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

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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 Earth models. An upper limit of 1.8 gm on the density jump was 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 (-), ); MCB = mantle core boundary, ICB = inner core boundary. outer core (=

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 density of iron is unlikely to exceed 13 gm at the pressure of the 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

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

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'

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 that (d+/dT), is always negative so that tmax = p/+(a+//ap),. The quantity [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 which should be a core give rather variable distributions of p/+(d~#~/dp), 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

R =

I

the rate of loss of internal energy 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 supplied, is (T,,, - Tmin)/Tmin. This differs from the corresponding Carnot engine efficiency of (T,a,y - Tmin)/Tmin because the ohmic heating remains 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, ~ that the effect helps drive the dynamo, and the sign of ( d ~ / d T ) ~is, such 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

+ Eo + E ,

-

:1

- dV

+

f

v-F dV

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 than about W. 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 , T is velocity required for a dynamo was of the order of ( T / T ) ” ~where 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 dx dx4

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

"

4

3

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 the actual topography dmO is given by (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) (Pm/Pc)[l + 6 ( 2 ~ a / A ) ~ l1

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. 6. AXISYMMETRIC BENDING 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

100

200 I

r, km

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).

RELAXATION 8. STRESS

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. For wider hills ( L 1 km and greater) the boundary layer thickness 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

RONALD B. SMITH

92

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

RONALD B. SMITH

100

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

3

-I -a

+ 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

- N 2 k 2 / l k 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)

r

and use the Fourier representation

(2.77)

G(x, z) = Re

Then W must satisfy

(2.78)

G ( k , z ) e i k Edk

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 disperf 2 ) the result will be a periodicsive part of the spectrum (i.e., k 2 looking 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 resulting vortex “streets” have been observed in the cloud patterns downstream of isolated islands. This subject has been reviewed by Chopra (1973). For more recent work on this phenomena the reader can refer to descriptions and laboratory results of Brighton (1978). Recently there have been a number of attempts to model numerically the three-dimensional flow over and around mountains, for example, Onishi (1969), Zeytounian (1969), Vergeiner (1975, 1976), Danard (1977), Mahrer and Pielke (1977), Warner ef (11. (1978), and Anthes and Warner (1978). A detailed description of the techniques and results of the studies would take us too the far afield-especially so because for the most part (1) the numerical models are full of complex and interrelated parameterizations; (2) the boundary conditions are not the same as in the theoretical studies; and (3) little attempt has been made to compare the numerical results with the earlier theoretical results. Still there seems to be rapid progress occurring which soon will make an impact on our understanding of three-dimensional stratified flows.

I28

RONALD B . SMITH

2.5. Large-Amplitiide Moiintains and Blocking

The earlier discussion in this section has been based on the assumption that the linearized equations of motion give a satisfactory description of the flow. This assumption may seem to gain some support from the fact that most of the Earth’s terrain features are rather gentle, that is, with small slope (hlL). In a stratified fluid, however, there are length scales other than the mountain width ( L ) with which the mountain height ( / I ) can be compared. It turns out that linear theory begins to break down when h becomes comparable to either the inversion height, if one exists, o r in a continuously stratified atmosphere, the vertical wavelength of hydrostatic disturbances A, = 27rU/N. In practice this usually means that any mountain greater than 0.5 to 1.0 km in height will produce disturbances too large for linear theory, even if the slope of the surface is quite small. The breakdown of linear theory is significant because it may be associated with new phenomena, for example, wave steepening and breaking and possibly blocking-the stagnation of low-level air ahead of the mountain. The present discussion of finite amplitude effects will be brief and rather cursory. This is partly because there have already been a number of reviews of this subject (Yih, 1965; Miles, 1969; Gutman, 1969; Kozhevnikov, 1970; Long, 1972) and partly because many of the available studies are of questionable relevance due to either a restriction to two dimensions or the imposition of a rigid-lid upper boundary condition. The recent interest in finite-amplitude mountain waves originated with the three papers of R . R. Long (1953, 1954, 1955). In the first and third of these papers, Long discussed the steady flow of an incompressible, continuously stratified fluid and pointed out that there is a special class of upstream profiles for which the governing equations become exactly linear. The simplest of these cases is when the dynamic pressure $ p U 2 and the vertical density gradient d p / d z are constant with height. Within the accuracy of the Boussinesq approximation this reduces to the case of constant wind and stability which was studied earlier using linear theory (e.g., Queney, 1947). This special case, together with the belief that the upstream flow can be specified a priori, constitutes ”Long’s model.” Long’s approach has been extended by Yih (1960) to widen the class of exactly linear flows and by Claw (1964) to allow compressibility, but most interest has centered on obtaining solutions for Long’s simplest case. The difficulty here is that while the equation for the interior motion is of a simple type, the boundary condition at the mountain surface is still of a difficult nature. Long (1955) obtained solutions using an inverse method and compared the theoretical derived flow fields with laboratory

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

I29

observation. The agreement showed by Long stands as one of the cornerstones of the subject even though the rigid upper lid, used in his experiments and theory, prevents direct application to the atmosphere. The rigid-lid problem, however, has continued to receive considerable attention as a fundamental problem in fluid mechanics (Yih, 1960; Drazin and Moore, 1967; Grimshaw, 1968: Davis, 1969; Benjamin, 1970; Segur, 1971: McIntyre, 1972; Baines. 1977). The basic thrust of this research has been to investigate the way in which Long’s model breaks down, either by the occurrence of instability and turbulence o r by the alteration of the upstream flow (i.e., blocking). These phenomena are of considerable importance for the atmosphere, but much of this work may have to be extended, by eliminating the rigid lid, before it can be applied to the atmosphere. Solutions to Long’s model in an unbounded atmosphere (using a radiation condition) were first obtained by Kozhevnikov (1968) and Miles (1968), and later by Huppert and Miles (l969), Janowitz (19731, and Smith (1977). The nature of these solutions are reviewed by Miles (1969). As the height of the mountain is increased, the basic structure of the flow field remains the same. In the regions of up motion, the streamlines steepen (see Fig. 1 1 ) more rapidly than predicted by linear theory. This steepening has been linked to the nonlinear lower boundary condition by Smith (1977). Eventually when the mountain height reaches a critical value the streamlines become locally vertical. Further increase in mountain height will cause overturning-regions where denser fluid is temporarily lifted above lighter fluid. This, it is thought, will allow convective instability to occur locally, and the rigid-lid experiments of Long (1955) and Baines ( 1977) seem to confirm this idea. The wave drag also increases more rapidly than predicted by linear theory (Miles, 1969). There does not seem to be a strong tendency for blocking of the flow upstream. The flow speed just ahead of the mountain is much reduced, just as in linear theory, but it resists going to zero until long after reversed flow regions have formed aloft (as an example, see Miles, 1971). The surface level flow thus does not seem to encounter a n y special difficulty in surmounting the obstacle. It would be a mistake to try to generalize these qualitative results of Long’s model. It is now widely recognized that the special cases for which the governing equations are exactly linear are not only mathematically special, they are also physically special. In flows with variable U (z ) and N ( z ) , new nonlinear effects may arise just as new phenomena appeared in the linear theory of lee-wave flow in structured atmospheres. The only other case for which analytic finite-amplitude mountain flow solutions are available is the situation where the incoming flow is com-

130

RONALD B. SMITH

-

FIG. 1 1 . The finite amplitude flow over a semielliptical ridge computed from Long’s model. The intense nonlinear forward steepening of the streamlines is due to the fact that the parameter h,l = h,N/CJ (= 0.93 here) is approaching unity. (From Huppert and Miles, 1969.)

posed of a finite number of layers (usually one or two), each of constant density (Long, 1954; Houghton and Kasahara, 1968; Houghton and Isaacson, 1970; Long, 1970). The effect of stable stratification is modeled by the decrease in density going upward from layer to layer. In the case of a single layer, this decrease in density (in the laboratory this is water to air) is extreme, leading to a free surface condition. The model is made tractable by assuming hydrostatic balance throughout each layer as they pass over the mountain. The hydrostatic assumption is not terribly restrictive as (i) many atmospheric flows are nearly hydrostatic and (ii) the qualitative nature of the Long’s model solutions discussed above are rather insensitive to this assumption. The major drawback of this approach is that with a finite number of layers there must always be a homogeneous uppermost layer of infinite thickness preventing any vertical radiation of wave energy. Thus all waves are totally reflected back toward the surface just as they were in the rigid-lid models discussed earlier. Knowing the importance of the radiation condition, this aspect of the layered models may seem fatal, but it can be argued that it is

THE INFLUENCE OF MOUNTAINS ON T H E ATMOSPHERE

131

equally important to model correctly the discontinuous nature of N ( z ) at, for example, a thin subsidence inversion, and its associated nonlinear effects. Thus for some upstream profiles the layer models may give a better representation of the flow than the unbounded Long’s model. Furthermore, some new phenomena arise in layered models such as upstream influence and internal hydraulic jumps. The internal hydraulic jumps (see Yih and Guha, 1955), in fact, play a central role in the layered models because lee-wave radiation and vertical radiation have been eliminated by the hydrostatic and the reflective conditions. Thus the dissipation within the jump is the only way the system can be irreversible and this in turn allows for flow asymmetry (e.g., strong downslope winds) and mountain drag. The hydraulic approach to the mountain flow problem has been pursued quite vigorously in the Soviet Union. Most of those studies have included the effect of Coriolis force (Khatukayeva and Gutman, 1962; Sokhov and Gutman, 1968; Gutman and Khain, 1975; Ramenskiy et al., 1976). The final and perhaps potentially the most powerful method for understanding nonlinear mountain flow is the direct numerical solution of the governing equations. Recent attempts to model the two-dimensional flow over a finite-amplitude ridge include Foldvik and Wurtele (1967), Granberg and Dikiy (1972), Furukawa (1973), Mahrer and Pielke (1975), Deaven (1976), Clark and Peltier (1977), Klemp and Lilly (1978), and Anthes and Warner (1978). Along with the advantages of this technique come a series of drawbacks which have plagued investigators. Because of limited computer memory and speed it is impossible to calculate the flow in a semi-infinite domain, leading to the necessity of specifying nearby boundary conditions-both inflow and outflow conditions and an upper “radiation” condition. The upper radiation condition can be directly applied only when the flow variables are expressed as additive Fourier components. In the finite difference models then, this condition can only be simulated by adding a “sponge” region above the region of interest in which the vertically propagating waves are dissipated-hopefully without reflection-by a gradually increasing viscosity (see Clark and Peltier, Klemp and Lilly, and Warner and Anthes). Other limitations on the numerical models are imperfect spatial resolution and the possibility of coding errors and numerical instability. Perhaps the strongest limitation is that while solutions and understanding often go hand in hand in analytical work, this is seldom the case with numerical simulation. The numerical solutions must be cleverly diagnosed to reveal the underlying processes. The recent numerical work of Clark and Peltier (1977) and Klemp and

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RONALD B . SMITH

Lilly (1978) have shed some light on the problem of what happens when a vertically propagating mountain wave becomes so steep that it causes a local reversal of the flow aloft. By parameterizing the small-scale turbulence, which apparently occurs by local instability, the larger scale mountain wave flow can continue to be calculated even after the critical breaking criterion is exceeded. Both pairs of investigators find that a region of slowly moving turbulent air is generated aloft and that this region seems to cause a partial reflection of the vertically propagating wave energy. This partial reflection increases the intensity of the mountain-induced disturbance in the lower atmosphere and increases the mountain drag. The details of this process are still unclear. It seems likely that this or some closely related phenomena may be the important factor in producing severe downslope wind storms such as reported by Lilly and Zipser (1972) and as shown in Fig. 8. The numerical models have also reproduced some of the aspects of low-level blocking, but the two-dimensional restriction probably prevents a true simulation of the blocking phenomena. The blocking of low-level air is one of the most important ways in which mountains affect the air flow. The tendency for the surface level flow to slow as it approaches a mountain is described by the linearized theories of mountain flow. It is probably fair to say that this windwardside slowing is due to the difficulty that the heavy surface air has in running upslope. This, by the same token, explains the large velocities on the lee side as heavy air runs downhill. At the same time we must remember that according to linear theory, this upslope-downslope asymmetry also requires the generation of waves that propagate away to infinity. Thus the blocking phenomena cannot be considered a strictly local phenomena. The linear theory cannot, of course, be used to investigate complete blocking as this immediately implies that the disturbance has become as large as the mean flow. This aspect of the finite-amplitude mountain flow problem has attracted a good deal of attention theoretically and in the laboratory. A brief list of the different types of “blocking” or more generally “upstream influence” is as follows: 1. Sheppard (1956) used Bernoulli’s equation to estimate the speed

that an incoming flow must have to overcome the background stability and reach the mountain top. To close the system of equations Sheppard had to assume that the pressure of a parcel as it rises is always equal to the environmental pressure far from the mountain. This leads to the approximate condition (2.102)

U > Nh

THE INFLUENCE OF MOUNTAINS O N THE ATMOSPHERE

I33

if the flow is to reach the top. It is interesting to note that the parameter h N / U appearing in (2.102) is the same parameter that enters as a measure of the nonlinearity in hydrostatic mountain flows (Smith, 1977; Miles, 1969). Miles finds that a lower value of h N / U = 0.67 as opposed to (2.102) marks the onset of overturning above a broad mountain of ellipsoidal cross section. Sheppard's condition (2.102) seems physically reasonable but there is little else to recommend it-especially as it contains no information as to what form the blocking will take. A further problem is that in practice it is difficult to determine the appropriate values of N and U as these are likely to vary quite strongly near the surface. 2 . One conception of the blocking phenomena is to consider it as the separation of the boundary layer or as the reversal of the slow moving flow in the boundary layer by the adverse pressure gradient upstream of the mountain (Scorer, 1955, 1978). 3 . In the fully three-dimensional flow near an isolated mountain or a ridge with ends or gaps, absolute blocking of the low-level flow is not possible. The layer of dense air may pile up slightly ahead of the mountain, but this can be relieved by airflow around the mountain or through gaps in the ridge. The tendency for the flow to go around is described both in the linear theory and in the low Froude number model of Drazin ( I96 I ) described earlier. 4. In two-dimensional flow with a rigid lid, there is the possibility that for specified upstream conditions [i.e., U ( z ) , N ( z ) ] there may be no steady-state solution to the governing equations. There is an analogy between this problem and the "choking" phenomena in the one-dimensional flow of a compressible gas into a strongly converging nozzle. In both cases a transient flow occurs in which a wave front moves upstream, altering the incoming flow in such a way as to make a steady state flow possible near the mountain. This situation has been investigated for continuous stratification by Long (1959, Drazin and Moore (1967), Grimshaw (1968), Benjamin (1970), Baines (1977), and in layered flows by Long (1954), Houghton and Kasahara (1968), Houghton and Isaacson (1970), and Long (1970). The "choking" seems to be associated with the reflective upper boundary condition and occurs when the Froude number of the flow is near unity. It is not then simply that the incoming flow is too slow to run up the mountain. 5 . The last type of blocking to be described here is the upstream influence that occurs naturally when the free stream is started from rest. Even if a steady state flow may have been possible with the intended upstream conditions, the start-up process can generate long waves which move upstream, altering the flow that approaches the mountain (Mc-

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RONALD B. SMITH

Intyre, 1972). Like the choking phenomena this type of upstream influence seems to depend on a reflective upper boundary condition and thus may not be important for the atmosphere. It has been observed in the laboratory by Baines ( 1977) and A. Foldvik (personal communication), although there is the possibility that the observed upstream wave may have been generated by viscous or turbulent redistribution of momentum near the mountain. 2 . 6 . The Observed Barrier Effect of Mountains-The

F d i n and Bora

The barrier effect of mountains is well documented in several parts of the world. In some regions the low-level flow is diverted horizontally around or through gaps in the mountains. As an example, the strong wintertime westerly winds in southern Wyoming (Dawson and Marwitz, 1978) probably represents air that was blocked by the Front Range. The mistral-a cold wind blowing off the continent in southern France-is able to avoid barriers by flowing down the RhGne Valley. Cool, moist Pacific air is able to move eastward through high passes in the Andes (Lopez and Howell, 1967). Low-level air originating in North America penetrates the central American highlands near Tehuantepec (Godske et al., 1957). In other situations the cold low-level air can be contained by an unbroken mountain chain for several days. Examples including the trapping of polar air north of the Brooks Range in Alaska (Schwerdtfeger, 1975), the containment of the Scandinavian anticyclone by the Scandinavian mountains, and the damming effect of the southern Appalachians (Richwien, 1978). It occasionally happens that after the barrier effect has persisted for some time, the large-scale pressure gradient will change, forcing the air from one side of the mountain to move over the crest and down the other side. When this happens, the lee slope environment may experience a sudden change in temperature and humidity as the old air mass is replaced by an air mass of different origin. Following Yoshino (1975, p. 393) we can classify these overflow events according to the change in temperature that accompanies the onset of the fall wind (i.e., heavy air moving downslope). "In my opinion . . . the definitions of the fohn and bora should be made in the simplest way as follows: The fohn wind is a fall wind on the lee side of the mountain range. When it blows, the air temperature becomes higher than before on the lee side slope. The bora is also a fall wind on the lee side of a mountain range, but when it begins, the air temperature becomes lower than before on the lee side slope." The fohn phenomenon is common in the Alps (see, for example,

THE INFLUENCE OF MOUNTAINS ON THE ATMOSPHERE

135

Defant, 1951; Godske et al., 1957; Yoshino, 1975; Vergeiner, 1976), and on the eastern slopes of the Rockies where it is called a chinook (see, for example, Brinkman, 1970, 1971; Holmes and Hage, 1971). In California it is an easterly wind and the local name is the Santa Ana (Serguis et al., I 962). The definition of the fohn as a warm downslope wind makes no attempt to distinguish the reason for its warmth. It could be ( I ) a warm source region, (2) warming by the release of latent heat as the air ascends over the mountain (i.e., the Hahn mechanism), or (3) the blocking of low-level air and the descent of higher potential temperature air from above (see Critchfield, 1966, pp. 130- 13 1). In many cases the condensation occurring during stable lifting over a mountain is not sufficient to explain the large temperature difference between the two sides of the mountain. This suggests either that blocking is occurring or that the condensation is increased by convection over the mountain, triggered by orographic ascent. In this regard the reader should refer to Section 4 on the subject of orographic rain. The most well-known occurrence of the bora is in the northern Adriatic near Trieste and south along the Yugoslavian coast (Yoshino, 1975, 1976). Air cooled over Eurasia spills over the highlands between the Alps and the Balkans and descends to the sea. The mistral flowing between the Pyrenees and Alps and the Tehuantepec fall wind south of the Sierra Madres in Mexico are also bora-type fall winds (see Godske et a f . , 1957).

2.7. The Influence of the Borrndrir?, L q e r on Mountain Flows The theories of airflow past mountains assume for the most part that the flow is inviscid, neglecting the presence of the thick turbulent atmospheric boundary layer. There have, however, been attempts to understand the nature of the boundary layer as it flows over simple topography (Counihan et al., 1974; Taylor and Gent, 1974; Jackson and Hunt, 1975; Deaves, 1976; Taylor et a l . , 1976; Taylor, 1977a,b). These studies have so far been restricted to small hills where the effects of buoyancy forces could be neglected. It follows that this type of flow may have something in common with the subject of potential flow. In order to model the flow in the boundary layer it is necessary to take into account (a) the shearing nature of the undisturbed flow; (b) the turbulence, both as it influences and is influenced by the mean flow; and (c) the degree of roughness of the surface (e.g., the roughness length z o ) . The complexity of this situation requires that some form of numerical computation be used to solve the governing equations.

