and is still available to drive convection. Backus (1975) gives a simple disk dynamo model driven by a Carnot engine that actually achieves the optimum thermodynamic efficiency. This efficiency is quite small in the core (0.2), and so heat is rather ineffective at generating magnetic fields. Notice, incidentally, that the can exceed unity if T,,, > “efficiency factor” (T,,, - Tmin)/Tmin 2Tmin;that is, the ohmic heating could exceed the heat supplied. This curiosity does not violate conservation of energy because the ohmic heating is “put back” into the system. A more important term from the geophysical point of view is E k , the entropy lost by conduction. Clearly in a compressible fluid core there will be an adiabatic temperature gradient that must be exceeded before convection can occur, and heat conducted away down this temperature gradient will not be available to drive convection. This effect is accounted for by the term E k in Eq. (3.18). About 1013 W of radioactive heat is required for the dynamo. Is there any evidence for such a large radioactive heating? This problem is discussed by Bullard (1949, 1950) and Elsasser (1950) who concluded that the radioactive content could be less than that of igneous rocks but must be considerably more than that found in iron meteorites. The present estimate for the required heating is several times bigger than Bullard’s value, and the problem is exacerbated. The idea of radioactive heating was revived by the suggestion of Lewis (1971) that the core contains a high proportion of potassium with its radioactive isotope K40. This radical view has caused considerable controversy, which is reviewed by Jacobs (1975). The arguments depend critically on the light alloying element in the core being sulfur. We do not describe the arguments further here except to note that our power requirements of 1013 W are near the upper limit set by Lewis (1971) and that his estimate of the power requirements of the dynamo is unreasonably low. Furthermore, the low half-life of K40

DRIVING MECHANISMS FOR THE EARTH’S DYNAMO

23

means that in the distant past, most of the Earth’s heat would have come from the core. All the heat must be carried away from the core mantle boundary or the core would become isothermal and convection would stop. Convection of the lower mantle is a very efficient means of transporting heat away, and the mechanism envisaged by Tozer (1969, where a temperature-dependent viscosity regulates the vigor of convection, would tend to maintain the core mantle boundary at a roughly constant temperature. Jones (1977) has taken the view that convection in the mantle is intermittent. In this case the temperature boundary condition for the core would be quite complex, but also the heat may not have reached the Earth’s surface yet. There may in fact be a large buildup of heat somewhere in the lower mantle that we know nothing about. Despite reservations about the applicability of thermal convection to the Earth’s core, this problem has received by far the most attention by theoreticians. This is because free convection is a well-understood phenomenon and the resulting fluid flows are likely to be similar to those driven by any buoyancy forces, be they thermal or compositionally driven. 4.2. Primordial Heat and Freezing of the Inner Core

Verhoogen (1961) revived the idea of the thermally driven dynamo by suggesting that heat released by cooling of the core was available to drive convection. Heat from the temperature drop would be augmented by latent heat of freezing as the outer core liquid froze, forming and enlarging the solid inner core. Referring to Eqs. (3.16) and (3.18), we drop radioactive heating and all boundary terms, neglecting the thermal contraction of the core and compressibility of the material. The magnetic Prandtl number of metals is small, and we neglect viscous dissipation in relation to that of ohmic heating. The result is similar to that for radioactive heating:

where the latent heat of freezing of iron is represented by a specific heat anomaly (Gubbins, 1977). It is clear from this equation that heat is more effective at generating magnetic field if it is released at a higher temperature, and since the latent heat contribution is released at the inner core boundary, this mechanism is slightly more efficient than uniformly distributed radioactive heating (Gubbins, 1976). Gubbins et al. (1979) esti-

24

D . GUBBINS AND T. G . MASTERS

mate that 5 x 10l2 W is required for a substantial magnetic field. This heat must be available for most of geological time, which would not be the case if the growth rate of the inner core were too fast. For the inner core to grow, the temperature of the surrounding liquid must drop to the local melting temperature. Assuming the liquid to be at or very close to the adiabatic gradient, the drop in temperature causing a given increase in inner core radius is dependent on the difference between the adiabatic and melting gradients. A minor error in Verhoogen’s paper is that he retains only the melting gradient. The new gradient gives a much smaller contribution from cooling of the liquid compared with that from the latent heat. The required growth rate of the inner core is consequently greater, and as a result Gubbins et al. (1979) find cooling an unlikely mechanism for driving thermal convection. As with radioactive heating, the heat must be carried away by deep mantle convection or by conduction. However, the cooling is presumably caused by a lowering of the temperature at the core mantle boundary, and this is only possible if all, or part, of the mantle cools also. This releases extra heat which may find its way to the surface of the Earth and which will be a very substantial contribution because of the high heat capacity of the mantle. If the whole mantle convects, then cooling will manifest itself even at the base of the crust. McKenzie and Weiss (1975) cite evidence from komatiites that the upper mantle has cooled through 200 K in 3 Ga, a figure that is similar in magnitude to the cooling rate envisaged by Verhoogen (1961) and Gubbins et ul. (1979). The mineralogy of the mantle may provide other constraints on the cooling; for example, the base should never have been above its melting point. If modest cooling rates of the whole Earth are important in magnetic field generation, then the dynamo should be considered as part of the thermal history of the Earth as a whole. A better understanding will come with improved models for convection and evolution of the mantle.

4.3. Gravitational Energy Release

Verhoogen ( 1961) went further to consider the gravitational energy released when the liquid contracted on freezing. By estimating the change in density as that due to pure iron freezing at high pressure, he found that the gravitational energy release was rather small. Seismological estimates of the density jump at the inner core boundary (e.g., Masters, 1979) are much larger than Verhoogen’s estimate and suggest that gravitational energy release may be a significant effect after all. More impor-

D R I V I N G MECHANISMS FOR THE EARTH'S DYNAMO

25

tantly, the observed density jump is so great that it can only reasonably be explained by a compositional difference between inner and outer core materials, and not by the freezing of pure iron (Masters, 1979). As the core cools, then a heavy fraction of the outer core mixture is likely to freeze out forming the inner core and leaving a lighter liquid fraction. This light material could rise up driving convection, with consequent release of gravitational energy. Braginsky (1963, 1964) was first to suggest that this mechanism could drive the convection responsible for the magnetic field. The present thermodynamic framework will be used to study this situation. First (3.16)-(3.18) are simplified by neglecting radioactive heating, precessional terms and the changes in kinetic and magnetic energy, and viscous losses. Two components are allowed in the mixture but for the moment we adopt the simplification that two derivatives of the chemical potential are zero: =

(the heat of reaction)

0

P.e

Effects arising from these terms are discussed in Section 4.4. It will be shown that the treatment is only consistent if the derivative (dp/dP)T,c is retained, because a Maxwell relation gives (4.3) The equations become (4.4)

Q=E+G+P

(4.5) Consider first the simplest possible case of a one-component liquid with no specific heat ( C , = 0) and zero coefficient of thermal expansion [(as/ 8 ~ = )01. ~This ensures that entropy is everywhere constant. The only physical effects lost are the heat source from cooling and adiabatic heating, but the gravitational energy loss due to a contracting core is reiained. Equation (4.4) gives (4.6)

Q

= -

"1 dt

pe dV

+

/

pv.V$ dV -

f

pv-dS

26

D. G U B B I N S A N D T. G . MASTERS

Here v represents the convective velocity of the liquid, but at the surface van = u.n, where u represents the slow shrinkage of the core from, say, freezing of the liquid. At the surface of the core, p can be approximated by hydrostatic pressure: vp = -pv+

and, using (3.81, the conservation of mass, and the transport theorem (4.7)

dt

I

p f dV =

1 ($+ p

v-V f) dV

(Malvern, 1969), (4.6) reduces to the remarkable result: Q = 0 . There is no heat generated at all! What has happened is that the gravitational energy has been taken up as work done against pressure forces. The energy of formation of the core is the work done against pressure forces in bringing every element of matter to its present position from a state of zero pressure. In contracting, the core loses gravitational energy but gains equal energy of formation, provided the pressure remains approximately hydrostatic. A similar point has been noted by Lapwood (1952) and Flasar and Birch (1973) in different contexts. It seems physically reasonable that the energy release from gravitational collapse should be ineffective at stirring the core if we bar adiabatic heating effects. This discussion shows that Verhoogen’s appeal to gravitational energy release from freezing of pure iron was ill-founded, and apart from some adiabatic heating, this energy source is not available to the dynamo. Now consider a two-component liquid, still with C , = a = 0 . If differentiation is taking place in the outer core and only a heavy fraction is freezing out, then some convection must be taking place and a description in terms of a slow radial contraction and hydrostatic pressure will be inadequate. Let the true fluid velocity be v. If we take a time average to eliminate the convective overturning, the core will appear to be contracting slowly with velocity u and independent differentiation or rearrangement of matter will take place simultaneously. The density change at any point due to the contraction can be defined through the continuity equation

($)

=

-V*(pu)

r

the remaining density change coming from rearrangement:

DRI VI N G MECHANISMS FOR THE EARTH’S DYNAMO

21

Equation (4.4) must be manipulated into a useful form, independent of v and dependent only on the overall contraction u . For example, we can write

=-Ip($+u.Ve)

d V - / e ( s )

r

dV

We are justified in making the assumptions u = v or V p = - p V $ on the boundary but not in the main body of the liquid where the convective velocity is important. Gubbins e( al. (1979) give full details on the reduction of (4.4) to Q = -

I (:7)r

$ -

dV+QU

The first term on the right-hand side is the gravitational energy release due to rearrangement (i.e., differentiation) alone, with no compressional effects. Q u is a complicated looking term representing the work done by pressure forces during volume changes associated with mixing of the components. It is very important to reduce the integrals to the correct form before putting in numbers to obtain numerical estimates. Here, for example, it is very tempting to ignore chemical potential effects and possibly pressure changes as well. This incorrect approach leads to either Q = 0 as before, or Q equal to the whole of the gravitational energy release, including that due to compression, which is also wrong. For sulfur this volume change is probably rather small, whereas if silicon is the light element in the core, the effect may be significant. Henceforth we take Q u = 0 and (3.18) becomes (4.8)

-1

$

($) dV

=

T o ( E o+ E k + E , )

r

This equation shows why recent authors have come to favor the gravitationally powered dynamo. It says that all of the gravitational energy released by rearrangement (but not that arising from compression or thermal contraction) is available to generate magnetic field. Contrast this with the thermal dynamos (4.1, 4.2) where only a small fraction of the heat is available for the dynamo. The physical reason for this greater efficiency is obvious. When light fluid is released at the inner core boundary, it must rise upward, stirring the core directly. It is this direct action that makes it so efficient. The magnetic field takes on the role of a

28

D. GUBBINS AND T. G . MASTERS

medium through which mechanical work may be converted into heat. All the energy must therefore be converted by one of the dissipative processes before it can escape the core as heat flowing across the core mantle boundary. Detracting from the appeal of the gravitational dynamo is the molecular diffusion term ( E J . The flux of light material generates heat and leads to a net gain in entropy in just the same way as do electric currents o r the diffusion of heat. In fact the diffusion flux vector, i , appears in E , in just the same way that the electric current vector enters the integral E o . This molecular diffusive process vies with ohmic heating as an alternative means of converting the available gravitational energy into heat, and must be assessed for its importance in the core. However, it is very dangerous to discuss its importance in isolation from the other chemical effects arising from the heat of reaction, and so this assessment is postponed until the next subsection. Now the additional complication of adiabatic heating may be added. The entropy is no longer constant globally or locally, and extra terms appear in both energy and entropy equations. Returning to the onecomponent system for simplicity, (3.18) gives TOE, =

I (::+ Ly

-

)(

U.VP

3

1 - - dV

=

To(E0 + E , )

Adiabatic heating enters the equations with low thermodynamic efficiency in the same way as other forms of heat. Gubbins et ul. (1979) claim the effect to be negligible in the core. When heat released by cooling is added ( C , # 0 ) , the effect is simply t o add on a source of the corresponding amount of heat. The relative contributions of gravitational and thermal terms to the energy budget depends on numerical values of the parameters in the core, but the greater efficiency of gravitational energy in generating magnetic field makes it a strong contender even though it may not dominate the energy flux. The gravitationally powered dynamo has been discussed by Braginsky (1964), Gubbins (1977), Loper (1978a), and Gubbins et al. (1979). The last two papers deal with the evolution of the inner core throughout geological time. Braginsky (1964) assumed that the light constituent in the outer core was silicon and appealed to the seismological evidence for the F region near the inner core boundary to support his claim for differentiation occurring there. His requirement for large magnetic fields of thousands of gauss would seem to tax the energy supply even from his gravitational dynamo. The latter authors tacitly assume sulfur to be the light constituent in accord with current fashion, and use ideal solution

DRI VI N G MECHANISMS FOR THE EARTH’S DYNAMO

29

theory to describe the properties of the mixture. Only the physical properties of the mixture are needed and these rough calculations are not very sensitive to the choice of light element. Loper (l978a) stresses the effect of density differences between the light and heavy components, while the approach of Gubbins et al. (1979) works from an assumed density jump at the inner core boundary and is less sensitive to the density of the light material. Jacobs (1975) has reviewed ideas on the composition of the outer core. A recent suggestion made by Ringwood (1977) is that it contains oxygen. The exact composition will influence the chemical effects to be discussed in the next section but is unimportant in determining the gravitational energy release itself. Schloessin and Jacobs (1979) have made the interesting suggestion that differentiation occurs across the core mantle boundary. The approximately equal size of the inner core and D” region of the lower mantle makes it tempting to try to connect the two. Perhaps the lowermost mantle is the ultimate destination of the rising light material, rather than it being diluted throughout the liquid outer core. The Schloessin and Jacobs case requires a little modification to our basic equations in order to allow for movement of mass across the mantle boundary. The chemical and thermal contact between core and mantle is clearly an important issue that will be studied further. The contraction of the Earth required by these dynamo models is too small to produce measurable changes in the Earth’s moment of inertia or radius. This may not be the case for other planets, and it will be interesting to see the results of thermal history calculations for planets like Mercury that have undergone significant contraction. 4 . 4 . Chemical Effects

We now relax all constraints on the properties of the liquid and allow the chemical potential to depend on concentration and temperature. Another Maxwell relation gives (4.9)

(%)

P.C

=-(3 P,T

The reaction is supposed to be exothermic (heat released as c increases). Differentiation at the inner core boundary will absorb heat and reduce the tendency of latent heat to drive convection, possibly even swamping it altogether. However, the light material rises and recombines at a higher level in the core and heat is released again. There is no net heat output

30

D. GUBBINS A N D T. G . MASTERS

from this heat of reaction (unless we adopt a feasible but highly elaborate model in which the heat of reaction varies with depth in the core), but because heat is released above and absorbed below, the net result is to inhibit convection. Of course, the reaction may be endothermic, in which case convection is assisted. The extra term in the entropy balance is E,

=

($+ u*Vc)( 1 $) dV

pQH

-

Yet another effect is the change in internal energy due to changing composition. The chemical potential varies with temperature and if light material moves from the high temperature at the inner core boundary to a lower one within the outer core, then energy will be released. The effect is tied with the heat of reaction which itself determines the temperature variations of p through Eq. (4.9). If the reaction is exothermic, the sign of ( d ~ / d T ) ~is, such ~ that the effect helps drive the dynamo, and because of the extra pV-i term in (3.17), all of this energy can be converted to magnetic dissipation, as for gravitational energy. Gubbins et al. (1979) show that for small temperature differences, this effect almost exactly cancels the influence of the heat of reaction term because of (4.9). The most important chemical effect is E,, the entropy gain due to diffusion of light material. It is positive definite. Braginsky (1964) discusses this point but his paper is not clear. He appears to call $i*Vp dV the “heat of diffusion” rather than $ i 2 / a T d V , and then estimates i from DVc. He claims that the heat of diffusion is very important, but does not state whether it helps o r hinders the dynamo. Gubbins (1977) attempted to estimate E , from i = DVc for sulfur and came up with a negligibly small estimate. Later Gubbins et al. (1979) decided that i = - ( p D K , / P ) V p was a better estimate and arrived at a larger value of E,, comparable with the other dissipative terms E o and Ek.This dominant influence of pressure on the diffusion is due to the force of gravity @ p = pg) causing the light material to settle upward, as envisaged by Anderson (1977). The effect of all the chemical terms, then, is to provide an extra, possibly large dissipative mechanism. Gubbins et al. (1979) show some calculations for a cooling core including all the effects of cooling, latent heat, gravitational energy, adiabatic heating, and chemical heating. The energy balance is usually dominated by the latent heat followed by cooling of the outer core and gravitational energy. Adiabatic heating and chemical effects only contribute a few percent of the total heat flux. By contrast, the entropy balance is dominated by gravitational energy release with latent heat a poor second in all the models, and so the gravitational effect is the most

DRIVING MECHANISMS FOR THE EARTH’S DYNAMO

31

important for the dynamo. Diffusive terms are much more difficult to estimate, and E D ,Ek,and E , could be comparable in size. 4.5. Rotational Energy: the Precessional and Tidal Forcing

The gravitational attraction of the Moon and, to a lesser extent the Sun, act on the Earth’s equatorial bulge and cause the Earth to precess about the normal to the ecliptic plane once every 25,800 years. The dynamical ellipticities of the core and mantle differ by 25% (Bullard, 1946) and the gravitational torques would therefore tend to make core and mantle precess at different speeds. This would be an intolerable situation because large shear flows would soon be set up across the core mantle boundary, and coupling between core and mantle would cause them to precess together. The coupling will involve fluid flow in the core and this flow could be responsible for generating the magnetic field. There is certainly enough energy in the Earth’s rotation to sustain the field throughout geological time if an efficient dynamo mechanism can be found. Well over 10l2 W is thought to be dissipated in the tides, and a comparable dissipation rate in the core would be adequate for quite large magnetic fields. The driving force from precession comes from shear stresses applied at the core mantle boundary. In an inertial reference frame the forcing has a nearly diurnal variation. It is therefore inappropriate to discuss the precessional dynamo in isolation from the other large diurnal influence, the tidal forcing. Tides will ’be set up in the outer core in the same way as in the oceans, and the resulting flow may be important for the dynamo. Again for simplicity assume that the core is a one-component fluid in a thermally steady state with no radioactive heating or gravitational energy release. Equations (3.16) and (3.17) give (4.10)

(4.11)

Q/To = E

jMv.(J x B)dV

k

-

:1

- dV

+

f

v-F dV

+ Eo + E ,

We assume that the mantle acts as a rigid body, which should be a good approximation at the diurnal frequencies considered, then v = o x r and the first mantle integral can be transformed:

32

D . GUBBINS AND T. G. MASTERS

where Mi, = BiBj- B2aii is the Maxwell stress tensor. These equations assure us that all the energy sources go directly toward generating viscous dissipation, generating magnetic fields, and maintaining the temperature gradients, in a highly efficient manner, like gravitational energy. The sources are losses in kinetic and magnetic energy in the core and all space, respectively, plus the rate of working by viscous and magnetic forces on the core mantle boundary and by tidal forces. The ohmic heat losses in the mantle must be subtracted from this. Equation (4. I I ) shows that in the absence of other heat sources, the precessional forcing must balance the entropy of conduction (Ek)as well as that due to ohmic heating ( E o ) .In the core, sufficient heat must be dissipated to attain the very steep adiabatic gradient before any field generating process begins to work. The diurnal forcing sets up fluctuating v and B which are not associated with appreciable heat fluxes because of the slow thermal response time. Only the time average of the terms on the right-hand side of (4.10) contribute to the heat flux. After time averaging, the magnetic energy loss will be that due to secular change in B which can be neglected. The surface stress terms draw their energy from the rotational energy of the mantle (see later). Conservation of angular momentum of the whole Earth requires that the loss of kinetic energy (KE) of the core ( K ) be restricted to about 1% of the loss in K E of the mantle, and hence K is fixed at 1% or so of the remaining stress terms. This ignores spin down of the core due to tidal friction in the oceans, but this is not thought to induce motions capable of generating magnetic fields. This leaves two possible driving terms for the dynamo: the viscous and magnetic stresses acting t o couple the core and mantle. The original formulation of the energy equation (3.16) brings out the explicit dependence of magnetic coupling on the electric currents flowing in the mantle, and therefore the mantle’s electrical conductivity. Where does the energy supply come from? The influence of tidal friction on the Earth-Moon system is well understood. Conservation of angular momentum applied along the normal to the ecliptic means that as the Earth spins down, the Moon must recede. The component of angular momentum in the plane of the ecliptic is conserved, and the Earth “tilts over” with an increased obliquity. Precessional torques act in the plane of the ecliptic, and so dissipation will reduce the Earth’s angular momentum in that plane. Total angular momentum is conserved, presumably by tilting of the lunar orbit with no change in radius, although this problem has never been properly examined. The net effect is to reduce the obliquity. Changes in the obliquity are discussed by Rochester (1976) who finds that the likely variation in the obliquity of 0.0001”/

D RI VING MECHANISMS FOR THE EARTH'S DYNAMO

33

cycle for either tidal or precessional effects is much less than the observational limit (0.05"lcycle by Duncombe and Van Flandern, 1976). For a rough estimate of the rotational energy available to the dynamo, we turn to observations in the change of the length of day. Astronomical observations of the period of the lunar orbit can be converted to a rate of spin-down of the Earth by assuming conservation of angular momentum of the Earth-Moon system. This gives only the lunar tidal contribution, and normally one adds a "fudge factor" to account for the solar tides (Munk and MacDonald, 1960; Hipkin, 1975). Tidal dissipation can be calculated directly, either from numerical models (Lambeck, 1975; Muller and Stephenson, 1975) or by satellite observations of the lag of the tidal bulge (Newton, 1968). Comparison of these two numbers gives an estimate of how much tidal energy could be dissipated elsewhere, either in the core or mantle. Table 111 shows values taken from Lambeck (1977), and there is clearly good agreement between theoretical and observed values, with no margin left for dissipation in the core. At most, tidal dissipation in the core is limited to 10-25% of the total, well below 10'2 w. Dissipation of precessional torques does not affect the lunar orbital period, but it does slow down the Earth. An independent observation of the spin-down is required. This important point does not seem to have been made in the literature. Two observations are available: timing of ancient eclipses (Newton, 1970) and growth rhythms (Rosenberg and Runcorn, 1975). The latter is not very helpful because it applies to averages over long stretches of geological time when the Earth-Moon distance and hence tidal friction would be quite different from what they TABLE111. VARIOUS ESTIMATES FOR OF

Astronomical estimate Satellite based estimate (tidal bulge) Numerical tide model Ancient eclipse data Growth rhythms

SECULAR CHANGEI N THE LENGTH DAY

THE

Change of lunar orbit ("/cycle')

Change in length of day (msec/cycle)

-28 t 3 -24 r+_ 5

-2.8 2 0.3 - 2 . 5 2 0.6

-30

-2.9 2 0.3 2.0 2.8

?

