ADVANCES IN
GEOPHYSICS
VOLUME 24
Contributors to This Volume PEEK V Hoi
RICHAKD PEI I I E R
Advances in
GEOPHYSICS Edifed by
BARRY SALTZMAN D e p a r h e n f of Geology and Geophysics Yale Unwersfty New Haven, Connecticut
VOLUME 24 1982
COI'YRlCliT @ 1982, B Y ACADEMIC PRLSS, INC. ALL RIGHTS RESERVED. N O PART O F THIS PUBLICATION MAY BE REPRODUCED OR TRANSMITTED I N ANY FORM OR BY ANY MEANS, ELECTRONIC OR MECHANICAL, INCLUDING PHOTOCOPY, RECORDING, OR ANY INFORMATION STORAGE AND RETKIEVAL SYSTEM, WITHOUT PERMISSION IN WRITING FROM THE PUBLISHER.
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LIBRARY OF CONCRLSCATALOG CARDN U M B E R5:2 - 12266 ISBN 0 -12-018824-4 PRINTED IN THE UNITED STATES OF AMERICA
82 83 84 85
9 8 1 6 5 4 3 2 1
CONTENTS vii
CONIRIBUIORS. . . . . . . . . . . . . . .
Dynamics of the Ice Age Earth Kl( tl I K I ) 1'1
1 TIER
............. 2 I . I n trod uct ion . . . . . . . . , . , , . . . . . . . . . . . . . . . . . . 2. Mantle Kheology: A Uniformly Valid I>inearV' Model . . . . . i2 34 3. The Impulse Response of ii h l i i ~ ~ eI kl l t h . . . . . . . . . . . . . . 59 4. Postglacial Variations of Relatibe Sea Level . . . . . . . . . . , . . , . . . . . . . . . . 75 I. 5 DeElaciation-Induced Perturbations of the Gravitational Field . . . . . . . . 90 6 . Deglaciation-Induced Pertui-liation\ of I-'l;inetat-y Rotation . . . . . . . . . . . 119 7. Glacial Iso\tasy and Clirnatic C'hiinge: A Theory of the Ice Age Cycle ........................ 133 8. Con c II Isi o n \ , . . . . . . . . . . . . . . References . . ....................................... 139 Planetary Solitary Waves in Geophysical Flows
I? MA1 ANOI
"I KlZZOLl
I . Introduction: Why Solitary Wives M a y He Important in Large-Scale ........................ 117 Geophysical Motions . . . . . , . . . . 154 2. Solitary Waves in One Dimension: A Short Synopsis . . . . . 3. The Existing Models for ILarpe-Scale Permanent Structures . . . . . . . . . . 160 173 4. Evolution of Coherent Structure\: The Initial Value Problem . 5 . Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 206 6. Further Investigations o n Coherent Structure\ . . . . . . . . . . . . . 211 ........................ 7. Conclusions . . . . . . . . . . . . , . . . . . . . . . . . . Keferences . . . . . . . _ . _ _ . _ . . . . . . . . . . . _ . . . . . . . . . . . . . . . . i _ . . . . . . . . 27 . .1 Organization and Structure of Precipitating Cloud Systems
ROBERTA . HOII/I.JK.
AND
PE-IER V. H o i m
...............................................
77 5
2. Extratropical Cyclones . . . . . . . . . . . . . . . . . . 3 . Midl at i t Lide Convective S y stem 4 . , . , . . , . . . . . . . . . , . . . . . . . . . . . . . . . . . .......... 4. Tropical Cloud Systems . . , , . . . , . . . . . . . . . . 5 . Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .... References . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
229
INDEX
317
247
2x7 303 305
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CONTRIBUTORS Numbers in puretithese\ indicate the pagcs on which the authors' contributions begin.
PETERV. HOBBS.Deptrrtmcnt of Atniospheric Sciences, Uniwrsity of Wuslzington , Seattle , Wushirigton 98 I 95 (225 1 ROBERTA. H o u z ~ JR., , Deptrrtrn~~nt of'Atmospheric Sciences, Unil-ersity of Washington, Seattle, Wu.riiingtoii 98195 (225)
F? MALANOTTE Rizzoi.~,D~prrrtrncntof Meteorology and Physical Oceanography, Mnsstrchrrsetts Iirstitrrte o f Technology, Cambridge, M~rsstrchrrsetts 02139 (1 47)
RicnAm PELTIER, Department o/'Pii.vsics, University of Rwonto, Kwonto, Ontario M 5 S IA7, Cantrdir ( I )
v II
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DYNAMICS OF THE ICE AGE EARTH RICHARDPELTIER Il~~purtinenr oJPhysics Univcwiij' 01 Toronlo 7 o r o n t o Onrario, Chnadu
Introduction Mantle Rheology A Uniformly Valid 1 inedr Viscoelastic Model 2 I The Generalized Burgers Rodv 2 2 Free Oscillations of d Hornopenews Spherical Burgers Body 2 3 Viscous Gravitational Relaxation ol a Homogeneous, Incompressible. and Sphencal Burgers Body 2 4 The PhenomenoloaLal Utililk ot the Generali7ed Burgers Body 3 The Impulse Response of d Maxwell 1 a r t h 9 I Thc Observed Elastic Structiiir 01 the Planet and Its Physical Interpretation 3 2 Formulation of the ViscoeldstiL Piohlern tor Models with Radial Heterogeneity 3 3 hormdl Modes of Viscous Ctravitdtional Relaxation 3 4 Love Number Spectra for Impulwe Forcing 3 5 Elastic and Isostatic Asymptotes of the I ove Number Spectra 3 6 Green s 1 unctions for the Surldce Mass Lodd Boundary Value Problem 3 7 Response to Simple Disk Load Deglaciation Histories Postgldcidl Variation5 of Relative Se'i I eve1 4 4 I An Integral Equation for Relativc Sea 1 eve1 4 2 Inputs to the RSL Calculdtioii The Deglaciation Chronology and Mdntle Viscosity Profile 4 3 Output from the RSL Calculdtion Crlobal Sea Level Histories 4 4 RSL Constraints o n the Mdntle Viscosih Profile When Initial Isostatic Equilibrium I > Aswmed 5 Degldciation-Induced Perturbations of the Gravitational Field 5 I Satellite dnd Surface Observdtions 01 the Gravitv Field over Deglaciation Centers 5 2 Disk Load ApprOXimdtionS and the bffect of Initial Isostatic Disequilibrium 3 1-ree-41r Anomalies from the \ell consistent Model 4 Gr,i\ity Field Constraints on thc Mantle Viscosity Profile 6 Dcglaciation-Induced Perturbation\ o f Planetai y Rotation 6 I rhe Ili\torical Records of Polar Motion dnd 1 o d Variation 6 2 The Theoi3 of Deglaciation-Forced Rotaliondl Effects 6 7 Polar Motion dnd I o d Con\traint\ on the Farth's Viscoelastic Stratification 6 4 Secular Instabilitv of the Rot'ition Pole 7 Gl'icial l\ost,isy and Climatic Change A Theoiy ol the Ice Age Cycle 7 I Oxkgen Irotope 5tratigrdphy and tlic Ohserved Spectrum of Climate Fluctuations on the 1 ime Scale IOJ-lOh Years I h c Milankovitch Hypothesis 7 2 4 Preliminary Model of the Plci\tiiccne C limdtic Oscillation 7 3 A Spectral Model with Isostatic Adlu\tment The Feedback between Accumuldtion Rate ~ 1 opographic Height and I L Sheet 7 4 An Analjsis ot the Propertie\ of 'I Reduced Form of the Spectral Model 8 Conclu\ions
I 2
2 12 14 19 2R 32 34 35 38 41 46 52 53 55
59 60 62 65 68 75 76 19
8X R9
90 90 93 114
I I5 I I9 I I9
123 12r 130 133
1 Copyright
4DV4NCES 1Y GtOPHYSICC. VOLUMF 21
Cr 1982 by Academic Press, Inc.
All rights of reproduction
any form roervcd. ISBN (I-12-0lR824-4
in
2
RICHARD PELTIER
1. INTRODUCTION
For at least the past lo6 yr, and most probably for the past 3 X lo6 yr, the Northern Hemisphere continents near the rotation pole have been subjected to a continuous cycle of glaciation and deglaciation. During this Pleistocene period, in which the first manlike fossils are to be found in deposits from East Africa dated at about 3.0 X lo6 yr B.P., vast continental ice sheets have waxed and waned on a regular time scale of about 10’ yr (e.g., Broecker and Van Donk, 1970). At each glacial maximum the mass contained in these transient ice complexes has been on the order of lOI9 kg, equivalent to a global drop of sea level of approximately lo2 m, which is itself approximately 1 part in lo6 of the entire mass of the planet. It should not be surprising, given the vast dimensions of this naturally recurring phenomenon, that it has inspired an intense interest not only among members of the geological and geophysical communities but also among atmospheric scientists, zoologists, botanists, and archaeologists, to name but a few of the disciplines whose members have made important contributions to its understanding. Although the Ice Age industry, begun by the Swiss zoologist Louis Agassiz over 150 years ago, has matured considerably since Agassiz’ time, it has yet to provide a fully satisfactory explanation either for the occurrence of the Ice Age itself or for the quasi-periodic life cycle of its major ice sheets. The purpose of this paper is to provide a geophysical perspective on the current state of this industry, to summarize what we “know” or think we know because of our employment in it, and to attempt to reveal as clearly as possible the issues which remain to be settled as effort continues. The geophysical importance of the Ice Age is connected both with the magnitude of the stress to which the planet was subjected in consequence of the growth of individual ice sheets on its surface and with the time scale over which these surface loads were applied. Because of the magnitude of the loads, the deformation of the planet’s shape effected by mutual gravitational attraction between load and planet was sufficiently large as to leave easily observable effects in the geological record. Because of the duration of individual loading events, on the other hand, the strains produced by loading are not entirely elastic in nature. Indeed, the 10’-yr time scale is such that the total deformation is dominated by an effectively viscous response to the gravitationally induced stress field, a viscous response through which the coupled system tends inexorably toward a state of isostatic (gravitational) equilibrium. Although the concepts of isostasy and isostatic adjustment are crucial to the understanding of a wide variety of geophysical observations (e.g., the lack of a gravity anomaly associated with the contrast between continents and oceans, the lack of a gravity anomaly associated
DYNAMICS OF THE ICE AGE EARTH
3
with large mountain complexes, and the observed ratio of the Bouger gravity anomaly over continents to the topography as a function of topographic wavelength), virtually all of these observations refer to the properties of the planet in the fully adjusted (isostatic) state. The observations associated with the phenomenon of glacial isostatic adjustment, on the other hand, are geophysically unique in that they not only provide evidence for the existence of this process but also provide us with quantitative information concerning the rate at which isostatic adjustment proceeds. If the elastic properties and density of the earth are considered fixed by the frequencies of its elastic gravitational free oscillations (e.g., Gilbert and Dziewonski, 1975), then the rate of isostatic adjustment is governed by a single physical parameter-the effective viscosity of the planetary mantle. Even in Wegener’s (1926) book “The Origin of Continents and Oceans,” in which the complete (for the time) range of arguments in favor of the hypothesis of continental drift was first put forward, the observation of delayed vertical motion associated with the Fennoscandian deglaciation was included as the central argumcnt in favor of the ability of mantle material to deform as a viscous fluid in spite of the fact that it has a seismically observed shear modulus approaching that of cold steel. Wegener argued that if such apparently viscous vertical motion could occur then similar horizontal motion should also be possible. Continents composed of relatively low-density granitic material surrounded by a more dense viscous sea of mafic or ultramafic material might then be considered analogous to blocks of ice floating in water. Although Wegener’s hypothesis was considered to be quite disreputable in many if not most geological circles until the early 1960s, a torrent of new geophysical discoveries beginning at that time (particularly in paleomagnetism and seismology) quickly enforced a new conformity to it. Today, of course, this hypothesis is firmly entrenched as the paradigm in terms of which most geological and geophysical research is organized. The importance of the study of glacial isostatic adjustment to the internal self-consistency of this paradigm remains what it was in Wegener’s time. It is the only reliable method we have of obtaining a direct rn sitzi measurement of the viscosity of the planetary mantle and of determining its variation with depth. This parameter is a crucial variable in modern thermal convection theories of the drift process itself (Peltier, 1980b; Jarvis and Peltier, 1982). The first quantitative attempt to infer the viscosity of the mantle from isostatic adjustment data was that by Haskell (1935, 1936, 1937), who was followed immediately in this work by Vening-Meinesz (1937). Both authors employed a Newtonian viscous half-space model of the earth with constant density and viscosity and inferred an effective viscosity from the observed recovery history of Fennoscandia following the deglaciation event which
4
RICHARD PELT'IER