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RONALD B . SMITH

The results of these computations (see, for example, Fig. 12) show a qualitative resemblance to potential flow. The wind speed (at a standard level) reaches a maximum near the top of the hill and the pressure has its minimum value there. This'is true in spite of the fact that Bernoulli's equation does not strictly hold. Similar computations done for flow across C

._ c

8 20" .-n

3 loo D

U

5

0"

0

Sl"< 40

-80 2.5

2.0

-n n.

%I

a

1.5

1 .o

0.5

.$lOoC D

N

C

-1om

-5 000

0

5000

10 000

15 000

x'z,,

FIG. 12. The computed flow over a small ridge, immersed in a turbulent Ekman layer. The skin friction (and the wind speed near the ground) reaches a maximum just upstream of the crestline while the pressure has a minimum just downstream. To a first approximation the flow resembles inviscid, irrotational flow (see Fig. 2). but upon close inspection, the influence of the preexisting thick turbulent boundary layer is apparent. (From Taylor, 1977b.3

THE INFLUENCE OF MOUNTAINS O N THE ATMOSPHERE

137

shallow valleys show a strong reduction in wind speed and a pressure maximum in the valley bottom. Qualitatively both the hill and valley results are in agreement with the measurements reviewed by Yoshino (1975, pp. 262-268) and with common experience; that is, hill tops are more exposed to the wind, while topographic lows are sheltered. A more quantitative examination of the computer results reveals, as expected, some deviation from potential flow behavior. There is a slight asymmetry to the pressure field on the ridge leading to a net drag. There is also a sheltered region which extends a considerable distance downstream. The reduction in surface shear stress in this region partially cancels the drag due to pressure forces as it affects the net areal drag. The numerical models have not yet been extended to include cases of abrupt topography, leading to separation. The great body of work on this subject in engineering aerodynamics may not be applicable here because of the lack of a thick turbulent boundary layer upstream. The laboratory model of the flow around Mt. Fuji by Soma (1969) did include this effect however. The modeling of the flow past a windbreak (see, for example, Seginer, 1972) is also relevant here. There has also been a good deal of speculation on the role of the boundary layer in flows where buoyancy forces are obviously important. One question is whether the rotor-a turbulent recirculating region found under the crests in a train of lee waves-is an example of boundary layer separation. This controversy is mentioned in Queney et al. (1960), and since that time several other observations have appeared in the literature (Gerbier and Berenger, 1961; Forchtgott, 1969; Lester and Fingerhut, 1974).

2.8. Slope Winds and Moirntain and Valley Winds Whenever the surface temperature differs from the temperature of the air above (due to radiative heating and cooling, or horizontal advection), heat will be transferred from one medium to the other. This will quickly establish a layer of air near the surface which, while more closely matching the soil temperature, will differ from the air still higher up. If this occurs on a sloping surface, the buoyancy forces associated with the temperature variations will cause the layer to accelerate up or down the slope. The acceleration will continue until the frictional resistance becomes equal to the buoyancy forces and a steady-state slope wind is established. A further requirement for steady state is that the rate of heating or cooling the air parcels must be matched by the rate at which these parcels move into regions of warmer or cooler environmental temperature so that their temperature anomaly remains constant. This is

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RONALD B . SMITH

possible if the lapse rate in the vicinity of the mountain slope is a stable one. These requirements for steady state can be written as (2.103a)

ei

a2u’

g sin(a) - + v -= 0 @ dZ2

and (2.103b)

-8s sin(a) u r + k

a26’

az

=0

after Prandtl (see Defant, 1951). In (2.103a) g is gravitational acceleration, 8 and 8’ are the background and perturbation potential temperature, u’ is the induced slope wind, v and k are the diffusivities of momentum and heat, S is the stability dlng/dz, and the z coordinate in (2.103) is directed perpendicularly to the slope a. If the temperature perturbation at the ground can be specified as AT, then the boundary condition is

@’=AT

at

z=O

together with the no slip condition ur=O

at

z=O

and the condition that the disturbance decay far above the slope Or,u’

vanish as

z + ~0

With all coefficients taken as constant, the solution is (2.104a)

e r ( z ) = ATepZl2cos z / l

(2.104b)

u ’ ( z ) = ( g / N ) ( k / v ) 1 ’ 2 A T /e8T Z nsin z / l

with N 2 = g S and I , a measure of the thickness of the layer of moving air, given by

(2.105)

1

4kv ( N 2 sin2 a )

This solution can be criticized on a number of grounds, for example, the slope of the terrain (a)has been presumed to be constant both in the downslope and cross-slope directions, and the transport of heat and momentum have been parameterized by specifying the eddy diffusion coefficients k and v. Nevertheless, the “slope wind” solution (2.104) is useful as it illustrates the type of momentum and heat balance that might be realized in nature. Probably the best direct application of the slope wind solution is to the nearly continuous katabatic winds which run down the slopes of the great

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

I39

ice domes of Greenland (Nansen, 1890) and Antarctica (Mawson, 1915). Lettau (1966) has compared (2.104) against observations of an antarctic katabatic wind and found reasonable qualitative agreement for suitably chosen values of k and v. The profile he studied was characterized by a wind maxima of 3 m/sec at 5 m above the surface corresponding to 1 = 7 m and a temperature anomaly at the surface A 7 = -10°C. The slope wind has also been observed locally in more complex terrain (see, for example, Bergan, 1969). When the slope changes in the downstream direction, the balances described by Eqs. (2.103) are altered and the horizontal advection of momentum and heat [terms not included in (2.103)] become important. Gutman (1969) has solved the nonlinear two-dimensional steady state equations in a region of changing slope and finds, among other things, that air is expelled from or drawn into the slope boundary current in the vicinity of slope changes. This must occur of course, because the mass flux in the fully developed boundary current depends on the local slope. Using these ideas it is possible to understand how the air in a closed valley can be cooled or warmed by a loss or gain of heat at the valley walls. The divergences in the boundary layer cause slow vertical motion in the interior which, because of the background stratification, results in slow cooling or warming. This situation is closely analogous to the “spinup” of a rotating fluid by Ekman layer pumping. If the change in surface slope occurs very abruptly and the katabatic wind is strong, the local flow may be dominated by advection of momentum and heat leading to nonlinear phenomena such as a hydraulic jump. Ball (1956) and Lied (1964) have investigated the abrupt transition that occurs when the antarctic katabatic wind reaches the edge of the continent. Locally they ignore the loss of heat and momentum to the surface by turbulent diffusion, thereby reducing the problem to simple hydraulics. He was able to show that the Froude number computed for the katabatic wind upstream is supercritical and thus the deceleration of the flow is expected to occur by means of an abrupt hydraulic jump-in agreement with the observed flow. In most mountainous regions the terrain is dissected by numerous river or glacially cut valleys. The slope wind solution may be applicable to some degree on the valley walls, but for the most part the flow is dominated by the tendency of the currents to concentrate in the valleys. The sides of the valley and smaller adjoining valleys then act as “tributaries,” swelling the current of cold air moving down the valley (the mountain wind) or the warm air moving up the valley (the valley wind). The dissected nature of the terrain is also important in decoupling the winds in the valley from the synoptic-scale winds aloft, presumably

140

RONALD B . SMITH

through the mechanism of separation. Thus, in many deep valleys the wind climate is almost totally determined by the mountain and valley wind phenomena. (See, for example, Gr8nBs and Sivertsen, 1970, for Norway; Jensen et al., 1976, for Greenland; MacHattie, 1968, for Alberta, Canada; Defant, 1951; Yoshino, 1975, p. 276, for several other areas.) Generally speaking it is the difference between the surface temperature and the air temperature that determines whether a mountain or valley wind will occur, and this in turn depends on the time of day, season of the year, and latitude. In the midlatitude Alps there is often a strong diurnal variation-mountain winds at night and valley winds during the day. Farther north in Norway, the mountain wind blows for almost the entire day in winter, while the valley wind dominates in the summer. In valleys with modern glaciers, the surface temperature is nearly always cooler than the air above, leading to the continuous "glacier wind" blowing downslope across the glacier o r through downward-leading crevasses and caves in the ice. Because of its influence on local climate, the mountain-valley wind cycle has received considerable attention, but there are still fundamental questions concerning (1) the existence of a reversed wind above the valley floor, and (2) the details of the transition between the valley wind and mountain wind which may occur, for example, at sunrise and sunset. A reversed current aloft could occur in many ways: (a) the weak return current in (2.104b) caused by eddy momentum transport and excess adiabatic cooling, (b) the continuation of the upper part of a deep valley wind after the wind near the surface has reversed, (c) a true "return" current in which the air flows one way in the mountain or valley wind and then returns aloft to satisfy the continuity equation, and (d) the synoptic-scale wind which may happen to be opposed t o the wind in the valley. Detailed models of the wind reversal at sunrise and sunset have been put forth by Defant (195 l), Urfer-Henneberger (1964), and Sterten (1963). The observations of Wilkins (1955) also bear on this question. The common point in these models seems to be that the slope winds on the valley sides respond rather quickly to the changes in solar insolation. The winds in the central valley, especially away from the surface, respond more slowly. This is shown in Fig. 13. Another unanswered question is why the warm valley wind or a warm upslope wind does not detach itself from the surface and rise vertically. Certainly it could release potential energy faster if it did so. The answer to this dilemma may be that the atmosphere above is stably stratified and air must continue to receive heat from the surface if it is to rise. On the

FIG.13. T w o views of the diurnal cycle of mountain and valley winds and the interaction with slope winds on the steep sides of the valley. The slope winds respond quickly to changes in solar insolation. while the wind in the valley responds more slowly. (a) The model proposed by Defant (1951) assumes the valley is uniformly heated by the sun: ( A ) sunrise, ( B ) forenoon, (C) early afternoon, (D) late afternoon, ( E ) evening, (F) early night, ( G ) middle of night, ( H ) late night to morning. (b) The model of Urfer-Henneberger ( 1964) takes into account the difference in insolation between the sunlit and shaded slopes.

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RONALD B . S M I T H

other hand, if the surface was strongly heated and of gentle slope, direct parcel ascent (i.e., penetrative convection) could occur.

3 . THEFLOWN E A R MESOSCALE A N D SYNOPTIC-SCALE MOUNTAINS

In this section we will consider the perturbation to the wind flow caused by a mountain of intermediate scale where the rotation of the Earth cannot be neglected. As an example, let us say that a mountain has a width of from 100 km to 500 km, so that an air parcel moving with the wind at say 10 m/sec will take lo4 sec (-3 hr) or 5 x lo4 sec (-15 hr) to cross the mountain. To estimate the relative importance of fluid accelerations relative to the Earth to those associated with the Earth’s rotation (i.e., the Coriolis force) we must compare the transit time of an air parcel with the rotation period of a Foucalt pendulum located at the appropriate latitude 7 = 27~/2flsin 4. For midlatitudes on the Earth this is about 12 hr, and we conclude that in the horizontal equations of motion both types of accelerations may be the same order of magnitude. Mountains in this size range are quite common on the Earth’s surface. Probably any surface irregularity that would be called a large mountain or a mountain range would be included. Examples include the Scandinavian mountain range (width 300 km), the Alps (width 250 km), and the Canadian 400 km). In all these examples the influence of the Rockies (width Coriolis force on the perturbed flow is too large to be ignored, yet too small to allow the assumption of geostrophic balance. Throughout this section we shall be working between two well-defined SO km) the Coriolis limiting situations. With small mountains (width force can be ignored while the hydrostatic assumption can still be considered as valid. The flow over synoptic-scale orography (width -1000 km) may be assumed to be nearly geostrophic. The types of flow occurring in these two situations are quite dissimilar, and one of the challenges of this section is to see if we can understand how the flow transitions from one type to the other as the Coriolis force becomes progressively more important. I n the first case (see Fig. 3) the flow is asymmetric (even for a symmetric mountain), the perturbation extends to great altitude, and there is a drag on the mountain. The perturbation caused by the broad mountain on the other hand (see Fig. 17) is symmetric (if the mountain is), decreases with height, and causes no drag on the mountain. The mountain wave situation is discussed in another section of this review, but the broad mountain, quasi-geostrophic limiting case must be discussed here in detail before we can proceed further.

-

-

-

-

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

I43

Consider the steady flow of a stratified rotating fluid over a mountain. This irregular surface of t h e ground will be assumed to be a stream surface to the flow. That is, the flow cannot penetrate the ground. We will assume also that the flow approaching the mountain is barotropic so that the potential temperature is constant along the ground. Then, if the potential temperature is constant following a fluid particle and if the surface of t h e mountain is completely covered by fluid that has come from upstream, it follows that the r?ioirntain srrrfuce ill coincide with the surface of corzstant poteritial ter?ipermtiire (see Fig. 14). This result is of crucial importance to the nature of the flow field near the mountain. Note that there are several ways that this condition could be violated: (a) baroclinic flow upstream, (b) transient effects or permanent blocking which prevent the particles with upstream properties from fully penetrating the region of interest, and (c) diabatic effects. Further, the observational evidence does not strongly defend this assumption. The lapse rate measured along a mountain slope is usually substantially less (in magnitude) then the adiabatic lapse rate (see Yoshino, 1975: Peattie, 1936, for a review of these observations), indicating that 8 =I= const along the surface. This criticism can be turned aside by arguing that this variation in 8 is due to heating of the air near the ground. There still might be a surface just outside the boundary layer which closely parallels the mountain shape and on which the condition 8 = const is satisfied. More disturbing are the aerological observations which occasionally seem to show isentropic surfaces, outside the boundary layer, intersecting a mountain. For the most part, however, the radiosonde network is not dense enough to determine the true shape of &surfaces in mountainous terrain. In the following analysis we will assume that the O-surfaces follow the terrain. It is clear, however, both that the validity of this

82 81 80 (a )

(b)

FIG. 14. Two possible configurations for the isentropic surfaces near a mountain. (a) The isentropic surfaces are pushed up. paralleling the ground surface. (b) The mountain penetrates up through horizontal O-surfaces. If there is flow, it must be going around the mountain. The former model is used as a lower boundary condition in most mathematical models of mountain flow.

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RONALD B . SMITH

assumption is doubtful and that the breakdown of the assumption will fundamentally change the nature of flow. The next important result involves the nature of the wind field near the mountain. If the flow is geostrophically balanced dP

(3. la)

pfu = - -

aY

(3.lb) The hydrostatic assumption is written as

To simplify the analysis we will treat the air as a Boussinesq fluid by neglecting its compressibility and neglecting the influence of density variations on the inertia [i.e., the left-hand side of (3. l)] of a fluid particle (see, for example, Batchelor, 1967). In the absence of compressibility the density of a fluid particle is constant

DplDt

(3.3)

=

0

If the background flow is stratified p ( z ) , Eq.(3.3) can be used to show that a local density perturbation p ‘ can be produced by raising or lowering a fluid particle a distance 7 into a different density environment. (3.4) Now combining (3. l ) , (3.2), (3.4) gives (3.5a)

-

au Pf-= aZ

ap S j p J

(3.5b) which is a Boussinesq version of the more general thermal wind equation. It displays the connection between the vertical wind shear ( d u / J z , d v / a z ) and the slope of the surfaces of constant p (or 0 in the atmosphere). This can be put in a more compact form by introducing the stream function (3.6)

q x = v,

qu=

-u

THE INFLUENCE OF MOUNTAINS ON T H E ATMOSPHERE

145

and the integrating (3.5) to obtain (3.7)

$2

=

-(N2/f)r)

where

Near the mountain surface the distribution of uplift of 0-surfaces

r)

(x,

y ) is just equal to the orographic height h (x, y ) so the pattern of vertical

wind shear is known immediately. (3.8)

$,

= - ( N 2 / f ) h ( x ,y

)

near the ground

To solve for the wind itself the entire flow field must be considered. We can reason intuitively as follows: Near a region of raised 0-surfaces in a stable geostrophical!y balanced atmosphere, Eq. (3.8) requires that we have either cyclonic motion which becomes stronger with height or anticyclonic motion which weakens with height. Intuitively we feel that a disturbance produced by a mountain ought to be strongest near the mountain, so it is natural to choose the latter alternative. Thus the flow around the mountain is identical to the textbook description of a "cold anticyclone," with the only difference being that it is the solid mountain surface, rather than the cold air near the ground, that is responsible for the upwarping of the 0-surfaces (see Fig. 15). The expression "mountain anticyclone" is chosen to refer to the qualitative aspects of the flow. To understand the detailed structure of the mountain anticyclone the concept of conservation of circulation is needed. Consider a column of fluid at point a confined between two isentropic surfaces 01, O 2 (Fig. 16). The absolute circulation r a -$pfi,b.dS, which can be written (6 + f ) A ,

FIG.15. The upwarping of isentropic surfaces near the ground caused by (a) a cold air mass at the surface or (b) a region of high ground. If the unwarping decreases with height and if the flow is geostrophically balanced, there must be an anticyclonic circulationeither (a) a cold-core anticyclone or (b) a mountain anticyclone.

146

RONALD B. SMITH

FIG. 16. Synoptic scale vorticity dynamics in a stratified fluid. As an air parcel moves from a region of weaker stability (a) into a region of stronger stability (b), its vertical dimension (6) decreases while its horizontal area ( A ) increases. The Coriolis force, acting during the horizontal divergence, produces anticyclonic vorticity.

is conserved, as is the volume AS (if the fluid is incompressible). Thus if the distance 6 between &surfaces decreases from point a to b, the horizontal area A of the fluid column must increase as it moves toward b and the absolute vorticity 5 + fmust decrease. Quantitatively 1

dA A

-- - - - d S

(3.9)

S

but the relative stretching of the column dS/S is due to the different vertical displacement 7 of the top and the bottom of the column so

--s dS

(3.10)

az

The conservation of absolute circulation can now be written

D Dt If

5

4

(3.11)

f,f

=

Dt

const, d A / A 4 1 and using (3.9), (3.10)

o=Dt

[ 5 - f - dd:]

= -D [q]

Dt

The quantity in square brackets in (3.11) is the “potential vorticity” (3.12)

It can be rewritten using the steam function defined in (3.6) (3.13)

Finally, using the geostrophic assumption in the form (3.7) and assuming

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

147

N 2 = const gives

(3.14) We shall call (3.14) the “geostrophic form of the potential vorticity.” In its more basic form (3.1 I), q will be conserved regardless of whether the flow is geostrophic or not. Using the geostrophic form (3.14) introduces a paradox which should be mentioned before going further. If the flow were exactly geostrophic, it could not undergo the horizontal divergence shown in Fig. 16. To see this, use (3.1) to evaluate V,,u = ( d u / d x ) + ( d v l d y ) . We know from observation that the large-scale winds are nearly geostrophic, but mathematically it can be shown that if the flow was exactly geostrophic, it could not do a n y of the interesting things the atmosphere is observed to do. The type of analysis used here and throughout much of dynamic meteorology is called quasi-geostrophic theory (see Holton, 1972). Conceptually, the flow is allowed to be slightly divergent (i.e., ageostrophic), but close enough to geostrophy so that an equation like (3.7) gives a sufficiently accurate description of the relation between the wind field and the density field. The simplest case of quasi-geostrophic flow over a mountain is the case of q = 0 upstream, for example, uniform wind approaching a mountain. If the entire flow field is filled with fluid that has come from upstream, then from (3.1 I ) and (3.14)

(3.15) Equation (3.15) must be solved subject to the boundary conditions at the ground (3.8), and at large z where the solution must be bounded. Because of the many simplifying assumptions (e.g., small perturbations, Boussinesq, quasi-geostrophy, constant N and f , q = 0 upstream) (3.15) has a simple form. If a stretched vertical coordinate is used i = ( N / f ) z ,(3.15) becomes Laplace’s equation (3.16)

This allows us to use all of the mathematical and conceptual techniques of potential theory while keeping in mind that the vertical scale of the motion is very much less (by a factor of f / N = 0.01) than the horizontal scale. It turns out that we can construct interesting mountain flow solutions, either in two or three dimensions, by analogy with the simplest potential flow solutions.