3

a Eclipse data should include precessional effects as well as lunar tidal effects, which the first three do not. Growth rhythms give an average over a long period of geological time and include epochs when lunar tides may have been very much greater than they are today.

34

D . GUBBINS AND T . G . MASTERS

are today. The eclipse data are more hopeful, and Lambeck (1977) claims it to be in agreement with lunar data and theoretical tidal estimates. The old “paradox” that the actual spin-down of the Earth was slower than the lunar effect has almost disappeared (see also Hipkin, 1975). Again there is no observational evidence of significant energy dissipation in the core from precession, but again it is impossible to rule out a loss of less W. than about The conclusion is that about 10l2 W of rotational power could be dissipated in the core. Equation (4.11) shows that this is adequate for the dynamo provided that the fields generated do not dissipate too much ohmic or viscous heating. It is not at all clear how diurnal forcing can produce steady magnetic fields without also generating large time-dependent fluctuations that also dissipate heat, either ohmically or viscously. The motion of an inviscid fluid contained in a precessing ellipsoidal cavity was studied by PoincarC (1910), who showed that pressure forces acting on the boundary could induce a constant vorticity flow in the core. Toomre (1966) has stressed the importance of this “inertial coupling” in the core. No energy is dissipated by this process and so no magnetic fields are generated, and the core fluid exhibits solid body rotation about an axis slightly different (by rad or so) from that of the mantle. The directions of the angular velocities of core (a,)mantle (a,) and precession (a)all lie in the same plane so that the core is “tilted over” slightly. No energy is lost and the flow cannot generate magnetic fields. The energy source for the dynamo relies on this inviscid flow being unstable. Reexamining Poincare’s (1910) problem for a viscous fluid turned out to be very difficult and defeated Bondi and Lyttleton (1953), who reopened the question of a precessional dynamo after Bullard (1949) and Elsasser ( 1950) had dismissed it. The mathematical difficulties were overcome in a linear boundary layer analysis by Stewartson and Roberts (1963) and Roberts and Stewartson (1963, but Busse (1968) showed that nonlinear effects in the boundary layers were important. Busse’s work was prompted by experiments of Malkus (19681, and the agreement between theory and experiment is striking. Busse (1968) also considers the additional influence of tidal forces as did Suess (1970). Although fluid flow is induced in the main body of the fluid, Busse concludes that the effect is too small to be noticeable in the Earth. All parties are agreed that viscous coupling, as studied by all of these authors, is unimportant in the core (Toomre, 1966). This leaves magnetic coupling, which is not modeled by the preceding experiments or theories. Stacey (1973) considered the relationship between magnetic core-mantle coupling and the precessional dynamo. If dissipative effects are im-

DRIVING MECHANISMS FOR THE EARTH’S DYNAMO

35

portant for the coupling then or, om, and fl will not lie in a plane, but o,will lag the plane of o m and fl by a small angle. By assuming the coupling torque to be proportional to this angle, he estimates the energy available for various coupling constants. Taking the value of coupling constant appropriate for the decade variations of the length of day gives an energy of over 1Olo W. Rochester et al. (1975) have criticized this choice of coupling constant because of the diurnal rather than decade variation of the precessional forcing and favor a much smaller value. Another criticism is that the proportionality between the coupling torque and lag angle will not be valid for turbulent core flow, and that the coupling constant will not be the same for tilt-over as for relative rotation. However, Stacey’s (1973) elegant and simple approach should be useful when more is understood about the core flow. Magnetic effects on the fluid flow driven by precession were first considered by Malkus (1963) who argued in favor of turbulent core motions. This paper has been criticized by Rochester et d . (1975) because it does not make clear the effects of the diurnal nature of the forcing, nor its dependence on the mantle conductivity, and it contains errors as well. The latter authors estimate magnetic coupling assuming laminar flow and find the surface stress terms totally inadequate for driving the dynamo. In an independent study, Loper (1975) analyzed the laminar boundary layers more rigorously and arrived at the same conclusion. In fact there is no need for such a rigorous study: It is clear that once one invokes laminar flow the precessional dynamo has been “assumed away.” By contrast it is not possible to use these arguments to eliminate a dynamo driven by turbulent flow. Gans (1970) tried to make a laboratory dynamo work from precession but he could not get a high enough magnetic Reynolds number. Turbulence might be expected to give a very poor dynamo, particularly if it were time dependent, and it may be possible to use bounding methods to eliminate the idea altogether. So far efforts in this direction have failed. 4.6. Other Effects

Larmor proposed the dynamo theory in 1919. There had been many earlier suggestions for generating the Earth’s magnetic field, all radically different. Stevenson (1974) lists ten such effects, most of which are totally inadequate for generating the observed field. These alternative theories are rediscovered from time to time; following are listed a few of those that have received recent attention and perhaps warrant further study: Earrhquakes. This is really a dynamo process. Seismic waves set off

36

D. GUBBINS A N D T . G . MASTERS

by earthquakes travel into the core and set up oscillations of the fluid which can generate magnetic field (Mullan, 1973; Crossley and Smylie, 1975). The theory seems adequate on energy grounds because the classical estimate for energy release by earthquakes is 1011-1012W, and this figure should be increased to allow for long-period waves from large earthquakes. However, most of the energy will be dissipated by anelastic processes in the mantle, and what remains is supplied to the core by rapidly fluctuating fluid motions. Gubbins ( 1975) has criticized this theory on the grounds that time-varying flows make inefficient dynamos. Battery Effects. Differentiation of the outer core mixture would lead to compositional differences between rising and sinking columns of fluid, and lateral variations in composition may lead to voltages being set up. Only very small potential differences are required to produce large current systems and magnetic fields. The mechanism has not received much attention and has never been assessed quantitatively, but the renewed interest in the gravitationally powered dynamo may provoke more interest in this idea. Non-Newronian Gravity. Braginsky ( 1964) has made the suggestion that general relativistic effects may be significant for convection in the core. His proposal was actually to find an observable phenomenon with which to test the theory of relativity rather than as a basic driving mechanism for the dynamo. The post-Newtonian approximation (Weinberg, 1972, p. 241) gives a correction to the Newtonian gravitational potential. Braginsky (1964) finds that the force balance suggests a magnetic field of 10 G from this effect, although he does not discuss the energy balance. The problem merits further study, particularly in the context of astrophysical magnetic fields. The Twisted-Kink Theory. This theory of AlfvCn (1950) is another variant of the dynamo theory. Strands of magnetic fields, twisted by differential rotation, may become unstable and form kinks which produce substantial radial fields. No quantitative models have been produced and the theory is still highly speculative.

5. IMPLICATIONSOF

THE

DYNAMICS AND CHEMISTRY OF

THE

CORE

Having established that there is sufficient energy available for a particular field generating process, the next stage is to work out a feasible mechanism for bringing about the regeneration. The power requirements of the dynamo are generally rather small compared with the total heat budget, and, as became clear in the last section, many possible driving sources meet the requirement. In this section some general points are

DRIVING MECHANISMS FOR THE EARTH'S DYNAMO

37

raised which may influence the efficacy of particular forms of fluid flow in generating magnetic field. They may also lead to observable phenomena thus giving some clues to the actual mechanism acting. It will never be possible to produce a complete description of the core flow, but there are some general questions that may, eventually, be answered. 5 . 1 . Results f r o m Dynamo Theory The question of how a particular fluid motion generates a magnetic field is called the kinematic dynamo problem. It is very easy for horizontal motions to stretch the dipole field into large toroidal magnetic fields. In order to complete the regenerative cycle the dipole field must be reinforced by radial motion acting on the toroidal field. Braginsky (1964) followed Bullard and Gellman (1954) in supposing that the radial motion was weak and that this was compensated for by the toroidal field being very large (several thousands of gauss). The toroidal field cannot be observed by us because of the insulating mantle, and so the ohmic losses of a Braginsky-type dynamo would be very much greater than would be expected from surface observations. Gubbins (1975) gives a lower bound on the ohmic heating in the core, based on the observed field, of about lo8 W. For toroidal fields of 50 G this figure would be closer to 10" W , and IOl3 W for 1000 G. It is worth bearing in mind that the important quantity is not the ohmic heating but the entropy gain as it appears in the relevant equation in Section 4, so that although IOl3 W of radioactive heat may be available, a field of 1000 G is quite out of the question because (3.18) shows that I O l 4 W or more would be needed. An alternative kinematic theory involves motions with small length scales (Steenbeck er a l . , 1966). Here again the small-scale field is shielded from our view. Although the fluctuating magnetic fields are small in magnitude, according to this theory, their associated electric currents are much larger than the global currents and will dominate the ohmic heating, which increases in inverse proportion to the length scale. For very small length scales such as those envisaged by Busse (1975a), the ohmic heating is again unreasonably large. Energetic considerations place a lower limit on the scale of magnetic variations of about 100 km (Gubbins ef a l . , 1979). The optimist may believe that simple global motions account for the magnetic field, and here the energy requirements are less severe. Kumar and Roberts (1976) and Gubbins et a l . (1979) report some ohmic heating calculations based on numerical kinematic dynamos in a sphere and arrive at values ten to a thousand times greater than the lower bound of 108

w.

38

D. GUBBINS AND T. G . MASTERS

One very important result is the radial motion theorem (Bullard and Gellman, 1954). The radial velocity cannot be zero for a kinematic dynamo to operate. Busse (1975b) has extended the theorem to a requirement of “sufficiently large” radial velocity. This would seem to demand some kind of buoyancy driven convection for the fluid flow rather than horizontal flow. Some authors have recently suggested that the fluid motions are driven by baroclinic instabilities arising from differential heating from radioactivity in the lower mantle. I t is difficult to imagine how such a flow could meet the requirements of the radial motion theorem, because heating from above would produce density stratification. Bullard (1949a,b), Bullard and Gubbins (1971), and Gubbins (1976) studied dynamo action of simple oscillating motions. In all cases the velocity required for a dynamo was of the order of ( T / T ) ” ~where , T is the decay time of the magnetic field and T the period of the oscillation. This result goes against the earthquake, precessional, and tidal dynamos [when ( d T ) l I 2 = lo3], although it is not sufficiently general to apply to turbulence. Backus (1958) showed that the fluid motion had to attain a sufficiently large stretching rate or magnetic Reynolds number (Childress, 1968) for dynamo action to occur. This result cannot be directly related to an energy source, but Proctor ( 1979) has found bounds on the minimum Rayleigh number required for a dynamo driven by thermal convection in a Boussinesq fluid. This work is crucial in understanding the dynamo, but the bounds are unlikely to affect heat flux estimates drastically because the heat conducted down the adiabatic gradient in the core may be so large as to swamp convected heat.

5.2. Freezing of the Liquid in the Outer Cure Higgins and Kennedy (1971) raised the possibility that the melting point gradient in the outer core might be less than the adiabatic gradient, and therefore in order for the outer core to be a liquid in equilibrium with a solid iron inner core, the temperature gradient must be subadiabatic. This would inhibit convection. This important suggestion sparked off much discussion. Both temperature gradients are very difficult to estimate in the core, and so it is impossible to make a convincing argument that one gradient is greater than, or less than, the other, although many authors have tried (for a review, see Jacobs, 1975). It seems more important to investigate the consequences of a possible subadiabatic melting gradient. Elsasser (1972), Busse (1972), and Malkus (1973) discuss two-phase convection in the core, in which solid and liquid phase coexist at the

DRIVING MECHANISMS FOR THE EARTH’S DYNAMO

39

melting temperature throughout the entire region. It is important to remember the distinction between phase and component. The local concentration of a component, as discussed in Section 4, can only change relatively slowly by diffusion, whereas the concentration of a single phase can change rapidly, in response to pressure changes for example, because it usually involves only melting of small solid particies. Busse ( 1972) considers a single-component, two-phase liquid with concentration of solid phase n and heat of fusion f.If the liquid is everywhere near its melting point, then a rapid adiabatic disturbance to a parcel of fluid will cause changes in temperature, pressure, and concentration. The temperature remains a t the melting point (provided the phase change can be accomplished sufficiently rapidly), and so the concentration change is determined by the isentropic condition. Ignoring differences in (Y and C , between the liquid and solid phases, and regarding s as a function of T , P , and n gives (5.1)

TdS=C,dT-

(F)

Tdp-fdn

P

for the isentropic condition. Taking T to be the melting temperature, (5.1) determines n as a function of pressure and hence depth in the core. This isentropic distribution must be exceeded in order for convection to occur. Gubbins et al. (1979) concluded that a two-component system could only be both neutrally buoyant and isentropic if the concentration were constant. This is not the case with phase changes which can occur rapidly. At equilibrium the phase rule assures us that n will be determined as a function of P and this equilibrium value may cause convection. Malkus (1973) has studied the problem in some detail and finds that the process is a very efficient method of transporting heat and is consequently poor at generating magnetic field. The energetics can be modeled with the present scheme by modifying the specific heat to allow for phase changes everywhere, giving similar results as for an ordinary thermal dynamo. Clearly this type of convection depends very sensitively on the size of solid particles and the equilibrium properties of the slurry. In fact, there is no reason to suppose that the outer core forms a slurry in the normal sense of the word: The solid particles may be many kilometers across. Malkus (1973) uses a simple terminal velocity argument to demonstrate that all the kinetic energy of the convection will be dissipated viscously if the particle diameter exceeds a micron or so. Two-phase convection does not seem to be important for the dynamo, but it may nevertheless occur in the core. Loper and Roberts (1978) have begun a program of research into convection of a binary alloy. In particular there may be layers where the temperature drops below the melting

40

D . GUBBINS AND T. G . MASTERS

point when slurries may be formed. Loper (1978b) has considered aspects of the convection that are influenced by various possible temperature gradients. Braginsky ( 1963) suggested that attenuation of seismic waves, caused by the outer core being a mixture of phases at the melting point, was observed near the inner core boundary. The point is followed up by Loper and Roberts (1978). The seismological evidence for an F layer around the inner core has now disappeared (Haddon and Cleary, 1974), and Gubbins (1978) argues that the rate at which the solid phase melts is too rapid to cause any attenuation. There is still hope for some direct evidence of slurries in the outer core.

5.3. Boundary Layers It is well known that convection at high Rayleigh number leads to almost adiabatic temperatures everywhere except near the boundaries. This is true for the core, where thermal boundary layers might be only tens or hundreds of kilometers thick. This raises a problem at the inner core boundary if substantial heat is conducted away, because the sharp temperature drop might imply that the temperature a few kilometers above the boundary would be below the melting point (Loper and Roberts, 1978). This apparent contradiction can be resolved by invoking a slurry with heat transported by two-phase convection. The thickness of the slurry layer would presumably be determined by the efficiency with which it transports heat and the particle size. This problem is discussed further by Loper (1978b). Schloessin and Jacobs (1979) have studied similar aspects of the evolution of the core mantle boundary. 5.4. Effects of Stable Density Stratification

The liquid core is often compared with the Earth’s atmospheres and oceans. Large regions of the latter two are stably stratified by temperature or salinity gradients, and the same may apply in the core. The acceleration of gravity varies by a factor of 2 between the bottom and top of the core, and this alone produces a big variation in the adiabatic gradient which may cause the upper regions to be stable. Density stratification in the core will influence seismic wave propagation, and in particular it will affect the normal mode eigenfrequencies. Masters (1979) has inverted normal mode frequencies for a model for the Brunt-Vaisala frequency in the core and has concluded that the present data are consistent with a neutrally stable core. The resolution is poor, however, and the data are also found to be consistent with a Brunt period of as low as 10 hr. A subadiabatic temperature gradient would affect our

DRIVING MECHANISMS FOR THE EARTH’S DYNAMO

41

calculations only in reducing the entropy gain due to thermal conduction ( E k )and lowering the overall heat requirements (Gubbins, 1976). Two-component convection has been studied extensively in the oceanographic literature because of the influence of both salinity and temperature. Clearly there are similarities between thermohaline convection in the oceans and the convection driven by sulfur rising through the core. When hot salty water lies above cold fresh water so that the whole column of fluid is stably stratified, then salt fingers are set up (Turner, 1973). The contrast in diffusivities between salt and heat in water is comparable to that between sulfur and heat in iron (Braginsky, 1964; Gubbins et u / . , 1979) and salt fingers may exist in the core. Interesting though this aspect of double diffusion may be, it is not very helpful in driving the dynamo because the motions are very weak and of small scale, and molecular diffusion dominates the whole process. The situation envisaged in the core is more one of vigorous convection driven by the light material in which heat plays an almost incidental role. Loper (1978b) and Gubbins el al. (1979) have noted that convection may occur, driven by differentiation, even though the overall temperature gradient is subadiabatic. The overturning would ensure that the temperature be almost adiabatic in the main body of the core with the boundary conditions being met through thin, strongly subadiabatic boundary layers. The overall effect of temperature is to stabilize the fluid in all of the calculations reported by Gubbins ef al. (1979). If the dynamo is driven gravitationally, then some of the energy supply (or strictly, entropy) must be used in overcoming the thermal stratification before magnetic fields can be generated. This explains the presence of the thermal conduction ( E k )term in the entropy equation (4.8). Loper’s (1978a) discussion omits the thermal conduction term, which would only be valid if the heat sources were sufficient to maintain the adiabatic gradient. It would be fortuitous indeed if the heat sources were just adequate for this purpose and no more. If the Earth’s dynamo is gravitationally powered, then it is most likely that it will be either assisted or hindered, quite strongly, by thermal effects. Numerical estimates of Gubbins (1977) and Gubbins et al. (1979) suggest that it is hindered, which explains the enormous difference between the magnetic fields they claim can be generated and that of Loper (1978a) (Loper greater by a factor of 20 in the ohmic heating). 6. SUMMARY

Although it has not been possible to eliminate rigorously many of the suggestions that have been made for powering the dynamo, gravitational

42

D . GUBBINS A N D T. G . MASTERS

energy release due to differentiation appears to be the most plausible. Much radioactive heating is needed if that is the major energy supply, and cooling of t h e core involves a rate of growth of the inner core that is rather too high. More persuasive is the fact that cooling will almost certainly be associated with some differentiation, and the relatively small amount of gravitational energy released wili be disproportionately important in driving the dynamo, swamping thermal driving. Precession suffers from not having an adequate theory to explain how it can generate magnetic fields, and even if Gans’s experimental dynamo had worked it would not be an easy matter to scale it to model the Earth. Much more work needs to be done on the magnetic coupling of the core and mantle, and the subsequent evolution of the Earth-Moon system before the precessional dynamo can be assessed or even understood. The gravitationally powered model can only be pursued further in conjunction with the thermal evolution of the Earth as a whole. Current ideas about mantle convection favor an adiabatic, convecting state throughout the mantle. Cooling rates for the mantle can be deduced from models of the convection (e.g., McKenzie and Weiss, 1975). Improvements in these models and in our understanding of the radioactive heat content of the deep interior (e.g., O’Nions et a / . , 1978) should lead to better models of the thermal evolution. Adding on the core contribution to such a calculation is a very simple matter. The heat flux from the core might be large enough to have a significant influence on the evolution of the mantle, and the magnetic field strength would play the part of an observational constraint on the evolution. Loper (1978a) and Gubbins et a / . (1979) have done calculations on the past history of the core, paying special attention to the growth rate of the inner core. Loper (1978a) considers a constant growth rate for the inner core, while Gubbins et a / . (1979) assume a constant temperature drop at the core mantle boundary. The latter boundary condition would be appropriate if the mantle cooled at a constant rate throughout. Core properties are estimated from present-day seismological observations, and an ideal binary solution is used to model outer core material. A surprising result from Gubbins et ul. (1979) is that less energy was available for the dynamo in the past when the inner core was smaller. This arises because the variation of g near the center of the Earth causes the temperature gradient to change as the inner core grows. Other boundary conditions suggest themselves: The heat flux into the mantle may be constant, for example. If the temperature of the mantle remains fairly constant, then heat may flow out of an initially adiabatic core and a stable region would gradually build up just beneath the mantle. Perhaps the most interesting development of all will be the application of these

DRIVING MECHANISMS FOR THE EARTH'S DYNAMO

43

ideas to other planetary bodies: The Moon, Mars, Mercury, and Venus ought to provide a good comparison with the Earth.

ACKNOWLEDGMENT This contribution was supported by Natural Environment Research Council Grant GR31 3475.

LISTOF SYMBOLS B c C, C , , C , , , C , , , Cue

Magnetic field Mass concentration of light component Specific heat at constant pressure Specific heat at constant volume: harmonic, anharmonic, and electronic contribution D Molecular diffusion coefficient e Internal energy d K , , J d p Pressure derivative of bulk modulus E Electric field E,, E c , E,, E k Entropy loss due to adiabatic heating, cooling differentiation, radioactivity, thermal conduction EkrE,, E,, E , Entropy gain due to thermal conduction, ohmic heating, molecular diffusion, viscous heating f Latent heat of fusion F Tidal force Fe (Subscript) pure iron g Acceleration due to gravity G Gravitational constant G I Gravitational energy release H Local radioactive heating i Diffusion flux of light material J Electric current vector k Thermal conductivity K Rate of change of kinetic energy K, Barodiffusion constant K,,c, K:.: Bulk modulus: imaginary part A Region occupied by the mantle M Rate of change of magnetic energy M Maxwell stress tensor n Mass fraction of solid phase n Unit normal vector p Pressure P Work done by pressure forces on the surface of the core q Heat flux vector Q Total heat flow QH Heat of solution Q u Work done by pressure forces with change in volume on mixing Q , , Q , Damping parameter for P, S waves

44

D. GUBBINS A N D T. G . MASTERS

R Total radioactive heating R G Gas constant Specific entropy Rate of working on core surface by magnetic forces Rate of working on core surface by viscous forces Temperature, melting temperature Temperature of core mantle boundary Rate of working by tidal forces Velocity of slow contraction Velocity of convection P, S velocities a Coefficient of thermal expansion aD,p Coefficients in the Onsager relations y Griineisen’s parameter yMCBy at core mantle boundary

h Constant in C ,

Chemical potential Imaginary part of Lame constant Permeability of a vacuum 4 Seismic parameter = V’, - 4 V8 J, Gravitational potential P Density r) Magnetic diffusivity of core Electrical conductivity of core, mantle t ‘ Deviatoric viscous stress tensor n Earth’s angular velocity p

P* PO

U,

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Palciauskas. V . V . (1975). The volume dependence of the Gruneisen parameter and crystal stability. J . f / r y s . C ' / r c , r r r . Solit/.\ 3h, 61 I-618. Poincare. H. ( I 9 10). Stir la precession des corps deformables. BrrIl. A.\/rori. 27, 321-356. Press, F. (1972). The Earth's interior inferred from a family of models. Irr "The Nature of the Solid Earth" (E. C. Robertson, ed.), pp. 147-171. McGraw-Hill. New York. Proctor, M. R. E. (1979). Necessary conditions for the magnetohydrodynamic dynamo. Gc,op/ry\. A.strop/ry.\. Flrrid f?\.rr. (in press). Qamar. A . (1973). Revised velocitie\ in the Earth's core. Brr11. Sc,i.\rrr. S[JC.Aur. 6 3 . 10731106.