began ca. 20 kyr B.P. The value of the viscosity which they inferred was very near lo2' Pa sec. Since no direct method of determining the ages of individual strandlines required to determine the history of uplift was then available, it is a testimony to the excellence of the varve-based chronology reported by L i d h (1938; see Morner, 1980) that application of modern direct methods of dating to the stratigraphic sequence has led to no significant alteration of the value of the upper mantle viscosity which is inferred from the data. The advent of the radiocarbon method of dating (Libby, 1952) has nevertheless had a profound effect upon the study of glacial isostasy by removing the necessity of possessing such detailed stratigraphic information to control the time scale. The first application of 14Cdating in the construction of relative sea level (RSL) histories of which I a m aware is that by Marthinussen (1 962) in a study of the shorelines of northern Norway. Elsen's ( 1 967) reconstruction of the history of North American glacial Lake Agassiz (which was centered on the present lakes Winnipeg and Winnipegosis in the Canadian province of Manitoba) using I4C dating is also a noteworthy early contribution. The 5730 (+40)-yr decay time for the beta disintegration of 14C (14C P- + I4N') makes the method perfectly suited to the study of the sea level record during the past 20 kyr since glacial maximum. Very recently devised accelerator-based techniques for 14 C dating, which count atoms directly (Litherland, 1980), are expected to gradually replace conventional @-countingin many applications. These new methods make it possible to obtain dates from much smaller samples (51 mg) than can by analyzed conveniently using the method developed by Libby and his co-workers. Since the pioneering studies of mantle viscosity by Haskell and VeningMeinesz there have been an enormous number of similar investigations reported by other workers using either the same data set or similar information from other geographic locations. The most important locations outside of Fennoscandia include the North American continent as a whole (the northern half of which, Canada, was covered by the huge Laurentide ice sheet at 20 kyr B.P.) and the region surrounding the much smaller scale glacial Lake Bonneville, which was located in what is now the state of Utah in the Basin and Range geological province. Taken together with the data from Fennoscandia, the observations from these regions provide information on the isostatic adjustment of horizontal scales of surface deformation ranging over a full order of magnitude from a few hundred to a few thousand kilometers. Since the horizontal scale of the adjusting region in part determines the vertical extent of the flow through which isostatic adjustment takes place, the existence of data covering such a wide range of spatial scales promises the capability of using it to infer the variation of viscosity as a function of depth in the mantle. Until rather recently, the scientific results
-
DYNAMICS OF THE ICE AGE EARTH
5
obtained through inversion of the combined data set have proved to be extremely controversial. The Newtonian viscous half-space model of Haskell and Vening-Meinesz predicts that a sinusoidal surface deformation with horizontal wave number kH should decay exponentially in time with decay constant r = 2vkH/pgq, in which v is the viscosity, p the density, and g, the surface gravitational acceleration. Van Bemmelen and Berlage (1935), working at the same time as Haskell and on the same data set, assumed that isostatic adjustment was confined to a thin channel near the surface of depth h. Their analysis showed the relaxation time for this model to be T = ( 12v/pgs,h3)( Ilk k) and thus to depend inversely upon the square of deformation wave number. Assuming h to be 100 km, they inferred a viscosity of Y = 3 X 10l8Pa sec [ l Pa sec (SI units) = 10 P (cgs units)]. Although the flaws in this thin-channel model (low near-surface viscosity in spite of low near-surface temperature, inability to fit relaxation amplitude and relaxation time simultaneously, etc.) are much more evident today than they were in the late 193Os, the model nevertheless disappeared from the early literature for some time. Its later resurrection may be attributed to Crittenden (1963). Crittenden had compiled data for the relaxation associated with the disappearance of Pleistocene Lake Bonneville and discovered that although the spatial extent of this region differed by an order of magnitude from that of Fennoscandia, the relaxation times for the two regions differed only slightly-both being on the order of 5000 yr. These rcsults seemed to Crittenden to support the idea of thin-channel flow, since Haskell’s model predicted that relaxation time should increase continuously with wave number whereas the channel model predicted a decrease. Even in 1963 there was therefore a body of informed geophysical opinion which considered that the effective viscosity of the earth was anomalously low in the coldest layer immediately adjacent to the surface! The controversy over simultaneous interpretation of the Lake Bonneville and Fennoscandia data inspired the work of McConnell ( I968), who realized that the increasing confinement of the flow to the near-surface region as wavelength decreased which these data seemed to demand did not require infinitely rigid material beneath some critical depth but could be accommodated by a model in which viscosity increased smoothly. His was the first attempt to use the 7(kH) information contained in the Fennoscandia data to directly constrain the v(depth) profile, and many of his conclusions remain valid today. The only novel feature which McConnell’s model contained was a “lithospheric” layer at the surface in which the viscosity was infinitely high but whose thickness was variable. He found that the relaxation spectrum 7(kH) extracted from the Fennoscandia data required the presence of such a layer and constrained its thickness to be near 120 km.
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RICHARD PELTIER
Although the required presence of such a layer agrees in a satisfying way with modern a priori expectation based upon knowledge of the temperature dependence of viscosity and the fact that temperature increases markedly with depth near the planetary surface, the picture is nevertheless rather different from that advocated in the model of Van Bemmelen and Berlage and supported by Crittenden. As demonstrated by McConnell, the presence of the thin surficial elastic lithosphere modifies Haskell’s model in such a way that for wavelengths shorter than a few times the lithospheric thickness, relaxation time is forced to decrease with increasing deformation wave number. Modern theory, discussed in Section 3, shows the effect to be such that the spatial scales of Fennoscandia and Lake Bonneville straddle the maximum in the 7 ( k H )spectrum and thus have comparable relaxation times though their spatial scales are separated b y an order of magnitude (see Fig. 13). The results obtained by McConnell ( 1968) for the viscosity stratification beneath the lithosphere have not proved as immune to later analysis as have his inferences concerning the lithosphere itself. McConnell found that the 7(kH)data from Fennoscandia, determined by the spectral decomposition of Sauramo’s (1958) shoreline diagram, required a n upper mantle viscosity which is very nearly constant and equal in magnitude to 10” Pa sec ( lo2’ P), the same value previously inferred by Haskell. He was aware that the 7(kH)data themselves were insensitive to viscosity structure beneath a depth of about 600 km, a fact which was later demonstrated by Parsons (1972) using the formal resolving-power analysis developed by Backus and Gilbert ( 1967, 1968, 1970). In order to constrain the viscosity profile beneath this depth, McConnell was obliged to invoke extra information. What he did was to assume that a so-called nonhydrostatic bulge existed; that is, that the polar flattening of the planet (produced by the centrifugal force) was in excess of that which would be in equilibrium with the current rotation rate, an idea which had been advocated by Munk and MacDonald (1960). He further assumed that the excess bulge was produced by glaciation. (Although this assumption is incorrect, we show in Section 6 that there are nevertheless important observable effects of deglaciation upon the earth’s rotation which can be employed to constrain mantle viscosity.) It followed to McConnell from the existence of the bulge that it must have been relaxing very slowly and therefore that the relaxation time of the degree-two harmonic was in excess of 7 = O( lo4) yr. This requires a high value for the viscosity beneath 670 km depth (the lower mantle), and McConnell described models in which the lower mantle viscosity was constrained to be in excess of lo2’ Pa sec ( loz4 P). McKenzie (1966, 1967, 1968) came to a similar conclusion which was also based upon the assumption that the nonhydrostatic equatorial bulge was a genuine characteristic of the planet and preferred models for the mantle viscosity profile in which the lower
DYNAMICS OF 'THE ICE AGE EARTH
7
Pa sec mantle value was greater than P). This argument for extremely high viscosity in thc lower mantle was completely undermined by Goldreich and Toomre (1969), who pointed out that the crucial nonhydrostatic equatorial bulge did not in fact exist! It had been inferred from a spherical harmonic expansion which was improperly biased to the degreetwo harmonic. The question of the magnitude of the viscosity in the deep mantle remained open. It was beginning about this time that data from I4C-dated shorelines in Canada began to become available in sufficient quantity and quality as to promise a considerable enhancement of depth resolution. Although major papers containing detailed observations on the rebound of the crust in North America began appearing in the early 1960s (Loken, 1962; Farrand, 1962; Washburn and Struiver, 1962; Bloom, 1963), it does not appear that any detailed efforts at geophysical interpretation were attempted prior to the paper by Brotchie and Sylvester ( I 969), who were followed by Walcott (1970). Brotchie and Sylvester were among the first to consider the isostatic recovery problem using a spherical model, but their work did not lead them to any particular interpretation of the mantle viscosity profile, although they did note that the observed relaxation times in North America were short (1000-1500 yr) and that they could fit them with their model. Such short relaxation times were also reported by Andrews ( 197O), based upon data from Baffin Island and other sites in the Canadian Arctic, who notedapparent relaxation times on the order of 2000 yr. The validity of this observation is also clear from the rather complete set of North American RSL data compiled by Walcott (1972). If one employs Haskell's half-space model to infer a viscosity from this relaxation time, for a deformation of Laurentide scale, one obtains a value for the viscosity of the mantle which is very nearly the same as that implied by the Fennoscandia data, i.e., -lo2' Pa sec. Because of the considerable increase of the horizontal scale of the Laurentide load over that which existed on Fennoscandia, the isostatic adjustment of Canada is sensitive to viscosity variations across the seismic discontinuity at 670 km depth, which marks the boundary between the upper and lower mantle. Since the viscosity inferred from the two data sets is the same, the implication is clearly that there is no substantial viscosity contrast across this boundary. This conclusion was enforced in work by Cathles (1975), Peltier (1974), and Peltier and Andrews (1976), who employed spherical viscoelastic models and showed that radiocarbon-controlled RSL data from the Hudson Bay region and from the eastern seaboard of North America could not be fit by models which had any extreme contrast in viscosity between the upper and lower mantle. This result has had considerable influence on the recent debatc concerning the style of the convective circulation in the mantle. It is a result which has proved to be controversial, however-as contro-
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versial as was the interpretation of the combined uplift data from Fennoscandia and Lake Bonneville. The reason for the controversy over interpretation of isostatic adjustment data from North America has concerned an apparent contradiction between the short relaxation time obtained from RSL data and the rather large free-air gravity anomaly which is observed over Hudson Bay, in very close apparent correlation with the topography of the Laurentide ice sheet at glacial maximum. A discussion of this difficulty first appeared in Walcott ( 1970), who presented a free-air gravity anomaly map based upon the work of Innes et al. (1968) which shows a clear elliptical anomaly trending NW with an amplitude of approximately -35 mGal. The zero anomaly contour on the map passes through the St. Lawrence Valley, the Great Lakes, lakes Winnipeg and Athabaska, Great Bear and Great Slave lakes, and Melville Sound; in other words, virtually coincident with the edge of the ancient Laurentide ice sheet (e.g., Bryson et al., 1969). As Walcott (1970) argues and later (Walcott, 1973, 1980) reiterates, it seems preposterous to suppose, as have O’Connell ( 197 1) and Cathles ( 1 975), that this gravity anomaly is unrelated to the currently existing degree of isostatic disequilibrium associated with the melting of Laurentide ice. Yet if one does ascribe the anomaly to deglaciation one is led to an impasse, since one then estimates an amount of uplift remaining to be Ah = Ag/2?rGp (where Ag is the observed gravity anomaly, G the gravitational constant, and p the density of the material displaced to form the depression), which gives approximately 250 rn.As Walcott correctly argues, this large remaining uplift is incompatible with exponential relaxation of the uplift with a time constant as low as 2 X lo3yr. Rather, he asserts that if the relaxation is purely exponential then the relaxation time must be “between 10,700 and 17,100” yr, which he (Walcott, 1970) recognized to be an order of magnitude greater than that implied by the sea level data. This is the contradiction which has fueled recent controversy. In his 1970 paper, Walcott suggests an empirical way out of this dilemma by showing that if the relaxation consisted of a superposition of two exponential decays, so that the remaining uplift (in meters) obeyed an expression like “h = 1 ~ O C ? / ’ ~y’r + 4 5 0 ~ . ~ ‘ / ~ Y‘,”then the impasse might be avoided. He suggests three possible physical effects which might support such behavior: (1) the presence of a lithosphere, (2) the presence of viscosity stratification with a low-viscosity “asthenosphere” overlying a high-viscosity lower mantle, or (3) non-Newtonian effects. As we show in Sections 3 , 4 , and 5 of this article, Walcott’s empirical idea turns out to be almost correct, but for none of the physical reasons he suggested! His own currently preferred explanation (Walcott, 1980) is that relaxation times in Hudson Bay are in fact in excess of 10,000 yr, even though this must be considered extremely unlikely given the weight of observational evidence to the contrary.