148 3.2.1. tions to -S/475-r strength

RONALD B. SMITH

The Flow over an Isolated Mountain. The two simplest soluLaplace’s equation in three dimensions are 4 = U x and 4 = which correspond physically to uniform flow and a source of S at the origin. With this as a guide we choose

4 4 , y , z> =

(3.17) where

r

=

UY - ( S / 4 7 4

[xz + y 2 + ( N 2 /fz)zz]3’/2

as a solution (3.15). Using (3.7) the corresponding pattern of stream surface (or O-surface) lifting can be determined

(3.18)

q ( x , y , z) = -

f

N2

S +z

(

= --

475-f

-312

x2

+ y2 + f“

zZ)

z

Using this form we can consider a whole family of bell-shaped mountains with circular contours by placing the “source” at a distance z o beneath the ground surface. Then for a mountain of shape

(3.19)

+

where R = (2 y z ) and R o = ( N / f ) z o is the measure of the mountain width. The O-surface displacement is

(3.20) The perturbation wind caused by the mountain blows around the mountain in the anticyclonic (clockwise in the northern hemisphere) direction with strength

(3.21) One important aspect of this flow is that the maximum vertical displacement of the isentropic surfaces [Eq. (3.20)] decreases with height, but the lifting becomes much more widely distributed so that the volume under the raised surfaces

149

THE INFLUENCE OF MOUNTAINS ON THE ATMOSPHERE

. .

FIG. 17. The vorticity dynamics in the stratified, quasi-geostrophic flow over an isolated mountain. The magnitude of the lifting of &surfaces aloft is less than the mountain height, but the lifting is more widespread. As parcels near the ground approach the mountain, they are first stretched producing cyclonic vorticity. Over the mountain, the parcels are shortened producing anticyclonic vorticity. The total amount of cyclonic and anticyclonic vorticities are equal at each level and, as a result. there is no far-field circulation. (After Buzzi and Tibaldi, 1977.)

equals the mountain volume at every level. The lifting of the stream surfaces aloft extends far from the mountain. Thus as a fluid column approaches the mountain it is first stretched due to the lifting of stream surfaces aloft, then shortened due to the mountain elevation (see Fig. 17). This behavior is discussed by Buzzi and Tibaldi (1977). The velocity field described by (3.17) is the vector addition of uniform flow of strength U and the mountain anticyclone [described by (3.21)] which weakens aloft (see Fig. 18). Near the ground z G zo, and far from the mountain R 9 R o , the perturbation velocity falls off like v8 1/R2. This decay is more rapid than in an irrotational vortex and accordingly the circulation around the mountain will decrease as the radius of the

-

FIG. 18. The streamline pattern in quasi-geostrophic stratified flow over an isolated mountain (see also Fig. 17). The incoming flow is distorted by the mountain anticyclone. The perturbation velocity and pressure field decay away from the mountain.

150

RONALD B . SMITH

circuit is increased. This rapid decay is another facet of the weak cyclonic relative vorticity surrounding the core of anticyclonic vorticity directly above the mountain. Note that this description of a mountain anticyclonic in a rotating unbounded stratified fluid is quite different from the flow over an isolated hill in a homogeneous fluid with a rigid upper lid. In this latter case the vertical displacement q and the relative vorticity 6 would be zero except directly over the mountain, the volume under the raised stream surfaces would decrease aloft, and the strength of the anticyclonic winds would be constant with height and decrease as 1/R horizontally away from the hill. A more general family of mountain shapes can be considered by using a combination of "sources" in or slightly below the z = 0 plane. To a large extent the qualitative nature of these more complicated flows can be determined graphically. In any case, far away from the orographic region the behavior will be as described above with the effective source strength [in (3.17), (3. IS)] being determined from the total mountain volume according to S = - N x [mtn. vol.]. As an example, we could consider qualitatively the flow over a long (but finite) narrow ridge. Near the center of the ridge the induced flow is parallel to the ridge as it would be for an infinite ridge, while far away from the ridge the streamlines for the induced flow become circular as in (3.17). One interesting feature of the flow over a finite ridge is that unlike the circular mountain or infinite ridge solutions, the induced velocity has a component across the height contours of the mountain. Thus, the vertical velocity near the surface which to first order is U ( d h / d x )can be strongly modified by the induced flow. This could have an important effect on the distribution of orographic rainfall along the windward side of a mountain range. In general, t h e perturbation stream function and therefore the perturbation pressure will be symmetric with respect to the topography. As a result the net horizontal force on the topography due to the perturbation pressure field (e.g., drag) vanishes identically. This result will, of course, be altered if we introduce a viscous Ekman layer, the p-effect, or if the Rossby number was large enough to allow the generation of mountain waves. Such forces, if they were present, would be proportional to the square of the mountain height [i.e., O(h2,)I. There is, however, an 0 ( h , ) force on the mountain due to the background geostrophic pressure gradient. Looking downstream, the isolated mountain finds itself in a pressure field increasing linearly to the right. Thus, according to Archimedes Law, the mountain feels a net pressure

THE INFLUENCE OF MOUNTAINS O N TH E ATMOSPHERE

151

force to the left given by

F

=

pUfv

where V is the volume of the mountain. This "lift" force acts perpendicularly to the mean flow, regardless of the shape of the mountain. This is true even for a long ridge because the large pressure difference at the ends of the ridge will be just what is needed to make the net for perpendicular to the mean flow direction, not the ridge. In the Boussinesq model described above, the air passing over the mountain does not respond to the lift force reaction (i.e., t h e force applied to the air by the mountain). Instead the force is passed upward from layer to layer without decreasing. This is so because the volume under each uplifted &surface and the cross-stream pressure gradient are both independent of height. The influence of compressibility on this result is discussed by Smith (1979b).

3.2.2. The Flow over mi Infitiite Ridge. To construct a solution for infinite ridge we can superpose a linear distribution of isolated mountains or simply use the well-known two-dimensional source solution to potential flow theory, 4 = (S/277)1n r . This latter method leads to a stream function of the form (3.23) where r =[x2 of shape (3.24)

$(x, z ) =

-

Uy

+ ( N Z / f Zz) ~ ] " ~This . h(x)

+ (S/277) In r is a solution for flow over a ridge

= h,na2/(X2

+

a2)

where a is a measure of the width of the ridge. The vertical displacement is (3.25) The induced velocity lies parallel to the ridge (3.26)

u(x, z ) =

h,Nxu x2

+ U2(Z/Z,

+ 1)2

As an example, the amplitude factor h,N might be lo3 m x 0.01 sec-l = 10 m/sec. This flow is in many respects similar to the three-dimensional flow over a mountain with circular contours, discussed earlier. As before, the

152

RONALD B . SMITH

general sense of the induced circulation is anticyclonic and its structure and strength is independent of the direction or strength of the incoming wind. The lifting of the 0-surfaces aloft q is again much more widespread than t h e orographic height h ( x ) . This means that as a fluid particle approaches the mountain it is first vertically stretched, producing positive relative vorticity, and simultaneously slowed to maintain constant mass flux between streamlines. From its vorticity, or from the fact that the slowed particle now feels a decreased Coriolis force, it is clear that the particle will curve to the left. When the particle is directly over the mountain, the 0-surfaces are closer together, the fluid moves faster and has negative relative vorticity, and the particle paths curve strongly to the right. Downstream of the mountain there is again stretching, slowing, and curvature to the left. When averaged over the mountain and the surroundings, the relative vorticity (or the circulation) is zero as the vortex stretching adjacent to the mountain exactly cancels the vortex shrinking over the mountain. Associated with this is the fact that the time necessary for a particle to traverse the whole flow field is the same as it would be without the mountain, thus the next impulse given to the fluid by the Coriolis force is zero. The force to the right acting on the fastmoving fluid over the mountain is exactly balanced by the force to the left acting on the slower fluid adjacent to the mountain. The result of this balance is that the infinire ridge causes no permanent turning of theflow. This can be seen from (3.26), as the induced velocity associated with the mountain anticyclone decays far from the mountain as u x - l . Having discussed the stratified, unbounded Boussinesq solution in some detail we are in a position to evaluate critically the models of quasigeostrophic flow over a ridge which have appeared in the literature. These are shown in Fig. 19.

-

(a) The flow of a homogeneous fluid with a rigid lid (see, for example, Batchelor, 1967, p. 573). There is a "permanent" turning of the flow (3.27)

A0

=

f * (cross-sectional area) tan-' HU

caused by vortex line shortening over the ridge. This is a useful model to illustrate the nature of vorticity, but it has no application to the strongly stratified atmosphere. (b) The flow of a stratified, unbounded fluid described qualitatively in the standard meteorology textbooks (see Haltiner and Martin, 1957, p. 357; Hess, 1959, p. 252; Holton, 1972, p. 70). The lifting of 0-surfaces occurs only above the mountain and the lifting decreases with height, presumably due to the stratification. This model is not a solution to the

THE INFLUENCE OF MOUNTAINS O N T H E ATMOSPHERE

153

governing equations of quasi-geostrophic stratified flow and should be discarded. This housecleaning will not be easy, however, as t h e textbook model has influenced a generation of meteorologists. (c) The flow of an unbounded stratified fluid (Queney, 1947) discussed in detail above. There is a local baroclinic disturbance but no permanent turning. (d) The flow of a stratified fluid with a rigid lid (Robinson, 1960; Jacobs, 1964: Hogg, 1973: Janowitz, 1975: Merkine, 1975; Merkine and Kalnay-Rivas, 1976: Mason and Sykes, 1978). This flow is characterized by both a local baroclinic distance and a barotropic permanent turning given by (3.27). The use of a rigid lid i s sometimes defended by suggesting that the great stability in the stratosphere will prevent vertical motion above the troposphere, but this argument is incorrect (Smith, 1979b). (e) The flow of a compressibie, unbounded stratified fluid (Smith, 1979b). In addition to the local baroclinic disturbance there is a barotropic permanent turning [given by (3.27) with H replaced by the density scale height] associated with the production of vorticity by volume changes as the air parcels lift over the ridge.

3.2. The Effect of lnrrtia on t h e Flair, m’rr Mesoscale Mountains The foregoing discussion was designed to show the relationship between the different quasi-geostrophic solutions to flow over a mountain. These solutions are applicable only to very large orographic features with their smallest horizontal dimension exceeding 1000 km or so. On this scale it is exceedingly difficult to find a meteorological situation that approximates a uniform steady wind approaching a mountain range. For this and several other reasons, it is appropriate to study slightly smaller mountains, with widths of a few hundred kilometers, which quite frequently produce identifiable steady state flow patterns lasting many hours or even a few days. To do this we must not assume quasi-geostrophy but allow inertial effects to be important. One attempt to do this is to determine the first effect of inertia. Merkine ( 1975) used the semi-geostrophic approximation (a slightly less restrictive version of the quasi-geostrophic approximation) to examine the rotating stratified flow over an infinite ridge with a rigid top lid. Merkine and KBlnay-Rivas (1976) examine a similar problem but for an isolated mountain. Buzzi and Tibaldi (1977) use an expansion in powers of the Rossby number to solve for flow over an isolated mountain in an unbounded fluid. These results show interesting differences with the quasi-geostrophic case, which increase as the Rossby number Ro = U / f L ap-

THE INFLUENCE OF MOUNTAINS O N THE ATMOSPHERE

155

FIG. 19. Five models of quasi-geostrophic flow over a ridge. (a) Homogeneous flow with a rigid lid. The flow is columnar. and relative vorticity is present only over the ridge. There is a "permanent" turning of the flow. (b) "Textbook" description of stratified unbounded flow. According to this model, the flow is similar to (a) with the vertical displacements decaying aloft due to the stratification. This is not a solution to the governing equations. (c) Stratified unbounded flow as given by (3.25) and (3.26). There is no "permanent" turning of the flow. (d) stratified flow with a rigid lid. In addition to the baroclinic disturbance near the mountain, there is a barotropic "permanent" turning caused by the rigid lid. (e) Compressible stratified unbounded flow. In addition to the baroclinic disturbance near the mountain, there is a "permanent" turning associated with an extra production of anticyclonic vorticity caused by volume expansion as the parcels rise over the ridge.

156

RONALD B . SMITH

proaches unity. The flow, however, remains symmetric about the y axis (i.e., windward vs. leeward side) and the disturbance decays rapidly with height just as in the quasi-geostrophic case. It is clear then that the mathematical techniques used in these studies (i.e., semi-geostrophic approximation or an expansion in Rossby number) are not sufficiently powerful to describe the propagation of inertia-gravity waves. We suspect physically that these waves will become more and more important as the Rossby number increases toward. and then exceeds unity. The much earlier work of Queney (1947, 1948) on the flow over a small-amplitude, infinitely long ridge is extremely valuable for clarifying 1) flows. Queney the role of inertia in these difficult mesoscale (Ro considered the two-dimensional rotating stratified flow over a mountain in an unbounded fluid. Using Fourier analysis he represents his solution as an integral over the contributions from the different horizontal wave numbers. For example, the vertical displacement of a fluid particle q ( x , z ) is given by

-

(3.28)

q ( x , z ) = esz

J

h ( k ) exp{i[kx

+ k,(k)z]}dk

0

where h ( k ) is the Fourier transform of the mountain shape h ( x )

(3.29) and the vertical wave number k , ( k ) is a function of the horizontal wave number k and the background wind, static stability and rotation rate according to

(3.30) where

k,

=

N/U

and

k,

=

f/U

Equation (3.30) is derived from the equations of motion, the thermodynamic equation, etc. The factor esz in (3.28), with S = -(1/2p)/(dp/az), describes the tendency for the disturbance amplitude to increase aloft due to the smaller density there. The solution (3.28) will depend on the nature of the function k , ( k ) in the range of k where h ( k ) has appreciable values. For rather wide mountains (say, L > 50 km) h ( k ) will be appreciable only for k 4 k, and so for the purposes of evaluating (3.28) we can simplify (3.30) to

-

(3.3 1)

k,(k)

=

kk,(k’ - kj)-”2

T H E INFLUENCE OF MOUNTAINS ON T H E ATMOSPHERE

I57

This is equivalent to making the hydrostatic approximation. To simplify the evaluation of (3.28) Queney chooses the familiar bellshaped ridge (3.32)

which has a particularly simple Fourier transform

h ( k ) = h,,,ae-ka

(3.33)

We are now in a position to investigate two interesting limiting situtions. In the case of a mountain range that can be crossed in a few hours by a fltiid particle, we can neglect the Coriolis force reducing (3.31) to k,(X)

(3.34)

=

k,

Then using (3.33) and (3.44) in (3.28) gives

LW

r l ( ~ Z, ) = h,aeSzeiks~

eikl--h-a

dk

p Iksz

=

h,,aesr a - ix

and taking the real part

(3.35)

q ( x , z) = h,aesf

a cos k , z - x sin k,z a 2 + x2

This describes a field of vertically propagating hydrostatic internal gravity waves excited by the mountain (Fig. 3 ) . The disturbance is asymmetric about the ridge and does not decay with height. There is a considerable wave drag [ D / I = ( ~ / 4 ) p U N h ~ ] . The other extreme case is when the mountain is so broad that h ( k ) is k,. We can then reduce (3.31) to appreciable only for k

*

(3.36) This is equivalent to the quasi-geostrophic assumption. In fact (3.36) could be derived immediately from ( 3 . 15). Now (3.28) becomes ~ ( x z,) = h,aeSz

which gives

h ,aesz

1

[ a + ( N / f ) z ]+ i x

RONALD B. SMITH

158

and taking the real part gives

(3.37) which is identical to the earlier a prior; quasi-geostrophic solution (3.25) if s is set equal to 1 consistent with the Boussinesq approximation (see Fig. 17). Of course, the method of sources and sinks used to obtain (3.25) is powerful since it can be used to determine the quasi-geostrophic flow over isolated mountains. In the present context, however, Queney's twodimensional formulation (3.28) is more useful as it allows us to investigate the flow over mesoscale mountains where the flow is not quasi-geostrophic. By using asymptotic methods to evaluate (3.28) [with (3.31) and (3.33)], Queney was able to plot solutions for intermediate cases akf = 1 where both inertia and Coriolis force are important. In particular, he computed the flow over a mountain with a half-width a = 100 which corresponds roughly to the scale of the Alps (Fig. 20). Note that the pattern of vertical displacement is distinctly wavelike and, judging from the pressure difference across the mountain, the waves are transporting considerable momentum. Just as in the nonrotating case, the phase lines tilt upstream with height but unlike the nonrotating hydrostatic case, the wave energy disperses considerably aloft. The long waves trail behind somewhat because of the influence of the Coriolis force on their group velocity. The influence of the Coriolis force is more evident in the horizontal projection of the streamlines and isobars. These do not coincide near the mountain as the flow is quite ageostrophic there. Far from the mountain the flow becomes nearly geostrophic and Queney computes the velocity induced along the mountain to be

(3.38)

,

U(X,

z = 0 ) == - h m a N / x

This is identical to the asymptotic behavior (3.26) which was derived under the assumption that the flow field was quasi-geostrophic everywhere, even over the mountain. Thus, the fact that a mountain is narrow enough to generate waves and wave drag has no influence on the far-field flow, which behaves as if the flow is quasi-geostrophic everywhere. This result follows immediately from the linearity of the small-amplitude equations as there is no interaction between the small-scale gravity waves and the large-scale Fourier components which are quasi-geostrophic and which dominate the far-field motion. This result is not of great use because in reality there can be a rather strong coupling between the gravity waves and the larger quasi-geostrophic scales of motion. Of the most obvious form for this interaction

159

T H E INFLUENCE OF MOUNTAINS ON TH E ATMOSPHERE

a

100KM ION

I1

-500

0 I

500

1000

KM

X

L,-L,-------i

I

I

iy

GROUND LEVEL

X

FIG.20. The stratified rotating hydrostatic flow over a ridge with the parameter a f / U In this case the flow is strongly influenced by the Coriolis force but no dominated to the extent that the flow is quasi-geostrophic. This flow is an intermediate type between Fig. 3 and Fig. 19c. The influence of the Coriolis force is evident in the lateral deflection of the streamlines (bottom of figure) and in the dispersive nature of the longer waves trailing behind the mountain. (From Queney, 1948.) = 1.

is the breaking of waves and the deposition of their momentum into the flow (see, for example, Bretherton, 1969). T o estimate the magnitude of this effect we must be able to predict the magnitude of the drag on the mountain, know where the waves will break, and understand how the mean flow will respond to the loss of momentum. If the wave drag is due to shorter wavelength components which are not affected by the Earth's rotation, then the ideas discussed in the preceding section can be used to estimate the wave drag and the location of breaking. The response of the mean flow to the momentum loss is appropriate for discussion here, but will be postponed until we investigate the possibility that the gravity

160

RONALD B . SMITH

waves may be long enough to be influenced by the Coriolis force (i.e., inertial-gravity waves) as, for example, in Queney’s intermediate-scale solution. Queney himself does not appear to have computed the mountain drag associated with his flow field solution. Blumen (1963, using linear theory, attempted to compute the flux of momentum in the disturbance over an isolated bell-shaped mountain [Eq. (3.19)]. Blumen evaluated the expressions X

but, as discussed by Jones (1967) and Bretherton (1969), the correct form for the momentum flux is

(3.39b)

F,

m

m

m

m

pu’w’

=

dx d y

-

f --m

-X

The second term in (3.39) accounts for the excess Coriolis force acting between the undisturbed and the lifted stream surfaces and is necessary so that (3.39) can be unambiguously interpreted as the mountain drag. Using the equations of motion and integrating (3.39) by parts m

(3.40)

m

Near the ground where q(x,y) = h(x,y), (3.40) is the pressure force on the mountain, while at any other level (3.40) is the horizontal force acting on each material layer by the layer above. Blumen’s formulation leads to the incorrect conclusion that the long, nearly geostrophic, nonpropagating wave components are responsible for a lateral drag force F, acting on the mountain. In fact the pressure field for these components is symmetric with respect to q or h and from (3.40), the drag forces are zero. As an example, the flow described by (3.20), (3.21) has a nonzero

THE INFLUENCE OF MOUNTAINS ON T H E ATMOSPHERE

161

-

u ‘ w ’ but Fy is zero as we know from the fact that the perturbation pressure field is distributed symmetrically around the mountain. The wave drag [Eq. (3.40)] on a mesoscale ridge with uniform incoming wind and stability has been computed by Smith (1979a) and is shown in Fig. 2 1. The effect of rotation is characterized by af/ U and as this parameter increases (i.e., progressively wider ridges) the wave drag decreases. There has been some further theoretical work to understand the influence of a background shear 8 U / a z on the vertical propagation of inertialgravity waves. Jones (1967) has shown that the momentum flux as defined by (3.39) or (3.40) will be independent of height in the absence of dissipation. Eliassen and Palm (1960) and Eliassen ( 1968) have investigated the relationship between the vertical fluxes of momentum and energy. With 0 = U ( z ) , the interesting possibility arises that the singularity k = k, = f / U ( z ) in (3.30) will occur only at a particular level (i.e., the critical level) for each wave component. Preliminary studies of the structure of this type of critical level has been carried out by Jones (1967) and Eliassen (1968). Observationally there is no question that large mountain ranges can experience significant drag due to the development of high surface pressure on the windward side and low pressure on the lee side. This phe-

FIG.21. The influence of the Coriolis force on mountain wave drag. As the parameter P, a f / U increases, gravity waves are suppressed and the drag F drops below its f = 0 value. Note, however. that even for broad mountains with low Rossby numbers, there is still some wave drag. The point at a f / U = 1 corresponds to the flow shown in Fig. 20. (From Smith. 1979a.)