Qamar, A , . and Eisenberg. A . (1974). 'l'he damping of core waves. J . Gcwp/r.v.\.K c . , . 7Y, 758-765.

Richards. P. G. (1073). Calculation BI' body waves, for caustics and tunnelling in core phases. Gc~ophys.J. R . A\/rorr. Sot.. 35. 243-246. Ringwood, A. E. (1977). Composition of the core and implications for origin of the Earth. G c , o c ~ / i c , f f r . J . I I , I I I- 135. Roberts. P. H.. and Stewartson, K . (1963).On the motion of ;I liquid in :I spheroidal cavity of ii pi-ecessing rigid body 11. Proc.. C ' f r u r h . f/ri/o.\. Sot,. 0 1 , 279-288. Rochester, M. G. (1976). The secti1;ii- decrease of obliquity due to dissipative core-mantle coupling. G c ~ o p / i y sJ. . K . Astrou. Sot.. 46 IOY- 126. Rochester. M. G.. Jacobs, J. A,. Sniylie. D. E . . and Chong. K . F. (1975). Can precession power the geomagnetic dynamo'? Gc,op/ry.r. J . K . A.\/rorr. Six.. 43, 661-678. Rosenberg. G. D.. and Runcorn. S. K . (1975). "Growth Rhythms and the History of the Earth's Rotation." Wiley. New York. Sacks. S. (1972). p structure of the innel' and outer core. t O S 7'1~rri.\.AGU 53, 601. Sailor. R. V.. and Dziewonski, A. M. (1978). Measurements and interpretation of normal mode attenuation. Gc~op/i!~.\.J . N . A . \ / r ~ ~ r.Sot,. r . 53, 559-581. Schloessin. H . H.. and Jacobs, J. A. (1979). Dynamics of a fluid core with inward growing boundaries. Submitted. Slater. F. D. (1939). "Introduction to Chemical Physics." McGrnw-Hill, New York.

50

D. GUBBINS A N D T. G. MASTERS

Spetzler, H., Sammis, C. G., and O’Connell. R. J . (1972). Equation of state of NaCI: Ultrasonic measurements up to 8 kb and 800°C and static lattice theory. J . f h F . \ . C ‘ / r i , r l r . Solid\ 33, 1727. Spetzler, H.. Meyer. M . D.. and Chan. T. (1975). Sound velocity and equation of state of liquid mercury and bismuth at simultaneous high pressure and temperature. fliglr 7<w1/7. Hi,
A D V A N C E S IN GEOPHYSICS. V O L U M E

21

FLEXURE DONALDL . TURCOTTE Depurtment of Geological Sciences Cornell University Ithacu. N e w York

1 .Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 2 . Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 3. Lithospheric Buckling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 55 4 . Flexure with End Loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 4.2 Ocean Trenches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 4.3 Hawaiian Islands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 4.4 Passive Continental Margins . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 4.5 Thickness of the Elastic Lithosphere . . . . . . . . . . . . . . . . . . . . . . . . 61 5 . Compensation of Loads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 6 . Axisymmetric Bending . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 66 6.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Islands and Seamounts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 6.3 Circular. Sedimentary Basins . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 6.4 Crustal Domes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 6.5 Flexure of the Lithosphere on the Moon and Mars . . . . . . . . . . . . . . . . . . 70 7 . Viscoelastic Flexure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 7.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 1.2 Application to Ocean Trenches . . . . . . . . . . . . . . . . . . . . . . . . . . 71 7.3 Application to Sedimentary Basins . . . . . . . . . . . . . . . . . . . . . . . . . 72 7.4 Time Dependence of the Thickness of the Elastic Lithosphere . . . . . . . . . . . . 73 8 . Stress Relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 9. Plastic Yielding during Flexure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 10. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 List of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

1 . INTRODUCTION

In terms of plate tectonics. the outer shell of the Earth behaves like a series of rigid plates that are in relative motion with respect to each other . These plates are defined to be the lithosphere; beneath the lithosphere the mantle behaves like a fluid on geological time scales . The lower boundary of the lithosphere is the isotherm associated with the transition from rigid to fluid behavior; this iostherm is 1200" ? 100°C. The thickness of the lithosphere beneath the ocean basins is near 100 km and beneath continents is near 200 km (Turcotte and McAdoo. 1979). There is extensive observational evidence that the surface plates be51

.

Copyright @ 1979 by Academic Press Inc . All rights of reproduction in any form reserved ISBN 0-12-018821-X

.

52

DONALD L. TURCOTTE

have elastically on geological time scales. Surface loads cause flexure of the lithosphere. Examples of such loads include mountains, islands and seamounts, the descending lithosphere at ocean trenches and continental collision zones, and isolated loads causing sedimentary basins. Theories for the elastic bending of plates have been applied to these problems. In order to obtain agreement between theory and observation the thickness of the elastic plate is a fraction of the thickness of the lithosphere. It is appropriate to define an elastic lithosphere which behaves elastically on geological time scales. Studies of the relaxation of elastic stress show that the boundary of the elastic lithosphere can be defined by the isotherm 600" f 100°C (Caldwell and Turcotte, 1979; Anderson and Minster, 1979). Due to time-dependent relaxation processes this thickness decreases somewhat with time. The concept that the lithosphere behaves like an elastic plate overlying an incompressible fluid is not new. In order to explain the absence of large gravity anomalies over mountain ranges it was necessary in the late nineteenth century to postulate low density roots. Mountain loading depressed the crust, displacing the denser mantle rocks. Smoluchowski (1909a,b,c) considered the buckling of an elastic lithosphere overlying a fluid mantle. His objective was to explain the folding in synclinal fold belts. Although the approach was not successful in terms of the problem studied, the formulation was identical to that used in most subsequent lithospheric flexure calculations. The flexure of the oceanic lithosphere at ocean trenches was studied by Gunn (1937, 1947). He also studied the flexure caused by the load of the Hawaiian Islands (Gunn, 1943a) and at passive continental margins (Gunn, 1944), and attributed major gravity anomalies to lithospheric flexure (Gunn, 1943b). The analysis given in those studies is essentially the same as that used today. However, the extensive measurements of bathymetry and gravity obtained in the last 20 years provide a data base for detailed studies of loading, lithosphere thicknesses as a function of age, and lithospheric rheology.

2. GOVERNING EQUATIONS

Many authors have applied thin plate theory to problems involving lithospheric flexure. The basic assumption in thin plate theory is that the wavelength A of the flexure is long compared with the thickness of the plate. In this limit the shear stresses due to vertical loading can be neglected compared with the bending stresses. This thin plate or shell approximation is only marginally valid to lithospheric flexure problems.

53

FLEXURE

Some authors have applied thick plate analysis, and others have applied the finite element numerical approach. Assuming that the outer shell of Earth (or any other planetary body) behaves like a thin elastic shell of thickness h, the vertical deformation w , measured positively downward, must satisfy the equation (Kraus, 1967) 12R2 h

24R2 h

1

v 6 w + 4 V 4 w + ~ ( I - v 2 ) V 2 w + ~ ( l - v ~ ) w

+ Ap g R 4 ( V 2+ 1 - V ) W = R 4 ( V 2+ 1 - V ) P where D = E h 3 / 1 2 ( l - v 2 )is the flexural rigidity ( E , Young’s modulus; v , Poisson’s ratio). The shell is assumed to be spherical with a radius R , that is, the Earth’s radius, h 6 R . The Laplacian operator V 2 is defined as

with 4 the polar angle (colatitude) and 6 the azimuthal angle (longitude). The last term on the left-hand side of (1) is the restoring force due to the fluid mantle. When this fluid is displaced, the result is a restoring force - A p g w . Usually a flexing lithospheric plate displaces mantle rock with a density p m . If the lithosphere is overlain with water, p = p w , then Ap - p m - p w and if it is overlain with sediments, p = p s , then Ap = p m - p s . The quantity p on the right-hand side of ( 1 ) is the load per unit area on the lithosphere. The formulation of the shell problem given in (1) includes both bending stresses and membrane stresses. For Earth with h / R = 0.02 the problems involving flexure and membrane stresses can be separated. For loads which have wavelengths of the order of R , ( 1 ) can be reduced to (2)

( E h / R 2 ) ( V 2+ 2 ) +~ Ap g(V2+ 1 -

V)W =

(Vz + 1 - ~ ) p

For these large-wavelength loads the bending can be neglected and the membrane stresses dominate. This type of problem has been considered by Turcotte (1974). For loads which have wavelengths small compared with radius R (but large compared with h ) , a flat Earth approximation is valid. For this case ( 1 ) reduces to (3)

DV4w +

Ap g w = p ( x , y )

Loads are supported by the bending rigidity of the plate. In the flat Earth approximation the total loading equals the total hydrostatic restoring

54

DONALD L. TURCOTTE

force: (4)

Ap g!

w dxdz

=

11

p dxdz

Although the separation of the membrane and flexure problems is a good approximation for Earth, it is a relatively poor approximation for the Moon. 3. LITHOSPHERIC BUCKLING

Smoluchowski (1909a,b,c) considered the buckling of a lithospheric plate with a hydrostatic restoring force as a possible explanation for geosynclines and anticlines. For the two-dimensional buckling of a plate under an end load per unit width S, the governing equation is d4w d2w D-+Sy+Apgw=O dx4 dx

The problem is illustrated in Fig. 1. This is an eigenvalue problem and no deformation will take place until a critical value of the end loading Scrit is reached. At the critical load catastrophic failure or buckling occurs. In order to solve ( 5 ) assume a solution of the form = w oe i Z ~ s / A (6) Substitution of (6) into ( 5 ) and solving for X yields

(7) For an infinite plate the wavelength must be real. The minimum value of S which will yield a real X is Scrit,and this is given by (8)

Scrit= ( 4 0 Ap g)"' sea level

FIG.1. Illustration of the lithospheric buckling problem. An elastic lithosphere with a thickness h is end loaded with a force per unit width S.

55

FLEXURE

Assuming that the maximum end load is the product of the strength of the lithospheric plate under compression IT, and the thickness of the plate (9)

Srrit= r , h

and using the definition of the flexural rigidity, the maximum thickness h,,, of lithospheric plate that will buckle is given by (10) Taking E = 10l2dyne/cm2, v = 0.25, g = lo3 cm/sec2, Ap = 3.4 gm/cm3, and IT, = 10 kbar as a maximum reasonable strength, the maximum thickness of a lithosphere that will buckle is h,,, = 0.8 km. This result led Smoluchowski ( 1909a,b,c) to conclude that lithospheric buckling could not occur. Goldstein (1926) using a similar analysis also reached this conclusion. Consistent with this result is the conclusion that reasonable horizontal loads do not influence the elastic flexure of the lithosphere. The elastic flexure of the lithosphere is the result of vertical loads and applied bending moments. However, when plastic bending of the lithosphere occurs the end loading can be important.

4. FLEXURE W I T H END LOADING 4.1. Theory

In the previous section it was shown that reasonable end loads cannot buckle the lithosphere. It follows that end loading has a relatively small influence on the elastic flexure of the lithosphere under loading. Therefore it is appropriate to set S = 0. In this section we also assume no distributed load so that p ( x ) = 0. The governing equation for the twodimensional flexure from (3) is (11)

D(d4w/dx4) + Ap g w

=

0

This case is illustrated in Fig. 2. The end load is made up of a vertical force Q and a bending moment M . In this section it is assumed that positive topography corresponds to positive w . It has been shown by Caldwell rt rrl. (1975) that a similarity solution can be obtained for this problem if the origin of the coordinate system, x = 0, is taken at the point of zero displacement, w = 0, nearest the load. In order to obtain a physically reasonable solution the displacement

56

DONALD L. TURCOTTE

L E V E L -

-SEA W

I t

pw

wbA+

'

n

I

iI

FIG.2. Schematic diagram of the bending of the elastic lithosphere under end loading. The thickness of the elastic lithosphere h and its physical properties are Young's modulus E , Poisson ratio v, and yield stress go.End loads include a vertical force Q and a bending moment M ; a self-equilibrating end load S is included. The magnitude w b and position x b of the highest point of the forebulge are shown.

must approach zero, w = 0, as x + 03. The general solution of (1 1) which satisfies these conditions is (12)

w = C sin(x/a)e-"'"

where a = (4D/Ap g)lI4is the flexural parameter. The flexural parameter is a measure of the horizontal extent of the flexure; from the definition of the flexural rigidity it is found that a h3I4. As illustrated in Fig. 2 the flexure of the lithosphere results in a region of positive topography known as a forebulge or outer rise. This outer rise is responsible for the uplift of islands adjacent to both ocean trenches (Dubois er al., 1974, 1975) and other islands (McNutt and Menard, 1978). The magnitude and position of the point of maximum positive deflection are denoted by wb and xb, respectively. The deflection profile (9) can be written in terms of w b and x b as

-

w

=

a w b sin

(z) [ 33 exp

(1 -

where Q! = 4Xb/n. The width of the forebulge is a direct measure of Q! and, therefore, the thickness of the elastic lithosphere. The resulting deflection is illustrated in Fig. 3a. At any point x the deflection given by (13) can be caused by a vertical force Q and an applied bending moment M. The vertical force is related to the deflection w by (14)

Q

=

-D(d3w/dx3)

57

FLEXURE

- 40

- a - 20

-X I I

-3

-2

.\I\

-3

.

,

c-

---4

/

-

FIG. Flexure of an elastic lithosphere under end ._fading. (a) Dependence (1- the nondimensional displacement w / w b on the nondimensional position x/xb. Results are also shown at an amplification of 10 to illustrate the structure of the forebulge. (b) Dependence of the nondimensional vertical force Q on the nondimensional position. Results are also shown at an amplification of 10 to 1 to illustrate the distribution in the vicinity of the forebulge. (c)Dependence of the nondimensional bending moment f i o n the nondimensional position.

58

DONALD L. TURCOTTE

It is appropriate to define a nondimensional vertical force Q by (15)

Q

=

Qxt/Dwb

Using ( 1 3) and (14) the nondimensional vertical force is

The distribution of the vertical force is illustrated in Fig. 3b. The bending moment M is related to the deflection w by (17)

M = -D(d2w/dx2)

It is appropriate to define a nondimensional bending moment (18)

fi by

M=hfX$/DWh

Using (13) and (17) the nondimensional bending moment is given by

The dependence of fi on x / x h is given in Fig. 3c. The deformation of an infinite plate by a line load is obtained by applying the vertical load Q at the point x L where d w / d x = 0. From (13) d w / d x = 0 when tan ?7xl/4xb = 1 or x L / x b = -3. The maximum deflection w L at the point where the load is applied is related to the load

-16 Qxb w,,= -7r3 D and the height of the forebulge w I ,is related to the load by

This solution is illustrated in Fig. 4. The deformation of a semi-infinite plate due to the application of an end load Q is obtained by applying the load at the point x Mwhere the bending moment is zero. From (19) the bending moment is zero when cos (7rx/4xb) = 0 or x / x b = -2. The maximum deflection w Mat the point where the load is applied is related to the load Q by WM

=

-32 Q x ; r3 D

--

59

FLEXURE X -

2

xb

3

4

5

L

0

r3Dw -~ 32 0x13 0.5-

1.0FIG.4. Flexure of an elastic lithosphere under a line load Q . The solution for a fixed end, d w / d x = 0, at x = 0, is valid for an infinite continuous lithosphere. The solution for a free end, M = 0 at x = 0 is valid for a semi-infinite or cracked lithosphere. The origin of the x coordinated is taken at the point of loading.

and the height of the forebulge wll is related to the load by

This solution is also illustrated in Fig. 4. The deflection of the lithosphere results in a gravity anomaly. Often the slope d w / d x is sufficiently small that the Bouguer formula is a good approximation; in this case the gravity anomaly A g is related to the deflection w by (24)

Ag

= 2rrG

Ap w

where G is the universal constant of gravitation. 4 . 2 . Ocean Trenches

An important example of the bending of the lithosphere occurs at ocean trenches. When subduction occurs the flat-lying lithosphere bends through angles as high as 70". During this bending process the lithosphere appears to maintain its coherence. Hanks (1971) first treated the deformation of the lithosphere in terms of the subduction process. He considered the elastic deformation of the lithosphere due to applied horizontal and vertical forces at the trench axis. In order to generalize the forces applicable at the trench axis it is necessary to include an applied bending moment also (Le Pichon et id., 1973; Parsons and Molnar, 1976; Molnar, 1977). The implications of the elastic bending in terms of the observed gravity anomalies have been considered by Watts and Talwani (1974). Seaward of the trench axis the bending of the lithosphere can be

60

DONALD L . TURCOTTE

considered as an end loading problem, and the similarity solution illustrated in Fig. 3 is applicable. In Fig. 5 the elastic similarity profile is compared with a bathymetric profile across the Mariana Trench from the Scan 5 cruise (Caldwell et a / . , 1976). For this comparison x h = 75 km and w h = 0.5 km. Taking E = 6.5 X 10" dyne/cm2, v = 0.25, and g Ap = 2.4 x lo3 dyne/cm3 yields a = 95 km, D = 5 x 1030 dyne cm, and h = 44 km. The maximum bending stress u B is 9.8 kbar. In general the bending stresses obtained from elastic solutions to the trench problem are large. Alternative models will be discussed in later sections. 4.3. Hawuiiar~Islands

The Hawaiian-Emperor island and seamount chain provides an approximately linear load on the Pacific plate. The load of the islands depresses the plate resulting in topographic depressions or moats adjacent to the islands. These are clearly evident in bathymetric charts. Gunn (1943a) proposed that these depressions were the result of lithospheric flexure. On the basis of more extensive data Walcott (1970a) considered the two models illustrated in Fig. 4. He concluded that the free end model fit the data better than the fixed end model. He suggested that the chain of volcanoes had weakened the lithosphere beneath them so that it was free to flex. Extensive studies of the Hawaiian-Emperor chain using both bathymetry and gravity have been carried out by Watts and Cochran (1974) and Watts (1978). A plot of bathymetry and gravity from a profile across Oahu island is given in Fig. 6. The lithospheric flexure is clearly illustrated. There are several problems involved in applying the end loading model directly to this problem. First, the model is valid only seaward of the island and sediment load. Second, the flexure is superimposed on a 2

I -100

I

--4

FIG.5. Bathymetric profile (corrected for sediment loading) across the Manana trench from the Scan 5 cruise (Caldwell er a / . , 1976) compared with the similarity solution for the elastic flexure of elastic lithosphere under end loading.

61

FLEXURE GRAVITY ANOMALY

zloCj 200

E

0

:]

TOPOGRAPHY

5

6 S

N

- O

too 200 km

FIG.6. Profiles of gravity and bathyrnetry normal to the Hawaiian chain taken across the island of Oahu (Watts, 1979).

broad topographic swell. This makes it difficult to establish a well-defined baseline.

4.4. Pii ssive Contin en tal Margins When the ocean floor is formed at mid-ocean ridges, it usually lies at a depth of about 2.5 km. As the ocean lithosphere cools it becomes more dense due to thermal contraction, and as a result of isostasy the ocean floor subsides to a mean depth of about 6 km. Since the ocean floor adjacent to a passive (Atlantic-type) continental margin was formed at an ocean ridge, it will subside about 3.5 km relative to the adjacent continent as it cools. Sediments deposited at the continental margin add an additional load, causing further subsidence. The flexure of the lithosphere adjacent to passive continental margins has been studied by Gunn (1944), Walcott (1972), Watts and Ryan (1976), and Turcotte et al. (1977). Finite element studies have been carried out by Neugebauer and Spohn (1978). Flexure calculations predict large gravity anomalies at passive continental margins; these large anomalies are not observed. The conclusion is that a large fraction of the differential sudsidence at these margins is accommodated on active faults. During the formation of passive margins at rift valleys, normal faults play an important role in the breakup of the preexisting continent to form an ocean. Apparently these faults remain active as the ocean expands, preventing significant flexure of the lithosphere in these regions. 4 . 5 . Thickness of the Elastic Lithosphere Utilizing the width of the forebulge, the elastic thickness of the oceanic lithosphere being subducted at a number of ocean trenches has been

62

DONALD L . TURCOTTE

obtained by Caldwell and Turcotte (1979). These thicknesses for the Mariana, Kuril, Tonga, Aleutian, Philippine, Nankai, and Middle American trenches are given in Fig. 7 as a function of the square root of the age of the lithosphere being subducted. Although there is considerable scatter, there appears to be a systematic increase in the thickness of the elastic lithosphere with age. The thickness of the elastic lithosphere has been determined from the flexure of the Pacific plate under the loading of the Hawaiian-Emperor chain by Watts (1978). His results are also given in Fig. 7 as a function of the square root of the age of sea floor at the time of formation of the seamount or island. The morphology of the older seamounts of the Emperor chain north of 40"N yield thinner values for the thickness of the elastic lithosphere than the seamounts and islands south of 40"N. The seamounts north of 40"N formed on young lithosphere. Again there is considerable scatter, but the results for the island-seamount loading are consistent with the results from flexure at ocean trenches. The observed dependence of the elastic thickness on age will be discussed in Section 8.