DYNAMICS OF THE ICE AGE EARTH
9
Virtually all authors who have attempted to infer the viscosity of the deep mantle from isostatic adjustment data and have come to the conclusion that the viscosity of the mantle is essentially constant have had to ignore the free-air gravity data in order to support their claims. Cathes (1979, for example, argues along with O’Connell ( 1 97 1) that there is no substantial deglaciation-related free-air anomaly associated with either the Laurentide or Fennoscandian depressions. Such correlations as apparently exist are considered by them to be merely coincidental. O’Connell’s argument for a low value of the deep mantle viscosity is particularly interesting. He attempted to infer a mean value for mantle viscosity by assuming, following Dicke ( 1969), that the cause of the nontidal component of the acceleration of the earth’s rotation was Pleistocene deglaciation (we prove by direct calculation in Section 6 that this assumption is correct). His analysis led him to believe that either one of two possible relaxation times for the degeetwo harmonic would allow him to fit these data, the two relaxation times being near 2 X lo3 and lo5 yr, respectively. To determine which of these times was appropriate, he compared the potential perturbation produced by the shift in surface load to the earth’s present anomalous gravitational potential as determined by the satellite data available to him at that time. Since he found no correlation between these fields, he concluded that compensation must be complete and therefore that the correct relaxation time was the smaller of the two, so that the mean mantle viscosity was low. Walcott (1980) accepts O’Connell’s claim of a lack of correlation and provides an argument (which does not seem reasonable to me) as to why this result should imply that the longer relaxation time must be preferred. It is quite clear from more modern satellite data (e.g., Lerch et al., 1979), however, that there is a very good correlation between the geoid anomaly and Laurentide ice topography, as clear as that shown by the surface freeair data. Several authors have used the apparent inability of Newtonian viscoelastic earth models to simultaneously satisfy the RSL and free-air gravity data as a point of departure from which to launch rather extreme theories for the rheology of the planetary mantle. Jeffreys (1973), for example, has mgintained his longstanding argument that this difficulty was due to the fact that the assumption of a steady-state Newtonian viscous rheology for the longterm behavior of the earth was fundamentally in error. He argues that any steady-state deformation of mantle material is impossible and as a corollary to this that thermal convection in the mantle cannot occur. Although somewhat less extreme in their views, Post and Griggs (1973) begin from the same point and argue (using data from Lliboutry, 1971, on uplift at the mouth of the Angerman River in Sweden) that the observed relation between the uplift remaining and the rate of uplift demands that the mantle
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RICHARD PELTIER
be non-Newtonian in its mode of steady-state deformation. Their conclusion is based upon the assumption that the present-day free-air anomaly over Fennoscandia produced by deglaciation is about -30 mGal. The actual anomaly is closer to - 15 mGal according to Balling (1980) (Walcott, 1973, accepts - 17 mGal), and this completely undermines the argument of Post and Griggs that the Fennoscandian rebound provides direct evidence for non-Newtonian behavior and for a power law exponent near n = 3. Cathles (1975), in his arguments in favor of uniform mantle viscosity, is forced to argue that the free-air anomaly over Fennoscandia associated with deglaciation is only -3.5 mGal, which is also, and clearly, incorrect on the basis of modern analyses of this field. Anderson and Minster (1979), following a suggestion by Weertman (1978), have adopted a position with respect to the rebound observations which is also rather difficult to defend. They have argued that postglacial rebound is in a transient creep regime rather than the regime of steady-state deformation and so are philosophically close to Jeffreys. Peltier et a/. ( 1980) have provided several arguments as to why this suggestion is unreasonable. These few citations should suffice to demonstrate that the apparent inability of Newtonian viscoelastic earth models to fit both gravity and sea level data simultaneously have caused many geophysicists to adopt extreme positions concerning the rheology. In Section 2 of this article we describe a simple linear viscoelastic rheology which appears capable of reconciling geodynamic phenomena with time scales which encompass the entire spectrum between those of the elastic gravitational free oscillations and those of mantle convection. The steady-state behavior of this model is Newtonian viscous, and the transient creep regime lasts only about 200 yr. Aside from demonstrating in the following sections of this article the way in which the modern theory of glacial isostasy is able to simultaneously reconcile observations of RSL and free-air gravity, we also show that certain characteristic properties of the earth’s history of rotation are attributable to Pleistocene deglacial forcing. The existence of such effects is particularly important since they depend only upon the degree-two harmonic of the glaciation-induced deformation and therefore provide optimal information on the viscosity of the deepest mantle. One of these effects is the observed nontidal component of the acceleration of rotation which is responsible for the observed nontidal variation of the length of day. Although this causeand-effect relation was previously assumed by O’Connell ( 1 97 1) following the suggestion of Dicke ( I966), we will discuss the recent results of Peltier and Wu (1982), who calculate this effect directly. The second of the deglaciation-related rotation effects discussed here is the observed secular drift of the rotation pole as recorded astronomically by the International Latitude Service (ILS) and the International Polar Motion Service (IPMS) over the
DYNAMIC5 OF- T H E ICE AGE EARTH
11
time period 1900- 1980. This drift is at a rate near 1'/lo6 yr toward Hudson Bay and as pointed out by Sahadini and Peltier (198 1 ) seems also to be due to deglacial forcing. When these rotation data are combined with the RSL and free-air gravity information, we are able to constrain the mantle viscosity profile to depths which extend well into the lower mantle. Although Sections 2-6 of this article deal exclusively with the theory and phenomenology of glacial isostatic adjustment, Section 7 has been included to provide one example of an application of the theory of glacial isostasy to an important problem in paleoclimatology. Recent analyses of the magnitude of the ratio of concentrations of the stable isotopes of oxygen ('80/160) as a function of depth in sedimentary cores taken from the deep ocean basins have revealed certain remarkable properties of the climatic oscillations which have characterized the Pleistocene period (Hays et al., 1976). Although it was initially believed (Emiliani, 1955) that this isotopic variability was a direct reflection of Pleistocene temperatures, it was subsequently established (Imbrie and Kipp, 197 1) that the isotopic ratio for the most part reflected the variation of Northern Hemisphere ice volume. Hays et ul. (1976) were able to transform the depth scale in some particularly long cores into a time scale by finding the depth corresponding to the Matuyama-Brunhes polarity transition of the earth's magnetic field, which is marked by certain faunal extinctions. Since the age of this polarity transition (-700 kyr B.P.) is known from the paleomagnetic time scale established on land, their assumption of constant sedimentation rate leads directly to a depth time transformation. Power spectral analysis of the resulting time series revealed remarkable periodicities to be present in the history of ice volume fluctuations. Statistically significant spectral peaks were found corresponding to periods near 19,000, 23,000, 41,000, and 100,000 yr. These are all astronomically significant periods, the first two corresponding to the period of precession of the equinoxes, the next to the period of changes in orbital obliquity, and the last to the period on which changes occur in the eccentricity of the earth's orbit around the sun. Hays et uf. (1976) of course interpreted their results in terms of Milankovitch (e.g., Imbrie and Imbrie, 1978) theory of paleoclimatic change, which attributed all major fluctuations i n climate to variations in the insolation received by the earth due to precisely these changes in the orbital parameters. An embarrassing difficulty with this interpretation, however, is that there is no significant variation of insolation predicted on the period of lo5 yr, yet it is at this period that well over 50% of the spectral variance in the ice cover record is found. In Section 7 we analyze a paleoclimatic model proposed by Weertman ( 196 1, 1976)and more recently elaborated by Birchfield (1977) and Birchfield et al. ( I98 I ) which essentially involves coupling of a model of ice-sheet flow with a model of glacial isostatic adjustment. These
-
12
RICHARD PELTIER
authors analyze the time variation of ice volume which would be produced by solar forcing of this nonlinear model and find that although there is a diffuse peak at lo5 yr in the fluctuation time series it is overwhelmed by peaks at the forcing periods associated with orbital precession and obliquity. As we will show, however, the model of isostatic adjustment those authors employ is completely inadequate to describe this physical process. When results from the modern theory of glacial isostatic adjustment are used to redesign this model, we find that it does support a free relaxation oscillation at the observed period. The process of glacial isostatic adjustment described in the main body of this article is therefore one which has rather wide-ranging importance in many fields of earth science. Indeed, the basic rudiments of the theory which has been developed to understand this collection of related phenomena (contained in Peltier, 1974, and later articles) has found application in the study of such diverse geophysical phenomena as the formation of sedimentary basins (Beaumont, 1978) and the relaxation of impact craters on Mars (Philips and Lambeck, 1980). It also contains within it, in a particular limit, a complete spherical viscoelastic description of the problem of lithospheric flexure which has proved useful in describing a number of other important geophysical observations (see the review by Turcotte, 1979). It is hoped that the following elaboration of these ideas may suggest further avenues of application. 2. MANTLERHEOLOGY:A UNIFORMLY VALIDLINEAR VISCOELASTIC MODEL
Perhaps the most important ingredient in the theory of glacial isostasy is the model which is employed for the rheology of the mantle. In a sense, all of the physical predictions of the theory follow immediately once the rheology is specified-given that some care is taken in solving the mathematical problem posed by the classical conservation laws. Since the question of the precise nature of the rheological law which governs the response of mantle material to an applied shear stress remains one of the most controversial in geodynamics, it is perhaps not inappropriate that we should begin by defending the particular form of the rheological law which will be employed in our subsequent analysis of glacial isostatic adjustment. Although it is well known that for short time scale seismic processes the earth behaves essentially like a Hookean elastic solid, even on these time scales this rheological model is inadequate in many important respects. It does not, for example, predict the observed spatial attenuation of propagating surface waves, nor does it predict the observed finite quality factors
DYNAMICS OF THE ICE AGE EARTH
13
(Q's) of the elastic gravitational free oscillations. In recent years there has developed a consensus among seismologists that it is possible to describe all of these and some other consequences of the departure of the mantle from perfect Hookean elasticity using the linear relations between stress and strain, which are those appropriate for anelastic materials. Several similar relations have been proposed, including the modified Lomnitz law preferred by Jeffreys (1972, 1973) and the very closely related absorption band models adopted by Liu et al. (1976) and Minster and Anderson (1980a,b). Since anelastic materials (Nowick and Berry, 1972) predictfinite strain in response to an applied shear stress which is maintained for a n infinite length of time, they cannot support continuous flow such as that associated with thermal convection. In order to support flow, the behavior of mantle material must become viscous in the long-time limit. As mentioned in the last section, early work on the problem of postglacial rebound was all predicated upon the assumption that the earth's mantle could be described as a Newtonian viscous fluid, and elastic effects due to loading were neglected entirely. This assumption may introduce substantial error, but to correct it we are forced to design a rheological model for mantle material which is uniformly valid in time. Its initial behavior must be that of an anelastic solid, while its final behavior must be that of a viscous fluid. Although the fact that mantle material does support a steady-state viscous mode of deformation has been very well established experimentally (e.g., Kohlstedt and Goetze, 1974; Durham and Goetze, 1977), the data from these high-temperature creep experiments suggest a nonlinear relation between stress and strain rate and therefore a non-Newtonian behavior in the steady-state viscous regime. This question of the linearity or nonlinearity of the rheological law in steady-state creep is the most outstanding issue which remains to be settled concerning mantle rheology. The experimental studies cited above cannot be considered definitive for two reasons. The first of these concerns the fact that most of the experiments have been conducted on single crystals of olivine, whereas the mantle is polycrystalline. The second reason is that the experiments must be performed at stress levels and strain rates which are orders of magnitude different from those which obtain in postglacial rebound or in convection (strain rates of sec-l are typical of the experiments, whereas 10 l 6 sec-' are typical natural strain rates). It could very well be that at natural stress levels near 10 Pa the rheological law governing steady-state creep could become linear as grain boundary processes become increasingly important. Recent studies by Twiss (1976), Berckhemer et a1 (1979), Greenwood et al. ( I 980), and Breathau el al. (1979) are all suggestive of this possibility. Given the plausibility that the rheological behavior of the mantle could
14
RICHARD PELTIER
FIG. I , Characteristic time scales of various geodynamic processes compared to the Maxwell time of the earth’s mantle. Phenomena with characteristic times shorter than the Maxwell time should be governed by anelasticity. whereas those with longer characteristic time should be governed by steady-state viscous deformation.