162

RONALD B . SMITH

nomenon is usually referred to as a "fohn nose" in reference to its association with the fohn wind and the characteristic "nose" shape to the surface isobaric patterns. Qualitative descriptions of the fohn nose can be found in Defant (1951) and Brinkmann (1970). It is difficult to determine precisely the pattern of surface pressure due to the wide spacing of surface microbarographs and the problem of reducing the measured pressure to a standard level, but approximate pressure differences across the mountain of 4 to 6 mbar are not uncommon in the Rockies, Scandinavian Mountains, and the Alps. The drag force associated with this A p is considerable and must act on the atmosphere somewhere. The response of the synoptic-scale flow to this loss of momentum represents a new facet to the dynamics which is not present in the linear theory of Queney. Theoretically little is known about the response of the atmosphere to a localized drag. The work of Eliassen (19.511, however, is a valuable conceptual guide. Eliassen used the w-equation (see Holton, 1972) to compute the response of a geostrophically balanced wind to a localized retarding force. He found that a secondary circulation (see Fig. 22) would be produced in the transverse plane as the flow attempts to restore itself to geostrophic balance. Qualitatively this transverse circulation has a strong component down the pressure gradient in the retarded region and a more widely distributed return flow. In this way the external force is balanced locally by the excess downstream Coriolis force there. The

a

b

FIG. 22. The cross-stream circulation caused by a local retarding force acting on a stratified geostrophically balanced stream. Such a force could be applied to a stream by the breaking of mountain waves. Locally the retarded fluid is pushed to the left by the pressure gradient force. The surrounding fluid is decelerated by the upstream Coriolis force acting on the return branches of the circulation: (a) barotropic mean flow: (b) mean flow with vertical shear and sloping isentropes: (c) force applied near the ground. (After Eliassen 195I .)

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weak return circulation has an excess Coriolis force acting upstream, and this serves to spread the localized imposed force over a much broader region of the flow. The secondary circulation is altered slightly by the presence of a nearby boundary or preexisting baroclinicity in the airstream. It is difficult to apply these ideas to the mountain flow problem for several reasons. For one thing it is difficult to predict where the mountain wave momentum will be deposited. Another problem is that Eliassen’s theory is restricted to two dimensions; that is, the external force acts everywhere along a line parallel to the mean flow. Observationally, Woolridge ( 1972), has noted that the equations of synoptic-scale motion involving acceleration, Coriolis force, and pressure gradient force do not balance in the upper troposphere and lower stratosphere over the mountainous terrain of Arizona and New Mexico. The extra force needed to balance the momentum equation is usually a retarding force and is interpreted by Woolridge to be the deposition of mountain wave drag. He defends his interpretation by noting the simultaneous appearance of wave clouds in the satellite photographs of the region.

3.3. Theories of Lee Cyclogenesis Mountains have been observed to influence the weather in many ways. The literature describing these observations has developed and remains quite separated from the theoretical ideas discussed earlier. The difficulty in connecting the observations and the theories is partly due to the complexity of the problem and partly to the limited training and experience of the investigators. The following discussion will also, unfortunately, fall short in this respect. There is a great need for detailed case studies of the weather in mountainous areas using closely spaced and frequent radiosonde releases, and with interpretation in terms of the fundamental concepts of fluid dynamics. The phenomenon of lee cyclogenesis has received far more attention than any other aspect of the synoptic-scale mountain flow problem, and for good reason. It is now known that a successful one-day or possibly a two-day weather forecast can be achieved by simply predicting the motion (using perhaps extrapolation or a barotropic numerical model) of the existing cyclonic storms. To go further it is necessary to predict the development of new cyclones, and from the statistical studies of Klein (1957), or Reitan (1974), and Radinovic (1965b), and others, it has become clear that the lee sides of the major mountain ranges are strongly pre-

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ferred sites of cyclogenesis. We note immediately that the usual explanation of cyclogenesis in terms of baroclinic instability (see Holton, 1972) makes no mention of orography. On the other hand, the simple prototype problems of the flow around mountains discussed earlier show no evidence of a developing lee-side cyclone. To explain the observed distribution of cyclogenesis will require new and more complicated ideas. The structure of cyclones developing in the lee of the Rocky Mountains has been analyzed by Newton (1956), Petterson (1956), Hess and Wagner (1948), Carlson (1961), Hage (1961), and Chung et al. (1976). Manabe and Terpstra (1974) and Egger (1974) have shown that numerical models can also be used to simulate lee cyclogenesis behind the Rockies and other large mountain ranges. The results of Chung et al. are the most comprehensive but do not differ greatly from the earlier studies. They find that it is important to distinguish between the weak depressions which form and then remain just to the lee (25% of cyclogenetic cases) and the stronger cyclones which form in the lee and then move away (75% of the cases). This is similar to the discussion of Speranza (1975) in which he emphasizes the difference between lee-side baric depressions and the actual production of cyclonic vorticity . The local baric depression could be associated with mesoscale mountain wave drag, whereas true cyclonic vorticity, because of its conservation property, could move away downstream as a migratory cyclone. The development of the true migratory cyclone can be described (following Chung et al.) as follows. An intense “parent” cyclone approaches the Rocky Mountain cordillera from the northwest. As it draws near, it turns slightly to the left and fills (i.e., weakens). Many hours later a cyclone is seen to form rapidly just downstream of the mountain range and initially move away to the southeast. At the same time the upper level trough which was associated with the parent cyclone has passed over the mountain and the eastern limb of the trough lies over the mountain lee side, the new lee cyclone is seen to develop (Fig. 23). There are two “classical” explanations for this behavior. The first would be called upper level or j e t stream control of cyclogenesis. According to classical theory, surface cyclogenesis (not just lee cyclogenesis) is associated with divergent flow in the upper troposphere. The upper level divergence is necessary both to cause the pressure to drop at the surface and to allow rising motion in middle levels which in turn produces cyclonic vorticity near the surface by low-level convergence. The region of upper level divergence is usually associated with positive vorticity advection, for example, a jet stream blowing out of a trough. According to the vorticity equation this positive vorticity advection aloft must be balanced by local divergence if the trough is only slowly moving.

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FIG.23. An observational model of lee cyclogenesis, triggered by an approaching "parent" cyclone. The parent cyclone approaches from the west and weakens as it encounters the high ground. Shortly thereafter a new cyclone forms and moves off toward the east. A number of theories have been proposed to explain these occurrences including simple conservation of potential vorticity and upper level "jet stream" control.

The argument of Newton (19561, Speranza (1975), and Chung et al. (1976) is just to say that lee cyclogenesis will occur where and when the low-level vortex stretching caused by divergence aloft is added to that caused by downward flow at the surface on the mountain lee side. This is precisely where the eastern limb of the upper level trough intersects the mountain lee side. One weakness of this type of argument is that it is difficult to tell whether the upper level divergence caused the surface cyclogenesis or whether there relationship is just a diagnostic association. A simpler explanation of the phenomena involves the conservation of potential vorticity. The potential vorticity of the parent cyclone is conserved as it crosses the ridge, but the relative vorticity is temporarily eliminated by vortex shortening while these air parcels are over the mountain. Upon leaving the mountain the parcels are stretched to their original length and the cyclone reappears. If the upstream flow is partially blocked, the lee cyclone can even be stronger than the parent. This theory also explains the curvature of parent cyclone path to the left as it approaches the mountain and the initial motion of the new lee cyclone to the right. The vorticity is moving with the fluid and therefore is advected by the mountain anticyclone described earlier. Of course the acid test of this theory is to determine if the lee cyclone is composed of the same fluid particles as the parent cyclone. Unfortunately this trajectory analysis has not been done. There has also been considerable interest in the influence of the Alps on the formation of cyclones in the Gulf of Genoa (RadinoviC, 1965a,b: Speranza, 1975; Egger, 1972; Buzzi and Tibaldi, 1977: Trevisan, 1976). There seems to be a consensus that the nature of Alpine lee cyclogenesis

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is different than the Rocky Mountain variety previously discussed. Whether this is due to the smaller horizontal dimension of the Alps, different orientation, more complicated geometry, nearby warm sea, or their different synoptic setting is not clear. The work of RadinoviC (1965a) and Egger ( 1972) indicates that lee cyclogenesis is associated with southward flow of cold air from central Europe and that the mechanism involves the blocking of cold low-level air by the mountains. The details as described by RadinoviC are rather complicated. None of these arguments are so compelling that the matter can be considered settled. The theoreticians have been going off in different directions proposing a wide variety of mechanisms for lee cyclogenesis. Some of these suggestions will be mentioned in the following: (a) Buzzi and Tibaldi (1977) note that the introduction of Ekman layer friction can alter the symmetric quasi-geostrophic flow discussed earlier to an asymmetric flow with a significant (but fixed) lee-side cyclone. The reason for this is apparently quite simple. The anticyclonic vorticity above the mountain tends to decay due to friction, but this represents an increase in potential vorticity. When the fluid columns leave the mountain, their increased potential vorticity is realized in the appearance of cyclonic relative vorticity. An extreme example of this is when the fluid particles have spent so long over the mountain that their relative vorticity has completely decayed. When dismounting the orography these particles will develop a full measure of cyclonic vorticity. In this sense, elevated regions with friction acting are generally sources of potential vorticity . This same mechanism operates in the laboratory study of slightly viscous flow of a rotating homogeneous fluid over an obstacle. This mechanism also seems to be acting in numerical weather prediction models. Whenever the mountain anticyclone is weaker than that required by constant potential vorticity, cyclogenesis will occur in the area where the air leaves the mountain. The weak anticyclone could be caused by friction in the model or by an incorrect analysis of the input data. (b) Another possibility is that the formation of a lee cyclone is a transient phenomena associated with rapid changes in the strength of the incoming flow. If initially there is little or no wind, the isentropic surfaces will not be parallel to the mountain surface but will be nearly horizontal. In this case there is no mountain anticyclone. A sudden increase in wind speed could cause the air near the mountain to be blown away downstream and replaced by upstream air. If the motion is adiabatic, the 8surfaces which intersect the ground must continue to do so; whereas in the vicinity of the mountain, the &surfaces lie parallel to the surface

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(Fig. 24). The intersection of the &surfaces with the ground comprises a warm-core cyclone caused by vortex stretching which moves off downwind. Such a “starting vortex” is commonly observed to be left behind in Taylor column experiments in homogeneous fluids when the obstacle is impulsively started from rest. This transient mechanism is particularly appealing as it agrees with the observation of Radinovie (1965a) that cyclones form to the south of the Alps soon after the onset of strong flow from the north. A difficulty with the theory is that it is hard to know when the 0-surfaces will intersect the mountain and when they will go over. A test of this theory would be to see if the lee cyclone is composed of air parcels that were originally over the mountain. (c) Merkine (19751, on the basis of a theoretical analysis of baroclinic flow over a ridge, has suggested that the effect of the ridge is to increase the baroclinicity of the atmosphere. This could lead to enhanced cyclogenesis through the classical mechanism of slantwise convection. (d) If the growth of a cyclone is viewed as a self-sustaining process which need only be triggered by low-level convergence then the lee side

anticyclone

warm-core cyclone

(b) FIG.24. The generation of a surface-intensified cyclone associated with the onset of a strong wind. (a) Initially there is no flow and the isentropic surfaces lie flat, intersecting the mountain. (b) With the onset of a strong wind, the air over the mountain is blown away and at the same time vertically stretched to form a traveling warm-core cyclone. The isentropes that initially intersected the ground continue t o do so. Near the mountain. the uplifted &surfaces indicate the presence of a mountain anticyclone.

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of a mountain would be a natural location for growth initiation. In a barotropic inviscid atmosphere, however, the lee-side convergence acts only to restore the vortex lines to their original length, not to create new cyclonic vorticity. This triggering mechanism seems to be implicit in several observational descriptions of lee cyclogenesis but apparently has not been investigated theoretically. (e) It is known theoretically from the work of Queney (1948) and Johnson (1977) among others that if the Coriolis force is taken to be a function of latitude, the generation of Rossby waves will introduce an asymmetry in the flow. In fact the first trough of the wave system will occur just downstream of the mountain. There is a certain resemblance between this flow and a lee cyclone except that this Rossby wave is a standing wave and the trough will not move away downstream. The effect of a nonconstant f(i.e., the p-effect) is not expected to be important unless the mountain dimension L approaches the Earth radius. The restoring force for Rossby waves is not just p, however, but the gradient in the background potential vorticity. Locally the horizontal gradients in wind speed and temperature (or thickness) can cause a restoring force much larger than that due to the variation of f.It follows that the influence of Rossby waves could be important for much smaller mountains. This may have had an influence on the numerical simulation of lee cyclogenesis by Trevisan (1976). (f) One simple way to explain lee cyclogenesis is to hypothesize that the mountain blocks the low-level flow. The flow aloft must then descend in a fohn-type wind producing warming and vortex stretching in the lee (Defant, 1951). This is related to, but is not the same as, the effect of blocking as described by RadinoviC (1965a) and Speranza (1975). (g) Another possibility is that the mountain drag may act in some way to create the lee cyclone. For example, near one flank of the mountain the variable drag could produce a torque on the atmosphere which in turn will produce vorticity. This list of mechanisms i s not meant to be inclusive but is merely intended to illustrate the kinds of possibilities that may have to be considered. These proposals are certainly less precise than the mathematical problems considered earlier in the section. The use of numerical models appears to be one promising method for closing the gap between the simple, but precise, analytical ideas and the reality of atmospheric flow. Several authors (Egger, 1972, 1974; Manabe and Terpstra, 1974; Trevisan, 1976) have reported success in numerically simulating lee cyclogenesis. This represents a great step forward, but as of yet it is not known exactly how these solutions can be used to understand the flow physi-

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cally. All of these models incorrectly model the dynamics of inertial gravity waves and so cannot be compared with the analytical solutions of Queney. Eliassen and Rekustad (1971) have taken more care to treat these waves by using greater resolution in the vertical and a radiation condition aloft. Their model, however, is restricted to two dimensions, has dangerously close inflow and outflow boundaries, and has been evaluated only for very special upstream conditions. Clearly more work is needed in this area.

4. OROGRAPHIC CONTROL OF PRECIPITATION

One of the most striking ways in which topography influences the weather is in its strong local control of the rainfall distribution. The most obvious examples are the rainfall maxima on the upwind side and the corresponding dry, rain-shadow regions in the lee of major mountain barriers at latitudes with consistent prevailing winds. The clearest case of this is the wet-dry contrast across the Andes in South America which reverse its orientation when passing south from the tropical easterlies to the midlatitude westerlies. On a much smaller scale, but equally striking, is the now well-confirmed observation that rainfall can often be a factor of 2 greater at the tops of small (50 to 100 m) hills than in the surrounding valleys. This has an especially large impact on the plant life of the region and also on the scientist trying to construct regionally averaged rainfall data. Of course we expect these two examples of orographic control of rain to be physically somewhat different, as between these two extreme scales there lie several natural "cloud physics" length scales-especially the distance of the downwind drift during (a) the lifetime of a cumulus cloud, (b) the formation of raindrops from cloud droplets, and (c) the fall of hydrometeors to the ground, as well as the natural scales N / U and f / U which affect the dynamics of airflow over the hills. One of the challenges of this section will be to investigate the effect of mountain size on the nature of orographic rain. Some aspects of this problem have been reviewed by Bergeron (1949).