5 . COMPENSATION OF LOADS

It has long been recognized that small-scale features such as hills or valleys are not compensated. That is, the gravity field measured over the topographic feature reflects the mass anomaly associated with the feature. However, a major topographic feature such as a mountain range is compensated. The free air gravity anomaly over the feature is near zero. This can be explained if the mountain range has a crustal root and is in hydrostatic equilibrium. This transition from no compensation for local topography to nearly complete compensation for large-scale topography can be explained by the flexural response of the lithosphere to loads. Since the response of a thin lithospheric plate to loading is a linear problem, it is appropriate to consider the response of the lithosphere to a two-dimensional sinesoidal load of the form

[

I:

p ( x ) = p c g d osin 2.rr -

The load is assumed to be made up of crustal rock with a density p c with a topographic amplitude do, and a wavelength A. The governing equation

63

FLEXURE 4

60-

40h,km

Emperor seamounts north

20-

0

0

400NY tiawalmn ridge

"' I

2

and Emperor seamounts south of 40'N

0

"

3

4

5 /AGE

6

7

8

9

10

II

12

(m y')

FIG.7. Dependence of the thicknesh of the clastic oceanic lithosphere on the square root of age. The points are data obtained from the flexure of the oceanic lithosphere as it is being subducted at an ocean trench (C;ildwell and Turcotte, 1979).The age is the age of the lithosphere being subducted. The boxes are data obtained from the flexure of the oceanic lithosphere due to the loading of the Hawaiian-Emperor island and seamount chain (Watts, 1978). The age is the age of the ocean lithosphere at the time of formation of the island or seamount. The solid curve is the predicted thickness of the elastic lithosphere assuming that elastic stresses relax at temperatures above 700°C and that the upper I S k m of the lithosphere is too weak to transmit stresses.

for the deflection of an elastic lithosphere in response to this load from (3) is -a 4 d'w

+ w=fidosin(2ii;)

4 dx4

P m

In this section M) is positive downward. Clearly the lithospheric deflection is also a sinusoidal function

whose amplitude is given by

Since the lithosphere deflects in response to the load, the amplitude of is given by the actual topography dmO (2%

dmO = do

- WO

The deflection of the lithosphere can be expressed in terms of the height

64

DONALD L . TURCOTTE

of the actual topography according to

For short-wavelength topography, h/a + 0, the flexural rigidity of the plate prevents any deflection, wo/dmo+ 0. For long wavelength topography, hla + 00, the flexural rigidity of the plate does not restrict the vertical displacement and we find that (31)

dmo ~c

= W o b m - ~

c

)

This is the condition of isostatic equilibrium. As long as the topography is relatively shallow the resulting gravitational anomaly A g , can be calculated using the Bouguer formula. Thus

A s , = 2~GpCdmo

(32)

However, the flexure of the lithosphere also displaces the Moho. The vertical deflection of the Moho is assumed to be equal to the deflection w of the lithosphere. However, the depth h , to the Moho (thickness of the crust) may be a significant fraction of the wavelength A of the deflection. Therefore, the surface free air gravity anomaly Ag, due to a displacement of the Moho is (33)

AgM = - 2 ~ G ( p , - p,>w exp(-2~h,/A)

Combining (30), (32), and (33), the total free air gravity anomaly A s f a due to topography of height d,, is

In the limit of short wavelength topography, A/a + 0, (34) yields Ag = 2rGdmop this is the gravitational contribution of the uncompensated topography. The Bouguer gravity anomaly A s B is only the Moho contribution as given by (33). Therefore (35)

As,= -

2 ~ G ( p m- Pc)dmo exp(-2~h,/X) ~ l1 (Pm/Pc)[l + 6 ( 2 ~ a / A ) -

A correlation of the Bouguer gravity anomaly with topography as a function of wavelength has been given by Dorman and Lewis (1972) for the continental United States and by McNutt and Parker (1978) for Aus-

65

FLEXURE

tralia. Their results are given in Fig. 8. From (35) we obtain (36)

-

A g B

2rGPcdmo

- [(P -

m -

P c . ) / ~ c le x p ( - 2 ~ hm / A )

+ i(2ra/X)*l - 1

( ~ m / ~ r ) [ 1

Taking p m = 3.4 gm/cm3, p c = 2.7 gm/cm3, and h , = 30 km this result is given in Fig. 8 for a = 5, 10, 20, and 50 km. From this correlation it appears that a = 20 km. Taking E = 6 x 10" dyne/cm2, v = 0.25, and g Ap = 3.4 X lo3 dyne/cm3, this gives D = loz8 dyne cm and h = 6 km. However, there is considerable scatter to the data. A correlation of this type with the data of Dorman and Lewis (1972) was given by Walcott (1976). However, he did not include the reduction of the Moho contribution to the gravitational field with distance. This effect was included by Banks et ul. (1977) who reached a conclusion similar to that given above. A similar correlation with data from Australia has been given by McNutt and Parker (1978). In order to interpret the results given in Fig. 8 it is necessary to discuss the origin of topography. Because of the rapid rates of erosion most topography is eroded in a few million years (m yr). Therefore a large fraction of the significant topography in the United States is in the western U.S. where active tectonics is occurring. Much of this area has high heat flow and active volcanism. Therefore it would not be surprising that the derived thickness of the elastic lithosphere is small. However, the correlation must be treated with considerable caution. The fact that the topography is being eroded nearly as rapidly as it is being formed has

0

A. ,km

FIG.8. The ratio of the Bouguer gravity anomaly to the topography as a function of wavelength for the United States (Dorman and Lewis, 1972) and Australia (McNutt and Parker, 1978) compared with (36) for several values of the flexural parameter.

66

DONALD L. TURCOTTE

not been included in this analysis. Although cross correlations of gravity and topography as a function of wavelength show considerable promise, much remains to be done before great reliance can be placed on quantitative results. In the analysis given, the depth to the Moho is taken to be a constant; in fact it is proportional to the height of the topography. The topography near the mid-Atlantic ridge is nearly two-dimensional, with ridges and valleys running parallel to the ridge. McKenzie and Bowin (1976) have carried out two-dinensional correlations of bathymetry and gravity from ship tracks crossing the structure. Using a thick plate model for the deformation of the lithosphere under a two-dimensional load, they obtained a plate thickness of about 10 km. They associate this thickness with the thickness of the elastic ridge near the ridge axis where the topography was probably created. It should be noted that this topography is not being eroded.

BENDING 6. AXISYMMETRIC 6.1. Theorv

Some problems of lithospheric loading are best treated with axisymmetric solutions. Examples are isolated seamounts and near circular sedimentary basins. Assuming that the loading p is a function only of the radial coordinate, the general equation for the vertical deflection of the lithosphere (3) can be written

D

(37)

d2 (dr

7

d 2 + -rI -dr) w + Ap gw

= p(r)

Again w is taken positive downward. The boundary conditions are that w 4 0 as r + 00 and w is finite at r = 0. Several solutions to this problem have been given by Brotchie and Silvester (1969). 1. Point load. The solution for a point load Q at the origin is given by

w = - Q f f 2 kei

(38)

27rD

(L)

2. Disk load. The solution for a disk load ( p

> a ) is given by w=*[?ker’(;) APg a (39) -P-[aber’(:) APg

a

b e r ( i ) -;kei U

ker(i)

U

-&bei

=

p o , r < a ; p = 0, r

(a)

bei(:)

)(:

kei(i)],

+I],

r < a

r>a

FLEXURE

67

where ker, kei, ber, and bei are Kelvin functions and primes denote derivatives with respect to the argument (Abramowitz and Stegun, 1965). 3 . Gaussian load. The solution for a Gaussian load [ p = p o exp(-r2/ a " ] is given by ffi

y exp(-y2a'/4a2)J,(yr/a) dY y4 + 1

where J , is the Bessel function of the first kind of zero order. Deflection curves for a disk load from (39) are given in Fig. 9 as a function of the ratio a / a . In the limit a / a + 0 the solution for a point load is obtained. In each case the total load r a 2 p , is the same. As the ratio a / a increases, the deflection near the center is reduced; as long as a / a is less than about two, the radial extent of the deflection is a good measure of the flexural parameter a .

6.2. Islands and Seiirnourits The flexure of the lithosphere under the load of an individual island or seamount can be modeled using the axisymmetric analysis. This has been done for the Great Meteor seamount in the northeast Atlantic by Watts et al. (1975). 6.3. Circular Sedimentup Basins The deformation of the lithosphere under axisymmetric loading can also be used to study the structure and evolution of sedimentary basins.

FIG.9. Dependence of the nondimensional deflection o n the nondimensional radius for a disk load with various ratios of load radius to flexural parameter.

68

DONALD L. TURCOTTE

The Michigan basin is the classic example of a near circular sedimentary basin. Haxby et a l . (1976) have hypothesized that the structure of this basin is the result of lithospheric flexure due to loading near the center of the basin. Structural contours of the Michigan basin for the top of the Middle Ordovician Trenton Limestone have been given by Hinze and Merritt (1969) and for the top of the Devonian Dundee Formation by Cohee and Underwood (1945). The difference in the depths of these two sedimentary layers gives information on the early evolution of the basin. This thickness from individual drill holes in the southwest quadrant of the basin are given in Fig. 10 as a function of the distance from the center of the basin. A good fit with (39) is obtained with a = 68.4 km and a = 95.7 km. Assuming Ap = 0.8 gm/cm3 the corresponding value of the flexural rigidity is D = 1.7 = 1030dyne cm and taking E = 6 x 10" dyne/cm2and v = 0.25, the thickness of the elastic lithosphere is 32 km. The origin for this flexure is 200 m below the present surface. This is attributed to an overfill of the basin between the Middle Ordovician and the Devonian. The depths of the Devonian Dundee Formation in the southwest quadrant of the basin are given in Fig. 1 1 . This contour gives information on the late development of the basin. A good fit with (39) is obtained with a = 110 km and a = 88 km. The corresponding value of the flexural rigidity is D = 1.17 x 1031 dyne cm, and the thickness of the elastic lithosphere is 60 km. The origin of this flexure is 250 m above the present surface. This is attributed to 250 m of erosion since subsidence of the

r, km

100

200 I

300 1

FIG.10. Difference in depths between the Middle Ordovician Trenton Limestone and the Devonian Dundee Formation in the southwest quadrant of the Michigan Basin. Points are from well data and the solid line from (39).

69

FLEXURE

Sea level

Depth, 0 km

1-1.5

FIG. 1 1 . Depths of the Devonian Dundee Formation in the southwest quadrant of the Michigan Basin. Points are from well data and the solid line from (39).

basin ended in the Carboniferous. This erosion is responsible for the systematic outcropping of younger rocks toward the center of the basin. Over the early evolution of the basin the thickness of the elastic lithosphere was about one-half the value late in its evolution. This difference can be attributed to the thermal evolution of the basin. Early in evolution the center of the basin was hot, the thermal gradient high, and the elastic lithosphere thin. As the basin subsided both the thermal gradient and the thickness of the elastic lithosphere increased. This is consistent with the cooling model which explains the rate ofwbsidence of the basin (Sleep, 1971). Since the subsidence of the basin is associated with loading near the center of the basin, it is observed, as expected, that the radius of the disk load is nearly constant through the evolution of the basin. The total flexure of the basin requires a loading p o = 458 bars. 6 . 4 . Crustal Domes

Crustal domes occurs on many scales. Domes are often associated with active volcanism either in isolated areas or along rift valleys. Finite element studies of the relationship between crustal doming and continental rifting have been carried out by Neugebauer (1978) and Neugebauer and Braner (1978). Some crust domes have horizontal dimensions that indicate lithos-

70

DONALD L. TURCOTTE

pheric flexure is an important aspect of the doming mechanism. An example is the Adirondack dome in New York. However, the erosion associated with doming makes it difficult to carry out quantitative studies of the process. 6.5. Flexure of the Lithosphere on the Moon and Mars One of the important discoveries of the Apollo missions to the Moon was the presence of local gravity highs over several of the lunar maria. These gravity highs are associated with topographic lows. Because of the implied excess mass within the lunar interior, these gravity anomalies are referred to as being caused by mascons. Kuckes (1977) has considered the flexure of the lunar elastic lithosphere under the localized loading of the mascons. He concluded that the thickness of the elastic lunar lithosphere is 50-100 km. Melosh (1978a) has also considered this problem. Thurber and Toksoz (1978) have considered the flexure of Martian elastic lithosphere under the volcanic loading of the Tharsus region. 7. VISCOELASTIC FLEXURE

7.1. Theory An important question is whether the lithosphere can sustain kilobar stress levels on geological time scales. Relatively small amounts of creep are required to relieve elastic stresses. There is conclusive observational evidence that stresses are not totally relaxed on geological time scales. The presence of topography and gravity anomalies requires the preservation of stresses. However, some stress relaxation is to be expected since solid state creep processes are always occurring in rock. The usual approach to stress relaxation in a medium is to treat it as a linear Maxwell viscoelastic solid. The assumption in one dimension is that the elastic and viscous strains are additive so that the stress-strain relation is given by

where the dot denotes the time derivative and where p is the Newtonian viscosity. If a strain increment E , , is applied instantaneously at f = 0, the stress as a function of time from (41) is given by (42) = Ee0 e - E t I @

71

FLEXURE

The stress relaxes exponentially and has decayed to l / e in a time PIE, which is known as the viscoelastic relaxation time. The flexure of a viscoelastic plate with a linear restoring force has been considered in detail by Nadai (1963). For cylindrical bending the governing equation is

(43) where the viscoelastic relaxation time

(44)

is

T , ~ ,

r,,, = 2 p / E

In order to write (43)it is necessary to assume that the elastic behavior is incompressible, that is, u = 8. Without this assumption a much more complex analysis is required since the two horizontal stresses are not equal. The solution for a viscoelastic plate under a constant line load Q applied at x = 0 and at t = 0 has also been given by Nadai (1963).The deflection of the plate is given by

where a 0 = (4D/Ap g)1/4and

Deflection profiles for various values of the ratio t / T , , are given in Fig. 12. It is seen that the amplitude of the deflection increases under the constant line load and the width of the region of deflection decreases. Considering the difficulties involved in quantitatively measuring lithospheric deflections, observations of viscoelastic deformation would require t / T , , > 50. In order for significant viscoelastic relaxation to occur in 100 m yr the viscosity would have to be less than 3 x poise (P). 7 . 2 . Application t o Ocemn Trenches A viscoelastic analysis of the flexure of the oceanic lithosphere seaward of ocean trenches has been given by De Bremaecker (1977). The resulting profiles are essentially identical to elastic flexure profiles. A viscoelastic

72

DONALD L. TURCOTTE x/a

W

-

FIG. 12. Viscoelastic relaxation of lithospheric flexure under a line load. The amplitude of the flexure is given as a ratio to the maximum initial elastic flexure.

relaxation time T , , = lo5 yr and a viscosity p = P were required. An essential question is whether the same analysis can be applied to loads that have been applied for much longer times. The typical time constant for loading at an ocean trench is about lo6 yr. Seamounts of the Hawaiian-Emperor chain have ages of up to 70 m yr. It has been shown in Fig. 7 that the behavior of the oceanic lithosphere under the loading of these seamounts gives similar results to those obtained at ocean trenches. This strongly favors an elastic rather than a viscoelastic solution. A combined elastic and viscous solution for the flexure at ocean trenches has been given by Melosh (1978b). 7.3. Applicution to Sedimentary Busiris

The viscoelastic theory for plate flexure has been applied to the Michigan basin by Sleep and Snell (1976). These authors attribute the systematic outcropping of younger rocks toward the central part of the basin to the type of relaxation illustrated in Fig. 12. They obtained a viscoelastic relaxation time T~~ = f x lo6 y r and a viscosity p = loz5P. As previously discussed in Section 6.2, the systematic outcropping of younger rocks can also be attributed to regional erosion. Viscoelastic theory has been applied to the North Sea basin by Beaumont (1978). The evolution of the North Sea basin has involved both

73

FLEXURE

flexure and faulting. Because of this complex evolution the applicability of a simple viscoelastic model must be questioned. 7.4. Time Dependence qf the Thickness of the Elastic Lithosphere

If viscoelastic theory is applicable to the lithosphere, than the apparent elastic thickness or flexural rigidity should decrease with the age of the load. Walcott (1970b) compared various data and concluded that the viscoelastic relaxation time is lo5 yr. More recently a similar comparison has been made by Beaumont (1978).

8. STRESS RELAXATION An alternative method of treating stress relaxation is to consider the local stress relaxation due to solid state creep (Caldwell and Turcotte, 1979; Anderson and Minster, 1979). Laboratory studies of olivine (Kohlstedt er a l . , 1976) have shown that deformation is controlled by thermally activated processes over a wide range of temperatures and stress levels. Since it is found that the strain is proportional to the third power of the stress, to a good approximation it was concluded that dislocation creep is the dominant process. It has been shown by Weertman and Weertman (1975) that the deformation law for thermally activated dislocation motions is given by = Bu3e-A/RT (47) where R = 1.98 caymole K is the universal gas constant, A is an activation energy, and B is a constant. It has been suggested by Murrell(l976) that (47) can be used to determine the relaxation of stress in the lithosphere. A similar approach using a different stress-strain relation has been proposed by Kirby (1977). In direct analogy with the viscoelastic stress relaxation given in (41), the stress relaxation due to thermally activated creep from (47) is given by 0

(48)

=

( c i / E ) + Bu3e-A'RT

Integrating with the condition u = u,,at u=

(49) The time

(-

1

=

0 gives

+ 2EBe-"IRTt)-

1/2

d T, that

takes the stress to relax to one-half its original value is

74

DONALD L. T'JRCOTTE

given by 7,. =

(50)

eAIRT ~

2EBug

For olivine Kohlstedt et al. (1976) have given the deformation law 122 kcal/mole where k is in sec-I and u is in bars. Using this result the absolute temperature can be related to the relaxation time by

T,

=

6.16 X 10" ln(2.62 X 109 u ~ T , )

with u o in bars, T~ in seconds, and T , in K. The temperature at which stress relaxation will occur is given as a function of time in Fig. 13 for several stress levels. It is seen that for geological time scales of lo6 to lo9 yr stresses relax at temperatures greater than about 600°C. Assuming that the temperature in the lithosphere has an error function dependence on depth the temperature distribution is given by (Caldwell and Turcotte, 1979)

(T,

(53)

-

T ) / ( T m - T , ) = erfc(l.16 y / d L )

where erfc is the complementary error function and d Lis the thickness of the lithosphere defined as the depth where T , - T = O.l(T, - T s ) .

800

f

600T,

9

O C

400-

01

Io6

10.0

I

10'

108

Io9

T, , y r

FIG. 13. Dependence of the stress relaxation temperature T , on time for several values of the initial stress uo.

FLEXURE

75

For oceanic lithosphere which is older than 80 m yr, it is appropriate to take d , , = 120 km. Using the relaxation temperature from (52) the thickness of the elastic lithosphere for ocean basins decreases from 45 km for loads that are 1 m yr old to 40.8 km for loads that are 100 m yr old (assuming u o = 1 kbar). This difference is probably less than can be reliably observed. Assuming a plate model for the lithosphere with a thickness of 125 km, that T , = 1350°C and T,, = 7W C , and that the upper 15 km of the lithosphere is too weak to transmit stress, the predicted elastic thickness is compared with observations in Fig. 7. Reasonably good agreement is obtained.

9. PLASTIC YIELDING D U R I N G FLEXURE Another important question is what happens when a load exceeds the strength of the lithosphere. Rocks are expected to either fracture or flow plastically. There are certainly cases in which the lithosphere fails along a relatively narrow zone which at the Earth's surface is recognized as a fault. These lithospheric failures often occur in the continents, particularly in regions with extensional tectonics. Many of these lithospheric failures in the continents are recognized to be the reactivation of preexisting fault systems. Lithosperic fracture is also recognized on the transform faults of the ocean ridge system. In the oceans this type of lithospheric failure appears to be restricted to young oceanic lithosphere. Old ocean lithosphere does not appear to fail on faults. There are examples of transitions from lithospheric failure on faults to lithospheric flexure. An example is the sedimentary basin in the North Sea. Early in the evolution of this basin subsidence occurred on faults. Subsequently, as the rate of subsidence slowed and the basin cooled, the faults locked. The late subsidence of the basin caused lithospheric flexure as in the case of the Michigan basin discussed in Section 6.2. At many ocean trenches the lithosphere bends through 45" or more. There is no evidence, however, that the lithosphere fails o n fractures at ocean trenches. In Section 4.2 an elastic theory for the bending of the lithosphere at ocean trenches was given. However, in a number of areas the curvatures and associated strains in the outer slope area of the trench exceed those which are compatible with the elastic analysis. Although there is ample observational evidence of crustal faulting in the outer slope area, the presence of a forebulge in essentially all cases indicates that the lithosphere has not failed and is still transmitting stress. In order to explain this behavior elastic-perfectly plastic bending has

76

DONALD L . TURCOTTE

been applied to the trench problem by Turcotte et uI. (1978) and McAdoo a / . (1978). The elastic-perfectly plastic rheology assumes an elastic behavior at stresses below the yield stress uo.If the material is stressed to the yield stress, it will flow plastically with no strain hardening, and if it is unloaded, it will recover elastically. This behavior is illustrated in Fig. 14. Experimental studies of the deformation of dunite at high confining pressures and strain rates give uniaxial, deviatoric stress-strain curves similar to this idealized behavior (Griggs et a/., 1960). The elastic-perfectly plastic bending of plates has been considered by Prager and Hodge (1951) and Kachanov (1974). Plastic behavior will be restricted to the vicinity of the outer trench slope. Approaching the trench from the seaward direction the bending will be elastic until the maximum stress reaches the yield stress u o ,this occurs at x = x,, (see Fig. 15). In determining this point it is necessary to take into account both the bending moment M and any axial force S (see Fig. 2). The deviatoric stress profile in the lithosphere for elastic bending is the sum of a uniform stress due to the axial force and a linearly varying stress due to bending

ef

S

M Y + 12-7 h h

ux(y) = -

(54)

The criteria for plastic yielding, either Tresca or Von Mises, is given by (55)

b x I = ~ o

10-

u0-

( J l

5-

kb

I

I

I

0.05

0.10 6

0.15

FIG.14. The elastic-perfectly plastic (solid line) rheology is compared with the experimental results (dashed line) of Griggs et ul. (1960) for Dun Mountain dunite at a confining pressure of 5 kb and a temperature of 300°C.

77

FLEXURE

Plastic

Zones

-Outer

Slope

1-

x=0

Outer

Rise

No V.E. O

!