very well be linear across the entire geodynamic spectrum, which includes the range of phenomenological time scales illustrated in Fig. 1, it is not at all unreasonable to inquire as to the form which a complete rheological law would then take. The development of this idea, which will be presented here, follows that in Peltier et al. ( 198 I ) and Yuen and Peltier ( 1982). Peltier et a/. (198 1 ) have argued that the simplest rheological law which is capable of describing the required transition from short-term anelastic to long-term viscous behavior is a model which they have called the generalized Burgers body. As is shown in what follows, this model behaves essentially as a Maxwell solid insofar as postglacial rebound is concerned if the parameters which determine the short time scale anelasticity are fixed by fitting the model to the observed Q’s of the elastic gravitational free oscillations. We can consider the process of fitting the observations of glacial isostasy with this model as a basic consistency test of the model assumptions. If the model can be fitted to the data then we may conclude that the data contain no characteristics which demand a non-Newtonian rheology to explain them. This would not, of course, imply that the mantle was Newtonian, although it would provide strong circumstantial evidence of the possibility. Much more work on the isostatic recovery of non-Newtonian models of the sort begun by Brennan (1974) and Crough (1977) will have to be done if we are to understand the diagnostic characteristics of such physical behavior properly. 2.1. The Generulizcd Burgcw Bodv In Fig. 2 we show a sequence of standard spring and dashpot analogs to several common linear viscoelastic rheologies. The analog shown in Fig. 2c
15
DYNAMICS OF THE ICE AGE EARTH
is the simplest linear model which exhibits the transition from short-term anelastic to long-term viscous behavior, which is characteristic of the planetary mantle. It consists essentially of the superposition of the Maxwell model shown in Fig. 2a and the standard linear solid shown in Fig. 2b. A three-dimensional tensor form of the rheological constitutive relation for the Burgers body solid has been derived in Peltier et al. (1981) and has the form
in which (Tk/ and ekl are the stress and strain tensors, dots denote time differentiation, p l and h l are the unrelaxed (elastic) Lami: parameters, and p2 is the shear modulus associated with the Kelvin-Voight element (parallel combination of spring and dashpot). For this model the elastic defect is A = p I / p 2 .The two viscosities and v2 are respectively the long and short time scale parameters which together control the range of time scales on which anelastic and viscous processes dominate (see Fig. 1). Constitutive relations for the simpler Maxwell and standard linear solids may be derived from the Burgers body expression (2.1). In the limit v2 co Eq. (2.1) becomes it1
-
2k/ + (/Jl/VI)(Gk,
~
hi,\h/) = 2PLIL)U +
ail
which may be integrated oncc in time to give the Maxwell constitutive
t4 - P z ' S ' - d
--7--g+$
-:+!. v2 (C)
L';
--V,is,--V;
(d)
FIG. 2. Spring and dashpot analogs or several conimon linear viscoelastic rheologies. (a) Maxwell solid, (b) standard linear solid. (c) Burgers body solid. (d) the "generalized' Burgers body.
16
RICHARD PELTIER
relation
+ (pI/Vl)(uk/
6d =
2PU,&/ + XI&
(2.2) which was introduced by Peltier (1974) in developing the linear viscoelastic theory of glacial isostasy. If in Eq. (2.1) we take the opposite limit vl oc, then we obtain UL/
- bJLA
6kl
+
%k/
+ [(PI + P 2 ) / V ? I ( & / =
2p1&+ XI&
- 3 L k
&/
6i/)
-t (2p1p2/w*)(&
- iPkk
6w)
which can also be integrated once in time to yield the rheologicai constitutive relation for the standard linear solid (SLS) as bk/
+ [(PI + P 2 ) / l 4 ( f f i / - t f f k k -=
2y,i‘,,
hi)
+ X 1 4 k 6 i / + (2pIp2/V2)(eA/- +kk
(2.3)
In the domain of the Laplace transform variable s, each of the linear viscoelastic constitutive relations (2.1 )-(2.3) may be written in Hookean elastic form by direct Laplace transformation to give Ck/ =
211(S)t’k/ $-
X ( s ) e k k &/
(2.4)
in which the moduli ~ ( s and ) h(s) are functions of the Laplace transform variable. Explicit forms of these moduli for the four cases are as follows: ( 1 ) Hookean elastic solid p(T) =
/.Ll
X(s)
=
A,
(2.5a)
( 2 ) Maxwell solid
(2.5b)
(2.5d)
17
DYNAMIC'S 01; T H E ICE AGE EARTH
It is important to note that all of these constitutive relations have been designed under the constraint that they have zero bulk dissipation, which is to say that the bulk modulus K = X(s) $L(s) is independent of s for every model. Dissipation is therefore realized only in shear and not in compression. In spite of the fact that the Burgers body described through constitutive relation (2. I ) displays the transition from initial anelastic to final viscous behavior which is required to understand the general behavior of the mantle, it is nevertheless incapable of fitting the full set of relevant observational data. The problem has to do with the single Debye peak representation of the anelastic behavior embodied in this model. As we will show through examples discussed below, when the simple Burgers body is employed to fit observations of the Q's of the elastic gravitational free oscillations, it is found to be incapable of delivering the weak dependence of Q upon frequency which is characteristic of the observations. In order to generate a viscoelastic model which does not suffer this deficiency, we are forced to consider constitutive relations which are not expressible in a simple differential form like Eq. (2.1 ). Generalized models such as are required to understand the short time scale viscoelastic structure of the earth require the superposition of a large number of distinct relaxation peaks, which are represented schematically by the chain of Kelvin-Voight elements in the analog shown in Fig. 2d. The necessity of using such models to describe the mantle makes this region of the earth very much like an amorphous polymer (Nowick and Berry, 1972) insofar as its rheology is concerned. As the number of Kelvin-Voight elements in the chain approaches infinity, it becomes advantageous to describe the resulting constitutive relation in terms of the notion of a continuous relaxation spectrum (Gross, 1947; Zener, 1948; Macnonald, 1961; Liu el a/., 1976). This in turn necessitates use of the integral representation of the stress-strain relation which follows from the Boltzmann superposition principle. The most general form of such a relation is
+
Uf,(t) =
J'
C'f,k/(t - 7)&/(7)
d7
(2.6)
where C,,, is a fourth-order tensor function for stress relaxation (Christensen, 197 1 ) and the convolution integral over 7 is to be regarded as a Stieltjes integral. For an isotropic material, Eq. (2.6) reduces to A(t
d7
- ~)Pkk(7)
+2
L
p(t
- T)k;,(7)
d7
(2.7)
in which X and p are the two stress relaxation functions required to describe an isotropic linear viscoelastic solid. Assuming that the viscoelasticity of the mantle is felt only in shear and not in compression, then Eq. (2.7) becomes
18
RICHARD PELTIER
+
where K = X $p is the elastic bulk modulus as before. If we first restrict our attention to the short time scale anelastic component of the rheology, it is useful to introduce the idea of a normalized relaxation function in describing the single parameter p ( t ) which is needed in the integral constitutive relation (2.8). We define (2.9)
p ( t ) = pRA$(t)
with pR the relaxed modulus of our generalized standard linear solid, A = ( p , - p R ) / p R the modulus defect of this solid, and $(t) the normalized relaxation function. Following standard work on linear viscoelasticity (Gross, 1953; Christensen, 1971), we may further relate IC# to the relaxation spectrum R through the integral transform @ ( t )=
si,'
R(7e)e-'/rcd(ln
(2.10)
7,)
In his recent work on transient wave propagation in an absorption band solid, Minster ( 1978) employed a relaxation spectrum which was parabolic in shape as (2.1 1) R(7J = (B/7,)H(7,- Tl)H(T2- 7,) with 7 , the relaxation time at constant strain, B a normalization constant, and H the Heaviside step function. Substitution of Eq. (2.1 1) into Eq. (2.10) and Eq. (2.10) into Eq. (2.9) followed by direct Laplace transformation of p(f) gives the analytic expression for the transformed shear modulus as (2.12) From Eq. (2.8) it is then quite clear that the Laplace transform domain form of the constitutive relation is just 6,,(.~) = M J ) e , ,+ [ K - $ . ~ ~ ) I Q L
a,,
(2.13)
which thus has the same Hookean elastic form as Eq. (2.4), which was obtained from the differential constitutive relations. Now expression (2.12) has been found to provide a reasonably accurate description of the high-frequency anelastic behavior of the mantle and is also a form which has been commonly applied in similar analysis of polymers (Ferry, 1980). Using Minster's geophysically reasonable values for the model parameters (Qnl = 250, TI = sec, and T2 = lo4 sec), we may estimate the elastic defect A = ( p , - p R ) / p R to be
DYNAMICS OF THE ICE AGE EARTH
19
so that the difference between the relaxed and unrelaxed shear modulus is only 3%. Even though Eq. (2.12) provides a good phenomenological description of high-frequency processes, however, it is incapable of supporting thermal convection as it does not possess long-term viscous behavior. In order to design a model which has both the correct long time scale and the correct short time scale behavior, we can appeal to the expression for the transformed shear modulus of the simple Burgers body given in Eq. (2.5d). Clearly, if we replace the expression in square brackets in the first equation of (2.5d) by the expression in square brackets in Eq. (2.12) we will obtain an expression for the transformed shear modulus which is uniformly valid in time. This is (2.15)
For
p,/ul
G s
< 1/T2,Eq. (2.1 5 ) becomes
which is the expression for the transformed shear modulus of a Maxwell solid, so that in the low-frequency limit the generalized Burgers body described by Eq. (2.15) will behave like a Newtonian viscous fluid. In the highfrequency limit s 5 1/T2, it will behave as a simple absorption band. In the following two subsections we will provide simple illustrations of the ability of the rheological modcl(2.15)to fit both high- and low-frequency geophysical data. 2.2. Free Oscillations of a Homogeneous Spherical Burgers Body
Our intention in this subsection is to demonstrate the way in which free oscillations data may be employed to constrain the parameters Qm,T2,and T , which are required to specify the absorption band part of the generalized Burgers body rheology Eq. (2. IS). We will restrict this discussion to consideration of a homogeneous, spherical, nonrotating, viscoelastic, and isotropic continuum which is perturbed from its hydrostatic equilibrium configuration by oscillations of infinitesimal amplitude. Such self-gravitating oscillations satisfy the following linearized equations of momentum balance and gravitation (see Gilbert, 1980, for a recent discussion):
v-u
-
V(pgu * ),;
~
pvQ,
+ gv - pugr = -ps2u
(2.17)
20
RICHARD PELTIER
024 = -4lrtiv ' pu
(2.18)
The scalar fields p ( r ) and g(r) are the density and gravitational acceleration in the hydrostatic rest state, respectively; is the stress tensor, 4 the associated perturbation of the gravitational potential, u the displacement vector, G the gravitational constant, and er an outward-pointing radial unit vector. The stress tensor in Eq. (2.17) is given by Eq. (2.13) with p(s) specified by Eq. (2.15). In general the Laplace transform variable s is complex, so that p(s) is complex also. In attacking the viscoelastic free oscillations problem in this fashion we are employing the so-called correspondence principle and in so doing following the same approach employed by Peltier (1974) in analysis of the isostatic adjustment problem. The method has very wide applicability. We lose no important generality by seeking solutions to Eqs. (2.17) and (2.18) in the form u
=
5 [U / ( r ,
S ) P / ( C O S 0%
/ 0
1
+ Vdr, s) ap/ (cos 0)eo + w/(r, s) (cos B)e,#, dB aP1 dB -
~