4. I . Observations of Rainfall Distribution 4.1 . I . Distribution of Annrrul Rair!fall with Respect to Elevation. A tremendous amount of information has been collected concerning the distribution of precipitation in mountainous areas. These data are for the

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most part concentrated in the hydrological, geographical, agricultural, and water resources literature-mostly in unpublished reports of government agencies. As a first attempt to organize these data it has become standard practice to correlate statistically the annual average precipitation (mm/yr) against station elevation (meters above sea level). In most cases it has been found that precipitation tends to increase with height. This is also consistent with Longley’s (1975) observation that precipitation decreases with depth down in isolated valleys. To easily represent this trend, the linear regression slope is (4.1)

dP( mm/year) ‘ a dz

computed where the coefficient a describes the rate of increase of annual precipitation with height. An alternative description (Ryden, 1972) is the relative increase of precipitation with height (4.2)

dlog P - 1 d P REdz P dz

Some typical examples of values for a and R are shown in Table I1 where the height increment is taken to be 100 m, and R is expressed in percent. The linear regression equations (4. I ) or (4.2) are somewhat misleading as in almost every case there is considerable scatter about the linear regression representation. This is to be expected as the precipitation should depend on many other factors, for example, the yearly pattern of weather type, ground temperature, the size and shape of the surrounding TABLE11. TYPICAL EXAMPLES OF VALUESFOR u

United Kingdom Bleasdale and Chan (1972) Pennines (UK) Chuan and Lockwood ( 1974) East Pennines West Pennines Western Canada. Mormot Creek Storr and Ferguson ( 1972) Northern Sweden Ryden (1972) Kamajokk ( 1967) Kamajokk (1968) Malmagen

AND

R

250

2S%

200 I90

40 25

60

10

18

7

7 22

6

9

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171

mountains, and the vertical profile of temperature, humidity, wind, speed and direction, and possibly the aerosol distribution. Furthermore, there are exceptions to the trend shown in Table 11. On very high mountains the rainfall may increase u p to a certain height and then decrease. From the wide variation of a between different regions (a factor of 40) it is clear that a is not in any sense a fundamental constant. The variation in R is somewhat less, indicating that regions with greater annual rainfall have greater variation of rainfall with height, but R is still clearly not a fundamental quantity. For the scientist trying to understand the nature of orographic rain, the data in Table I1 are of little use. It cannot be interpreted to mean that in a single rainfall event the precipitation increases with height, which might, for example, be explained by the evaporation of raindrops before hitting the ground. This method also fails to describe the upslope rain-rain shadow contrast which we know is important for broad mountains. If the region being considered has a consistent prevailing wind, this latter effect is well represented by the areal distributions of annual precipitation. 4.1.2. Rainfall distribution with Respect t o Wind Direction and Weather Type. In order to gain more information about the orographic control of precipitation, while retaining the statistical approach, it is necessary to classify the data according to weather t y p e , wind direction, or both. Some examples of this kind of analysis will be given here. Wilson and Atwater (1972) studied the distribution of rainfall in the state of Connecticut, a region of low (<300 m) hills. They restricted their study to rainstorms of the large-scale stratiform type and classified the cases according to wind direction. They conclude that the major part for the spatial variation of rainfall can be explained by the influence of topography. Generally there was greater precipitation in the hills, but with the maximum shifted toward the upwind slopes. Bergeron (1968, 1973) studied the rainfall distribution in the region of low hills (<60 m) near Uppsala, Sweden. For the most part Bergeron used monthly averaged data with each month classified according to weather convective showers or continuous stratus rain was predominant. With convective showers (typically summertime) the rainfall distribution is characterized by swaths of rainfall corresponding to convective clouds passing over the area. Topography appears to have little effect. In the fall months, with primarily stratus rain, Bergeron found a remarkably strong dependence of rainfall in height. For example, between Lake Ekoln and Lunsen Hill, 60 m higher and only 5 km away, the rainfall nearly doubled. Bergeron (1968) presents a conceptual model for this effect which will be described later.

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On a larger scale, Nordo and Hjortnaes (1966) and Andersen (1972) considered the precipitation distribution over the broad mountains (width 250 km, height 1500 m) of southern and central Norway. Nordo and Hjortnas statistically relate the rainfall distribution to the component of geostrophic wind directed against the mountain during the same period. Andersen, on the other hand, uses monthly mean precipitation, with each month being classified as to prevailing wind direction and weather type. In spite of the differing technique the results of both studies are broadly similar. The rainfall is strongly controlled by topography with the maximum occurring near or slightly upwind of the steepest surface slope. At the divide the rainfall is a small fraction of the upslope maximum and the lee side is remarkably dry. There are some important exceptions to this simple picture, for example, the strong precipitation maximum that sometimes occurs in the wintertime on the southeast coast of Norway. Bergeron (1949) noticed this same anomaly in a case study and has put forward a possible kinematic explanation (see also Smebye, 1978). Andersen (1972) and Utaaker (1963) appear to find that the smaller scale mountains in Norway exhibit a quite different rainfall pattern with the maxima occurring at mountain top o r in the lee. The use of rain gauge data, even from special networks, has so far proved unable to provide a clear picture of the structure of orographically induced cumulonimbus rain. This is not surprising as these cloud elements appear rapidly and apparently randomly over mountains terrain on warm summer days and seem to move away downstream. Biswas and Jayaweera ( 1976), using satellite observations, concluded that there can be strong topographical control of air mass thunderstorms in the Alaskan region. Kuo and Orville (1973) studied the climatology of convective clouds in the Black Hills using radar. They also found strong topographic control with the maximum cloud development appearing downwind of the mountain peaks. Skaar (1976) indicates that in the summer over Norway the prevalence of convective rain results in a much broader distribution of rainfall although in detail, much more complex. This sampling of observational results seems to suggest the importance of two factors:

-

-

1 . The weather type (or season) which can determine the relative importance of stable versus convective rain. 2. The size of the mountain which determines whether the orographic rain will occur on the upwind slope with a rain shadow in the lee (i.e., larger mountains L > 100 km), o r with the maxima more nearly centered on the mountain (i.e., smaller mountains L < 20 km).

There appear to be three rather independent mechanisms of orographic

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rain. These are (see Fig. 25): 1 . Large-scale upslope precipitation. Orographically forced vertical motion or convection triggered by smooth orographic ascent brings the air to saturation and after some delay, raindrops form and fall to the ground. 2 . Enhancement of rainfall over small hills (after Bergeron, 1968). Rainfall from preexisting clouds (either frontal or orographic) is partially evaporated before hitting low ground, but over hills amplification occurs by washout of cloud droplets from low-level pannus clouds. 3. Orographic control of the formation of cumulonimbus clouds in a conditionally unstable airmass. Heating of the mountain slopes by insolation causes upslope winds leading to thermals above the mountain peak which trigger the formation of convective clouds.

In the following sections 4.2, 4.3, and 4.4, we will examine each of the above mechanisms. 4 . 2 . The Mechanism of Upslope Rain 4 . 2 . 1 . A Prototype Model of Upslope Rain. We will derive a simple model of upslope rain which will serve as a focal point for further discussion. Consider a reference volume V = 1 m3 at some height z above the surface in a saturated region of the atmosphere. The rate of condensation (kg/sec) in the volume will be the rate at which the saturation water vapor density pw, = r s p a i rdecreases following the parcels flowing through the volume,

FIG.25. Three mechanisms of orographic control of precipitation: (a) broad-scale upslope rain, (b) small-scale redistribution of rain by hills, and (c) orographic-convective showers.

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RONALD B. SMITH

If the decrease in pn, is caused by adiabatic lifting, then (4.3)

If precipitation-size particles (raindrops or snowflakes) can form immediately from the cloud droplets, and if these hydrometeors fall directly to the ground with no downwind drift, then the rate of precipitation at the ground (kg/m2 sec) is just the vertical integral of (4.3).

(4.4)

Equation (4.4) is still rather general in that w in (4.2) could be associated with (a) large-scale frontal uplift, w 10-50 cm/sec; (b) orographic uplift, w 10-50 cm/sec; or (c) convective updraft, w 1-5 m/sec. In all of these cases the precipitation rate will be closely associated with the strength of the updraft. The intensity of orographic uplift can be estimated by assuming the the flow at al! levels is parallel to the sloping mountain surface. Then

-

-

(4.5)

-

w ( z ) = U(z)ff

where a is the surface slope and U ( z ) is the horizontal wind. With this simplification (4.2) becomes (4.6)

R

(A) = afomU(z) m2/sec

I

dz

2

dz

dpu

ad

To obtain a representative estimate for the intensity of orographic precipitation we can consider the special case (1) U ( z ) = U = const and (2) the environmental temperature 7 ( z ) lies along a moist adiabat, so that (4.7)

This is nearly true in many cases of orographic rain (see, for example, Douglas and Glasspole, 1947). With these, (4.6) becomes

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175

and as pn, + 0 aloft (4.8) In this special case the general dependence of (4.6) on the temperature and humidity structure reduces to a simple dependence on r,(o), the mixing ratio at the ground. As an example, choose a = 1/100, U = 10 m/sec, r , ( P = 1000 mbar, T = 15°C) = 11 grn/kg = 0.011 kg/kg, pair = 1.2 kg/m3. Then R(kg/m sec) = R(mm/sec) = 1.3 x lo+, or R = 4.7 m d h r . This value is not atypical of observed rates, but the analysis above is intended only to illustrate the kind of assumptions which are often used. 4.2.2. Efficiency of Release. A number of authors have made estimates of the efficiency of precipitation release during orographic lifting. These estimates require (1) rawinsonde profiles of the incoming wind, humidity, and temperature; (2) a way to estimate the amount of lifting that occurs at each level; (3) rain gauge measurements of precipitation over the upslope area. Sawyer (1956) compared six examples of measured precipitation on the windward slopes in Wales with the amount calculated from an equation like (4.6). He found that when conditions are favorable for heavy orographic rain, the efficiency (i.e., observed rainfallkomputed condensed water) is nearly 100%. On the other hand, when only shallow layers of moist air is present, so that less total condensation occurs, the efficiency is reduced t o 30-5096, and thus the rainfall is greatly reduced. Browning rt al. (1975) have computed the efficiency during four cases of pre-cold-frontal upslope rain in the same area, i.e., Wales. They as-

---

FIG. 26. Two possible mechanisms of upslope precipitation: (a) stable upglide and (b) orographic triggering of closely packed convective showers.

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sumed that the lifting was independent of height [as in (4.6)], but included only the condensation occurring below z = 3 km, arguing that the liquid water formed above that level would be blown downstream out of the upslope control region. In two cases the efficiency was quite large, close to 70%. These cases were characterized by (a) a strong, moist low-level jet directed against the mountain slope; (b) incident airstream already near saturation: (c) some condensation occurring above the freezing line allowing ice-phase process to aid in the release of precipitation. The two other cases examined by Browning et al. (1975) had much lower efficiencies-approximately 10% and 30%. These cases were characterized by initially unsaturated air requiring some finite ascent before condensation begins, and in one case, little or no condensation above the freezing line. A number of papers have been written concerning the efficiency of orographic rain on the western side of the California mountain ranges (Myers, 1962; Elliot and Shaffer, 1962; Elliot and Hovind, 1964: Colton, 1976). Myers computed an efficiency of 70% for the large Sierra Nevada Range using a modified form of (4.6). Instead of taking a to be constant with height, he assumes that (Y drops off linearly from the ground to a presumed “nodal surface” at about z = 5 km. This somewhat reduces the computed condensation and increases the efficiency. Elliot and Hovind (1964) used a similar nodal surface assumption in the study of rainfall on the smaller San Gabriel and the still smaller Santa Ynez mountains. They classified each case according to the conditional stability of the incoming air (Table 111).They conclude (looking also at Myer’s data) that efficiency tends to increase with mountain size. The increased efficiency for unstable flows over the Santa Ynez could either be caused by an improved release mechanism or by an increase in condensation over that given by (4.6) due to the strong vertical motions in local convective elements. TABLEI11 Mountains

Number of cases

Efficiency

San Gabriel Stable Unstable Total

31 8

39

26% 27 26

21 22 43

26 22

Santa Ynez Stable Unstable Total

17%

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Colton (1976) compared computed and observed rainfall rates on the Sierra Nevada (130 km wide) and the Smith River Basin (70 km) using the assumption of full release. 'The use of a flow model with w going to zero at about z = 5 km, gives remarkably good results, suggesting nearly complete release. As in the other studies, however, it is difficult to sort out the efficiency of raindrop formation (a cloud microphysical process) from errors in the assumed w-field. We have seen that the precipitation efficiency depends both on the assumed field of lifting and on the microphysical processes leading to the formation of hydrometeors. In later sections we will discuss the former problem both with regard to the mean flow and to the possibility of smallscale convection triggered by the orographic lifting. Some insight into the cloud physics problem associated with orographic precipitation is provided by Young (1974a,b). Using a specified flow field and upstream sounding, Young computed the distribution of precipitation over the Front Range in Colorado. His model included a wide variety of microphysical processes and the downwind drift of hydrometeors. The parameterization of artificial seeding by silver iodide was also allowed for. The efficiency found by Young is natural conditions with optimum seeding

0.04% 20.00%

The natural efficiency found by Young (.04%) is so much smaller than the observations discussed previously, that some attempt must be made to explain it. Either Young's model is incorrect, or the mechanism of Front Range orographic rainfall is entirely different than that of California and Wales. The primary differences between these mountain flows are: (a) Unlike the mountains of Wales and California, the Front Range is set well inland and even its base is 1500 m above sea level. In the case studied by Young, the bottom of the moist layer is at about 3400 m with T = 0°C. Thus the amount of available water is much less than in the case of a coastal range. We have already seen the suggestion (from Sawyer and Browning et ul. ) that a decreased moisture content can lead to a decreased efficiency. (b) Another important difference is that the Front Range is only 40 km wide-significantly narrower than the mountains of California or Wales considered previously. One result of this is that the distribution of precipitation computed by Young is nearly symmetric with respect to the ridge crest instead of having a maximum on the windward slope. Possibly, by considering narrower mountains such as the Front Range we have stepped into a new regime where the time for hydrometeor formation and

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fallout is the same or longer than the time for the air to pass over the mountain. The condensed water aloft is never realized as precipitation at the ground. The narrower mountains can cause precipitation only by the introduction of seeding, either natural or artificial. Young finds that the efficiency can be significantly increased by artificial seeding and this agrees with results of the Climax experiment (Mielke et a / . , 1971), which found significant increases in wintertime precipitation by artificial seeding. The other possible way to get small-scale orographic precipitation is that described by Bergeron (l960a, 1968), namely natural seeding from above by a larger scale cloud system. This possibility will be discussed in a later section. Even if we accept the idea that large-scale orographic lifting a deep warm moist air current can cause some release, it is still surprising, in light of the difficulties in forming precipitation-size particles, to find release efficiencies of 7% to 10% such as reported by Sawyer (1956), Myers (1962), and Browning et a / . (1975). Is it possible to convert such a high fraction of the condensed water into precipitation? If not, the simple method of computing the vertical motion field (4.3) must be considerably in error. Either the mean streamlines aloft are lifted by an amount greater than the mountain height or the orographic lifting triggers deep convection. 4.2.3. The Controversy concerning Stable versus Unstable Upslope Rain. In a lecture before the Royal Meteorological Society, L. C. W. Bonacina (1945) surveyed the regions of intense orographic rain around the world and the possible mechanisms. He concluded that orographic rain does not occur every time an airstream impinges on a mountain, but rather that the airstream must have been conditioned by the prevailing synoptic situation. In particular he emphasized the importance of convective instability for the generation of intense orographic rain. Two years later Douglas and Glasspoole (1947) refuted Bonacina’s contention by examining several cases of warm-sector orographic rainfall in the British Isles. They noted that the upstream surroundings showed slight conditional stability rather than instability. Further, they showed that simple orographic lifting could (assuming 100% release) account for the observed rainfall amounts. They concluded that, at least in their cases, the orographic uplift was a stable well-ordered process without convection. The interested reader should also notice the controversial discussion that follows Douglas and Glasspoole’s paper. To a certain extent Douglas and Glasspoole’s arguments seem to have carried the day

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as many subsequent authors have flatly assumed stable lifting in their models. Recently, however, a much more detailed study of warm-sector orographic rain in this same geographical area has been reported by Browning et al. (1974). From frequent rawinsonde releases, a network of continuously recording rain gauges, and radar observations of the precipitation-bearing clouds they were able to construct a remarkably complete picture of precipitation mechanism. The distribution of precipitation rate (mm/hr) in space and time is shown in Fig. 27. It is clear that while there is continuous rain over the mountain, the rain is really due to a coalescence and intensification of the rain from moving convective cells (called MPA’s-mesoscale precipitation areas, by the authors). This structure is also shown in Fig. 28. These MPA’s form upstream in a middle-level layer of potentially unstable air-presumably triggered by far-reaching orographic lifting. They move downstream at a speed characteristic of the wind speed at rnidlevels. Note on Fig. 28 that the vertical extent of the MPA’s is about 2000 rn, and this is probably a fair estimate for the ascent distance of air parcels within these convective elements. This represents a great increase in lifting over the stable ascent hypothesis as the mountain in this case is only 300 meters high. Browning et al. also found small-scale convection occurring in a 2-km thick, potentially unstable layer near the ground. Again the lifting in these clouds probably far exceeded the height of the mountain, leading to considerable condensation. If it were not for these low-level clouds, the precipitation reaching the ground would have been considerably reduced and distributed more widely downstream. These low-level clouds are responsible for the apparent coalescence and intensification of rainfall over the mountain. The low-level clouds probably have insufficient time to form precipitation by themselves before the lee slope descent begins. Thus they would produce no rain. In the presence of precipitation from above, however, their condensed water can be efficiently washed out, greatly increasing the rainfall at the ground. This redistribution of rain by low-level clouds will be discussed again later. The contrast between stable and unstable rain has also been considered in the frequent orographic rain on the west coast of Norway. In two papers Spinnangr and Johansen (1954, 1955) attempted to use conventional synoptic and radiosonde data to describe, and to distinguish between, cases of stable and convective orographic rain. Their 1955 paper describes the approach of Maritime Polar (MP) air toward the mountains. They show that the rainfall begins when the sounding just upstream has become slightly unstable with respect to saturated lifting. They argue that the cold MP air has been destabilized by its passage over the warm Gulf Stream and the orographic lifting then triggers the shower activity.

I80 c

1-

2 E

W-

I

E

1

FIG.27. A xf diagram showing the rainfall rates (mm hr-') and the movement of "mesoscale precipitation areas" (MPA) over the hills Of South Wales. The rainfall begins ahead of the cold front and continues in the warm sector. The rainfall rate over the hills is continuous. but variable. and closely associated with the passage of convective clouds (MPA's) aloft. (After Browning er ul., 1974.)

-1

I

a

E

J

1

t

8

LowLEVEL PI

- a h upw*D

-loo

0

mLSn Hus

+loo

+200 k n DowrwlNo

FIG. 28. A schematic cross section of the rain clouds in a warm sector over hills of South Wales. The moving. orographically triggered. convective clouds aloft produce precipitation which is locally enhanced by the low-level convective clouds over the hills (see Fig. 27). The slope of the hydrometer trajectories changes abruptly at the freezing line as snow changes to rain. (After Browning et d.,1974.)

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Occasional showers out at sea give additional evidence of the unstable state of the atmosphere. In spite of this, the heavy orographic rain inland was described as contiriiroiis rtriri by local observers-/lot twin s l i o w r s . The showers had apparently “been packed together and had amalgamated into continuous rain.” This is similar to the observations of Browning rf a / . (1975) discussed earlier. The intense rainfall concentrated just inland from the coast where t h e topographic slope is the greatest, with little rain reaching the divide 100 km inland. This rapid decrease of rain downstream is attributed by the authors to the short lifetime of convective clouds. In the second paper Spinnangr and Johansen (1954) consider cases of southwest flow of warmer Maritime Tropical (MT) air toward the west coast of Norway. This air was contained in the warm sector of a developed cyclone just as in the case studies of Douglas and Glasspoole and Browning r t a / . discussed earlier. They argue that the northward flow of this tropical air should generate a stable air column. The soundings taken well upstream (in England) are noticeably, but not strongly stable against saturated lifting. The observed intense orographic rainfall was distributed spatially in a similar way to their 1955 study of instability showers. The observations of rain type was also similar with a predominance of coiltiniioiis r(ii17 (although of temporally variable intensity) and reports of twin shobr*crs near the beginning and end of the rain period. Spinnangr and Johansen also show, using arguments similar to those of Douglas and Glasspoole, that the observed intense rainfall can be explained by stable upglide if 100% release is assumed. Upon rereading these two studies by Spinnangr and Johansen ( 1954, 1955), one is struck by the great similarity between them and with the studies of Browning r t (11. (1975) and Douglas and Glasspoole (1947). Certainly the reports of rain type cannot be used to distinguish reliably between the two types of orographic rain as even the rain showers tend to coalesce as shown in Browning c’t t i / . and Spinnangr and Johansen (1955). The spatial distribution of precipitation may also be similar. Even the observation of slight conditional stability in the approaching air, as in Douglas and Glasspoole, and Spinnangr and Johansen (19541, does not rule out instability showers because (as will be shown later) the orographic lifting aloft upstream of the mountain acts strongly to destabilize the column. It seems possible then that even the events which were presumed to be cases of stable rain [i.e., Douglas and Glasspoole (1947) and Spinnangr and Johansen (1954)] may have been in fact instability showers such as described in the more detailed observations of Browning of a / . (1975). This interpretation also offers an explanation of the rainfall intensity found in these cases without requiring the assumption that all of the condensate reaches the ground as rain.