O

Irm

FIG. 15. Illustration of the elastic-perfectly plastic bending of the lithosphere. In region 1 (x > x,) the bending is fully elastic. At x = x, the lower edge of the elastic lithosphere begins t o deform plastically (assuming a compressional load). In region I1 (x, < x < xo) the shaded section of the lithosphere deforms plastically. At x = x 1 the bpper edge of the lithosphere begins to deform plastically. In region I11 (x < x,) elastic unloading occurs. Stress distributions at cross sections BB to EE are illustrated in Fig. 17.

where u oequals or slightly exceeds the uniaxial yield stress depending upon which criteria is adopted. Yielding will first occur at the top of the elastic lithosphere if the axial force is tensional and at the bottom of the lithosphere if the axial force is compressional. Evaluating (54) at y = + h / 2 and substituting into (55) gives the yield criteria

where S* = h u o and M* = ;t h 2 u o . The locus of first yielding in the S - M plane is given in Fig. 16. When the stress is equal to the yield stress throughout the lithosphere, u I = +uo,this is the loading limit or collapse state. If the load exceeds this value, a plastic hinge will develop. This limiting stress is given by (57)

where y = z0 is the position of change in sign of the stress. For a tensional axial stress l ois negative and for a compressional axial stress t ois positive. The corresponding values of the axial load and bending

78

DONALD L . TURCOTTE

moment are (58)

(59)

S = -2toS*/h M = M,(1

-

4t2,/h2)

Eliminating t ofrom (58) and (59) gives the limit or collapse condition

The collapse condition is plotted in Fig. 16. If T and M lie between the first yield condition and the collapse condition, elastic-plastic deformation is occumng. Stable bending is not possible if T and M lie outside the collapse condition. When the locus of points in the T vs. M region obtained from the elastic solution reaches the first yielding locus, the lithosphere becomes partially plastic, that is, x = xo. For the compressional case the lower part of the lithosphere will become plastic first; this is region IIa in Fig. 15, x 1 < x < x,,. A typical stress distribution in this region is illustrated in Fig. 17c. Subsequently, the upper part of the lithosphere may also become plastic; this is region IIb in Fig. 15,x, < x < xl,where x , is the point of maximum bending moment. A typical stress distribution in this zone is illustrated in Fig. 17b. In addition to defining y = t oat c r z = 0, we also define y = t 2to be the lower boundary of the elastic zone where u X = v 0 .The resulting

.- M/M,

First-Yield Locus

61Elastic Elastic-mastic

FIG. 16. Locus of first yielding and collapse in the axial force (S) and bending moment ( M ) plane.

79

FLEXURE

ty

Y=F h

+ ty

-

y=o

h

y=

-5

(C)

-Y=C,

+U

FIG. 17. Typical distributions in the lithosphere at cross sections BB through EE in Fig. IS: a compressive axial stress is assumed. (a) Stress distribution during elastic unloading in region 111, section EE. (b) Two plastic zones in region Ilb, section DD. (c) Single plastic zone in region Ila, section CC. ( d ) Elastic stress distribution in Region I , section BB.

stress distribution at any cross section within region I1 is given by (61)

u z = uo,

(63)

u r = -(To,

6 , sy

h 2

I-

h 2

- - 5 y 5 t 2

The stress distribution within the elastic core is given by (62). Note that in region IIa either t 1or e2 is greater than h / 2 . In regions of elasticplastic bending the plate curvature is obtained using small deflection theory with the result d'w - 2 ~ 0 ( 1- v') (64) dx at1 - 5 2 ) Details of how a solution is obtained in the elastic-plastic region have been given by McAdoo et (11. (1978). Once the point of maximum bending moment has been reached, at x = x,, unloading of the lithosphere begins. In plasticity problems, un-

80

DONALD L. TURCOTTE

loading is an elastic process. However, the stress distribution in the lithosphere depends upon the prior plastic deformation. The decrease in stress in the unloading region is proportional to y ; but the initial stress distribution at x = x, is not a linear function of y so that the actual stress distribution in region I11 is not linear in y. A typical stress distribution is given in Fig. 17a. Careful studies of the distribution and focal mechanisms of earthquakes in the descending lithosphere beneath Japan (Hasagawa et al., 1978a,b) show two well-defined planes of seismicity. Focal mechanism studies of earthquakes in the upper plane indicate downdip compression; focal mechanism studies of earthquakes in the lower plane indicate downdip tension. This distribution is consistent with the elastic unloading of a plastically deformed lithosphere. A comparison of alternative flexure models with a Kuril trench profile (Watts and Talwani, 1974; profile C1207) is given in Fig. 18. Included are ( I ) an elastic solution with axial loading (dashed-dotted line), (2) an elastic-plastic solution without axial loading (dotted line), and (3) elasticplastic solution with axial loading. The elastic thickness is the same in all the cases, and the axial load is the same in cases (1) and (3). It is clear from this comparison that the deflection on the outer trench slope is too large to be elastic. If the elastic profile were fitted to the outer trench slope, the predicted amplitude would be a factor of two or three too large. The axial compression load is required to move the plastic hinge, xo, sufficiently seaward. In all cases E = 6.5 x 10" dyne/cm2, Y = 0.25, and g Ap = 2.3 x lo3 dyne/cm3. For the best fitting elastic-plastic profile the elastic thickness is h = 46.7 km, the applied compressive stress is T / h = 4.5 kbar, the yield stress is cro = 6.5 kbar, xb = 80 km, x o = 65 km, x 1 = -5 km, x, = -35 km, and w b = 0.5 km. The fit of this model to the corresponding gravity profile is given in Fig. 19. Excellent agreement is found. The elastic-plastic model for the flexure of the lithosphere explains the relatively sharp curvature observed on many outer trench slopes. A layered elastic-plastic model has been applied to the flexure problem at ocean trenches by Chapple and Forsyth (1979). The behavior of the subducting lithosphere after it passes through the trench axis is a problem of considerable interest. The effect of the accretional prism of sediments on the flexure of the lithosphere has been considered by Karig et nl. (1976). The geometry of the descending lithosphere beneath many ocean trenches has been given by Isacks and Barazangi (1977). The curvature of the descending lithosphere is very large; for elastic bending the bending stresses would be of the order of 100 kbar. Although this stress is very high, so is the overburden pressure.

81

FLEXURE

M(dynes x

I

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

lorr)

1

t

A

..

-100 100

..... ............:.

X ( km)

FIG. 18. Bathymetric profile (corrected for sediment loading) across the Kuril trench (Watts and Talwani, 1974; profile C1207) compared with an elastic model (dashed-dotted line), an elastic plastic model with no axial loading (dotted line) and with an elastic-plastic model with a compressive axial load (dashed line).

82

DONALD L . TURCOTTE

- 100 I

100

,

=/-

I

I

I00

I

km

I

200

I

I

-

300

mgaIsL FIG. 19. The gravity profile across the Kuril trench corresponding to the bathymetric profile given in Fig. 18 compared with the elastic-plastic model including axial load.

There is little or no evidence of plastic hinging except on the seaward trench slope. Plastic buckling of the lithosphere as a cause of geosynclines has been considered by Vening Meinesz (1955). An elastic-plastic analysis has been applied to the flexure of the lithosphere under the load of the Hawaiian islands by Liu and Kosloff (1978). 10. CONCLUSIONS

Most surface features that have wavelengths of several hundred kilometers can be attributed to the flexure of the lithosphere under loading. Examples include ocean trenches and the loading of islands and seamounts. Many sedimentary basins can be attributed to lithosphere flexure. Examples include circular basins such as the Michigan basin and linear basins such as the Appalachian basin. Some domed structures can be attributed to flexure. The bending stresses associated with the elastic flexure are, in general, large, approaching 10 kbar. It is doubtful that near surface rocks can accommodate deviatoric stresses of this magnitude particularly in tension. However, as the overburden pressure becomes large, deviatoric stresses as large as the overburden pressure should be possible as long as the temperature is sufficiently low that creep processes do not relieve the stress. At ocean trenches the oceanic lithosphere bends through angles of 45" up to 70". In these cases it is not surprising that the yield stress of the lithospheric rock is exceeded. Elastic-plastic theories appear to explain the deformation of the lithosphere in these cases. An important question regarding the flexure of the lithosphere is stress relaxation. Many authors have approached this problem by using a linear viscoelastic analysis. Although the viscoelastic theory provides mathematical simplifications, there is little or no physical justification for its application to the cold rocks of the lithosphere. A preferred approach is to consider the relaxation of elastic stresses in the lithosphere by accepted creep processes. When this is done it is found that the thickness of the elastic lithosphere is about one-half the thickness of the thermal lithosphere associated with plate tectonics and increases only slightly with time.

FLEXURE

83

ACKNOWLEDGMENTS The preparation of this review article has been supported in part by the Division of Earth Sciences of the National Science Foundation under Grant EAR 76-82457. This paper is contribution 641 of the Department of Geological Sciences of Cornell University.

LISTOF

SYMBOLS

A Activation energy a Radius of applied vertical load B Constant C Constant D Flexural rigidity d o Amplitude of topographic load dmOAmplitude of actual topography E Young's modulus G Universal constant of gravitation Ag Gravity anomaly Bouguer gravity anomaly Free air gravity anomaly Thickness of the elastic lithosphere Thickness of the crust (depth to Moho) Bending moment Nondimensional bending moment Reference bending moment Load on lithosphere Vertical force Q Nondimensional vertical force R Radius of the Earth R Universal gas content r Radial coordinate S Horizontal force S * Reference horizontal force S,,,, Buckling force T Temperature T,, Stress relaxation temperature T , Mantle temperature T , Surface temperature w o Vertical displacement of the lithosphere w,, Height of the forebulge w,, Displacement of position of loading w Maximum amplitude of displacement x Horizontal coordinate x b Position of the forebulge y Vertical coordinate z Horizontal coordinate a Flexural parameter E Strain p Dynamic viscosity 0 Longitude h Wavelength

84

D O N A L D L. TURCOTTE

Poisson’s ratio Position where u r = 0 Position where u r = u,, 51 5 2 Position where u, = - u o P Density AP Density difference P c Crustal density P m Mantle density Ps Sediment density P w Water density U Stress uo Yield stress ur Stress due to axial load Viscoelastic relaxation time (7Ye U Colatit ude V

5 0

REFERENCES Abramowitz. M., and Stegun, I. A . (196S).“Handbook of Mathematical Functions.” Dover, New York. Anderson, D. L., and Minster, J . B. (1979). The frequency dependence of Q in the earth and implications for mantle rheology. Phys. Etrr. Pltinet. I t i f . (submitted). Banks, R. J . , Parker, R. L., and Huestis, S. P. (1977). Isostatic compensation on a continental scale: local versus regional mechanisms. G i w p h y s . J . R. Astron. Soc. 5 I, 43 1-452. Beaumont, C., (1978). The evolution of sedimentary basins on a viscoelastic lithosphere: theory and examples. G w p h y s . J . R. Asrron. Soc. 55, 471-497. Brotchie, J . F., and Silvester (1969). Crustal Flexure. J . Geopl7y.s. R r s . 74, 5240-5242. Caldwell, J . G . , Haxby, W. F., Karig, D. E., and Turcotte, D. L . (1976). On the applicability of a universal elastic trench profile. G i r t h Pltrtier. Sci. Let. 31, 239-246. Caldwell. J . G . , and Turcotte, D. L. (1979). Dependence of the thickness of the elastic lithosphere on age. J . Gctphys. R r s . (in press). Chapple, W. M . , and Forsyth, D. W. (1979). Earthquakes and bending of plates at trenches. (submitted). Cohee, G . V., and Underwood, L. B. (194.5). Lithology and thickness of the Dundee Formation and Rogers City Limestone in the Michigan Basin, U.S. Geol. Survey, Preliminary Chart 29, Oil and Gas Investigations Series. De Bremaecker, J . X . (1977). Is the oceanic lithosphere elastic or viscous? J . Ceoph.v.c. R u s . 82, 2001-2004. Dorman, L . M . , and Lewis, B . T. R. (1972). Experimental isostasy. 3. Inversion of the . 77, 3068-3077. isostatic Green Function and lateral density changes. J . G r o p h y . ~Res. Dubois, J . , Launay, J., and Recy, J . (1974). Uplift movements in New Caledonia-Loyalty Islands Area and their Plate Tectonics Interpretation. Twto/iop/i.v.s.24, 133- 1.50. Dubois, J . , Launay, J . , and Recy, J . (197.5). Some new evidence on lithospheric bulges close to Island Arcs, Tc.cto/wphy.s. 26, 189- 196. Goldstein, S. (1926). The stability of a strut under thrust when buckling is resisted by a force proportional to the displacement. Proc. Ctimh. Phil. S o c . 23, 120- 129. Griggs, D. T., Turner, F. J . , and Heard, H. C. (1960). Deformation of Rocks at 500°C to 800°C. I n “Rock Deformation,” Geol. Soc. Am. Mem. 79, pp. 39-104. Gunn, R. (1937). A quantitative study of mountain building on an unsymmetrical earth. J . Froti/,li/i / / 7 s t . 244, 19-53. Gunn, R. (1943a). A quantitative study of isobaric equilibrium and gravity anomalies in the Hawaiian Islands. J . FrutiAliti. / t i s t . 236, 373-390.

FLEXURE

85

Gunn, R. (1943b). A quantitative evaluation of the influence of the lithosphere on the anomalies of gravity. J . F‘r(rn/,/iti ln.\r. 236, 47-65. Gunn, R. (1944). A quantitative study of the lithosphere and gravity anomalies along the Atlantic Coast. J . f ‘ r ( r d / i n ln.\r. 237, 139- 154. Gunn, R. (1947). Quantitative aspects of juxtaposed ocean deeps, mountain chains, and volcanic ranges. Gcophgsic.~12, 238-55. Hanks, T . C. (1971). The Kuril Trench-Hokkaido rise system: large shallow earthquakes and simple models of deformation. Goophgs. J . R . Asrron. S o c . 23, 173-189. Hasegdwa, A,, Umino, N., and Takagi, A. (1978a). Double-planed structure of the deep seismic zone i n the northeastern Japan arc. Ti,ctonophgs. 47, 43-58. Hasegawa, A,, Umino, N., and Takagi. A. (1978b). Double-planed deep seismic zone and upper mantle structure in the northeastern Japan arc. Geophys. J . R . Asrron. SOL.. 54, 28 1-296. Haxby, W. F., Turcotte, D. L., and Bird, J. M. (1976). Thermal and mechanical evolution of the Michigan basin. Tc.cronophg.s. 36, 57-75. Hinze, W. J . , and Merritt, D. W. (1969). Basement rocks of the southern peninsula of Michigan. Mich. Basin Geol. Soc. Am. Field Excursion, pp. 28-59. Isacks, B. L., and Barazangi, M. (1977). Geometry of Benioff zones: lateral segmentation and downwards bending of the subducted lithosphere. In “Island Arcs, Deep Sea Trenches, and Back-Arc Basins” (M. Talwani and W. C. Pitman, eds.), Maurice Ewing Series I , pp. 99-1 14. American Geophysical Union, Washington, D.C. Karig, D. E., Caldwell, J . G., and Parmentier, E. M. (1976). Effects of accretion on the geometry of the descending lithosphere. J . Geophys. Res. 81, 6281-6291. Kirby, S. H., (1977). State of stress in the lithosphere: inferences from the flow laws of olivine. PWCJAppl. Gcwphys. I 15, 245-258. Kohlstedt, D. L.. Goetze, C., and Durham, W. B. (1976). Experimental deformation of single crystal olivine with application to flow in the mantle. I n “The Physics and Chemistry of Minerals and Rocks” (R. G. J. Strens. ed.), pp. 35-49. Wiley, London. Kuckes, A. F. (1977). Strength and rigidity of the elastic lunar lithosphere and implications for present-day mantle convection in the moon. f h g s . Etrrth. PlrrntJr.Inr. 14, 1-12. Liu, H. P., and Kosloff, D. (1978). Elastic-plastic bending of the lithosphere incorporating rock deformation data, with application to the structure of the Hawaiian Archipelago. T c c fonophys. 50, 249- 274. McAdoo, D. C., Caldwell, J . G., and Turcotte, D. L. (1978). On the elastic-perfectly plastic bending of the lithosphere under generalized loading with application to the Kuril Trench. Gcwphys. J . R . Astron. Sot.. 54, 11-26. Mckenzie, D.. and Bowin, C. (1976). The relationship between bathymetry and gravity in the Atlantic Ocean. J . Gwphg.s. R e s . 81, 1903-1915. McNutt, M. K., and Parker, R. L. (1978). Isostasy in Australia and the evolution of the compensation mechanism. Scionw 199, 773-775. McNutt, M. K., and Menard, H. W. (1978). Lithospheric flexure and uplifted atolls. J . Geophys. R e s . 83, 1206- 1212. Melosh. H. J . (197th). The tectonics of mascon loading. /‘roc.. Lrrntrr f / t r n c , r . Sci. Conf. 9rh, 3513-3525. Melosh. H. J . (1978b). Dynamic support of t h e outer rise. Giwphgs. R e s , Let. 5, 321-324. Molnar, P. (1977). Gravity anomalies and the origin of the Puerto Rico trench. Giwplr.vs. J . A’. Asfron. Soc. 5 I, 701-708. Murrell, S. A . F. (1976). Rheology of the lithosphere-experimental indications. Tectonopi/>..\. 36, 5-24. Nadai, A. (1963). “Theory of Flow and Fracture of Solids,’’ Vol. 2. McGraw-Hill, New York. Neugebauer, H. J . (1978). Crustal doming and the mechanism of rifting. Part I: Rift formation. Tecronophys. 45, 159- 186. Neugebauer, H. J. and Braner, B. (1978). Crustal doming and the mechanism of rifting. Part 11: Rift development of the upper Rinegraben. Tecronophys. 46, 1-20.

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Neugebauer. H . J . , and Spohn. T. (1978). Late stage development of mature Alantic-type continental margins. Tecfonophys. 50, 275-305. Parsons. B.. and Molnar, P. (1976). The origin of outer topographic rises associated with trenches. Geophvs. J . R. Asfron. Soc. 45, 707-712. Prager. W.. and Hodge, P. G . (1976). ”Theory of Perfectly Plastic Solids.” Wiley, New York. Sleep, N . H . (1971). Thermal effects of the formation of Atlantic continental margins by continental break up. Geophys. J . R. Asfron. Sor. 24, 325-350. Sleep, N. H . . and Snell, N. S. (1976). Thermal contraction and flexure of mid-continent and Atlantic marginal basins. Geophys J . R. Astron. Soc. 45, 125-154. Smoluchowski. M. ( 1909a). Uber ein gewisses stabilitatsproblem der elastizitatslehre und dessen beziehung zur entstehung von faltengebirgen. BttII. / / i f . A ~ t lS. C ~ill,. Crtrroi-ie. 3-20. Smoluchowski, M. (I909b). Versuche uber faltungserscheinungen schwimmender elasticher platten. Bull. / / I / . A c i d . Sri. dc C‘rtrc.oi.ic,. 727-734. Srnoluchowski. M. (1909~).Kilka Uwag o Fizyczuych podstawach teoryi gorotworczych. K o s m o s 34, 547-579. Thurber. C. H., and Toksoz. M. N. (1978). Martian lithospheric thickness from elastic . Lrr. 5, 977-980. flexure theory. G ~ o p l r y . ~Rrs. Turcotte. D. L. (1974). Membrane tectonics. Gcophys. J . R. A.sfro/i. SOC. 36, 33-42. Turcotte, D. L., Ahern, J . L.. and Bird. J . M. (1977). The state of stress at continental margins. Twfo/iop/iy.\. 42, 1-28. Turcotte, D. L.. and McAdoo, D. C. (1979). Thermal subsidence and petroleum generation in the southwestern block of the Los Angeles basin. California. (submitted). Turcotte. D. L.. McAdoo, D. C . . and Caldwell, J . G . (1978). A n elastic-perfectly plastic analysis of the bending of the lithosphere at a trench. Ti,c,/o,ioph\,.s. 47, 193-205. Vening Meinesz, F. A. (195.5). Plastic buckling of the earth’s crust: the origin ofgeosyncline. Goo/. Soc,. Am. Spec. Pup. 62, 3 19-330. Walcott, R. I . (1970a). Flexure of the lithosphere at Hawaii. T w f o t i o p h y s . 9, 43.5-446. Walcott. R. E. (1970b). Flexural rigidity, thickness and viscosity of the lithosphere. J . Geoph.vs. Ras. 75, 394 1-3954. Walcott, R. I . (1972). Gravity, flexure, and the growth of sedimentary basins at a continental edge. Geol. S o r . A m . B d l . 83, 1845-1848. Walcott, R. I. (1976). Lithospheric flexure, analysis of gravity anomalies, and the propagation of seamount chains. / / I “The Geophysics of the Pacific Ocean and its Margin” (G. H. Sutton, M . H. Manghnani, and R. Moberly. eds.). American Geophysical Union Monograph 19, pp. 431-438. American Geophysical Union, Washington. D.C. Watts, A. B. (1978). An analysis of isostasy in the world’s oceans- I . Hawaiian-Emperor seamount chain. J . Geophys. Res. 83, 5989-6004. Watts, A. B., and Cochran, J . R. (1974). Gravity anomalies and flexure of the lithosphere along the Hawaiian-Emperor seamount chain. G r o p h y . ~J. . H. A.\fr.ori. S o c . 38, 1 19141. Watts, A. B.. Cochran. J . R., and Seltzer, G. (1975). Gravity anomalies and flexure of the lithosphere: a three-dimensional study of the Great Meteor seamount. northeast Atlantic. J . Geophy~.R1J.s. 80, 1391-1938. Watts, A. B., and Ryan, W. B. F. (1976). Flexure of the lithosphere and continental margin basins. 7~iwouopliy.s.36, 25-44. Watts, A. B.. and Talwani, M. (1974). Gravity anomalies seaward of deep sea trenches and their tectonic implications. Gc,ophy.\. J . R. AL\t,o/i. Soc. 36, 57-90. Weertman, J . , and Weertman, J . R. (1975). High temperature creep of rock and mantle viscosity. A n w . Rei,. Etrrfh P l m , f . Sci. 3, 293-3 IS.