(2.19a) (2.19b) in which and 6* are unit vectors in the 0 and 4 directions, respectively, and PI is the Legendre polynomial of degree 1 and order zero. Substitution of Eq. (2.19) into Eqs. (2.17) and (2.18), assuming that the physical properties of the earth model p , p, and K are constant, reduces the field equations to two decoupled sets of first-order ordinary differential equations of the form d Xldr = BX (2.20a) d Y l d r = AY (2.20b) QJ', and A and B are in which X = (U',, T4Jr, Y = ( U / , L'/, Trl,THI,4/, reduced forms of the 2 X 2 and 6 X 6 matrices of coupling coefficients given by Gilbert (1980). The r- and s-dependent coefficients Tr/,To/,and T,, are those which appear in the spherical harmonic expansions of the err, are, and ur4 components of the stress tensor, and the Q, are those which appear in the expansion of the auxiliary variable 4 = d4/dr + ( I + 1)4/r 4lrti/u,. The differential systems (2.20a,b) respectively govern the toroidal and spheroidal free oscillations. With I = 0 the spheroidal system describes the radial modes of free oscillation for which V0 = Too = 0 so that (&, Qn) decouple from U(,. T, and Eq. (2.20b) reduces to the second-order system
+
dZldr where Z
=
(U,?,TrJTand
TI
=
CZ
is used to label the radial eigenstates.
(2.20c)
DYNAMICS OF THE ICE AGE EARTH
21
Complex eigenvalues s = sr + I S , for the homogeneous systems (2.20) are determined from simultaneous zero crossings of the real and imaginary parts of the secular functions D,(s, I ) associated with each set of equations. The secular functions are themselves determined by the boundary conditions at the earth’s surface. From Takeuchi and Saito (1972), the secular function for the toroidal system (2.20a) is DdS, I )
=
(I
~
I)h(ka ) - @/+l(ka )
(2.2 1 )
where k , = [ p u / p ( s ) ] ” z , a is the earth’s radius, and po is the average earth density. For the spheroidal system (2.20a) the characteristic equation is of the form
(2.22)
where the superscripts 1, 2, and 3 denote the three linearly independent solutions regular at the origin, each of which consists of a combination of two spherical Bessel functions I,(:) with different complex arguments z and a polynomial in r of degree 1. Explicit expressions will also be found in Takeuchi and Saito ( 1 972). The secular function for the radial system is also given in this reference as
where
(2.23)
Complex eigenspectra for the homogeneous earth model with properties listed in Table I have been calculated by Yuen and Peltier (1982) using the numerical methods developed for hydrodynamic stability analysis by Davis and Peltier ( 1 976, 1977, 1979). We will discuss a small subset of their results in order to illustrate the points which most concern us here. Figure 3 shows free oscillation frequency s, = w as a function of angular degree I and overtone number n for the radial modes, spheroidal modes, and toroidal modes of a homogeneous model with either the simple Burgers body rheology, which has p(s) given by Eq. (2.5d), or the generalized Burgers body rheology, which has p(s) given by Eq. (2.15). The parameter u , is fixed as v I = lo2’ Pa sec in both models. For the simple Burgers body, the short time scale viscosity u2 has been fixed by the requirement that the frequency of oS2does not deviate substantially from the elastic frequency and that the
22
RICHARD PELTIER
TABLEI. PHYSICAL PROPERTIES OF THE HOMOGENEOUS MODEL Property
Symbol
Average value
Units
Density 3-Wave velocity p-Wave velocity Shear modulus Comp. modulus Surface gravity Radius
PO
5517 5130 10798 1.4519 X 10" 3.5288 X 10" 9.82 6.371 X lo6
kg/m3 m/sec m/sec N/m2 N/m2 rn/sec'
11%
V'P
P
x R% U
m
Q of oS,, where Q = sr/2s,, is equal to 200. This requires that v2 = 1016Pa sec (10" P); p2 is chosen such as to make A = 0.03. The parameters of the prototype absorption band have been fixed at Q,n = 250, T , = lop2 sec, and T, = lo4 sec. With such low intrinsic dissipation, the free oscillation frequencies are not significantly affected by the deviation from perfect Hookean elasticity, as is well known. The necessity of choosing v 2 = 10" Pa sec in order to fit observed modal Q's (which are on the order of a few hundred; Anderson and Hart, 1978; Buland et a1 , 1979; Sailor and Dziewonski, 1978) explains the value for this parameter which is marked on Fig. 1. In the case of absorption band anelasticity we can think of a continuous spectrum of short-term viscosities centered on this characteristic value. Although the simple and generalized Burgers bodies are not distinguishable from one another through their predictions of free oscillation frequency, they are strikingly different in their predictions of modal Q. Figure 4 shows Q = sr/2s, for a selection of radial, spheroidal, and toroidal modes for each of these models. The simple Burgers body with v2 = 10l6 Pa sec predicts rapidly increasing Q with increasing modal frequency, whereas the absorption band model eliminates this extreme variation. Since observations of free oscillation Q's for the real earth cited above show no extreme variations of modal Q to exist, the single Debye peak of the simple Burgers body is not a valid description of mantle anelasticity and the generalized Burgers body must be employed. The predicted variations of Q with angular order and overtone number for this model shown in Fig. 4 are remarkably like those which have been observed for the stratified real earth. We note the sharp drop in Q along the radial-mode sequence from the fundamental mode to the first overtone, which is also characteristic of the observations (Buland et al., 1979). We note also the local Q maxima along the spheroidalmode overtone sequences, which are due to the relative partitioning of energy in the modes between shear and compression. This is also char-
D Y N A M I C S 01, T H E ICE A G E EARTH
23
ANGULAR ORDER ( 1)
FIG. 3. Free oscillation frequency as a function of angular degree I and overtone number n for a sequence of (a) radial (,&J, ( h ) spheroidal (,+Sf), and (c) toroidal (,TJ modes for a homogeneous spherical Burgers body modcl of the carth.
24
RICHARD PELTIER
ANGULAR ORDER ( 1 )
ANGULAR ORDER ( 1)
FIG. 4. Normal-mode Q as a function of angular degree / and overtone number n for the simple (a. c, e) and generalized (b, d, f ) Burgers body rheologies for radial (,,So),spheroidal (,J,), and toroidal (J,) modes, respectively.
DYNAMICS OF THE ICE AGE EARTH
25
actenstic of the observations (Anderson and Hart, 1978). The interested reader will find a more detailed discussion of these calculations in Yuen and Peltier (1 982). For our present purposes the above abbreviated discussion suffices to establish that an absorption band description of the anelasticity of the mantle seems to be required to explain observations of free oscillation Q. Because of the magnitude of the long time scale viscosity v I in the generalized Burgers body, it produces no influence upon the attenuation of high-frequency seismic oscillations. This is demonstrated in Table 11, where we show a sequence of calculations comparing the Q’s and periods of the mode 0S2 for the absorption band model, with p(.s) defined in Eq. (2. I2), with those for the generalized Burgers body, with p(s) given by Eq. (2.15) as a function of the long time scale viscosity v l . Not unless u , is less than Pa sec is the influence of its presence significant. Seismology is oblivious of the fact that the eventual behavior of the mantle is viscous. As discussed in Peltier et al. (1981) and Yuen and Peltier (1982), the generalized Burgers body supports not only the weakly damped oscillatory modes of the free oscillation family, but also two families of quasi-static modes which lie on the negative real axis of the complex s plane. They are normal modes of viscous gravitational relaxation and exhibit exponentially decaying rather than oscillatory behavior in time. In Fig. 5 we show two schematic diagrams of the complex s plane, one for the simple and one for the generalized Burgers body. Both plots show the spheroidal modes of degree 1 = 2 for the prototype rheologies and include all of the free oscillations up to overtone number n = 5 . The arrows show the displacement of the free oscillation eigenvalues off the imaginary s axis due to finite anelasticity. For the simple Burgers body (Fig. 5a), there are two additional modes on the negative real s axis. The first has a decay time of 15.94 hr and is supported by the anelastic component of the rheology, whereas the second has a decay time of approximately lo3 yr and is supported by the viscous component. The first of these modes is marked by a solid circle and the latter by a cross near the origin s = 0. Because of the magnitude of their relaxation times, inertial forces play no role in the dynamics of these modes, and I therefore refer to them as quasi-static. The complex s plane for the generalized Burgers body (Fig. Sb) differs from that for the simple model only with respect to the nature of the short time scale quasi-static mode. For this generalized model, no such distinct mode exists, but rather there is a continuum of them ranging in relaxation time from the short time scale cutoff of the absorption band T I to the long time scale cutoff T2.This is illustrated in Fig. 5b by the branch cut connecting the points s, = -1/Tl and s, = -1/T2. The quasi-static mode associated with the viscous response is imperceptibly shifted from its location for the simple Burgers body. All
26
RICHARD PELTIER
TABLE11. COMPARISON OF PERIOD AND Q OF THE MODE$2 FOR THE GENERALIZED BURGERSBODY RHEOL~CY( T ~ Q,) , WITH THOSE FOR THE ABSORPTIONBAND RHEOLOGY ( l o Qo) ,
lo2’ I 020 1019 10lX
10” loi6
-6 -I -2 -8 -3
X
I x 10-5
x x
3 x 10-5
10-3 10-3 X lo-’
x lo-’ - 4 x 10
I
x
4 x 6 X lo-’ 2 x lo-’
of the quasi-static modes are viscous gravitational in nature and are supported by the density contrast across the free outer surface of the model. As we will see in the detailed discussion of the quasi-static viscous modes provided in Section 3, additional density discontinuities which exist in the radial structure of realistic earth models lead to the appearance of additional quasi-static modes. This turns out to have extremely important physical consequences in the theory of glacial isostasy, which is formulated in terms of the quasi-static viscous modes, and in fact explains the ability of our theory of this phenomenon to explain RSL and free-air gravity data simultaneously. The discrete quasi-static modes supported by the anelasticity of the simple Burgers body and the continuum present in the generalized model have yet to be exploited in the understanding of geodynamic phenomena. It is in terms of these modes that the explanation of the phenomenon of postseismic rebound might be found, for example, though they have yet to be employed in this context. Much of the recent literature on this problem may err seriously, in my view, by attempting to describe the observed relaxation of the surface above the slip zone in terms of models which have Maxwell rheology below the lithosphere (Rundle and Jackson, 1977; Thatcher and Rundle, 1979; Cohen, 1980a,b). These models seem to require much lower values for the long time scale viscosity than those required to fit glacial rebound data. However, when SLS rheology is assumed to invert the postseismic uplift data, values of the short time scale viscosity v 2 which are about 10l6Pa sec appear to be required (Yamashita, 1979; Nur and Mavko, 1974). This accords with the value required to fit seismic Q and that which fits the observed Q of the Chandler wobble (Scheidegger, 1957; Smith and Dahlen, 1981). As illustrated in Fig. 1, all geodynamic phenomena with time scales less than the Maxwell time of the mantle (-200 yr) should see only the anelastic component of the rheology. It is important to keep in mind that the short time scale and long time scale viscosities v2 and v , are actual
DYNAMICS OF T H E ICE AGE EARTH
27
material properties of the planet and not merely “factors” which appear in equally convenient Maxwell and SLS rheologies, which both predict exponential relaxation of harmonic loads. Although it is possible to fit postseismic rebound data with a Maxwell model by choosing an appropriate low value of the viscosity, this does not mean that this phenomenon is actually controlled by steady-state creep.