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Two recent technological developments are providing new insights into the orographic rain problem. The first is the use of cloud seeding to enhance orographic precipitation (Chappell et a l . , 1971; Mielke et al., 1971; Grant and Elliot, 1974). The success of this seeding is itself an indication the natural efficiency of precipitation release must be significantly less than 100%. More direct evidence comes from visible and infrared imagery of orographic clouds from satellite (Reynolds et al., 1978: Reynolds and Morris, 1978). As a frontal system approaches, the precipitating orographic cloud is often obscured by a high-level layer of cirrus. Later in the storm, the cirrus moves off downwind and the orographic cloud is revealed. The top of the orographic cloud appears very irregular indicating vigorous convection within the cloud layer (see Fig. 29). 4.2.4. The Interaction of Frontal and Upslope Rain. It often happens that frontal rain appears to be intensified and prolonged in mountainous areas. To a certain extent this could be explained as a sequential ap-

FIG.29. Satellite photograph of a broad-scale orographic cloud over the Sierra Nevada range east of San Francisco, California. The rectangle has dimensions 68 km north-south and 42 km east-west. The irregular top of the cloud layer indicates the presence of vigorous small-scale convection. (From Reynolds and Morris, 1978.)

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pearance of prefrontal “pure“ orographic rain followed by the frontal passage. As an example, the warm-sector orographic rain discussed by Douglas and Glasspoole (1947) and Browning et a / . (19741, ended with the passage of the cold front. They found, however, that the frontal rain itself was not much increased by the orography. Browning Pr a / . (1975) also discussed the interaction between a cold front and the Welsh hills. A somewhat different view is expressed by Petterssen (1940, pp. 298302) and Bergeron (1949) who describe the orographic influence on frontal characteristics. The most detailed case study of mountain-front interaction is that of Hobbs et a / . (1975). Using aircraft measurements, frequent soundings, and a network of automatic rain gauges, they were able to piece together a fairly complete picture of the passage of an occluded front over the Cascade Mountains in Washington state. They find a definite influence of the mountain on the front. The timing of the precipitation is strongly influenced by the front, but the location of the rainfall is almost entirely on the windward slopes. This is shown in Fig. 30.

185

148

I1I

74

37

Time (hours) and distance (kilometers) from frontal passage

FIG. 30. The passage of an occluded front over the Cascade Range in Washington state: (a) the distribution of clouds 1 hr after the front crossed the mountain crestline: (b) the precipitation rate (lo-’ m d h r ) plotted as a function of the distance from the crestline (km) and the elapsed time from local frontal passage (hr). Frontal precipitation occurs even before t h e front reaches the mountain (region A on the figure). The mountain greatly increases the frontal precipitation (region D), but this seems to “dry out” the front as frontal precipitation stops as the mountain is left behind (region F). In the upslope region, rain continues for many hours after the passage of the front (region E). (From Hobbs et a/.,1975.)

FIG.30. (Continued.)

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The relationship between orographic precipitation and synoptic situation in the central Rockies has been studied by Williams and Peck (1962). 4 . 2 . 5 . Tho D.vtianiic.s of' Air:flo\\, in Coririectioii with thc P~-ohlcniqf Upslopr Rrriri. The general dynamical problem of airflow over mountains has been discussed in the other sections. Still there are some special aspects to the problem which are associated with upslope rain and they deserve attention here. First, there is the question of the dynamical influence of the latent heat of condensation. Second, the results of the earlier sections should be re-examined to understand their implications for the generation of orographic precipitation. For example, if conditional instability is important in orographic rain, we should see what destabilizing influence the mountain may have. It has become quite common to assume that the vertical velocities above a mountain are confined to the region directly over t h e mountain and that the slope of the streamlines is either constant with height [as in Eq. (4.511 or decreases to zero at some midtroposphere level (see Fig. 31). Dynamically such models are clearly incorrect as are the attempts to derive them from the governing equations. Myers (1962) defended such a solution using arguments taken from hydraulics-the study of discrete fluid layers. Such arguments are not applicable to the continuously stratified atmosphere, however, as in this latter case there is no maximum long-wave speed. Atkinson and Smithson (1974) use a correct set of governing equations (without moisture) to derive such a flow field, but their solution violates the important upper radiation condition and is therefore incorrect. On the other hand, the use of such a simplified model as in Fig. 31 probably does not introduce very much error in the compution of total condensation if the air is passing smoothly and stably over the mountain. This is especially true in cases when most of the moisture

I

w=o

FIG.31. The simple model of airflow used by several authors in their models oforographic rain. The lifting of the air is confined to the region directly over the mountain, and the slope of the streamlines decreases upward to zero at an undisturbed level. Such an airflow pattern probably would not occur. but it is not known whether the use of such a model would introduce appreciable error.

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is in t h e lowest layers as these layers must follow t h e ground regardless of what model is used (unless there is blocking). The primary point here is that all actual solutions to flow over an isolated mountain or ridge show that the lifting aloft begins well upstream of the mountain. This is true for hills and mountains of all scales regardless of whether the flow is dominated by inertia, buoyancy, or rotation, but i t is probably most marked for mountains from 10 to 200 km wide, generating gravity waves which tilt strongly upstream. For example, the flow over a I-km high mountain could well experience 500 m of lifting aloft, say, at 600 mbar, before the ground surface begins to rise. In a model of stable upslope rain, this effect would move the condensrrtioii i ? ~ m i i i i i i i nsome distance out ahead of the point of steepest surface slope. Of course the delays associated with the formation of hydrometeors and their fall to the ground could well push the vmiiifXl inciximiiin back to or even beyond the steepest mountain slopes, depending on t h e horizontal scale of the mountain. The lifting of middle-level air ahead of t h e mountain has a stronger implication if we consider the destabilization of the air by the mountain. Consider the case where the incident airstream is slightly undersaturated (relative humidity slightly less than 100%) and slightly conditional stable (8, increases slowly with height). As the air approaches the mountain there is lifting aloft, but not at the surface. The lifting by itself is important, of course, as it brings t h e air closer to saturation, but, the fact that the column is vertically stretched, may be just as important in the generation of showers. The lifting aloft is at first along a dry adiabat (see Fig. 32) and acts strongly to destabilize the column. Then, when saturation is reached, t h e column is conditionally unstable even if it was conditionally stable way upstream. Such a destabilization cannot be described by the simple models with an undisturbed nodal level as only colilmn shortening occurs in such a model.

FIG.3 2 . The destabilization of a nearly saturated. nearly conditionally unstable air mass due to lifting aloft upstream of the mountain. The air parcels aloft rise dry adiabatically thus decreasing the stability of the air column against moist convection.

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There have been a number of studies in which the governing equations for fluid motion have been used to determine the airflow pattern over a mountain for use in a orographic rain model. As a first step, Sawyer ( 1956) discussed the application of the adiabatic linear theory solutions to the qrographic rain problem. He also estimated the amount of "spillover" of rainfall into the downslope area due to the delays in droplet coalescence and fallout. A more complete model of this sort has beer, presented by Gocho and Nakajima (1971). There must be some question as to the validity of their flow solutions, however. as their streamline patterns do not agree with those of other investigators. Their computation of spillover shows that most precipitation falls downstream of the crest, but this follows naturally from their model as the trajectory slope (droplet fall speedispeed) was approximately S m sec-'/20 m sec-I = I14 and the mountain width was only 10 k m . Actually, the small-scale orographic rain pictured by Gocho and Nakajima would be unlikely without vigorous seeding from above (Bergeron. 1960a). A more recent treatment is Gocho ( 1978). To form a consistent model of orographic rain, the latent heat released by condensation should be included in the dynamics. Within the methodology of linear theory this added heat can be accounted for by the use of the saturated adiabatic lapse rate, in the definition of static stability. at the levels that are saturated. This method was used by Sarker (1966) in his model of flow over the Western Ghats in India-an area noted for its torrential monsoonal rains (Ramage, 1971, p . 11 I ) . In Sarker's case the observed lapse rate was equal to t h e saturated adiabatic lapse rate r5 over a deep layer. Thus when r is replaced by r, in the expression for the Scorer parameter P(Z) =

-g[(aT/az) TU2

r]

1

u2

a2u az2

the buoyancy term (i.e., the first term) vanishes. Sarker concludes that the restoring force due to the curvature in t h e mean velocity profile, a term which is usually quite small, must dominate. This leads to a rather peculiar flow field solution, partly because the profile curvature must be evaluated from a sounding near the mountain which is strongly altered by the mountain. Some alterations in the model are described in Sarker (1967). One problem with the use of linear the0i.y is that in a nearly saturated atmosphere, even slight lifting will bring the air to saturation and thus change the effective stability of the fluid. This problem has been avoided in the nonlinear models of Raymond (1972) and Fraser et a l . (1973). Fraser (it ul. introduced nonlinear corrections at the interface between

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saturated and unsaturated layers. Raymond used a more straightforward model with a local heating function prescribed as a function of the vertical velocity. A few iterations are necessary before a consistent solution is found. Raymond finds that the heating makes only minor changes in the flow field-a result which suggests that the adiabatic flow-field solutions might be applicable, at least qualitatively, to the orographic rain problem. This view is strongly contested by the observations of Hill (1978) who measured precipitation-induced vertical motion in winter orography storms in the Wasatch Range, Utah. In particular, Hill's observations suggest that the role of heat release i n condensation and cooling during evaporation is to generate small-scale convective motion. These motions locally control the rainfall intensity. A fully numerical, finite difference solution to the two-dimensional, orographic rain problem has been presented by Colton (1976). The computed rainfall distribution over the Sierras and the Smith River Basin in California agrees well with observations, but it is difficult to understand completely why this is so. One would t h i n k that his assumptions of 100% release of condensate with no delay and a reflective upper boundary condition at 1 I km would degrade his results. Such an incorrect upper boundary condition requires the flow (except for the action of friction) t o be of a standing wave form with a level of no disturbance and with lifting confined to the region directly over t h e mountain. Colton appears to find no evidence of convective instability. but his coarse grid and frequent numerical smoothing of the flow field probably rule out such a possibility ( I priori. The remaining dynamical problem concerning orographic rain is the question of whether an incoming airstream will pass over t h e mountain, or whether due to the mountain's disturbing influence, it will slow, turn aside, and either flow completely around the mountain or pass over at a more convenient spot. Such a turning would surely influence the horizontal distribution of precipitation. Two examples from the Scandinavian mountains will serve to illustrate the point. When a large-scale westerly flow crosses the west coast of Norway, there is usually a local disturbance to the pattern of reduced sea level pressure (see Spinnangr and Johansen, 1954). This pressure perturbation has the form of a high at, or upstream of, the ridge crest and a trough in the lee. This description roughly coincides with the theoretical solution for the flow over a mesoscale mountain (see Section 3 ) . The pressure perturbation does not seem to be s o strong as to suggest a complete blocking of the flow, but at the same time the surface winds just upstream are observed to be northward, parallel to the mountainous coastline. The occurrence of upslope rain indicates that there is lifting aloft, so that the

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RONALD B. SMITH

THE INFLUENCE OF MOUNTAINS O N THE ATMOSPHERE

b

191

C

FIG.33. Examples of the horizontal redistribution of precipitation by the blocking influence of the mountain. (a) The maxima of SE Norway occurring with SE winds. (After Andersen, 1972.) Two possible models of the foehn wind; (b) with flow over the mountain and upslope rain, and (c) with blocking and descent from aloft.

blocked air must be confined to a shallow layer (perhaps only the lower part of boundary layer) near the ground. When the wind approaches southern Norway from the east or southeast, the surface winds may also turn to the left and run down the coast. In this case, however, there appears to be an alteration to the horizontal distribution of precipitation with a maxima occurring in southeast Norway (see Fig. 33a). This maxima appears in case studies described by Bergeron (1949) and Smebye (1978) and in monthly averaged rainfall patterns described by Anderson (1972). According to the linear theory of flow over mesoscale mountains, this local intensification could be explained by the increased wind at the left end of a mountain range by the mountain anticyclone (see Section 3). The horizontal deformation of the wind field is also important in the warm, dryfiihn wind that blows down the lee slopes of mountains such as the Alps (Defant, 1951; Yoshino, 1975), and the Rockies (Brinkman, 197 I ) . There seems to be two possible explanations for the appearance of the dry, warm air. The first, shown in Fig. 33b, is connected to the theory of orographic rain. If the incoming air is nearly saturated and if it goes directly over the mountain, it will rise along a saturated adiabatic to the top. If the condensed moisture has rained out, the air will be warmed adiabatically as it descends-ending up warmer and with lower relative humidity. There are many “ifs” in this “textbook” explanation and sometimes it is difficult to explain the extreme dryness of a fohn in this way. The consideration of [instable upslope rain may be of some help in this regard. The other possibility (Fig. 33) is that the low-level air has been blocked upstream by the mountain or has been diverted to the sides. Then the potentially warm, dry air from aloft must descend generating a fohn effect. This mechanism can produce drier air than the classical explanation as the air need not be saturated at mountain top level. The general relationship between the three-dimensional perturbation of the flow by orography and the distribution of orographic precipitation

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is little understood. There seems to be a need for more synoptic studies of this problem.

4.3. The Redistribution of Rainfall by Small Hills

Recent studies of the small-scale distribution of precipitation using dense networks of precipitation gauges have revealed an important effect due to small hills. Examples include Bergeron (1960b) for the hills around Uppsala, Sweden: Skaar (1976) for the Sognefjord district of western Norway: Wilson and Atwater (1972) for the state of Connecticui: Changnon et al. (1975) for southern Illinois. In general these studies reveal a strong positive correlation between altitude and precipitation, although in the case of a sudden rise to a plateau (Wilson and Atwater, 1972) the precipitation decreases further in on the plateau in the downwind direction. The correlation is most marked in the case of stable frontal rain from nimbostratus clouds, but Browning et a / . (1974) found a similar correlation in unstable rain. This type of orographic control is clearly different than the larger scale processes considered earlier. First, this control by small-scale hills results in a rainfall maximum near the hill top instead of in the upslope region. Second, the small size of the hills means that by themselves they could not produce precipitation. As the air passes over the mountain, there is insufficient time for raindrops or snowflakes to form from cloud droplets. The simplest explanation for this would be called “differential evaporation.” The raindrops falling from cloud base to the ground experience evaporation in the drier air beneath the cloud. This is sometimes visible as virgae (i.e., fall-streaks) which occasionally terminate before reaching the ground. Mason (1971, p. 313) estimates that drizzle drops falling from 1750 m through the air with 90% relative humidity would decrease in radius from 250 pm to 100 pm. The rainfall (mm/hr), being proportional to the radius cubed, would decrease even more strongly. Thus even if a hill does not disturb the flow, it will experience a higher precipitation rate than the surrounding valleys as it reaches u p t o intercept the droplets before they evaporate. The full implications of this model have apparently never been computed, but Bergeron (1968, p. 5) suggests that by itself the “differential evaporation” cannot account for the strong increase of precipitation with height. Bergeron (1968) suggested that there may actually be an enhancement of precipitation over hills by washout of cloud droplets which have formed a s the low-level f l o w lifts over the hill. Thus over the valleys the drops evaporate as they fall, but over the hills the drops expand as they

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fall through low-level pannus or skud clouds. The large-scale rain cloud aloft-perhaps a frontal stratus cloud or an orographic upslope cloud mass-plays a crucial role as ( 1 ) a moistener of low-level air bringing the boundary layer near to saturation and (2) a seeder cloud providing a mechanism for release of the low-level condensate. This is shown in Fig. 25b. Recently Storebo (1976) and Bader and Roach (1977) have estimated the efficiency of the washout of low-level cloud droplets by raindrops falling from above. These estimates are still rather crude, but they provide a feeling for the magnitude of the effect. In a sample calculation Bader and Roach considered the lifting of 1500-m thick layer of saturated air onto a 500-m hill. The rainfall rate from above is prescribed and the growth of the raindrops as they fall is computed using a radius-dependent collection efficiency. They find that the local precipitation rate can be doubled or tripled by low-level washout. There is, however, one difference between Bergeron’s conceptual model and Bader and Roach’s computation. Bergeron makes the consistent (although perhaps incorrect) assumption that the seeder cloud aloft is unaffected by the hill. Bader and Roach, on the other hand, use an ad hoc upper boundary condition in which the rainfall from above is specified as constant along a surface which parallels the irregular terrain. This means that the incoming rainfall aloft varies along a level surface which could only occur if the upper cloud is influenced by the hill.

4 . 4 . Orogruphic-Convective Prtvipitu tion

The final mechanism of orographic control of rainfall to be discussed is the local generation of cumulonimbus in a conditionally unstable air mass. The classical description of this process is similar to that given by Henz (1972) in his study of thunderstorms in the Rockies. Beginning in the morning the mountain slopes are heated by the sun. The air near the surface is heated by conduction and small-scale convection and begins to rise buoyantly-first directly u p the slopes and later, in a more organized way, u p the valleys (Defant, 1951). At the mountain tops the warm air breaks away from the surface to form rising thermals. When these thermals reach the lifting condensation level, visible cumulus clouds are formed, which can grow into cumulonimbus if the air mass existing in the area is conditionally unstable. These developing clouds will generally be carried with the local gradient wind: thus any showers that occur will be located downwind of the mountain which triggered that particular cloud. It is of course true that thermals, cumulus clouds, and convective

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RONALD B. SMITH

rain can occur on warm days over flat land without the disturbing influence of mountains, but it is clear that such development tends to occur sooner over mountains. The transient nature and small scale of this type of orographic rainfall means that the conventional rain gauge networks and radiosonde soundings are of little use. The most useful methods for the study of this phenomenon are: (a) Aircraft measurements. Silverman (1960) and Braham and Draginis (1960) have reported aircraft flights near the mountains of southwest North America. Braham and Draginis were able to confirm the supposed link between the thermally driven upslope winds and the observed cumuli. (b) Satellite photography. The infrared motion pictures, which are now being produced from geostationary satellite photography, show clearly the diurnal cycle of cumulus activity in mountainous regions. In the summertime over the Rockies, for example, the cumulus appear over mountain peaks almost every morning. Biswas and Jayaweera (1976) have combined satellite and synoptic observations in the study of thunderstorms in Alaska. (c) Radar. The spatial and temporal distribution of precipitation-size particles can be determined from the intensity of radar return. Ackerman (1960) and K u o and Orville (1973) have used this method in the study of orographic convection in Arizona and South Dakota, respectively. Harrold et al. (1974) discusses the general problem of using radar to estimate rainfall in hilly terrain. (d) Sirrface wind measurements. The onset of orographic convection is closely connected to the dynamics of slope and valley winds at the surface. Defant (1951) reviews the observations of the diurnal cycle of slope and valley winds. Examples of more recent observations include MacHattie (1968) for Alberta, Canada, and Sterten (1963) for the fjords of western Norway. (e) Theory and numerical simirlation. The dynamical problem of the rising of fluid along heated boundaries was treated first by Prandtl (see Defant, 1951). A more general treatment of buoyancy driven flow is given by Turner (1973). Fosberg ( 1967) and Orville (1968) have specifically attacked the orographic convection problem by using finite difference techniques to solve the governing equations in two dimensions. Fosberg concentrates on the formation of slope winds and convective plume above the ridge, while Orville follows the development of the cumulus clouds.