A D V A N C L S I N GEOPHYSICS. V O L U M E 21

THE INFLUENCE OF MOUNTAINS ON THE ATMOSPHERE RONALDB . SMITH Drpurtincnr

OJ Gcwlogv

und Geophysics

Y d r Uniiwsity N r n Hu1.c.n. Connecticut

I. An Introduction to Mountain Effects . . . . . . . . . . . . . . . . . . . . . . . . . . 2 . The Flow over Hills and the Generation of Mountain Waves . . . . . . . . . . . . . . . 2.1 Buoyancy Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 The Theory of Two-Dimensional Mountain Waves . . . . . . . . . . . . . . . . . . 2.3 Observations of Mountain Waver . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 The Three-Dimemional Flow over Isolated Hills . . . . . . . . . . . . . . . . . . . 2.5 Large-Amplitude Mountains and Blocking . . . . . . . . . . . . . . . . . . . . . . 2.6 The Observed Barrier Effect of Mountains-The Fohn and Bora . . . . . . . . . . . . 2.7 The Influence of the Boundary Layer on Mountain Flows . . . . . . . . . . . . . . . 2.8 Slope Winds and Mountain and Valley Winds . . . . . . . . . . . . . . . . . . . . 3 . The Flow near Mesoscale and Synoptic-Scale Mountains . . . . . . . . . . . . . . . . . 3. I Quasi-geostrophic Flow over a Mountain . . . . . . . . . . . . . . . . . . . . . . 3.2 The Effect of Inertia on the Flow over Mesoscale Mountains . . . . . . . . . . . . . 3.3 Theories of Lee Cyclogenesis . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 . Orographic Control of Precipitation . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 . I Observations of Rainfall Distribution . . . . . . . . . . . . . . . . . . . . . . . . 4.2 The Mechanism of Upslope Rain . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 The Redistribution of Rainfall by Small Hills . . . . . . . . . . . . . . . . . . . . . 4.4 Orographic-Convective Precipitation . . . . . . . . . . . . . . . . . . . . . . . . 5 . Planetary-Scale Mountain Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 A Vertically Integrated Model of Topographically Forced Planetary Waves . . . . . . . 5.2 The Vertical Structure of Planetary Waves on a P-Plane between Bounding Latitudes . . 5.3 Models of Stationary Planetary Waves Allowing Meridional Propagation and Lateral Variation in the Background Wind . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

87 89 90 92 117 121 128 134 135 137 142 143 153

163 169

169 173 192 193 195 197 201

209 217

1. A N INTRODUCTION TO MOUNTAINEFFECTS

It is often said that if the Earth were greatly reduced in size while maintaining its shape. it would be smoother than a billiard ball . From this viewpoint the mountains on our planet seem insignificant. and it makes us wonder how they manage to have such a strong influence on our wind and weather . One answer to the dilemma is that the atmosphere itself is 87 Copyright @ 1979 by Academic Press. Inc . All rights of reproduction in any form reserved. ISBN 0-12-018821-X

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RONALD B. SMITH

very shallow-a density scale height of about 8.5 km-so that many mountains reach to a significant fraction of its depth. This argument, however, underestimates the mountain effect. The real answer is that our atmosphere is exceedingly sensitive to vertical motion-and for two reasons. First, its strong stable stratification gives the atmosphere a resistance to vertical displacement. Buoyancy forces will try to return vertically displaced air parcels to their equilibrium level even if such restoration requires a broad horizontal excursion o r the generation of strong winds. Second, the lower atmosphere is usually so rich in water vapor that slight adiabatic ascent will bring the air to saturation, leading to condensation and possibly precipitation. As an example, the disturbance caused by a 500-m high mountain (i.e., a very small fraction of the atmospheric depth) could well include (a) broad horizontal excursions of the wind as it tries to go around rather than over the mountain, (b) severe downslope winds as air that has climbed the mountain runs down the lee side, and (c) torrential orographic rain on the windward slopes. The intention of this article is to review the meteorological phenomena that are associated with topography. This topic is of course only one of several subdisciplines within physical and dynamical meteorology, but it is one which scientists should be familiar with as they struggle to understand the workings of the atmospheric system. This review will not supersede previous works such as “The airflow over mountains” by Queney et al. (1960) as it is in many ways less detailed. On the other hand, the range of phenomena and scales included here is broader than in previous reviews. Because of the lack of detail, the experts in the various areas will probably find little for them except a widened awareness which comes from seeing their specialty placed into a broader setting. The study of airflow past mountains is complicated by the wide range of scales that must be considered. The nature of the disturbance caused by a narrow hill will be quite different from that caused by a broad plateau, even if the terrain height and other factors are the same. This is so because there are several natural length scales in the atmospheric system with which the mountain width (say, L ) can be compared. These include (with scale increasing): 1. the thickness of the atmospheric boundary layer 2. the distance of downwind drift during a buoyancy oscillation 3. the distance of downwind drift during the formation and fallout of precipitation 4. the distance of downwind drift during one rotation of the Earth 5 . the Earth radius

THE INFLUENCE OF MOUNTAINS ON THE ATMOSPHERE

89

The ratios of the mountain width to each of the natural length scales are important in determining the physical regime of the flow. This idea is emphasized in the present article by treating the effects of boundary layers and buoyancy (length scales I and 2) in Section 2, the effect of the Earth’s rotation (length scale 4) in Section 3, and the effect of the Earth’s curvature (length scale 5 ) in Section 5 . The section on orographic rain (Section 4) includes mountain scales both shorter and longer than the natural cloud physics length scale (length scale 3). In the past several years there have been some remarkable advances in the understanding of mountain flows and yet there are many outstanding problems. A partial list of these will serve to illustrate the vigor and breadth of the subject. Theoretically and observationally the distinction between trapped lee waves and vertically propagating mountain waves is now clear. In most cases, however, it has still proved impossible to predict these waves accurately. In spite of recent work on large-amplitude waves and wave breaking, the connection with downslope wind storms is still unclear. The studies of flow around realistic mesoscale mountains ( L 100-500 km) has just begun. New theoretical results have invalidated the *‘textbook” descriptions of vortex stretching over broad mountains. The strong variation in rainfall amount between the top and bottom of small hills now has been demonstrated but the physical cause is undetermined. There is mounting evidence that much of the presumed stable orographic rain may in fact be generated by closely spaced convection, triggered by orographic lifting. Nunierical simulations indicate that the general circulation of the atmosphere may be strongly influenced by the major mountain systems. The midlatitude westerlies are deflected into standing waves which produce zonal variations in climate and, by transporting heat northward, reduce the frequency and strength of cyclonic storms.

-

2. THEFLOWOVER HILLSA N D THE GENERATION O F MOUNTAIN WAVES The flow over small-scale mountains or hills (100 m to 50 km wide) can be considered without including the Coriolis force. One would expect however that the influence of buoyancy forces and the turbulent Reynolds stresses acting in the boundary layer will be important. Two natural length scales thus arise: first, the distance of downwind drift during a buoyancy oscillation (U,”) = 1 km, and second, the thickness of the boundary layer, 6 = 300 m. Of course, all these parameters U , N , 6 depend ultimately on the Coriolis force, but the local problem is usually

90

RONALD B . SMITH

considered to be well posed if these parameters are known. For a narrow hill (say, L = 100 m @ U / N ) the characteristic time for the flow, that is, the time it takes an air parcel to cross the hill, is much less than the buoyancy period, and the buoyancy forces can be neglected. If there were no boundary layer or if the boundary layer were very thin, this “narrow hill” flow would be closely analogous to the irrotational (i.e., potential) flow much studied in engineering aerodynamics. In fact, in the atmosphere, there is a thick preexisting boundary layer, and the disturbance caused by a narrow hill is for the most part confined within the boundary layer. Thus the aerodynamic analogy is not a good one. 1 km and greater) the boundary layer thickness For wider hills ( L becomes smaller in relation to the scale of the flow. At the same time, the buoyancy force associated with the atmosphere’s static stability becomes more important. The flow outside the boundary layer cannot be considered irrotational, and internal gravity waves become possible. For still wider mountains ( L 10 km) the boundary layer seems vanishingly thin, and the buoyancy forces dominate to the extent that the vertical accelerations become relatively small and the pressure field is nearly hydrostatic. Finally, for mountains wider than about 50 km (depending on the wind speed) the Coriolis force begins to become important. This added complication will not be considered in this section but will be treated in some detail in the following section on mesoscale flows. The cornerstone of this subject seems to be the huge body of literature on the theory of inviscid mountain waves. There have in fact already been several reviews of this subject (e.g., Queney et ul., 1960; Miles, 1969; Musaelyan, 1964; Eliassen, 1974; Vinnichenko et ul., 1973; Kozhevnikov, 1970), and the existence of these somewhat lessens the demands on this section. In response to this we will back off and attempt a less detailed overview of the subject. A more detailed discussion will be necessary only on matters of continuing controversy or points that have only recently been clarified. It is also important to keep in mind that a theoretical understanding of the inviscid problem is probably not sufficient. The interaction of the inviscid flow with the turbulent boundary layer and the turbulent regions of wave breakdown aloft could completely change the flow just as in aerodynamics where the separation of a thin boundary layer completely invalidates the potential flow solution. More on this point later.

-

-

2. I . B i r o y n c y Forces

In order to understand the transition from small-scale irrotational flow to larger scale, buoyancy-dominated hydrostatic flow, we examine the

THE INFLUENCE OF MOUNTAINS O N T H E ATMOSPHERE

91

vertical momentum equation for an inviscid fluid Dw Dt

dp

p-=---

dZ

Pg

Consider a parcel of fluid of density p F ,surrounded by a fluid of density p . If the surroundings are not accelerating, the pressure field will be hydrostatic (2.2a) (2.2b)

P

=

P o - Pgz

where p o is an arbitrary reference pressure. Any object placed in such a linear pressure field feels a net force equal to its volume times the pressure gradient [this follows from the derivation of (2.1) or from Archimedes Law]. The net force on the parcel is the sum of the net pressure force and the gravitational force (2.3)

p g = PPg = g(P - PP)

and this is the so-called buoyancy force. If the parcel is less dense than its surroundings, the buoyancy force will act upward and vice versa. One important property of the buoyancy force is that it can create vorticity in the fluid by applying a torque about the center of mass of a fluid parcel. It is well known that the gravitational force cannot create vorticity as it is a “potential” force field and always acts through the center of mass. The buoyancy force, however, includes a pressure force term and it is this, that can create vorticity. The alert reader will have already noticed a weakness in the above parcel agreement. If there is a net buoyancy force acting on the parcel then it will be accelerating and the fluid nearby the parcel will also be accelerating to keep out of the way. Thus our assumption that the surrounding fluid is hydrostatic is incorrect and the clear definition of buoyancy force fades slightly. If the parcel could be persuaded to hold its shape (e.g., a sphere), the effect of the accelerations could be easily accounted for by using the hydrodynamic theory of “added mass” but this too is an abstraction. The best way to resharpen our view of buoyancy is to consider its effect on a consistent field of fluid motion, and this will be done later when we consider the motion field induced by mountains. The generation of density differences can occur in two ways, either by a local heating or cooling, or by moving an air parcel from one environment to another. In the latter case, the resulting density variation can be

92

RONALD B. SMITH

associated either with the differing properties of the "source" environment or with density alterations during the displacement, for example, adiabatic expansion. For either of the above mechanisms (i.e., heating/ cooling or displacement) the resulting density variation can be computed easily by assuming that the parcel has come to pressure equilibrium with its new environment. Any pressure imbalance would be quickly eliminated by the generation and propagation away of acoustic waves. The potential temperature (2.4)

0 = T(P/Po)"

( P ois an arbitrary reference pressure, K is the ratio of R / c , for the gas in question) is especially useful as it is conserved during adiabatic lifting or sinking. Any parcel A having a higher potential temperature than some remote environment B (0, > 0,) will, when brought into environment B (i.e., brought to the pressure P,),be warmer ( T , > T B )and less dense ( p A < p,) than its environment and will experience an upward buoyancy force. This parcel argument can also be used to show (as is done in most meteorology texts) that an environment in which potential temperature increases with height ( d 0 / d z > 0) is stable since any vertical displacement will result in a restoring buoyancy force.

2.2. The Theory of Two-Dimensional Mountain Waves 2.2.1. The Governing Equations. The theoretical description of mountain waves begins with the following set of equations (restricted here to two dimensions): the horizontal momentum equation:

the vertical momentum equation:

Dw dp P z = - z = - P g the equation of continuity: (2.7)

DP -

-pv.

Dt

a n equation describing adiabatic, reversible changes (derived f r o m the first law of thermodynamics, the definition of specijk heat, and the

T H E INFLUENCE OF MOUNTAINS O N T H E ATMOSPHERE

93

perfect g a s law):

(2.8)

-DP =

Dt

c 2D - ,p Dr

c2=yRT

the perfect gas law:

(2.9)

p = pRT

With these equations we can examine the small perturbations produced by a mountain on a basic state of horizontal hydrostatic flow described by o(z), T(z)

specified

W ( z )= 0 and p ( z >,p ( z ) determined from (2.10)

dpldz

=

-pg

and (2.11)

p = pR?'

Each variable will be represented as a sum of the basic value and a perturbation

(2.12)

u(x,2 )

=

V(Z)+ u ' ( x , z )

w(x, 2 )

=

0

P ( X , 2) =

+ w ' ( x ,z )

Is(z) + P ' k z )

P(X, 2) = P(Z)

T ( x ,t ) =

+ P ' b , z)

T ( z )+ T'(x,z )

We have assumed further that the flow will eventually come to a steady state and it is this state that we seek. Upon substituting (2.12) into the governing equations and linearizing, we obtain the following equations for the perturbation quantities: From (2.5): (2.13) From (2.6): (2.14)

94

RONALD B . SMITH

From (2.7): (2.15)

From (2.8): (2.16)

which can be written as

This last equation warrants special attention as it describes the formation of density anomalies which in turn [through Eq. (2.14)] produce buoyancy forces. The left-hand side (a) is the rate of change of density encountered by an observer moving horizontally downstream at a speed U ( z ) . The right-hand side represents the causes for the observed density variations. Term (b) represents the lifting of denser air into a less dense environment, but this is strongly modified by term ( c ) , the adiabatic expansion of the parcel as it is lifted. Terms (b) and (c) can be combined to read (2.18)

1 d6

6 dz - 0

which is a measure of the static stability. The final term (d) is a correction for any lack of pressure equilibration. For fast acoustic waves this term is of central importance, but for lower frequency motions, such as considered here, it is negligible. That is not to say that there are no pressure variations but only that these variations are not important in the generation of density anomalies. As shown by Queney et al. (1960) the retention of term (d) substantially complicates the analysis, but if U 2 E2(e.g. lo2 6 3002) it many be neglected. In this limit the continuity equation is simplified [substitute (2.17) into (2. IS)] to (2.19)

so that the divergence in the velocity field is clearly associated with the adiabatic ascent of air parcels. The previous equations can be reduced among themselves to obtain a

THE INFLUENCE OF MOUNTAINS O N T H E ATMOSPHERE

single equation for the vertical velocity w (x, z )

(2.20)

w:,

95

t,)

+ w ; , - s w ; + (as -2 + m -U U

wt=o

with the subscripts denoting differentiation. The coefficient by

is defined

(2.21) and has been called the heterogeneity by Queney (1947). is not related to the generation of buoyancy forces as it does not include the effect of adiabatic expansion but instead describes the effect of density variations in the divergence of the velocity field [Eq. (2.19)] and the vertical variation in inertia in the momentum equations (2.13) and (2.14). The first term involving in (2.20) cannot be neglected even if is small as it will always result in an amplification of the disturbance in the far field. It can be neatly accounted for by introducing the new dependent variable

(2.22)

B = [p(z)/p(0)]”2wt

The square of this new variable is proportional to the energy of the wave disturbance (see Eliassen and Palm, 1960). Equation (2.20) then becomes

(2.23)

B ,,

+ B,# + P ( z ) B = 0

with pg su 1 p ( z ) = T+ 2- - s2 u 2 u 4

u,, + -1 s, v

2

U

This equation is the central tool for theoretical studies of small-amplitude, two-dimensional mountain waves. In practice, the coefficient P ( z ) is usually dominated by the first term, that is, the buoyancy force term p g / U 2 : although in regions of strong shear the term U J U may become important. The neglect of the terms is equivalent to making the Boussinesq approximation-that density variations are only important as they effect the buoyancy. In this case Eq. (2.23) may be easily interpreted as a vorticity equation with (upon multiplying through by 0 ) U ( w , , + w,,) being the rate of change of vorticity following a fluid particle; p g w / U being the rate of vorticity production by buoyancy forces: and -Urzw being the rate of production of perturbation vorticity by the redistribution of background vorticity ( - U z ) . In the case of P ( z ) = 0, we have

(2.24)

V2B

=

0

which states that the flow is irrotational.

96

RONALD B . SMITH

2.2.2. The Flow over Sinusoidal Topography. It is helpful at this point to examine the flow of a semi-infinite stratified atmosphere over sinusoidal topography (see Queney, 1947). The choice of sinusoidal topography simplifies the solution, thereby clarifying the underlying physics, and also paves the way for the use of Fourier methods in the next section. Consider the flow of uniform incoming wind speed U ( z ) = U and Brunt-Vaisala frequency N 2 = fig, over terrain described by z = h ( x ) = h, sin kx, where h, and k = 2rr/wavelength describe the height and spacing of the ridges. At the ground the flow is assumed to follow the terrain so that the streamline slope equals the terrain slope (2.25)

W

u

W‘

-

-

U + u’

dh dx

at z

=

h(x)

For small amplitude topography (and disturbance u ’) this simplifies to (2.26)

w ’= G

=

U(dh/dx)

at z

=

0

so

G(x, 0) = h,k cos kx

(2.27)

We will look for solutions (here using real functions and variables) of the form (2.28)

G(x, z ) = & ( z ) cos kx

so the equation for these

#IJ

(2.29)

I#Jzz

+ I#J2(z)sin kx

functions becomes from (2.23)

+ (1’

-

k2)+ = 0

The k in (2.29) will be the same as the terrain wave number in (2.27), in order to satisfy (2.27). In this equation the sign of the term in parentheses is of central importance. We must consider two cases: first, the case of closely spaced ridges (together with weak stability and high wind speed) such that (2.30)

k2> l2

For example, if N and U are chosen to have typical values of 0.01 sec-’ and 10 d s e c respectively, then condition (2.30) will be satisfied for topographic wavelengths A = 2rr/k < 6.3 km

In this case the term in parentheses in (2.29) is negative and the solution is (2.31)

+(z) = A exp[(k2 - 12)1’2z]+ B exp[-(k2 - 12)1’2z]

T H E I N F L U E N C E OF MOUNTAINS O N T H E ATMOSPHERE

97

The first term represents an unlimited growth of the disturbance energy away from the terrain, which should be considered as source of the disturbance. This runs counter to our intuition and to laboratory experience and must be regarded as unphysical. Therefore we set A = 0 and the solution becomes [from (2.3 I ) , (2.28), (2.27)]

(2.32)

G(x,

z)

=

h,k exp[-(k2 - f2)’/*z] cos kx

The other variables u ’ , p ’ , T , p can be easily determined from &(x, z ) by using the original equations (2.13)-(2.17). The streamline pattern corresponding to (2.32) is shown in Fig. la. The other possibility is

k2 < f2

(2.33)

which corresponds to more widely spaced ridges or alternatively, greater stability and slower wind speed. Here the term in parentheses in (2.29) is positive and the solution is

(2.34)

+(z)

=

A sin(f2 - k2)1/2z+ Bcos(f2 - k2)1’2z

Combining this with (2.28) and using a trigonometric identity gives the

a

.

FIG. I . The steady inviscid flow over two-dimensional sinusoidal topography. (a) Little or no influence of buoyancy, U k > N . The disturbance decays upward with no phase line tilt. (b) Strong buoyancy effects, U k < N . The disturbance amplitude is constant with height while the lines of constant phase tilt strongly upstream.

98

RONALD 9 . SMITH

vertical velocity in convenient form +(x,

Z) =

C cos[kx

+ (1’

-

k2)”2~]

+ D cos[kx - (1’ - k’)”’ 2 1 + E sin[kx + (1’ - k2)”2z] + F sin[kx - (Z2 - k2)”’z]

(2.35)

where the upward and downward propagating waves are now recognizable. The lower boundary condition (2.27) requires E + F = 0 as well as

C + D = h,k

(2.36)

and as before, the remaining indeterminacy between E and F , and C and D requires the use of an upper boundary condition. As shown by Eliassen and Palm (1960), disturbance energy can be transferred from one region of the fluid to another by doing work at the boundary between the two regions. In an inviscid electrically neutral fluid, as we have here, this work is done by pressure forces acting together with displacements across the boundary. Thus the vertical flux of energy across a horizontal surface is (2.37)

Eliassen and Palm also show that for U 2 e C2 the vertical flux of energy, averaged over a wavelength, can be written in terms of the field of vertical velocity (2.38)

A comparison of this with the solution (2.35) shows that the C and E terms, with phase lines tilting to the left, produce an upward energy flux (i.e., p ’ in phase with w’).The opposite is true for the D and F terms. Physically we regard the irregular terrain as the source of the disturbance energy so in the absence of energy production or reflection aloft, terms D and F must be regarded as unphysical and we set D = F = 0. This gives E = 0 and C = h,k so that (2.39)

+(x,

Z) =

h,k cos[kx

+ (1’

- k2)”’~]

The streamlines for this are shown in Fig. lb. Note that in the above argument it is not sufficient to require that the net energy flux be upward, as this only demands C + E > D + F . Instead we must require that

T H E INFLUENCE OF MOUNTAINS O N TH E ATMOSPHERE

99

there be no components of the flow which radiate energy downward. This “radiation condition” will be discussed again in the next section. The difference between the two types of flow k 2 > l 2 or k 2 < l 2 is striking and deserves discussion. In the case of closely spaced topography the flow is qualitatively similar to irrotational flow (i.e., I * = 0 ) in that the phase lines are vertical and the disturbance decays with height. This flow is inherently nonhydrostatic as the w r s s term in (2.20) carries the influence of the vertical accelerations and it is this term which allows the disturbance to decay vertically. In fact there is a crude balance between the vertical decay of the pressure disturbance and the vertical accelerations. As k 2 + l 2 this balance is increasingly altered by buoyancy forces and the vertical penetration of the disturbance increases. When k 2 < l 2 the intrinsic frequency of the motion U k (i.e., the frequency experienced by a fluid parcel moving through the stationary pattern) is less the N , the Brunt-Vaisala (or buoyancy) frequency, and internal gravity waves are therefore possible. These waves propagate vertically and thus the disturbance does not decay upward. The phase lines tilt forward into the mean wind, and as we have seen this is connected with the propagation of energy vertically away from the topography that produces the wave. Vertical accelerations play only a modifying role in this wave motion. For the case l 2 B k 2 we can ignore the k 2 term in (2.29) and this is equivalent to making the hydrostatic assumption in (2.14). In this case the vertical wave number of the disturbance is (2.40)

(12

-

k2)1/2

1

1

which depends only on the characteristic of the airstream, not on k . Note that with l 2 > k 2 the flow near the hills is asymmetric with low speed and high pressure on the windward side and high speed and low pressure on the leeward side. Thus there is a pressure drag on the ridges and this momentum is transported vertically by the waves. The amplitude of this “wave drag” averaged over a wide area is (2.41)

p u)w)= p U 2 k ( i 2- k 2 ) 1 / 2 h &per unit area

2 . 2 . 3 . Isolated Topography. Although much of the interesting physics of mountain waves was captured in the foregoing constant 12, sinusoidal terrain model, there are still new concepts which arise in the flow over a single ridge and still more when we allow P ( z ) to vary. Following Lyra (1943), Queney (1947), and Queney et a / . (1960), we express all disturbance variables in terms of a one-sided Fourier integral. For example, the

100

RONALD B. SMITH

vertical velocity is written as (2.42)

@ ( k , z ) e f f k Zdk

$ ( x , z ) = Re 0

substituting this into (2.23) gives an expression for $ ( k , z )

w,,+

(2.43)

[12(z)

- k2]@ = 0

The lower boundary condition (2.26) becomes

~ ( k0), = U(O)iki;(k)

(2.44) where

( k ) is the Fourier transform of the mountain shape,

.