( day-‘ )
FIG.5. Complex s plane for spheroidal modes .SZ for both the simple (a) and generalized (b) Burgers body rheologies. The earth model is homogeneous and the solid circles denote complex normal-mode frequencies supported by the anelastic component of the rheology. The crosses near the origin denote the imaginary frequency of the normal mode of viscous gravitational relaxation supported by the steady-state viscous component of the rheology.
28
RICHARD PELTIER
2.3. Viscous Gravitational Relaxation of a Homogeneous, Incompressible, and Spherical Burgers Body Because the eventual behavior of the generalized Burgers body is both viscous and incompressible, we may approximately determine the nature of the quasi-static viscous part of the complete normal mode spectrum of the earth model by neglecting compressibility at the outset. For such a model, the constitutive relation which replaces Eq. (2.13) is rJ,,
= R
6,,
+ 2pe,,
(2.24)
-
Making use of the fact that V u = 0, the stress tensor can be shown to have divergence V - a = v7T - pv x v x u (2.25) when the shear modulus p is independent of spatial position. Since s is small for the quasi-static viscous modes (see Fig. 5 ) , we may neglect the inertial force on the right-hand side of Eq. (2.17) and rewrite it, using d,po = 0 and V * u = 0, in the form
-V(po4, where w
=
+ pogou
*
er - x) - p V X w = 0
(2.26)
V X u. The divergence of Eq. (2.26) is (PO/CL)O*(~I + gOu * er
-
R/P~)
0
1
(2.27)
In the incompressible limit when the density is constant, Eq. (2.18) also reduces to the form V2@,= 0 (2.28) Spheroidal solutions to the system (2.27, 2.28) can be constructed by expanding u and @ as in Eq. (2.19) and R and w as (2.29a) (2.29b) where H , = V, them to
+ VJr - U//r.Substitution in Eqs. (2.27) and (2.28) reduces of(@/ + ( r u , - 7r7/po) = 0
(2.30)
v:@/= 0
(2.3 1)
where C7f(r2U/)= ( d 2 U / / d r 2+) (2/r)(dU//dr)- l(1 + l ) ( U / / r 2 )The . radial component of Eq. (2.26) is also required and is
29
DYNAMICS OF T H E ICE AGE EARTH
Now the solutions to Eqs. (2.30) and (2.31) are
a/ + tI'('/
(2.33) (2.34)
= c'31'/
+/
(/.dfh,)~'lI'/
( 7 T / / { J , , )=
Substitution of Eq. (2.34) into Eq. (2.32) then gives
,'C
=
[G',//2(2/+ 3 ) ] I ' / + l + C'?I'/ '
(2.35)
Given a/. L'/. and x/from Eqs. (2.33)-(2.35) in terms of the three unknown constants C?. C3 we may compute the tangential displacement amplitudes I;. the normal stress amplitudes Tr/ = s/ + 2pU'/, the tangential stress amplitudes = p( I I '//I' -t L'//I') and the amplitudes Q! = + ( I + I)@,/I' 47rQpoUl of thc auxiliary variable Q related to the radial derivative of the potential. In terms of a solution 6-vector Y = ( U / , I ;, 7;/, I;,/. a/. QI)' the complete solution may then be represented as
',
+
1
2
1' =
(2.36)
CT1yl
I I
where /).it
Yi=
-
(2w
I
+ 3) ' -
(I + 3)r" I ~ 2(2/ + 3 ) ( / i
rJ
,,,,[/I'I+~
t 2pI'/(/2 - / 2(2/ 3 )
+
*
+ 3) (2.3723)
(2.37b) (2.37c) The quasi-static spectrum for the incompressible homogeneous model may now be determined by applying homogeneous boundary conditions at the outer surface of the model. We insist that the tangential stress and the normal stress vanish and that the gradient of the potential perturbation be continuous. which give the boundary conditions Tr/(u= ) 0
'/;!/(u)
=
Application of these conditions at I' = mogeneous algebrdic equations for the MC
where the matrix M is
=
0
N
0
Q/(u) = 0
(2.38)
leads to three simultaneous hoin the form
(2.39a)
30
RICHARD PELTIER
For a nontrivial solution we clearly require that the secular condition det M = 0 be satisfied. Since det M =
2P(s)(1 - 1)(21+ 1 l(1 + 1)(21+ 3 )
)
~
[pot/
+ p(s)(2j2 + 41 + 3)]-2
(2.40)
the eigenvalues of the homogeneous problem are either solutions of p(s) =
-pol[a2/(212 + 41 + 3)
(2.41)
or are the values of s which make p ( s ) = 0. Using p ( s ) for the homogeneous Burgers body given by Eq. (2.15), the nonzero eigenvalues are obviously solutions of s
+ PI/VI
(2.42)
Since the Maxwell time T,,, = v I / p l is such that Tm % T I(T2)there will be (2.43) on the negative real axis of the complex s plane (where pr is the relaxed elastic modulus). Since the modulus defect is small, p J p I N 1, and s' may be amroximated as (2.44) A fundamental property of the earth is that
(2.45) for sufficiently large 1. Since the factor on the left-hand side of the inequality (2.45) is just the relaxation time for a homogeneous Newtonian viscous sphere (Peltier, 1974), we therefore see that the reason why the earth may be approximated as a Newtonian viscous fluid for the purpose of analyzing
31
DYNAMICS OF THE ICE AGE EARTH
TABLEIII. COMPARISON OF s AND s-"' FOR VARIOUSVALUESOF 1 AND A HOMOGENEOUS EARTH s-v/
I
S-I
2 4 6 8 10 50 LOO 500
-0.9 165 -0.7 I96 -0.5766 -0.4784 -0.4080 -0.1023 -0.0527 -0.0108
1.1458 -0.8537 -0.6597 -0.5342 -0.4479 -0.1046 -0.0534 -0.0108
-
Percentage difference 25 18 14 10 10 2 1
t o .1
the rate at which surface deformations relax under the gravitational force is due to the small value of the nondimensional parameter
P=
Polgoa pI[2l2+ 41 + 31
(2.46)
which clearly does not involve the viscosity. In Table I11 we compare(s')-' with (sv') ' for various values of / and a homogeneousearth with po = 55 17 kg/m3, c1 = 6.371 km, go = 9.82 m sec-*, p I = 1.4519 X 10" N/m2. Inspection of this table shows that the elastic correction to the viscous decay time is at most 25% and obtains for the smallest value of 1 shown. For 1 = 10 the difference is reduced to 10%. These results show that although elastic-viscous coupling contributes significantly in determination of the rate of decay of harmonic surface irregularities, the viscous approximation is nevertheless not an unreasonable one insofar as the computation of decay times is concerned. We will conclude this subsection by demonstrating the way in which Eq. (2.44) may be employed to estimate the steady-state viscosity of the planetary mantle. Figure 6 shows a photograph of a flight of raised beaches located in the Richmond Gulf of Hudson Bay near the center of the Laurentide rebound. The relict beaches remain very well exposed at this site, as they do throughout much of the Canadian Arctic and sub-Arctic. For this reason they are in many ways much easier to collect data from than is found to be the case in Europe, where much of the region of uplift is quite heavily populated. Figure 7 shows the RSL data for this site, with the individual data points shown as crosses and dots and with the height of each beach in the sequence plotted as a function of its age in sidereal years (corrected I4C dates). The solid curve on this figure is the prediction of a model which will be discussed later. In Fig. 8 the data are replotted on a
32
RICHARD PELTIER
FIG. 6. Flight of raised beaches located in the Richmond Gulf of Hudson Bay.
log-linear diagram on which they would appear as a straight line if the relaxation were perfectly exponential. The solid line on this figure is the best-fit straight line to the data for the last 5000 yr, during which time we can be fairly sure that the vertical motion was essentially free decay, since most of the surface ice had by that time disappeared. The slope of this straight line gives a relaxation time at this site of approximately 1760 yr. Since the Laurentide ice sheet had a radius of approximately 15", its relaxation spectrum is dominated by harmonic degree 1 = 6. If we substitute (sl)-' = 1760 yr and 1 = 6 in Eq. (2.44) and solve for the viscosity, we obtain v , = lo2' Pa sec ( P), which demonstrates in a completely uncomplicated way how this number is obtained. 2.4. The Phenomenological Utility ofthe Generalized Burgers Body
The brief discussion in the preceding subsections should be seen as an argument in favor of the phenomenological utility of a particular viscoelastic rheology which appears to be uniformly valid in time. The generalized Burgers body has 5 parameters (p,,ulr T , , T2,and Qm), which along with the density p are to be determined by fitting the model to geophysical observations such as the elastic gravitational free oscillations and postglacial rebound. In this phenomenological approach, what one hopes to do is fix
DYNAMICS OF THE ICE AGE EARTH
33
all of the parameters of the model by fitting a subset of the totality of geodynamic observations and then check the validity of the model so determined by predicting other geophysical observables. An example of this approach applied to the short time scale parameters is to fix T I , T2, and Q,(v) using free oscillation data and then use the model to predict the Q of the Chandler wobble; an analysis of this sort has been completed by Smith and Dahlen (1981). A second example applied to the long time scale parameters is to use postglacial rebound to fix v , ( r ) and then use v I in mantle convection models of the sea-floor spreading process. A discussion of this idea has been given by Peltier (1980b). It should be recognized, however, that the above approach is purely phenomenological in that there is no guarantee that the constitutive relation which we employ can be given rigorous microphysical justification. What we would eventually like to accomplish is a direct derivation of this relation from solid-state physical principles concerned, for example, with the dynamics of dislocations. The macroscopic approach which we have elected to take will nevertheless provide rather clear guidelines which will have to be accommodated by any successful microphysical model. One might make the obvious analogy here with simple liquids. Although one cannot, for liquids, directly derive the Navier-Stokes equations which describe their macroscopic behavior from .s.irnple statistical-mechanical first principles (which can be done for gases), this does not make the Navier-Stokes
FIG. 7. Relative sea level curve obtained from radiocarbon-dated beach material in the sequences shown in Fig. 6. Ages have been correcfed to give proper sidereal age; the solid curve is the prediction of a theoretical model.