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5. PLANETARY-SCALE MOUNTAINWAVES The availability of a hemispheric network of regularly reporting radiosonde stations has made it possible to construct monthly averaged upper air charts. From these time-averaged maps it is clear that in addition to the transient disturbances in the atmosphere, there are planetary-scale waves which remain stationary with respect to the Earth (see, for example, Van Loon et af., 1973: Saltzman, 1968). It has also become clear that these stationary planetary waves are partly responsible for the variation in climate around a latitude circle. Reiter (1963), for example, has compiled a good deal of evidence relating the meridional position of the jet stream to the stormy cyclogenetic belts near the ground. The relationship is particularly evident when the stationary wave shifts its position causing an anomalous pattern of climate around the globe. This mechanism has been invoked by Namias (1966) to explain the drought in the northeastern United States during 1962-1965 and by Wagner (1977) to describe the abnormally cold 1976-1977 winter in the same region. There has also been some success using monthly averaged upper air patterns to predict the following months' weather (Ratcliffe, 1974). It is obvious that the existence of stationary disturbances must be related in some way to the irregularities on the Earth's surface. It is much more difficult to determine whether it is the geographical distribution of ground elevation, surface roughness, or surface thermal properties which is most important. The first theories of the stationary waves by Charney and Eliassen (1949) and Bolin (1950) suggested that largescale orography, primarily the Rocky Mountains Cordillera and the Tibetan Plateau, could cause the observed disturbance. Soon after, Smagorinsky ( 1953) showed that nonuniform heating could also produce such a disturbance. Now, 25 years later, the numerical models of the atmosphere suggest that both orography and heating are influential but their relative importance is still in doubt. Following the theme of this review we will describe only the forcing due to orography. Modcds of Topographicully Forcc.d Planetary Waves The theoretical study of planetary-scale waves requires consideration of two new aspects which were not included earlier, namely:

I . The influence of the spherical shape of the Earth. This requires either the use of a spherical coordinate system or, if a local Cartesian coordinate system is used, the waves must belong to a discrete spectrum with an integer number of waves around the globe. Further, in such a

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RONALD B . SMITH

Cartesian coordinate system the Coriolis parameter must be considered a function of latitude to retain the dynamical effect of the Earth's curvature. 2 . Unlike the mesoscale disturbances which has a vertical length scale of a few kilometers, synoptic- and planetary-scale motions can have a vertical scale equal to a scale height H or greater. Therefore full consideration must be given to the variation of density. The Boussinesq approximation is no longer suitable. These new factors complicate the analysis. On the other hand, the simple quasi-geostrophic theory, which was questionable for mesoscale flows, should accurately describe this larger scale of motion.

A further complication is that we can no longer consider the effect of the mountain on a uniform steady current. The description of the stationary planetary wave given, for example, by Van Loon et al. (1973), is a time-averaged picture. Occurring at the same time are energetic smaller scale eddies and storms. The incorporation of these smaller scale motions in the planetary wave model is discussed by Saltzman (1968). For the most part in this review we will ignore this problem and concentrate on the direct topographic forcing of planetary waves. By neglecting the forcing due to nonuniform heating, nonuniform friction, and smaller scale storms, it is impossible for us to compute quantitatively valid results. Instead we will review what is known about the natural resonances and the response characteristics of the midlatitude westerlies. In any case, it seems a bit premature to discuss the quantitative results of the different published studies as there is still little agreement concerning the basic nature of the physical system (for example, whether it is appropriate to consider the atmosphere as vertically bounded). A natural starting point for this discussion is the perturbation equations of quasi-geostrophic flow as given by Charney and Drazin (1961)

(5.2)

(5.3)

The first equation states that the rate of change of vorticity is due to the Coriolis torque caused by divergent motion. The influence of vortex

T H E INFLUENCE OF MOI'NTAINS ON T H E ATMOSPHERE

197

twisting and baroclinic generation of vorticity are not important for these large-scale, nearly horizontal flows. The second equation is a rcduced form of the continuity equation where the terms ilpidt + V , . V p , which can be important for higher frequency motions. have been eliminated. The third equation, along with the definition

is the hydrostatic form of the thermodynamic equation. I t states that temperature changes are caused by vertical motion in the presence of a variable background e ( z ) in accordance with the conservation of potential temperature. Equation ( 5 . 4 ) is the statement of geostrophic balance. This set of equations is not exact. Equation (5.2) is the anelastic form of the continuity equation and ( 5 . 3 ) is exact only in the limit of y . the ratio of specific heats, going t o one. 'The nature of these assumptions has been clarified recently by White (1977). 5 . I. A VerticuIIy Ititcgrvrtecl Motl~lof' ~opogrcrpliic,rll~ Forced PlCl nrttrry Wtr I ' C S

If there is no need to understand the vertical structure of the disturbance, then a reduced set of vertically integrated equations can be used. Substituting (5.2) into ( 5 . I ) gives (5.6)

which relates vorticity changes to the vertical motion field; either vortex stretching f ( ii ~ , / id)

or to the expansion of the particles as they rise .fit) a p _ -

p

Multiplying by

p

dz

and integrating vertically

(5.7) If either the vertical velocity or the background density (or both) approaches zero as z -+ m , then there is no contribution from the upper limit of the integral. Taking M' at the ground to be U o ( d h / d x )gives

198

RONALD B . S M I T H

To help interpret the left-hand side of ( 5 . 8 ) , define a mass-velocity weighted average vorticity according to

where p ( z ) and U ( z ) are background density and wind speed. Then, in the steady state, (5.8) becomes

(5.10)

In the simple case of U ( z ) = const hand side of (5.10) is

=

U o , the denominator on the right-

where H is the density scale height and (5.10) is (5.11)

a (( ax

+f )=

-

f ah --

H ax

The mathematical form of (5.11) is similar to the equation describing columnar motion of a homogeneous fluid with a rigid lid at z = H . This analogy helps to visualize the solution to (5.11) [or (5.lO)l but is also dangerous. The actual flow field may vary strongly with height and the vertical motion probably does not vanish at z = H . A s an example, the integrated equation (5.10) or (5.11) continue to be valid even when the disturbance takes the form of a vertical propagating planetary wave with strongly tilted phase lines. Thus, while the integrated equations are easier to solve than the full equations, the results are sometimes difficult to interpret. A vertically averaged perturbation velocity, vorticity, and stream function can be defined in a similar way to (5.9) so (5.12)

199

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

Then with the p-plane approximation f becomes

=

fo + p y , py

G f o , (5.1 I )

(5.13) Charney and Eliassen ( 1949) and Bolin (1950) used a n equation of this type in their pioneering studies of the influence of orography on the midlatitude westerlies. Charney and Eliassen assumed that the perturbation would vanish at certain bounding latitudes determined by the meridional extent of the mountains. If the mountains are chosen of the form (5.14)

Iz(x,

jt) =

h,,, cos l y cos k x

where h,,, is the amplitude of the topography, then d l must be chosen as the distance between the bounding latitudes, and k must be chosen according to ( 5 . IS)

n(2rrlh)

=

nA = L

so that an integer ( n ) number of wavelengths ( A ) fit exactly into the circumference of a latitude circle L = 277 a cos 4, where a and 4 are the Earth’s radius and latitude. The solution to (5.13) with (5.14) is (5.16) $(x, Y ) =

[ zri”,, k2

cos l y cos k x

for

37

-

2

ly

77

-

2

Here p = (2R/rr) cos 4 and U is chosen to represent the strength of the mean westerly current in rnidlatitudes. Mountains with a sufficiently small longitudinal and meridional scale have k 2 + l 2 > p / U . The bracketed coefficient in (5.16) is positive in this case and the streamlines will be displaced northward over the regions of high ground-southward over lower ground. When the mountain’s horizontal scale is so large that p/ U > ( k 2 + 1 * ) , the response is reversed with southward displacement over the high ground (see Fig. 34). In both cases the absolute vorticity 5 + f i s decreased over the high ground. In the so called ”long” waves, k 2 + l 2 > p / U , the decrease in absolute vorticity is brought about by the generation of negative relative vorticity. In the “ultralong” waves, p / U > k 2 + i2, the decrease in absolute vorticity is associated with the movement of air parcels southward to a region of smaller planetary vorticity. Near p / U = k 2 + 1’ these two tendencies nearly cancel and the amplitude given in (5.16) becomes exceedingly large. This singularity associated with the fact that there is a free solution to (5.131, that is, a solution with the forcing h = 0. Physically this is a standing Rossby wave with its westward directed

200

RONALD B . SMITH N

w s

1,) \ Y

r S

, 1,

ultra-long

E

/

long" waves

0

FIG. 34. The orographic perturbations to a westerly wind according to the vertically integrated model. For k 2 = k 2 + l 2 > p / U , the pressure disturbance and meridional streamline displacement are in phase with the orographic height (so-called long-wave behavior). For longer waves (k' < p / U ) there is southward displacement and low pressure over the mountain regions (so-called ultralong wave behavior).

phase speed exactly balanced by the eastward advection by the basic current. The actual topography of the Earth around a particular latitude circle can be represented as a sum of Fourier components each similar to (5.14). According to linear theory the atmospheric response is a superposition of disturbances like (5.161, some acting like "long waves" and some like "ultralong waves." It is unlikely that there will be any forcing at the singular wave number

(5.17)

k2

i'I =

PI fJ

as the possible wave numbers are strongly restricted by (5.15). Nonetheless. the flow may be dominated by the components which most closely satisfy (5.17). This near resonance would presumably also be important for other types of forcing, for example, heating. Charney and Eliassen (1949) used t h e orographic distribution of 45"N to compute the steady-state perturbation to the January westerlies. Considering the simplicity of their model, t h e results (Fig. 3 5 ) are remarkably close to the observed pattern. Bolin (1950), using the same integrated equations (5.13), considered the perturbation to the westerlies by a large isolated mountain on an infinite p-plane. H e did not demand that disturbance vanish at "bounding latitudes" nor that the solution join itself smoothly after once around the Earth. He found that the mountain generates a standing Rossby wave in its lee. Bolin found good agreement between his theory and the upper air trough and ridge system observed to the east of the Rockies in winter. To understand more about these topographically generated disturbances and to understand the success of the vertically integrated models

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

-

-lo--

7-

6--

20 1

h(1000f t )

5 --

4-3 -2 -I -0 1

I,

FIG. 35. The topographic profile (bottom) and the observed (solid line) and computed (dotted line) 500-mh heights at 35"N ( t o p ) . (After Charney & Eliassen, 1949.) Mean zonal wind is chosen as 17" longitudeiday and the north-south distance between bounding latitudes is 33".

we must investigate t h e full structure of these motions using the complete set of governing equations.

In this section we will investigate the vertical structure of topographic disturbances following the treatments of Charney and Drazin ( 196 I ) and

202

RONALD B. SMITH

Hirota (1971). We will retain the assumption that the disturbance is contained between two bounding latitudes on a P-plane and cannot propagate meridionally. After describing this type of solution we can show the connection to Queney’s (1948)work by making the Boussinesq approximation and the connection to the work of Saltzman (1965,1968)and Sankar-Rao (1965a,b) by imposing a reflective top lid on the system. The quasi-geostrophic perturbations on a zonal flow can be described using Eq. (5.1)-(5.4)given earlier. Combining (5.1) and (5.2) and assuming steady state (didt = 0) flow on a P-plane gives

(5.18) Equation (5.3) can be rewritten as

u-d X

(5.19)

(”.)

:,;

- +-w=o

At the ground (5.20)

so the lower boundary condition can be written in terms of qI as (5.21)

dqI N2 -+-h=O

z=O

at

JO

az

To construct a simple prototype problem we will treat U ( z ) and N ( z ) as constants. Then eliminating w between (5.18) and (5.19)gives (5.22)

YJ

1 $zz--$r+-

H

v;+-

$=O

The troublesome second term in (5.22) can be eliminated by introducing the new dependent variable (5.23)

4 = (P/P,)1’2$

so that (5.22) becomes

(5.24)

+zz

+ /l*+

=

0

with (5.25)

n 2 = N2 7 ( ~ ; + $ )

-- 1

4H2

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

203

The transformation (5.23) can be motivated physically as the magnitude of + 2 is directly proportional to the kinetic energy density of the disturbance. Because of (5.23) the lower boundary condition becomes slightly more complicated (5.26)

If (5.27)

h(x, y)

=

h, cos l x c o s k x

where k and I are constrained by (5.15). The solution to (5.24), (5.25), and (5.26) can be written as +(x, y , z )

(5.28)

=

Re{&(z)cos lye‘””}

If (5.29) is positive, for example with weak westerly winds and very broad orog( z ) is given by raphy,

4

(5.30) where a positive sign has been chosen in the exponent to satisfy the radiation condition aloft. The substituting (5.30) and (5.28) and using (5.23) gives

Equation (5.3 I ) describes a vertically propagating planetary wave with phase lines tilting westward with height. The phase of the disturbance relative to the orographic highs and lows is determined by the relative importance of the two terms in the curly brackets. The first term represents a pattern of high and low pressure which at the ground is exactly opposite to the orography. This is similar to the “ultralong” behavior discussed earlier. There is no vertical energy flux associated with this term as the pressure and the vertical velocity are 90” out of phase. This term vanishes in the Boussinesq limit.

204

RONALD B. SMITH

The second term in (5.31) represents a pattern of high and low pressure which, at the ground, is shifted one-quarter of a wavelength westward relative to the orography. The high pressure regions, for example, with streamline displacement to the north, occur on the westward (i.e., upwind) slopes of the mountains. From this we can see that the vertical velocity will be in phase with the pressure and there will be an upward flux of wave energy. If the mountains have a smaller horizontal scale or if the winds are easterly, n 2 , given by (5.29), will be negative and the solution to (5.24) and (5.26) with (5.8) is (5.32) where the negative sign in the exponent has been chosen to keep the solution bounded. Putting (5.32) into (5.28) and using (5.23)

This describes a perturbation which decays exponentially with height. The phase is constant with height and is determined by the relative magnitudes of the two terms in the denominator of t h e square brackets. For smaller scale mountains, L - 1000 km or so, the I n I term will dominate. In this case the square bracket is positive and the high and low pressure regions are in phase with the highs and lows of the orography. This is similar to the “long wave” dynamics discussed earlier. This would always be the case if we made the Boussinesq approximation in (5.33) by taking H + =. For broader mountains (but keeping n 2 < 0) the solution changes sign. The high pressure areas lie over the orographic lows and vice versa. The situation is similar to the “ultralong waves’’ discussed earlier. When k 2 + l 2 - p/CJ = 0 the denominator of (5.33) vanishes and the solution is infinite. This singularity clearly has the same physical origin as the singularity in the vertically integrated equations, namely, the existence of a free standing Rossby wave characterized by a balance of relative and planetary vorticity. This free wave has I)( w , y , z ) independent of height, while $ ( x , y , z ) decreases as -c--ri2H and so has a finite total kinetic energy. T o summarize this complicated variety of cases we can fix p, H , CJ, f, N in our minds and consider the wave number 1 k I = ( k 2 + f 2 ) l i Z as a variable (see Fig. 36). In 1948, Queney discussed the effect of a nonzero p on the flow over

-

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

evanescent vertical propagation ultra-long"

205

"long" wave

FIG.36. The influence of wave numhcr k' = k Z + 1* o n the phase and vertical structure of the orographic disturbance. Short wave> ( k P > p / U ) are in phase with the orography (i.e., "long-wave" behavior) and are evanescent. Longer waves ( k 2 < p / U ) are out of phase ("ultralong wave" behavior) and evanescent. Still longer waves ( n z > 0) are vertically propagating with westward tilting phase lines. At k' = p / U there is a singularity (S) in the response.

an isolated mountain. He obtained a simple solution by using the Boussinesq approximation (see also Iohnson, 1977, in this regard). To understand this solution we must see how this approximation changes the above result. Looking back t o (5.6) see that changes in absolute vorticity can be caused either by a term j ; )(a M ~ / dz )

which would be interpreted as the vertical stretching of a vortex line, or by a term

which is best described as the effect of volume expansion as the fluid particles move upward t o a region of lower density. In the Boussinesq approximation the volume expansion is neglected. The effect of this assumption is easily seen by taking the scale height H to be very large in t h e above analysis. The most striking simplification is that the boundary between propagating and evanescent behavior now coincides with the Rossby wave singularity. The range of scales where the solution is evanescent, with "ultralong" behavior, is gone (see Fig. 37). Unfortunately, the vertical length scales associated with planetary motions are the same order as the scale height H and the Boussinesq approximation is not valid. Vorticity generation by volume expansion can play an important role. A number of authors have attempted to model stationary planetary waves using a system of equations similar to that used in the preceding, but with the use of a rc:flecti\~c 11ppot. hoirtid(iry contlitioti. Examples include Saltzman (1965) and Sankar-Rao (1965). while Saltzman (1968) has reviewed several others. To understand these models we must investigate the influence of reflective condition aloft as opposed t o the

206

R O N A L D B . SMITH

evanescent

+

propagation long" wave K-

P/U FIG. 37. The influence of wave number on the phase and vertical structure of the orographic disturbance in a Boussinesq fluid. Unlike a compressible fluid (Fig. 36) the evanescent "long" waves go directly over to vertically propagating waves a s the wavelength increases.

radiation condition used in Eq. (5.30). Out of a number of possibilities the arbitrary choice of u = 0 at the top, or equivalently u = 4 = $ = 0, has been made by Saltzman, Sankar-Rao, and others. This is quite a strong condition as it requires the horizontal flow, not just the vertical motion, to vanish at the top of the domain of interest. Applying this condition at z = D gives instead of (5.30) and (5.32) (5.34)

&(z) =

for the case n 2 ( 1

-(N2/J.o)~,

- n cos n D

I

+ ( 1 / 2 H )sin n D

, N , U , H , p. .f > 0 and

1

sin n ( D - z )

for the case n 2 < 0. The nature of the flow for a given wave number and the location of the singularities of (5.34) and (5.35) in wave number space depend on the height D at which the upper boundary condition is applied. The singularities in (5.34) and ( 5 . 3 5 ) , respectively, occur at (5.36)

tan nD

=

2nH

and (5.37)

tanh n D

=

2nH

Qualitatively it is particularly important to know whether D is greater or less than 2 H . For the sake of illustration it will suffice to describe just one case, and the D < 2 H case seems to bear the closest relationship to the solutions of Saltzman (19651, Sankar-Rao (1965), and others. In this case there are no solutions to (5.37) so all the singularities occur with n 2 > 0. Referring to Fig. 38, note that there is no longer a singularity at k 2 + / 2 = / 3 / U . This is explained by the fact that the upper boundary

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

207

F I G . 38. The influence of wave number on the nature of the orographic disturbance in a model with a particular reflective upper boundary condition ($ = 0 at z = D < 2 H ) . The small diagrams represent the variation of & z ) according to (5.34)and 5.35). The solutions could be described a s standing waves in the vertical with no phase line tilt. Positive implies "long-wave" behavior (L). while negative 4 means "ultralong wave" behavior (UL). The response characteristics of this unbounded model are fundamentally different than the unbounded model (Fig. 36). The Rossby wave singularity is gone, while other singularities (S) arise due t o reflection at the upper boundary.