,m

(2.45) For the case of l 2 = const, the solution to (2.43) is for k 2 > 1'

+(k, z )

(2.46)

=

w ( k , 0 ) exp[-(k2

- 12)1/22]

and for k 2 < l 2 (2.47)

+ ( k , z ) = w ( k , 0) exp[i(12 -

k2)1/22]

In each case the arbitrariness in the sign of the exponent allowed by (2.43) has been eliminated by using an upper boundary condition. In the evanescent case the solution is assumed to decay rather than grow as z + 00, while in the propagating case the positive sign is chosen in the exponential so that the phase lines tilt upstream and energy is propagated upward. In this case of constant U , it is slightly more convenient to solve directly for the vertical displacement of a streamline 7 ( x , z ) where (2.48)

w' = U(dq/dx)

Then using (2.42), (2.44), (2.46), and (2.47) 1

q ( x , z ) = [ p o / p ( z ) ] 1 ' 2Re[

(2.49)

+

h ( k ) exp[i(12 - k 2 ) 1 ' 2 z exp(ikx) ] dk

m

h ( k ) exp[-(k2

- 12)1/2z]exp(ikx)

dk

For the purpose of illustration it has become standard practice (after Queney, 1947) to consider a bell-shaped mountain described by (2.50)

K ( x ) = h m u 2 / ( x 2+ u 2 )

T H E INFLUENCE OF MOUNTAINS ON TH E ATMOSPHERE

101

which has a particularly simple Fourier transform

h ( k ) = h,ue-kQ

(2.51)

The height of the ridge is h,, and N is a measure of its width. The function h ( k ) happens to be real because the ridge is symmetric. The type of flow predicted by (2.49) with (2.51) depends on the dimensionless quantity a / which is proportional to the ratio of the time it takes for a fluid particle to cross the ridge, to the period of a buoyancy oscillation 2 r / N . Even with the choice ( 2 . S O ) the exact evaluation of (2.49) is readily done only for the two limiting cases N / 9 1 or a / e 1 . For the narrow mountain, weak stability, and strong winds, a1 is small and the first integral in (2.49) becomes small; while the second becomes

(2.52)

[ I,’

~ ( x z,) = [po/p(z)]1/2 Re h,a

e-ka e - k r e i k x

and taking the real part 112

(2.53)

h,a(a

+ z)

Note that as z + 0, 7 (x, z ) -+ h ( x ) given by (2.50) as it should. In this a1 G 1 limit, the buoyancy force is not important and the flow is irrotational. This same flow field could have been constructed from potential flow theory by placing a doublet slightly below the ground. The streamline pattern is shown in Fig. 2. In the opposite extreme case, a1 + 1, buoyancy effects dominate and the flow is hydrostatic. Mathematically, h ( k ) is small in the range 1 <

FIG.2. The steady flow of a homogeneous fluid over an isolated two-dimensional ridge, given by (2.53). From the spacing of the streamlines, it is evident that the highest speed and the lowest pressure will occur at the top of the ridge.

102

RONALD B . SMITH

k < w, so the second integral in (2.49) makes no contribution. Then ~ ( xz) , = [ p , , / p ( ~ ) ] Re "~ (2.54) = [ ~ o / p ( z ) l " ~ h mRe a

taking the real part

(2.55)

V ( X , Z) =

s]

[

1/ 2

hma

-I -a

3

+ i(lz + k x )

( a cos l z a2

- x sin fz)

+

x2

This flow (shown in Fig. 3) is best described as a field of nondispersive

-50

0

50 KM

X

FIG. 3. Buoyancy-dominated hydrostatic flow over an isolated two-dimensional ridge, given by (2.56). The disturbance is composed of vertically propogating internal gravity waves of the sort shown in Fig. Ib. The evident upstream tilt of the phase lines indicates that disturbance energy is propagating upward away from the mountain. The maximum wind speed and minimum pressure occur on the lee slope of the ridge. The mountain height h , = 1 km, the half-width a = 10 km, the mean wind speed U = 10 misec, the BruntVaisala frequency N = 0.01 sec-', and the vertical wavelength L , = 2 n U / N = 6.28 km. (From Queney, 1948.)

T H E I N F L U E N C E OF M O U N T A I N S O N T H E ATMOSPHERE

103

vertically propagating waves. The flow is periodic in the vertical so that at z = 7r/l the streamline shape is the inverted ridge shape h(x) and at z = 2 r / l the ridge is reformed upright. This is true for any ridge shape, not just h (x) given by (2.50). The upstream tilt of the phase lines, which we required of the individual Fourier components, is still very noticeable in the composite flow. This asymmetry to the flow, which is associated with the vertical propagation of wave energy, has a number of implications. From the distance between the streamlines in Fig. 3 it is apparent that the wind speed is low on the windward slope of the ridge and faster on the leeward slope. From Bernoulli’s equation, this requires a pressure difference across the ridge-high pressure upwind and lower pressure downwind. The primary reason for windward-side high pressure is the thickened layer of dense cool air just above the mountain, but this in turn is related to the radiation condition aloft. The pressure difference results in a net drag on the mountain which can be computed either as the horizontal pressure force on the mountain

(2.56)

D

=

1-1

P ’ ( x ,z

=

dh 0) - dx dx

or equivalently, as the vertical flux of horizontal momentum in the wave motion

(2.57) For the bell-shaped mountain, the drag per unit length is (using 2.55 and 2.56)

This momentum is transferred to a level where the wave breaks downa process no! included in the linearized model. Mountain wave drag is discussed in more detail by Sawyer (1959), Eliassen and Palm (1960), Blumen (1965), Miles (1969). and Bretherton (1969) among others. Direct measurements of mountain drag using (2.56) or (2.57) have been attempted by Lilly (1978) and Smith (1978a). The increase in wind speed on the lee slope is an especially interesting facet of the model. It has been invoked as an explanation for severe downslope winds found occasionally in the lee of mountain ranges. Frequently the strong lee-side winds are warm and dry, replacing colder air, and in this case it is proper to call the wind a fohn o r foehn (chinook is the local name ir? the northwest United States: it is called the Santa Ana

104

RONALD B . SMITH

in California). There has been a good deal of work concerning the relationships between the fohn, the downslope wind storm (these two are not necessarily the same), and the generation of mountain waves. Descriptions of the fohn phenomena can be found in Defant (1951), Stringer (1972), Godske r t al. (1957), Brinkman (1970, 1971), Holmes and Hage (1971), and Beran (1967). The application of mountain wave theory to this problem can be found crudely treated in Holmes and Hage (1971), Beran (1967), and in more complete form in Vergeiner (1971, 1976) and Klemp and Lilly (1975). Other theories of the generation of strong leeside winds, such as the hydraulic theories of Kuettner (1958) and Houghton and Isaacson (1970) and the trapped lee-wave theories (see Holmes and Hage, 1971), appear to be less well founded-but will be discussed later. It is clear from the discussion above that the qualitative description and interpretation of mountain wave theory hinges crucially on the use of the correct radiation condition aloft. In 1949, R. Scorer wrote what is now considered to be a classic paper describing the physics of trapped lee waves (a subject that will be treated later in this review). As a sidelight to his treatment he computed the vertically propagating wave field-just as we have done here-but using an incorrect radiation condition. In the controversy that followed (see Scorer, 1958; Corby and Sawyer, 1958; Palm, 1958a) the physical arguments of Lyra (1943) and Queney (1947) involving friction, of Palm (1958a) concerning the approach to steady state, and of Eliassen and Palm (1954, 1960) concerning the vertical flux of energy, clearly carried the day. The use of the radiation condition prohibiting downcoming energy is now standard among researchers of this subject. Unfortunately there is still some confusion on this point among researchers in other fields who wish to use the results of mountain wave theory. One of Scorer’s (1949) figures, showing downstream (and therefore incorrect) phase line tilt has been reproduced in reviews such as Queney et al. (1960), Vinnichenko et al. (1973), Kozhevnikov (1970), Stringer (1972), and Scorer (1978). Confusion on this point is also evident in studies of the fohn (Beran, 1967; Holmes and Hage, 1971), orographic rain (Atkinson and Smithson, 1974), and blocking (Scorer, 1978). Recently the wider understanding of the concept of group velocity and the favorable comparison of theoretical solutions with numerical and laboratory experiments have eliminated most of the remaining confusion. Scorer (1978), however, seems to cloud the issue by discussing the radiation condition as if it were only one of several equally plausible choices. The calculation of the inverse Fourier integral (2.49) for the solution of stratified flow over a bell-shaped ridge is more difficult in the case of

T H E I N F L U E N C E O F M O U N T A I N S ON T H E ATMOSPHERE

105

af - 1. In this case buoyancy forces are important but they do not dominate to the extent that the flow can be considered hydrostatic. This problem has been studied in detail by Queney (1947) using the asymptotic properties of Bessel functions and by Sawyer (1960) using numerical integrations. In this section we will apply directly the asymptotic technique of stationary phase to determine t h e nature of the flow far from the mountain. The second integral in (2.49) containing the evanescent components will rapidly tend to zero as z + because of the exponential decay of the integrand. At large 1 x 1 this integral will again tend to zero as the rapid oscillations in eikxwill cause cancellation of the contributions from the different wave numbers. The same kind of cancellation will occur in the first integral in (2.49) with one important difference. This integral can be written as

(2.59)

I

=

loffi

h(k)e'@"k'd k

where the phase function

(2.60)

$(k)

(f2

+ kx

- k2)1'2z

For the most part, with either large I x 1 or z , $ ( k ) is a rapidly varying function, but there is an obvious exception to this. Taking

(2.61) shows that $ ( k ) is approximately constant in the region near k * defined by (2.61)

(2.62) Thus, far from the mountain, at any specified point ( x , z ) , there is a range of wave numbers near k * given by (2.62), whose contributions to the disturbance will not cancel themselves out. The noncanceling wave number k * is constant along a straight line with slope given by (2.62), emanating from the origin where the mountain is located. The physical interpretation of this is that waves of that wave number k * are generated by the mountain and are propagating away in the direction given by (2.62). Note from (2.62) that as z > 0, k > 0 waves will only be found for x > 0, that is, downstream of the obstacle. To evaluate the integral (2.59) we expand $ ( k ) in a Taylor series near k *

(2.63)

$(k)

=

$(k*) +

3$kkk2

I06

RONALD B . SMITH

where k = k - k*. The only contribution to the integral (2.59) comes from k near k * so approximately (2.64)

I = h(k*) exp[i4(k*)]

I

exp[(i/2)4 ,d2]dk

--m

The definite integral in (2.64) can be simplified by the change of variable (this is equivalent to using the method of steepest descent) (2.65) with the result that

(2.66)

then

together with (2.45), (2.62), and 4 k k = - l 2 z / ( l 2 -k*2)3’2 . For the symmetric bell-shaped ridge (2.50), (2.67) becomes

e-k*a

L

cos ( l :

with k*(x, z ) given by k* = I / [ ( z / z ) + ~ 1 I 1 I 2 . This approximate form is not useful near the mountain or directly above the mountain (large z/x and small k ” ) because of the assumptions used to obtain (2.68). It does however clarify the nature of the nonhydrostatic waves which trail behind the vertically propagating hydrostatic waves discussed earlier in the a1 9 I case (see Fig. 3). As we look up along a sloping line of fixed z / x we find waves of constant wave number and decreasing amplitude. As we look further downwind at a given level (fixed z , decreasing z / z ) we find shorter waves with k approaching 1 (i.e., + 2 7 r U / N ) . The upstream tilt of the phase lines is obvious just as it was for the long hydrostatic waves, but the tilt decreases as we move downstream to shorter waves. The decrease in wave amplitude with height l / f i is associated with the fact

T H E INFLUENCE OF MOUNTAINS ON THE ATMOSPHERE

107

that the wave energy is progressively dispersed over a wider horizontal area. These results are consistent with the concept of group velocity. In a stagnant stratified fluid the dispersion relation for (time varying) internal gravity waves is (see, for example, Turner, 1973) (2.69)

cr2

- N2k2/lk

12,

I k I = ( k 2 + m2)1’2

where u,k , m are the frequency and horizontal and vertical wave numbers. The horizontal phase speed is (2.70)

CIB,= +N/(k

I

and the group velocity, which describes the direction and efficiency of wave energy propagation, is (2.71) In the study of steady mountain waves we are interested in the waves that first, have a component of their group velocity directed upward away from the mountain and second, have a phase velocity (relative to the fluid) equal and opposite to the mean flow U (see Fig. 4). Only in this way can the waves stand steady against the stream. This second condition requires that (2.72)

Cp, =

-u

FIG.4. A diagram illustrating the nature of steady mountain waves. The upstream phase speed of the wave is exactly equal and opposite to the freestream speed. The wave energy propogates upward and upstream relative to the air, but is advected into the lee by the mean wind. Relative to the mountain, the disturbance energy propagates upward and downstream.

108

RONALD B . SMITH

With this choice, and going into a mountain fixed reference frame, the vertical and horizontal components of the group velocity become

k U-

(2.73)

m

and

k2 U , m

The ratio of these two expressions is the slope along which a packet of waves produced at the mountain would propagate

(2.74)

slope = m / k

and using m = ( 1 2 - k2)1'2 this is exactly what was derived using the method of stationary phase (2.62). The purpose of the preceding analysis was to give a physical interpretation to the train by nearly periodic waves found aloft (Fig. 5). It is appropriate to consider these as a "dispersive tail" of nonhydrostatic waves with k less than, but not much less than 1. If the mountain is too broad and smooth to create any of these shorter

GROUND LEVEL PRESSURE AND WIND 1

:

X

FIG.5. The flow over a ridge of intermediate width ( a l = 1) where the buoyancy forces are important, but not so dominant that the flow is hydrostatic. The dispersive character of the nonhydrostatic waves (k less than, but not much less than N / U ) is evident as they trail behind the ridge. Parameters are as in Fig. 3 except a = 1 km. (From Queney, 1948.)

THE INFLUENCE OF MOUNTAINS ON T H E ATMOSPHERE

109

waves (af 9 1 ) then the flow will be as pictured in Fig. 3. It seems fair to call these trailing waves "lee waves" as they appear downstream of the mountain, but as we shall see there is another type of mountain wave for which this name is more suitable. 2.2.4. Trapped Lee W a i ~ s Scorer . (1949) pointed out that the theoretical description of the "dispersive tail" bears little resemblance to the long trains of waves which often cause lee-wave clouds in the lower atmosphere (see Fig. 6). By contrast with Fig. 5, these observed lee waves do not decay downstream but the amplitude of the disturbance does fall off rapidly with height. Scorer showed that under conditions

FIG.6. Photograph of clouds forming in the crests of trapped lee waves over northern Mexico, southern Arizona, southern New Mexico, and western Texas. Clouds originate at the Sierra Madre Occidental Mountains in Mexico. Wavelength near central cross is 21 km. Taken from TIROS VI orbit at 1706 GMT, 15 November 1962. (From Nicholls, 1973.)

110

RONALD B . SMITH

where the parameter 12(z) decreased rapidly with height, the atmosphere could support a new kind of mountain wave-the trapped lee wave o r resonance wave. This realization triggered a decade of active theoretical research on the effect of l 2 variations and provided a framework for understanding the atmospheric observations. As before, we consider the equation governing the field of vertical motion w ( x , z ) = [ p O / p ( z ) ] ’ ” % ( x , z )

(2.75)

citss

+ G Z Z+ P ( z ) G = 0

with

(2.76) and use the Fourier representation

(2.77)

G(x, z) = Re

r

G ( k , z ) e i k Edk

Then W must satisfy

(2.78)

w,,+ [ P ( Z ) - k2]W = 0

The linearized lower boundary condition is

(2.79)

W ( k , 0) = ikU0I?(k)

where I? ( k ) is the Fourier transform of the mountain shape. We wish to consider waves that do not propagate vertically but instead obey the upper boundary condition

(2.80)

G(k,z)+O

as

2-03

Combining (2.77) and (2.79) gives

where ratio W ( k , z ) / W ( k , 0) is to be determined from (2.78) and (2.80). In general, with a nonconstant 1 2 ( z ) , W ( k , z ) is so complicated that it is impossible to perform the integral (2.81) exactly. Nonetheless it is still possible to obtain an expression for the flow far from the mountain by using asymptotic techniques. In fact we can proceed quite far without having to specify the mountain shape h (x) or the structure of the incoming atmosphere N (z), I / ( z ). According to the Riemann-Lebesgue lemma the integral (2.81) will go to zero as x -+ 30 if the kernel is well behaved. It follows that the only nonvanishing disturbance as x + = must come

T H E I N F L U E N C E OF M O U N T A I N S O N T H E ATMOSPHERE

111

from portions of the integral nearby to where w ( k , 0) = 0. Just as in the previous section, this allows (2.8I ) to be approximated by

where k , is any of the special wave numbers such that

(2.83)

G ( k , , 0) = 0

and nearby to k,, w(k, 0) has been expressed as a Taylor series

(2.84) with k = k - k,. The integral in (2.82) can be evaluated by closing the path of integration at infinity and then computing the residues of the singularities inside. For x > 0 the contributions from the extended path will vanish if the path is taken upward enclosing the first and second quadrants. For x < 0 it must go downward enclosing the third and fourth quadrants. The final result can now be seen to depend crucially on whether the integral path along the real axis is taken just over or just under the singularity at k,. The former choice will result in waves upstream of the mountain and none downstream. The latter choice will put all the waves downstream. Scorer correctly chose to put the wave downstream, and the physical basis for this choice will be discussed shortly. We have then, at large I x 1

(2.85)

w(x, z) = 0

or taking the real part

upstream

downstream

112

RONALD B. SMITH

where

I h(k,)l

(2.88)

=

[h(k,) * h(kn)*]”2

The phase angle $ given by

4

(2.89)

= -tar1(&

r)

is zero if the mountain is symmetric making h ( k , ) real. The contributions to the amplitude (i.e., term in braces) in (2.87) are physically clear (see Corby and Wallington, 1956) except for the (dw/ d k ) factor. This term can be reexpressed by multiplying (2.78) both for k and k,, by w ( k , ) and w ( k ) , respectively, and integrating from z = 0 to m. Integrate by parts once, subtract the two equations, and take the limit as k + k , to obtain (2.90)

= k”

2 k , 3~,

i3 =

lo

w 2 ( k , , z) dz (a w/az 1,=o)2

In this way the sensitivity of the airstream to topographic forcing from below is represented solely in terms of w (k,, z ). Computational experience has shown (see Smith, 1976) that the length Z is roughly the distance from the ground to the height where the lee-wave energy is concentrated, which in turn is close to the level of maximum f 2 ( z ) .Thus lee waves associated with a stable layer near the ground are sensitive to topographic forcing. Other interpretations of Z are also possible. The remaining part of the theoretical lee-wave problem is the determination of the lee-wave wavelength (or k , ) and the associated vertical structure function G (k,, z ). The simplest cases in the literature are the two-layer model of Scorer (1949), the Couette linear shear flow model (1 l / z ) of Wurtele (see Queney ef al., 1960), the exponential model ( I P )of Palm and Foldvik (see Foldvik, 1962), and the sharp inversion case (see, for example, Smith, 1976). With the advent of computers, more realistic P ( z ) profiles can be treated (see Sawyer, 1960; Vergeiner, 1971; Smith, 1976). Theoretically there is one obvious requirement for the existence of true trapped lee waves. Looking at (2.78) we note that at any level where k > I the vertical accelerations dominate over the buoyancy forces and the function k ( k , z ) will behave exponentially. Conversely, where k < 1, buoyancy dominates and k ( k , z ) will be trigonometric-always curving back toward zero. If k < f at high levels, the upper boundary condition a ( k , a) = 0 cannot be met as the solution will continue to oscillate. If k > 1 at all levels, the exponential form of the solution cannot satisfy the 6 ( k , z) = 0 condition at both z = 0 and z = m. Thus the necessary

-

T H E INFLUENCE OF MOUNTAINS O N THE ATMOSPHERE

113

(but not sufficient) condition for a wave number k to be a k, with W ( k n , 0) = 0, is that k 2 > 1 2 ( z ) aloft but k 2 < P ( z ) over some lower levels. This is the basis for Scorer’s condition fi)r the existence o j lee W U I Y ’ S , namely, P ( z ) decreasing strongly a l o f t . In practice this is associated with [looking now at (2.76)] a strongly stable layer in the low atmosphere and/or strongly increasing wind speed with height. There are two helpful ways to think about lee waves. The first is to consider the analogy with the standing waves found downstream of an obstacle in a running river or stream. The stable air-water interface is equivalent to the stable layer required by lee-wave theory. With this analogy in mind we can attempt to understand why the waves are found only downstreurn of the obstacle. Just as in the vertically propagating waves, the key is to consider the obstacle as the source of the wave disturbance. For water waves it is known that the phase velocity (i.e., speed of crest motion) is greater than the group velocity (rate of energy propagation). Now in a standing wave the phase velocity must be equal in magnitude and opposite in direction to the mean stream U . It follows that the speed U will be greater than the group velocity and the transport of wave energy will be dominated by the advection due to the mean velocity U . Then, with the obstacle acknowledged as the source of the wave energy, the wave energy must be found downstream. Note that this last result is not universally true. Some types of wave motion, for example capillary waves on the surface of a liquid, have a larger group velocity than phase velocity, and standing waves of this sort will be found upstream of the obstacle. Another way to help understand the lee wave is to consider the movement of wave packets in an atmosphere with a decreasing 1 2 ( z ) (see Bretherton, 1966). In the stable lower atmosphere the generated wave (with k 2 < 1 2 ) propagates up and to the right as discussed before. Eventually it reaches a level where k 2 > 1 2 . The wave cannot propagate in such a region, and the wave energy is totally reflected back toward the Earth. The wave energy bounces up and down between the ground and the low 1 2 ( z )region aloft, forming a standing wave pattern in the vertical (i.e., no phaseline tilt). Both the vertically propagating waves and the trapped lee waves can occur together. An example of this is shown in Fig. 7. The clear distinction between these two wave types begins to fade as we consider atmospheres with more complicated structure. Two examples of mountain waves with intermediate qualities were found in the solutions of Sawyer (1960), and both of these are associated with the presence of a stable stratosphere aloft. If l 2 in the stratosphere is greater than anywhere below, Scorer’s criterion cannot be satisfied and trapped lee waves, in

I

I

1

N

-

)

y

\

.. .. ... .... ... .,.. ... ... .... .......e . ....