34
RICHARD PELTIER
FIG. 8. Log-linear plot of the RSL data shown in Fig. 7.
equations less useful for describing the macroscopic motion of liquids. Neither does it make the measured viscosities of liquids less useful physical parameters. In the next section we shall go on to apply the low-frequency limiting form of our general rheological model to develop the theory of glacial isostatic adjustment.
3. THEIMPULSE RESPONSEOF A MAXWELL EARTH In the limit of low frequency, the generalized Burgers body developed in the last section reduces to a Maxwell solid with frequency-dependent Lam6 parameters given essentially by Eq. (2.5b), although the parameter p l which appears in the numerator of the expression for p ( s ) should be replaced by the relaxed shear modulus pR. Since the elastic defect is small, however, which it must be to fit seismic observations, we may safely neglect this effect in constructing a model for glacial isostatic adjustment. This argument justifies use of the Maxwell constitutive relation UlJ =
X(s)e,,
a,, + 2pL(s)e*,
(3.1)
DYNAMICS OF T H E ICE AGE EARTH
with X(s)
=
p(s) =
AS s
+ pK/u + y/u PS
~
.Y
+ p/u
35
(3.2a) (3.2b)
where we have now dropped the subscript 1 on X and p, which denotes the instantaneous elastic value, and on u, which distinguishes it as the long time scale parameter. In constructing our model of glacial isostatic adjustment we will assume that the elastic Lami parameters X and p are known functions of radius determined by the systematic inversion of body wave and free oscillations data and that u ( r ) is to be determined by fitting the model to isostatic adjustment observations.
3. I . The Observed Elastic Striicturc>of'thePlanet and its Phj:sical interpretation In Fig. 9a we show a representative spherically averaged elastic earth model which fits a large fraction of the seismic data set. The parameters which describe this model are X and y and the density field p , which in the
FIG.9. (a) Radial elastic structure of model 1066B of Gilbert and Dziewonski (1975). Note the presence of discontinuities of the elastic parameters at depths of 420 and 670 km associated with solid-solid phase transformations. (b) Several of the mantle viscosity models which are discussed in the text.
36
RICHARD PELTIER
figure have values equal to those in model 1066B of Gilbert and Dziewonski (1975). Actually, we have not shown X and p individually in this figure, but of longim and V, = rather have given the velocities V, = i tudinal and transverse elastic waves, respectively. Inspection of this figure clearly reveals the major regions into which the planetary interior may be divided: the small solid inner core; the liquid outer core, in which V, = 0; the lower mantle, beneath 670 km depth, throughout which V,, V A ,and p increase smoothly; the transition region marked by the presence of two discontinuities in V, and V, at about 420 and 670 km depth (the deepest of which has a somewhat larger associated density jump); the upper mantle, between about 30 and 420 km depth, in which the physical properties also change smoothly; and finally the crust, which extends to about 30 km depth (above the so-called MohoroviCic discontinuity), in which the density is low and seismic wave speeds are slower than in the underlying mantle. The physical explanations of the major divisions of the interior have been well understood for some time. The most important division, that between core and mantle, is clearly chemical in origin, the mantle being essentially a mixture of iron and magnesium silicates and the core consisting of a mixture of iron alloyed with some lighter element (e.g., Jacobs, 1975). The density jump between the solid inner core and the liquid outer core contains a small contribution due to the fact that the light alloying element is expelled into the melt as the solid inner core freezes; this idea is useful to dynamo theorists, since the process is expected to drive a compositional convection which would provide an extremely efficient energy source for the geomagnetic field (Braginski, 1963; Loper and Roberts, 1978). Only rather recently has a fully satisfactory explanation of the seismic discontinuities at 420 and 670 km depth in the mantle been provided. Although Ringwood and Major (1970) showed by direct high-pressure experiment that the 420-km boundary was due to a solid-solid phase transition of the low-pressure phase olivine to the high-pressure spinel structure, only very recently (Yagi et al., 1979)has high-pressure diamond anvil technology advanced to the extent that the regime of higher pressures (and greater depths) could be assessed directly with experiments in which thermodynamic equilibrium prevails. In Fig. 10 we show a phase equilibrium diagram from Jeanloz (198 1) for the system (Mg, Fe)2Si04as a function of pressure at a fixed temperature of T = 1000°C. Inspection of this figure clearly shows that the 670-km discontinuity is also due to a phase change, in this case from spinel to a mixture of perovskite magnesiowustite. Data discussed in Yagi et al. (1979) show that the increase in density which occurs in this transition is just that which is required to explain the seismically observed increase. As explained in Jeanloz (198 l), the olivine spinel and
+
-
DYNAMICS OF THE ICE A G E E A R T H
37
FIG. 10. Phase equilibrium diagram for the system (Mg, Fe)pSiO, as a function of pressure (depth) (from Jeanloz, 1981.) The composition corresponding to the earth’s mantle is shown on the figure.
38
-
RICHARD PELTIER
+
spinel perovskite magnesiowustite transitions are fundamentally different. The former involves no change in the atomic packing configuration, whereas the latter is a true high-pressure transformation in the sense that such a change of coordination does occur. In my view, these new data may severely undermine the idea which has been prevalent in the geophysical literature for some time (e.g., Anderson, 198 1 ) that there is a significant change of mean atomic weight (i.e., chemistry) across this boundary. Most of what we know about the mantle suggests that it is very nearly homogeneous chemically. The only apparent exception to this is the recent information on the degree of mantle mixing which has been derived from studies of the Rb-Sr and Nd-Sm isotopic systems which seem to suggest that the mantle consists of two fairly distinct isotopic reservoirs, one of which is essentially “primitive” in its content of radioactive elements and the other of which is essentially depleted. Given preexisting ideas in the literature to the effect that the 670-km discontinuity is a chemical boundary, it is perhaps not surprising that geochemists have tended to associate the primitive reservoir with the lower mantle and the depleted reservoir with the upper mantle, although their data give no direct information concerning the location of these apparently required reservoirs. The last of the boundaries evident in the spherically averaged model shown in Fig. 9, that between crust and mantle, is clearly influenced to a nonnegligible degree by lateral heterogeneity connected with differences between oceans and continents. Such lateral heterogeneity of the near-surface elastic structure should not be too important to viscoelastic relaxation, however, since all of this heterogeneity is contained in the low-temperature, high-viscosity lithosphere in which flow may occur only on extremely long time scales.
3.2. Formulation cf the Viscoelustic Problem for Models with Rudiul Heterogtxeity In order to describe the viscoelastic response of realistic earth models with the rather complicated elastic structure shown in Fig. 9, we are forced to extend the discussion in Section 2.3 considerably, both to include strong radial heterogeneity of p and p and to include the effect of finite X (compressibility). The mathematical problem is that posed by the linearized versions of Eqs. (2.17) and (2.18) in the quasi-static limit, which we will rewrite for convenience as
V .CJ - V(pgu. e,) - poVdl - gop,C,= 0 V24, = 4irGp, where the density perturbation equation as
pI
(3.3) (3.4)
is obtained from the linearized continuity
DYNAMICS OF THE ICE AGE EARTH PI
=
-PUV * u - u * (drPo)gr
39 (3.5)
The momentum equation (3.3) has been linearized with respect to perturbations from a background hydrostatic equilibrium configuration (PO, PO, $o) which satisfies V P =~ -PogOCr (3.6a) v2&= 4&po (3.6b) In Eqs. (3.3) and (3.4) the gravitational potential perturbation cbl will in general be the sum of two parts, & and &, which are respectively the potential of any externally applied gravitational force field (the load) and that due to the internal redistribution of mass effected by the load-induced deformation. We will require solutions to Eqs. (3.3) and (3.4) which describe the deformation of the radially stratified planet induced by surface loading. Since the response to an arbitrary load can be obtained by convolution with an appropriate point-load Green’s function and since symmetry considerations demand that this response depend only upon r, s, and the angular distance from the point load 8, fundamental solutions are spheroidal and have the following vector harmonic decompositions: oc
u=
C (U/(r, S)P/(COS
8)ir
/=0
ap1 + V / ( r ,s) (cos 8)g8) a8
(3.7a) (3.7b) (3.7c)
Substitution of Eq. (3.7) into Eq. (3.3) reduces these field equations to the following set of three simultaneous second-order equations: (3.8a)
0
+ p o g ~ x d(goU)/dr+ d(Xx + 2 p U ) ) / d r + (p/r2)4rU 4U + 1(/ + 1)(3V U - r p )
= --PO&
- PO
-
0:-
po@ -
pogoU
-
(3.8b)
+ Xx +
+ (l/r)[5U + 3rll’
-
I/ -
21(1
+ 1)V]
(3.8~)
in which the dot denotes differentiation with respect to r and
x = u. +2r- u - -1(Z+r
1) I/
(3.8d)
40
RICHARD PELTIER 20
+ (2/4?0
=
(3.8e)
4TGPO
in all of which it is understood that U , V, @, x, A, and p stand for U/(r, s), V/(r, s), @/(r,s), x/(r, s), X(r, s), and p(r, s), respectively. Equations (3.8ac) may be rewritten as a set of simultaneous first-order equations in terms of a vector Y, the elements of which are
Y
=
(u/,V / , 7,/,TH/,
Q/Y
(3.9)
where
(3.1 1 )
(3.12a) (3.12b)
(3.12~)
where
P
=
X(r, s)
+ 2p(r, s)
(3.13a) (3.13b)
Now the solution 6-vector which solves Eq. (3.1 1 ) may be represented quite generally as a linear superposition of six linearly independent solutions. The combination coefficients in this linear superposition are determined by the r Ia. Three of boundary conditions at the endpoints of the domain 0 I the required six boundary conditions are that U, V, and @ be regular at the origin r = 0. The remaining boundary conditions depend upon the physical
DYNAMICS OF THE ICE AGE EARTH
41
conditions which obtain at r = (I. In Section 2.3 we assumed homogeneous boundary conditions Eq. (2.38) at r = a and deduced an analytic expression for the relaxation spectrum of the incompressible model with constant physical properties. In the next subsection we will discuss the properties of the relaxation spectrum of realistic earth models with elastic structure fixed to that shown in Fig. 9.