4

condition & = I) = 0 has eliminated the free barotropic Rossby wave. The first singularity occurs at a much longer wavelength where the form of $ ( z ) demanded by (5.24) has become oscillatory. Near this value the amplitude of the solution becomes very large and upon crossing over the singularity (to smaller wave numbers) the sign of the solution changes from "long wave" behavior to "ultralong wave" behavior. At smaller wave numbers the horizontal motion acquires a node at a certain height. Above the node the high pressure regions lie directly over the topographic lows (i.e., "ultralong behavior"), while near the ground the pressure and h ( x , y ) are in phase. At still smaller wave numbers another singularity will be encountered. Clearly the use of a reflective upper boundary condition has significantly altered the physical characteristics of the system. It is still true that each singularity can be associated with a standing free Rossby wave. These free waves, however, are not the naturally trapped barotropic wave with k 2 + l 2 = /3/ CJ, but baroclinic waves trapped by the reflective upper boundary. The description just given agrees remarkably well with the more detailed computations of Saltzman and Sankar-Rao. Such a comparison is not completely straightforward, however, as those computations were carried out using pressure coordinates. The difficulty with this is that pressure coordinates become strongly stretched at high altitudes, while the vertical scale of the disturbance (e.g., the vertical wavelength) remains about the same. Thus in pressure coordinates the disturbance appears to oscillate rapidly at high levels. For this reason it is advantageous to use Cartesian or log pressure coordinates (see Holton, 1975) which do not have this unwanted stretching. To avoid this problem

208

RONALD B . SMITH

Saltzman and Sankar-Rao use the somewhat unphysical device of choosing an N 2 ( z ) which becomes vanishingly small at high altitudes. This causes the vertical scale of the disturbance to increase with height and thus to remain well behaved when described in pressure coordinates. We have seen that a reflective upper condition can radically change the response characteristics of the atmosphere. As an ad hoc hypothesis such a condition should probably be avoided. There are, however, valid reasons for interest in such a system. First, the numerical models of the atmosphere invariably use a reflective condition aloft (usually w = d P / d t = 0 ) . If we wish to understand the results from these models we must know about the influence of a reflective lid. Second, and more to the point, there are conditions that will naturally cause the reflection of planetary waves. If the wind speed increases with height, then, as discussed by Charney and Drazin (1961), waves which can propagate at low levels [ H ’ > 0 in ( 5 . 2 5 ) ] become evanescent ( n z < 0) aloft. In this situation the wave energy will be totally reflected and resonant response such as described by (5.34) and (5.35) becomes possible. The validity of $ = 0 as an upper boundary condition cannot be defended by arguing that the disturbance is absent high in the atmosphere. This could occur either because the disturbance is dissipated or reflected at lower levels. In the former case a radiation condition is appropriate, while in the latter a reflective condition at the correct altitude is appropriate. Since the study of Charney and Drazin (1961) there have been a number of theoretical studies concerning the eventual fate of vertically propagating planetary waves. Dickinson (1968a) treated the problem in spherical coordinates with the background westerly wind field assumed to be in solid body rotation. He found that Charney and Drazin’s estimate of the critical wind speed [i.e., the speed beyond which the waves become evanescent, see Eqs. (5.29)], may be too low. In 1969 Dickinson considered the decay of vertically propagating waves by preferential cooling at the warm regions by radiation to space. Geisler and Dickinson (1975) use a realistic vertical profile of background wind to determine the possible free waves of the system. They found an “external” wave, which closely corresponds to the natural free Rossby wave discussed previously, and four “internal” modes which are associated with reflection from levels of high wind speed. Another interesting possibility arises if the background wind U ( z ) decreases to zero at some level and then becomes easterly. Clearly from (5.29) the wave cannot propagate above this critical level so it must either be absorbed or reflected. From the analogy with small-scale mountain waves (Booker and Bretherton, 1967) we note from (5.29) that as U ( z )

T H E INFLUENCE OF MOUNTAINS ON T H E ATMOSPHERE

209

+ 0, the vertical wave number 11 approaches infinity, making the disturbance susceptible to viscous and radiative dissipation or to small-scale instability. A further insight into this problem is given by Dickinson (1970) and Beland (1976) who investigated the time development of a Rossby wave critical level. The interested reader should also consult Holton's ( 1975) recent review of stratosphere and mesosphere dynamics.

Throughout the previous section we have retained two strong assumptions about the meridional structure of t h e mean flow and the perturbations. First, we assumed that the mean flow was independent of latitude (i.e., y ) . Second, we eliminated the possibility of meridional propagation by requiring that the disturbance vanish at the walls of a fictitious zonal channel. Both these assumptions are incorrect and they are clearly the severest ad hoc hypothesis remaining in the model. A more general three-dimensional approach has been tried by Dickinson (1968b), Matsuno (1970). Simmons (1974),and Schoeberl and Geller (1976), and some of the results of these theories have been compared with observation by McNulty (1976). A brief description of Matsuno's model will serve to explain some of these new results and concepts. Matsuno ( 1970) derived the steady, linearized quasi-geostrophic potential vorticity equation in spherical coordinates and log pressure coordinates 1

#(b'

where (1

R N

radius of the Earth rotation rate of the Earth Brunt - Vai salii frequency angular speed of the basic flow perturbation height field - H I n ( p / p o ) vertical coordinate longitude latitude

The physical meaning of each term in (5.38) is easy to establish. The

210

RONALD B . SMITH

square bracket is the perturbation potential vorticity and is composed of the perturbation relative vorticity (the first two terms) and the Coriolis torque caused by horizontal divergence. The last term in (5.38) is the meridional advection of potential vorticity which is proportional to the local gradient in background potential vorticity dc)/de. Matsuno gives

This can be compared with the same quantity expressed in local Cartesian coordinates on a P-plane (Simmons, 1974)

(5.40)

The terms marked (a) in (5.39) and (5.40) represent the gradient in planetary vorticity: those marked (b). the gradient in background relative vorticity; and those marked (c), the gradient in background static stability written in terms of the vertical shear. The last term in (5.381,together with (5.39) or (5.40), is of crucial importance as it represents the restoring force for Rossby wave motion. In our previous models with U ( y , z ) = const this restoring force was due soley to P, the gradient in planetary vorticity, and indeed this is the classical view of Rossby wave dynamics. Matsuno uses the observed Northern Hemisphere wintertime distribution of zonal winds (shown in Fig. 3Y) to compute d q / d O from (5.39). The most striking result of this computation is that the contribution o j t h e basic wind to d q / d O is comparable to or larger than P. This suggests that the propagation of Rossby waves is not necessarily associated with the variation of f with latitude. It is also interesting to note that the pattern of d q / M is quite nonunifoTm with maxima just to t h e south of the tropospheric jet stream and the highlatitude polar night jet. I t is not surprising that it never goes negative in this seasonally averaged picture as a negative value would lead to baroclinic instability which would tend to restore the stability of the system. Matsuno avoided the question of what drives the stationary waves by using the observed pattern at SO0 mbar as his lower boundary condition. Then. Fourier transforming in x and using (5.38) and (5.39), he computed the flow field aloft and the pattern of wave energy flux in the y,z-plane.

THE INFLUENCE OF MOUNTAINS ON T H E ATMOSPHERE

I

I

I

I

I

I

I

21 I

I

LIllTUM

FIG.39. The basic state zonal wind distribution (m sec-’) in the winter Northern Hemisphere used by Matsuno (1970) in his model of stationary planetary waves. Note the jet stream near the tropopause at 30-40” and the strong polar night jet.

Generally the flow of energy was found to be upward and southward, but in detail strongly controlled by the nonuniform field of d q / d O . In particular the wave energy was drawn into regions of high mean wind (Fig. 40). At high latitudes the wave energy is channeled upward along the axis of the polar night jet until the mean velocity becomes so large that the wave is reflected. Thus the distribution of d i j / d O forms a partial resonant cavity, and Matsuno finds the strongest response for a zonal wave number between I and 2. These results of Matsuno together with those of Dickinson and Simmons have provided a more complete and possibly a more correct view of the response characteristics of the atmospheric system. Matsuno’s description of the “resonant cavity” may provide some qualitative justification for approach of Saltzman ( 1965) and Sankar-Rao (1965) who used zonal walls and a reflective upper boundary condition. At the same time it points out the arbitrariness of such ad hoc assumptions. The

212

RONALD B . SMITH

f

1

I

I

I

!

I

I

I

t

I

T t I

FIG.40. The computed distribution of energy flow in the meridional plane associated with the longest (i.e., one wavelength around the globe) stationary wave. Due to the nonuniform distribution of U and d q / d O , the wave energy flux is guided into regions of high wind speed-particularly the polar night jet. (After Matsuno, 1970.)

precise nature of the resonant activity, if indeed one exists, cannot be specified (I priori, but will depend, perhaps sensitively, on the structure of the atmosphere at the time being considered. Many questions still remain concerning the effects of nonlinearity and the relative importance of forcing by topography, heating, and migratory storms. Andrews and McIntyre (1976) and Holton (1976) have investigated the nodinear interactions between planetary waves and the mean flow. A more comprehensive treatment, including the vastly complicated forcing and dissipation in the troposphere, requires the use of a numerical model. The past few years have witnessed rapid progress in the numerical simulation of planetary-scale atmospheric motions. The models range from two-layer systems (e.g., Egger, 1978) with specified forcing, to multilayer models with much of the forcing determined parametrically within the model (e.g., Kasahara and Washington, 1971; Kasahara el al.,

THE INFLUENCE OF MOUNTAINS ON T H E ATMOSPHERE

213

1973: Manabe and Terpstra, 1974). In these models the primary goal of the investigator is to simulate the atmosphere, not to explain its behavior. The analysis of model output is, however, significantly easier to diagnose than is observational data since the spatial and temporal coverage is complete and the investigator is free to do controlled experiments. The primary difficulty with this approach is that, like the atmosphere itself, the models are so complicated that the connection between the fundamental laws of physics and the model results cannot be clearly traced. Unlike the atmosphere, there are also the uncertainties involving the finite difference representation, the parameterization of subgrid scale processes, and the arbitrary choice of a restricted vertical domain. Probably the most valuable simulation with regard to the influence of mountains on the atmosphere is the work of Manabe and Terpstra (1974). This study is notable both for the comprehensiveness of the model and for the detail in which the results are compared with observations. The global climate under perpetual January conditions was computed both with and without mountains. In this way the influence of the mountains can be clearly identified. It would take too long to give a detailed description of Manabe and Terpstra’s model and results. We will mention only three interesting points. (a) The structure ofthe stationary disturbance. Both in the simulations with (M) and without (NM) mountains the distribution of time mean meridional winds is qualitatively similar to the observed winds (Fig. 41). In the presence of mountains, however, the disturbance to the zonal flow is much stronger and its dominance over the thermally induced disturbance increases with height. In both models the phase of the disturbance tilts westward with height, although it is not clear whether this is due primarily to vertical propagation or to nonadiabatic effects associated with the northward transfer of sensible heat. Manabe and Terpstra point out that the description of stationary disturbances given by their model is much more realistic than in the linear theory model of Sankar-Rao (1965a). They suggest that Sankar-Rao’s model may have been unduly influenced by the phenomenon of resonance. In the highly nonlinear and dissipative atmosphere, they argue, resonances may not be important. On the other hand, they have made no attempt to determine whether their solutions were influenced by quasi-resonance. Furthermore, the distribution of zonal winds, which is determined internally in the Manabe and Terpstra model, is much stronger than the observed distribution. Referring back to the work of Charney and Drazin (1961) and Matsuno (1970), we know that this can have an important influence on the structure of the stationary disturbances.

38 8

a

4

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FIG.41. The longitude-height distribution of the time mean meridional component of wind (m sec-l) along the 45" latitude circle: (top) observed (Oort and Rasmusson. average over five Januaries, 1959- 1963): (middle) the computed distribution with mountains (i.e., the Mmodel): (bottom) the computed distribution without mountains (NM-model). (From Manabe and Terpstra. 1974.)

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(b) The distribrition of cyclogenesis. Manabe and Terpstra's numerical model is quite successful at simulating the increased frequency of cyclogenesis in the lee of the Rocky Mountains and the Tibetan Plateau. As in the atmosphere, the reason for this increase is not clear. One possibility is that the increased baroclinicity in the southeastern side of the first trough of the orographic planetary wave provides a preferred site for frontal instability. Cyclogenesis in the Alpine region is not simulated in the model as the Alps are too small to be represented in the 250-km grid system. (c) Meridionul heat trunsfrr and midlatitude storms. The most dramatic difference between the mountain and no-mountain computer runs is in the nature of the northward transfer of heat and the conversion of available potential energy to kinetic energy in midlatitudes . Without mountains the northward transport of heat is accomplished by transient waves, especially wave numbers 5 , 6, and 7. These transient waves are also responsible for much of the conversion of available potential energy into the kinetic energy of the winds. With orography in the model the situation is quite different. The heat transport is dominated by the meridional motions associated with the stationary planetary wave, primarily with wave number 2. To a certain extent this heat transfer is caused by orographically induced north-south motion with air parcels gaining heat at low latitudes and losing heat at high latitudes. At the same time, however, these standing waves are responsible for the major part of the conversion of potential energy to kinetic energy. Thus the standing waves are in part driven by the meridional temperature gradient. Apparently then the Earth's orography is large enough to change completely the nature of the midlatitude dynamics. Without mountains the midlatitudes would have many more energetic transient storms associated with baroclinic instability of an intensified north-south temperature gradient. With mountains the meridional heat flux is partially accomplished by the stationary disturbances. The temperature gradient is weaker and the transient storms less frequent and intense. The stationary waves are now an interesting mix, being forced partly by orography and partly by baroclinic instability. Further insight into this problem is afforded by the much simpler twolevel model of Smith and Davies (1977). Without mountains, the midlatitude flow is characterized by the recurring buildup of a strong meridional temperature gradient and breakdown into transient baroclinic waves. With mountains present in the model, large-amplitude standing waves occur and the amplitudes of the transient disturbance are decreased.

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ACKNOWLEDGMENTS My introduction t o mountain flow dynamics came through the lectures of F. P. Bretherton and R. R . Long at The Johns Hopkins University and A . Eliassen at the 1974 Summer Institute a t the National Center for Atmospheric Research. Conversations with W. Blumen, D. K . Lilly, J . Klemp. B. Gjevik, and M . Lystad among others, have also been informative and stimulating. Most of this review wiib written while the author was a visiting lecturer and scientist at the Institute of Geophysics at the University of Oslo under Fulbright-Hays and N A T O fellowship support. During this time, frequent discussions with Professor A . Eliassen provided much needed encouragement and advice. The review was completed at Yale University with support from National Science Foundation grant ATM-7722175. The help of S. R.-P. Smith in organizing the references. E. Moritz in drafting t h e figures. and B . Dabakis in typing the manuscript is gratefully appreciated. Several authors and journals were kind enough t o allow their figures to be reproduced in this review. Unfortunately. because of the breadth of this review and the limited time available for its completion, many excellent research papers have been omitted from the foregoing discussion. I apologize to those authors whose work I did n o t include.

REFERENCES Src./io/i 2. The FloLi' oi'er Hills rrnd r l i r G o t i ~ r . o / ; o tof' i Motriitrrin Wrr\.e.s Abe. M. ( 1932). T h e formation of cloud by the obstruction of Mt. Fuji. Grophys. M n g . , 6, 1-10,

Anthes. R . A , . and Warner. T. T . (1978). Development of hydrodynamic models suitable for air pollution and other mesometeorological studies. Mou. Wetrtlirr. K r i . . 106, 10451078. Atkinson. B. W . . and Smithson. P. A. (1974). Meso-scale circulations and rainfall patterns in an occluding depression. Q. J . H . Motc,oro/. S o c . 100, 3-22. Baines. P. G. (1977). Upstream influence and Long's model in stratified flows. J . Flrtitl Mei.h. 82, 147- 159. Ball. F. K . (1956). T h e theory of strong katnbatic winds. Atist. J . Phys. 9, 373. Benjamin. T. B. (1Y70). Upstream influence. . I . Flrritl Mec4. 40, 49-7Y. Beran. D. W . (1967). Large amplitude Ice wiives and chinook winds. J . A p p l . M ( ~ t ~ o r o6,l . 865-877. Bergan. J . D. ( 196Y). Cold air drainage on ii forested mountain slope. J . A p p l . M e t r o d . 8, 884-895. Berkshire. F. H . . and Warren. F. W. C . ( 1970). Some aspects of linear lee wave theory for the stratosphere. Q. J . K . M ~ t e o r o l S. o c . 96, 50-66. Blumen. W . (1965). A random model of momentum flux by mountain waves. Grofi.s. Prihl. 26, 1-33. Blumen, W . , and McGregor. c'. D. i IY76). Wave drag by three-dimensional mountain lee waves in nonplanar shear flow. Tc//tt,\ 28, 287-298. Booker, J . R.. and Bretherton. F. P. ( 1967). The critical layer for internal gravity waves in a shear flow. J . Flrritl Medi. 27, 513-53Y. Breeding. R . J . ( 1971). A nonlinear investigation of critical levels for internal atmospheric gravity waves. J . Niritl Mcch. 50, 545-563. Bretherton. F. P. ( 1966). T h e propagation of groups of internal gravity waves in a shearflow. Q../. K . M r t r o r o l . Soc.. 92 i394), 466-480. Bretherton. F. P. ( 1969). Momentum tranbport by gravity waves. 0 . J . R . Meteorol. Soc. 95, 213-243.

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Bretherton. F. P., Thorpe, S. A.. and Wood. I . R. ( 1967). Appendix to a paper by P. Hazel. J . Fluid M c ~ h 30, . 78 1-784. Brighton. P. W. M. (1978). Strongly stratified flow past three-dimensional obstacles. Q. J . R . Mrtrorol. Sot,. 104, 289-307. Brinkmann. W. A. R . ( 1970). The chinook at Calgary (Canada). Arch. M o t c w d . Geophys. BioL-/imo/o/.( B ) 18, 279-286. Brinkmann, W. A. R. (1971). What is a foehn? Wewtlr(,r 26 (61, 230-239. Chopra. K . P. ( 1973). Atmospheric and oceanic flow problems introduced by islands. Ad,.. Grophys. 16, 297-42 I . Clark, T . L., and Peltier, W. R. (1977). On the evolution and stability of finite amplitude mountain waves. J . Atnios. Sci. 34, 1715- 1730. Claus. A. (1964). Large-amplitude motion of a compressible fluid in the atmosphere, J . Fluid M c J c , / i . 19, 267-289. Colson, D. Lindsay, C. V . , and Hand, J . M. (1961). Radar echo of a mountain wave on February 15, 1960. Mori. Wcwtlwr R ~ T 89, . 17-19. Conover. J . H. (1964). The identification and significance of orographically induced clouds observed by TIROS satellites. J . Appl. M o t c w d . 3, 226-234. Corby, G. A. (1957). A preliminary study of atmospheric waves using radiosonde data. Q. J . R . Meteorol. Soc. 83, 49-60. Corby, G . A,. and Sawyer, J . S . (1958). The airflow over a ridge: The effects of the upper boundary condition and high-level conditions. Q. J . R . Mrtoorol. So(,. 84, 25-37. Corby. G. A , , and Wallington, C . E. (1956). Airflow over mountains: The lee-wave amplitude. Q. J . R . Mrtcwrol. Soc.. 82, 266-274. Counihan. J.. Hunt, J . C. R., and Jackson, P. S. (1974). Wakes behind two-dimensional surface obstacles in turbulent boundary layers. J . Huid M r c h . 64, 529-563. Crapper, G. D. (1959). A three-dimensional solution for waves in the lee of mountains. J . Flriicl M ~ c h 6, . 5 1-76. Crapper, G. D.. (1962). Waves in the lee of a mountain with elliptical contours. Philos. Trrrris. R . So(,. Lo/it/o/i ( A ) 254, 601-623. Critchfield. H . J . ( 1966). "General Climatology." Prentice-Hall, Englewood