0 N

0 D

9

0

I I

E

k

355 N

U

D

W

0

N

4 9

THE INFLUENCE OF MOUNTAINS O N THE ATMOSPHERE

115

the strict sense, are impossible. Still, if there is a thick middle-upper tropospheric layer with small f 2 above a stable layer, it is possible to have a partially trapped or “leaky” lee wave with a structure very similar to that obtained without a stratosphere (see Corby and Sawyer, 1958; Bretherton, 1969). Such a wave will, however, decay slowly downstream, and in the stratosphere the disturbance will have the form of a smallamplitude vertically propagating wave with a nonzero energy flux. Mathematically the singular wave number k, has moved slightly off the real axis, and this allows for the decay downstream and for the vertical propagation of wave energy through the low l 2 middle-upper troposphere. The second way in which a stratosphere can result in a wave with intermediate characteristics does not require a low-level stable layer. If the change in stability across the tropopause is abrupt, a vertically propagating wave will be partially reflected back toward the Earth. The reflected wave energy will continually rebound between the Earth and tropopause, losing a certain fraction of its energy each time, because the downward reflection is only partial. If this process occurs in the dispersive part of the spectrum (i.e., k 2 f 2 ) the result will be a periodiclooking lee wave which decays downstream.

-

2.2.5. Other Ejyrcts of’ Vciriciblr 1 2 ( z ) . One effect of P ( z ) variation has been described in the previous subsection-the trapping of waves in a high f 2 waveguide. Others will be considered in this subsection. If the Scorer parameter P ( z ) is a slowly varying function of z, then we can expect to find a solution to (2.78) in the form (2.91)

G ( k , z ) = a ( k , z)eim(k*Z’

where 4 is the phase function and a ( k , z ) is a slowly varying amplitude function (Bretherton, 1966). Substituting (2.91) and (2.78) gives, for the rapidly varying part

(2.92) nr

-a

(2.93)

+ ( k , z)

*

4;

+ [P(Z)

-

k2]a = 0

lo L‘

=

[ P ( z ’) - k2]1’2dz

FIG.7. The flow over a ridge where the background wind speed and stability vary with height. High above the mountain, the disturbance is composed of vertically propagating waves with tilted phase lines as in Fig. 5. In the lower atmosphere, trapped lee waves are evident extending well downstream. These waves have no phase line tilt. Trapped lee waves occur in this case as the Scorer condition-that P ( z ) decrease strongly with heightis satisfied by the incoming flow. (From Sawyer, 1960.)

116

RONALD B . SMITH

and for the slowly varying parts (2.94)

2a,4, +

4 z 2

=

0

or

(2.95)

a 2 4 z= a 2 [ 1 2 ( z )- k2]1’2 = const

For long hydrostatic waves the above relations are simplified to

I, 2‘

(2.96a)

+(z) =

l ( z ’ ) dz’

and (2.96b)

a21 = const

An alternative way to derive these relationships is to use the result of Eliassen and Palm (1960) that the vertical flux of energy is proportional to U ( z ) , together with the expression for the energy flux in a upwardgoing wave in a uniform medium [see the expression for the group velocity (2.71)]. From this second derivation, the special characteristics of the slowly varying medium are more clearly revealed. Locally, the wave must behave as it would in a uniform medium, and the changes in l 2 must be so gradual that no down-going waves are produced by reflection. A further condition for the validity of (2.96) is that l 2 > k 2 everywhere. If l 2 drops below k 2 , strong reflection (and trapping) can occur even though l 2 is slowly varying. The solution in the neighborhood of these l 2 = k 2 “turning points” can be expressed in terms of Airy functions. Qualitatively [from (2.96b)I the amplitude of the vertical velocity a in a mountain wave is reduced in regions of strong static stability (e.g., the stratosphere) and increased in regions of high wind speed (e.g., the jet stream). The amplitude of the vertical displacement [ - a / U(z)] is reduced in regions of high wind speed as the fluid particles spend less time in the updraft and downdraft regions. This behavior is evident in the measurements of streamline patterns over the Rockies by Lilly and Zipser (1972). The wave activity appears strong above and below the jet stream and weakest in the jet core, in spite of the fact that the vertical energy flux is largest there. When appreciable changes in 1 2 ( z ) occur over a height comparable o r smaller than a vertical wavelength, partial reflection will occur. Eliassen and Palm (1960) computed the fraction of energy reflected by discontinuous changes in 1 2 . Blumen (1965) and Klemp and Lilly (1975) have shown how partial reflections from the tropopause can considerably alter mountain drag and the severity of the lee side downslope winds. These phenomena will either be amplified or attenuated depending on the height

T H E I N F L U E N C E O F M O U N T A I N S O N T H E ATMOSPHERE

1 I7

to the tropopause in relation to the vertical wavelength of the mountain wave. A most interesting and important situation occurs when the component of the mean wind perpendicular to the ridge [i.e., U ( z ) ]reverses above some height z = D. Near z = D the mean wind U ( z ) approaches zero, sending the Scorer parameter to infinity. Booker and Bretherton (1967) were able to show that a small-amplitude wave would be absorbed at such a critical level if the local background Richardson number were large. Bretherton rt a l . (1967) showed experimentally that little, if a n y , wave energy reaches the region of reversed flow aloft. Jones and Houghton (1971) compute the time development of the mean flow as it is influenced by the absorption of wave momentum. The acceleration of the mean flow near z = D appears to decrease the Richardson number locally. Breeding (1971) and Geller rt al. (1975) have studied the local structure of the critical levels. Their results, and the estimates of Smith ( 1977) using the "slowly varying" solutions (2.96), suggest that nonlinearity may be important at or just before the critical level even if the incident wave is of small amplitude. The possibility of getting significant reflection from a critical level was suggested by Breeding (1971), and this has been confirmed by the numerical experiments of Klemp and Lilly (1978). This whole issue must be considered unsolved, and the simple results of Booker and Bretherton (1967) cannot be accepted yet as representative of real flows.

2.3. Obseriations of'Moiintriin Wai1r.s There have been a large number of observations of mountain waves and in particular lee waves. Reviews of some of these observations can be found in Queney Pt ul. (1960), Nicholls (1973), Musaelyan (1964), Vinnichenko et al. ( 1 9 7 3 ~and Yoshino (1975). A partial list of some of the most easily available studies is given in Table I. A number of authors have attempted to verify aspects of linear theory by comparison with observations. In regard to lee waves, the use of Scorer's criterion has proved successful, at least in a statistical sense, for predicting the occurrence of lee waves. On a case-to-case basis, however, there are still many discrepancies. The lee-wave wavelength has been used as a basis of comparison by many authors (see Corby, 1957; Corby and Wallington, 1956; Wallington and Portnall, 1958; Sawyer, 1960; Foldvik, 1962; Pearce and White, 1967; Berkshire and Warren, 1970; Vergeiner, 1971; Smith, 1976; Cruette, 1976). Such comparisons have been generally successful but not completely convincing because

118

RONALD B . SMITH

TABLEI . OBSERVATIONS OF MOUNTAINWAVES Lee

Wril~es

England Corby (1957) Cruette (1976) Foldvik (1962) Manley (1945) Starr and Browning (1973) United States (Appcr/achians 1 Colson c’t d . ( 1961) Fritz (1965) Fritz and Lindsay ( 1964) Lindsay (1962) Smith (1976) (Cascuc/e,s1

Fritz (1965) (Rockies)

Vergeiner (1971) Vergeiner and Lilly (1970)

(Sierras) Holrnboe and Klieforth (see Queney rt a / . 1960) Nicholls (1973) Viezee et a / . (1973) Scandinavia DOOs (1961) Foldvik (1954) Larrson ( 1954) Smirnova ( 1968) Alpine Region and France Cruette ( 1976) Forchtgott (1957, 1969) Gerbier and Berenger ( 1961) Kuettner (1958) South America DOOs (1962) Fritz (1965) Sarker and Calheiros ( 1974) Soviet Union Kozhevnikov ei a / . ( 1977) the Urals See Vinnichenko ei a / . (1973) Middle East and India De (1971) Doron and Cohen ( 1967) Mars Leovy ( 1977) (continued)

T H E INFLUENCE O F M O U N T A I N S ON T H E ATMOSPHERE

1 I9

TABLEI.--C'ontinuc~d Larger Scule Vertically Propcrgiiritig Wut,es

United States (Rockic,~) Lilly et a / . (1974) Lilly and Kennedy (1973) Lilly and Lester (1974) Lilly (1978) For other areas in the United States see Nicholls (1973) Canada (Rockies) Lester (1976) Orographic Clortds (in C e t ~ c ~ r r r l )

Conover ( 1964) Hallet and Lewis ( 1967) Ludlarn (1952)

the lee-wave wavelength seldom varies by more than a factor of two or three (i.e., 6 km to 20 km) and it is always difficult to obtain a 1 2 ( z ) profile at the same place and time of the lee-wave observation. The general tendency for the wavelength to increase with wind speed has been confirmed by observation (see Corby, 1957; Sawyer, 1960; Fritz, 1965; Cruette, 1976) and is now being used to estimate wind speed from satellite pictures of lee-wave clouds. Trains of lee-wave clouds have also been observed behind craters in the atmosphere of Mars and may eventually be used to estimate the wind speed and stability of that atmosphere. The comparison of predicted and observed lee-wave amplitudes is more difficult. The prediction requires knowledge of the size and shape of the mountain that is generating the wave, and in mountainous terrain this is not always obvious (see Smith, 1976). The empirical estimation of lee-wave amplitude cannot normally be done from wave cloud observation alone, but requires direct aircraft or balloon measurement. Such a comparison has been completed by Holmboe and Klieforth (see Queney et d.,1960), Vergeiner (1971), Foldvik (1962), and Smith (1976). The first two of these studies concerned very large mountains (Sierra Nevada and Front Range) where the linear theory would not be expected to hold. Foldvik studied the waves over complex terrain, and the wave source could not clearly be identified. Smith measured the waves produced by a low straight section of the Blue Ridge in the Appalachians, and reported amplitudes four times larger than theoretically predicted. This discrepancy is confirmed in the laboratory and explained by the early onset of nonlinearity in I2(z ) profiles with thin, strongly stable layers. Certainly the most extensive measurements of the longer, vertically

120

RONALD B . SMITH

propagating waves are those by Lilly and collaborators (1973, 1974, 1978) in the Rocky Mountain Front Range region. Using aircraft measurements (Fig. 8), the qualitative predictions of linear theory-that is, penetration of the disturbance to great height, forward tilt of the phase lines, downward flux of momentum-have been confirmed. Some observations of the breakdown of waves to turbulence have also been described. The great vertical extent of the disturbance is also evident in the formation of orographic cirrus (Ludlam, 1952) and mother-of-pearl clouds (Hallet and Lewis, 1967). The outstanding qualitative questions concern the degree to which the flow is two-dimensional and steady and the degree of upstream low-level blocking. The question as to the steadiness and the three-dimensional structure of the wave field can now be treated using the remote sensing methods of Reynolds et a / . (1968), Stan- and Browning (1972), and Viezee et a / . (1973). The theory of mountain waves also predicts mountain drag. Lilly and Kennedy (1973) have indirectly measured this drag by computing the vertical flux of momentum in the observed waves. They find that during i

FIG. 8. Cross section of the potential temperature field ( K ) along an east-west line through Boulder, as obtained from research aircraft on 1 I January, during a downslope windstorm in Boulder. To the extent that the flow is steady and adiabatic, these isentropes are good indicators of the streamlines of the air motion. Note that while the predicted vertically propagating nature of the disturbance is evident from its great vertical extent and from its tilted phase lines, the amplitude is much larger than predicted from linear theory (see Fig. 3). (From Lilly and Zipser, 1972.)

THE INFLUENCE OF MOUNTAINS O N THE ATMOSPHERE

121

severe downslope wind events, the mountain drag on the Colorado Front Range can be an appreciable fraction of the total surface drag around the Earth in a latitude belt. Direct measurements of mountain drag have recently been accomplished (Lilly, 1978; Smith, 1978) by recording the surface pressure on each side of a mountain. 2 .4, Tlz e Thw e - Dim ens ion a I Flo H o I sc r Iso luted Hills 1

2.4.1. Three-Dimensional Vertically Propagating Wat9es. All of the foregoing discussion has been concerned with the two-dimensional problem of flow over an infinitely long ridge. Most mountains on the Earth, however, are of a more irregular shape, and even the naturally occurring long ridges do have finite length. Furthermore it appears that there may be some fundamental theoretical differences between the two- and threedimensional flows. Thus for both practical and theoretical reasons, we must attempt to understand the three-dimensional mountain flow problem. There has been much less theoretical and experimental work on the three-dimensional mountain wave problem, and at the present time there are still a good many unanswered physical questions. One approach to the problem has been to extend the small-amplitude linearized theory to three dimensions. This approach has been used by Wurtele (1957), Scorer (1956), Crapper (1959, 1962), Trubnikov (1959), and Blumen and McGregor (1976) to study the orographic disturbance in an atmosphere with little vertical structure. Probably the most straightforward analysis is the study of Wurtele (1957; see also pp. 88-9 I in Queney et al., 1960). The field of vertical velocity w ( x , y , z ) is written as a double Fourier integral according to

(2.97)

w(x, y , z )

=

Re

II

% ( k , I , z)eMkZ+lu) d k dl

--m

For the case of constant Scorer parameter k , (2.98)

iv(k, I , z )

=

=

N/U

% ( k , I , O)eimr

where magnitude of the vertical wave number m is given by -.2

and the correct sign is chosen to prevent downward radiation; sgn(m) = sgn(k). As in the two-dimensional theory, the vertical velocity near the ground can be written in terms of the mountain profile z = h (x, y )

122

RONALD B . SMITH

according to

(2.99)

w(x, y , z

=

0)

=

U(dh/dx)

Wurtele simplified his problem by choosing topography in the form of semi-infinite plateau of height h with narrow width 2 b in the cross-flow (i.e., ” y ” ) direction. In this case the vertical velocity vanishes near the ground except near the origin x = y = 0 and

(2.100)

w(x, y , z

=

0 ) = 2 U h b 6(x) 6 ( y )

Even with this choice, the integral (2.97) is still intractable, and Wurtele resorts to the method of stationary phase to investigate the region far from the mountain; k , x , k , y , k , z all large. At the level k , z = 2 , the theory predicts that the regions of updraft take on a horseshoe shape and are located some distance downstream of the mountain. Wurtele points out the relationship between this result and the observation of horseshoeshaped wave clouds in the lee of Mt. Fuji (Abe, 1932). Crapper (1959, 1962) has extended the work of Wurtele by allowing for more realistic mountain shapes but is again forced to use asymptotic techniques which are valid only in the far field of flow. Crapper (1962) finds that the nature of the far-field flow depends in an intricate way on the presence of curvature in the mean velocity profile, but no clear physical explanation for this is evident. Trubnikov (1959) has a similar approach to the three-dimensional mountain flow problem, but his results are expressed only formally, in terms of unevaluated integrals. Scorer’s (1956) solutions for the stratified flow over an isolated mountain should probably be disregarded as the incorrect radiation condition was used. One fundamental difference between the two- and three-dimensional problems is the direction in which wave energy propagates away from the mountain. It was shown earlier that in two dimensions, as the mountain becomes wider and the flow more nearly hydrostatic, the group velocity (relative to the mountain) becomes directed vertically with the result that the wave energy is found directly above the mountain. This result does not carry over to three dimensions. Some of the hydrostatic waves generated by an isolated mountain lie downstream of the mountain and to the side, tending to form trailing wedges of vertical motion. This is shown in Fig. 9 .

2.4.2. Three-Dimensional Trapped Waves. The three-dimensional problem, just like the two-dimensional problem, changes considerably when the Scorer parameter decreases with height rapidly enough to permit the existence of trapped lee waves. This situation was first studied

T H E INFLUENCE OF MOUNTAINS O N T H E ATMOSPHERE

123

by Scorer and Wilkinson (1956) and later by Palm (1958), Sawyer (1962), Crapper (1962), and more recently by Gjevik and Marthinsen (1977). The three-dimensional development of trapped waves in the atmosphere is similar in many respects to the occurrence of surface waves behind a ship moving in calm water. The wave pattern is generally contained within a wedge with apex at the mountain. The waves within the wedge are of two types. The transi'ersr waves lie approximately perpendicular to the flow direction and physically are composed of waves that have attempted to propagate approximately into the wind but have been advected to the lee. These waves are analogous to the trapped lee waves found in the two-dimensional problem. When, for example, the wind is faster than the phase speed of the fastest trapped wave, lee waves disappear in the two-dimensional problem and the transverse waves will disappear from the three-dimensional problem. The other type of wave-the diverging wave-has crests that meet the incoming flow at a rather shallow angle. These crests are composed of waves that have not attempted to buck the stream but have propagated laterally away from the mountain and have been advected into the lee. At high wind speeds these waves continue to exist and respond only by aligning their wave crests more closely with the free stream direction in order to keep their upstream phase speed equal to the free stream velocity. Both types of waves are evident in Fig. 10 which shows the cloud patterns associated with flow past Jan Mayen. 2.4.3. Tl~ree-Dirnc.nsiona1Flow at a LOW Froiidr Number. Another approach to the problem of stratified flow past an isolated three-dimensional mountain is to consider the limit of very slow speeds and strong stratification so that the Froude number

(2.101)

Fr = U / ( N . L )

is small. Intuitively it is clear that in this limiting situation there will be little vertical motion and the fluid particles will deflect horizontally to move around the mountain while remaining in horizontal planes. As the Froude number is increased, vertical deflections will occur, and Drazin (1961), using an expansion in Fr, has devised a method of computing these vertical deflections. According to Drazin, the cause of the deflection is the vertical difference in the pressure field associated with the two-dimensional potential flow occurring in each horizontal plane. On an object with fore-aft symmetry, the potential flow pressure field will be symmetric (i.e., high pressure of the front and back, low pressure on the

b

a

*.----I

-. \

z . 0

'.

T H E INFLUENCE OF MOUNTAINS O N T H E ATMOSPHERE

125

f

/ I I I I

\ \

\ \

FIG.9. The buoyancy-dominated, hydrostatic flow over an isolated mountain with circular contours [ h ( x , y ) given by (3.19)].(a) The topographic contour of the mountain (solid lines) and the surface pressure field (dashed lines) C , e p ' / p U N h , = - u ' / N h , . (b)-(e) Contours of the vertical displacements of the isentropic surfaces at the heights l z = v/S, ~ / 4 ~, / 2 ,T ,determined by evaluating (2.97) numerically using a two-dimensional Fast Fourier Transform algorithm. (fj A schematic representation of the disturbance far from the mountain at l z = 2v, determined by asymptotically evaluating (2.97) using the method of stationary phase. The wave crests and troughs (solid lines) point back at the mountain. while the wave energy at each level is confined to the region near the parabola y 2 = ( k , R , z ) x (shown dashed). This parabola becomes progressively wider at higher levels. (From Smith, 1980.)

FIG. 10. Trapped gravity waves behind an isolated mountain. (a) The computed wave crests for a case with both diverging and transverse waves present. (b) Satellite VHRRphotography showing the wave pattern (primarily diverging waves) induced by the island Jan Mayen. (From Gjevik and Marthinsen, 1978.)

T H E INFLUENCE OF MOUNTAINS ON T H E ATMOSPHERE

127

sides), and this is also true of t h e vertical deflection computed by Drazin’s method. Riley ef al. (1975) have checked these theoretical predictions in the laboratory by slowly towing obstacles through a stratified fluid. They find that as soon as the Froude number is increased enough that vertical deflections are noticeable, those deflections are strongly asymmetric. Slight lifting is observed upstream and strong sinking in the lee. There are two possible explanations for this discrepancy between theory and experiment. Riley Pt a / . suggest that one should take into account the fact that in a slightly viscous fluid the potential flow solution is often replaced by separated flow with low pressure in the lee. The lee-side low pressure causes drag on the obstacle (the resolution of’ D’Alembert’s paradox) and in the stratified model would cause a drawing down of flow surfaces in the lee. The other explanation is just to realize that the laboratory observation is qualitatively consistent with the predictions of linear theory for inviscid flow over small-amplitude topography, as discussed earlier. In that case the lee-side low pressure and downward deflection are associated with the generation of mountain waves and wave drag. The inviscid wave drag mechanism is not described by Drazin’s Froude number expansion. One way to expiain this is to note that mathematically the drag turns out to be exponentially small for small Froude number and thus cannot be described by a power series expansion. Alternatively, note that while the full equations can describe wave motion, the equations generated from the expansion in Froude number cannot. There is one other interesting phenomenon that can occur when the Froude number is low-the periodic shedding of vortices. The