3.3. Normal Modes of’ Viscoir.\ Gravitational Relaxation When the effect of radial stratification of the earth model is included, the elements of the secular matrix M in Eq. (2.39b) must be determined numerically. In order to do this, we proceed by integrating the system of equations (3.11) for each harmonic degree 1 from the center of the earth to its surface. We employ a standard “parallel shooting” method, in which the system is integrated from depth to the surface using three linearly independent starting vectors determined from the three linearly independent solutions which solve Eq. (3. I 1 ) for a homogeneous compressible sphere. These solutions are given explicitly in Wu and Peltier (1982a). As 1increases, the starting depth is decreased and the properties of the homogeneous sphere which determine the starting solution are taken equal to those at the starting depth. Propagation of each of the linearly independent solutions to the surface r = u generates values of Tj,, T&,,and QJI where j = 1 , 3 denotes the number of the “shot.” For homogeneous boundary conditions (2.38), it then follows from the linearity of the differential equations that the secular condition is given by
D&, I )
=
0
(3.14)
where D2(s,I ) is as defined in Eq. (2.22). The similarity between the problem of viscous gravitational relaxation and the problem of spheroidal free oscillation is therefore further reinforced. As pointed out previously, the totality of the normal-mode frequencies are represented as points in the complex s plane. In such a representation the quasi-static modes appear as points on the negative real s axis as shown previously in Fig. 5a,b and therefore have frequencies which are purely imaginary. Figure 1 1 shows a plot of the secular function Dz(s, 1) for I = 6 and for an earth model whose viscosity profile is shown in Fig. 9b as model 1 and whose elastic structure is as in Fig. 9a. The entire core is taken to have zero viscosity, the viscosity of the mantle is lo2’ Pa sec P, v is in poise in the figure) throughout, and the model has a 120-km-thick lithosphere in which v is infinite. Inspection of Fig. 11 shows that for fixed 1 there are several modes for a given value of 1 rather than one as was found for the homogeneous earth model.
42
RICHARD PELTIER
FIG. 1 I . The secular function for the spheroidal system for angular degree I of the Laplace transform variable s.
=
6 as a function
Relaxation diagrams for all three of the viscosity models shown in Fig. 9b are shown in Fig. 12a,b,c (from Wu and Peltier, 1982a). On these diagrams we have plotted on a log-log scale the inverse relaxation time of each mode as a function of spherical harmonic degree 1. Relaxation times have been nondimensionalized with a nominal time of lo3 yr, so that where log(-s) = 0 the relaxation time is lo3 yr, whereas where log(-s) = -1 the relaxation time is lo4 yr. Visible on each of these three plates are six modal branches, which are marked LO, MO, M1, M2, CO, and Cl. These distinct branches are analogous to the various body wave and surface wave branches which are visible on the dispersion diagram for the elastic gravitational free oscillations (e.g., Gilbert and Dziewonski, 1975) in that they owe their existence to specific physical properties of the radially stratified viscoelastic earth model. The MO branch is the fundamental mode of the mantle, which corresponds to the single mode which exists in the homogeneous earth model discussed in Section 2.3. However, along this branch the relaxation time does not increase continuously with increasing spherical harmonic degree as predicted by Eq. (2.44) for the homogeneous model. Rather, for angular order 1 greater than about 30, relaxation time decreases with increasing 1. This effect is due to the presence of the lithosphere and was first demonstrated by McConnell ( 1968) for half-space models. McConnell discovered this characteristic behavior in his deconvolution of Sauramo’s (1958) shoreline data and used it to measure the thickness ofthe lithosphere.
FIG. 12. Relaxation diagrams for viscosity models I , 2, and 3 shown in Fig. 9b are shown in parts (a), (b), and (c), respectively. The elastic structure is that of model 1066B shown in Fig. 9a.
44
RICHARD PELTIER
As well as altering the fundamental-mode relaxation curve in this way, the presence of the lithosphere also introduces a second modal branch, which is labeled LO in the figures. As discussed in Wu and Peltier (1982a), the modes along this branch are not efficiently excited in general and so play no substantial role in viscoelastic deformation. At large 1 the MO and LO modal lines converge, and this is simply a mathematical manifestation of the physical fact that for sufficiently short wavelength (large I ) all viscoelastic relaxation is suppressed, since such short-wavelength disturbances are completely controlled by the perfectly elastic lithosphere. The CO branch on each of the modal diagrams is supported by the density contrast across the core-mantle boundary, and inspection of the relaxation diagrams shows that the MO, LO, and CO branches are more closely interleaved for the uniform viscosity model (Fig. 12a) than they are for the other models which have moderately high lower mantle viscosity (Fig. 12b,c). In fact, in both the models with high lower mantle viscosity (models 2 and 3 of Fig. 9b) the relaxation times along the CO branch are very nearly one order of magnitude larger than they are in the uniform viscosity case. This may be simply understood in terms of the variational principle derived in Peltier (1976). Since the core mode is sensitive only to lower mantle viscosity and since the lower mantle viscosity is one order of magnitude greater in models 2 and 3 than it is in model 1, the relaxation times for this mode are increased by one order of magnitude according to the variational formula. The remaining modes M 1, M2, and C2 on this diagram have considerably larger relaxation times than do the modes MO, LO, and CO. Modes M1 and M2 are supported by the density jumps across the 670-km and 420-km discontinuities, respectively, whereas C 1 is supported by the density contrast between the inner and outer cores. Of this sequence of long relaxation time modes, M1 is by far the most important, since it is the most efficiently excited by surface loading (Wu and Peltier, 1982a). It is at least in part a consequence of the excitation of this mode that models with uniform mantle viscosity are able to reconcile free-air gravity and RSL data simultaneously, as we shall see. In Fig. 13 we have plotted relaxation time versus angular order for the fundamental mantle modes MO of each of the previously discussed viscosity models numbered 1-3 in Fig. 9b. Also shown is the corresponding modal curve for a model numbered 4, which differs from 3 only in that the viscosity in the sublithospheric low-viscosity zone is lo2' Pa sec ( lo2' P) rather than l O I 9 Pa sec (lo2' P). Superimposed upon these modal curves are hatched regions denoting observational estimates by various authors of the relaxation times for specific spatial scales. Also included is the ~ ( k "spectrum ) deduced by McConnell ( 1968) from Sauramo's shoreline data, which must, however, be considered unreliable at both the longest and shortest relaxation times.
DYNAMICS OF T H E ICE AGE EARTH
45
FIG. 13. Relaxation time versus angular degree for the sequence of viscoelastic earth models discussed in the text. The hatched regions represent estimates of relaxation times for specific horizontal scales by the authors noted.
The greatest disagreement evident in this diagram is that between Andrews (1970) and Walcott (1980) concerning the relaxation time for the 1 = 6 harmonic, which is deduced from the observed uplift of Hudson Bay after removal of the Laurentide load. Andrews’s estimate of the relaxation time is based upon the shoreline data from the Ottawa Islands and some other locations and is between 1700 and 1900 yr. Our analysis in Section 2.3 of the Richmond Gulf data of Hillaire-Marcel and Fairbridge (1978) gave a relaxation time of 1760 yr and therefore agrees with Andrews’s. Walcott’s (1980) estimate is from a site near the Ottawa Islands and is based upon radiocarbon ages of shells and shell fragments of a single species (Mytilus edulis). He claims that the shortest relaxation time allowed by the data is about 4000 yr but that the actual relaxation time could be very much longer (i.e., > 10,000 yr). Inspection of Fig. 13 shows that the minimum difference of a factor of 2-4 between Andrews’s and Walcott’s estimates of the relaxation time of the 1 = 6 harmonic is the difference between the model with uniform mantle viscosity and the model whose lower mantle viscwity is higher than lo2’ Pa sec by at least one order of magnitude. It should be quite clear on the basis of the complete relaxation diagrams for several viscosity models shown in Fig. 12, that attempts to constrain the mantle viscosity profile by comparing one modal branch of these diagrams to crude estimates of relaxation time for specific spatial scales of defor-
46
RICHARD PELTIER
mation, such as is done on Fig. 13, is an inaccurate process at best. There is a good analogy which can be profitably drawn here between the interpretation of isostatic adjustment data and the interpretation of body wave seismic data. Very considerable advances in the latter area have been achieved by interpreting not only the times of arrival of the various phases but also the amplitudes of the waveforms themselves-which requires the construction of synthetic seismograms (e.g., Aki and Richards, 1980). We may consider the arrival times for specific body wave phases to be analogous to the relaxation times for specific horizontal scales. In order to improve the accuracy of our inferences of mantle viscosity from isostatic adjustment data, we are obliged to develop a theory for the equivalent in glacial isostasy of the synthetic seismogram in seismology. A s we will see in Section 4 , when we require the theory to predict isostatic adjustment amplitudes as well as decay times, then much of the ambiguity of interpretation evident from Fig. 13 may be removed. In the next subsections we will begin to develop the theoretical apparatus required to construct the synthetic “relaxograms” of glacial isostasy.
3.4. Love Number Spectra for Impulsive Forcing In order to calculate the viscoelastic deformation of the planet’s shape produced by glacial loading, we have simply to solve the field equations (3.3) and (3.4) with the appropriate boundary conditions. What we do in practice is to consider first the response of the planet to a surface pointmass load which is applied as a Dirac delta function in the time domain. To determine the appropriate boundary conditions for this problem we treat the point-mass load y as a uniform disk load of vanishingly small radius a. Expanding y in a Legendre series then gives a
=
C r l ~ , ( c oe)s
(3.15)
I=O
in which the
rl are given by (Longman, 1963; Farrell, 1972) 1 rl = 21+ ha2
(3.16)
~
-
in the limit a 0. The surface boundary conditions (Longman, 1963; Farrell, 1972) are that V42-Z, change by 4 a y across r = a, that (V& + 47rGpu) * Z, be continuous, that the normal stress balance the applied load so that a,(a) = -yg, and that the tangential stress vanish so that a,,,(a) = 0. When these boundary conditions are expanded in spherical harmonics the expansion coefficients are forced to satisfy Trl(u)=
-gr/
TBI(u) = 0
Q / ( a )= - 4 ~ G r /
(3.17)
DYNAMICS OF THE ICE AGE EARTH
47
which are the conditions which replace Eq. (2.38) for the inhomogeneous problem under present consideration. Since the surface load is assumed to be applied as a delta function in the time domain, boundary conditions (3.17) are independent of the Laplace transform variable in the domain of which the field equations (3.3) and (3.4) must be solved. These boundary conditions suffice to completely determine the three other elements of the solution 6-vector Y = ( U l , V,, T,/, To/,a/, Ql)f at the surface r = a and throughout the earth. By analogy with the surface loading problem for an elastic sphere it is convenient to describe these remaining elements in terms of a triplet of dimensionless scalar Love numbers (h/, //, kl) which are functions of r, I, and .r defined through the relations
::;::) :[@
VI ( r ,
4
1 1= +2 / ( r )
h/(r, sYg0 I/ (r, d/go k / ( r ,I)]
(3.18)
Using Eq. (3.18) we may write the total potential perturbation as @I./
=
- @p?/
+ +3,/
=
-
+ k,)
(3.19)
where the coefficients in the expansion of the potential of the surface load a2,/are obtained from the definition (3.20) in which integration is over the earth's surface. If we substitute for y(r) in terms of its Legendre expansion Eq. (3.15) and use the addition theorem for spherical harmonics (e.g., Jackson, 1962) we find (3.21)
so that (3.22) where Me is the mass of the earth and go the surface gravitational acceleration as before. In Fig. 14 are shown example Laplace transform domain Love number spectra for an earth model with viscosity structure similar to that of model 1 in Fig. 9b and with the elastic structure shown in Fig. 9a. These spectra are found by direct integration of the simultaneous ordinary differential equations (3.1 1) subject to boundary conditions (3.17) for a sequence of values of s along the positive real s axis in the complex s plane. Two alternative representations of the Love number spectra hl(a, s) at the earth's surface are shown in Fig. 14, one an x-y plot of hl(a, s) for various values
FIG. 14. Love number spectra h, (a, s) for realistic viscoelastic earth models. The spectra shown on the left are for a model with the viscous structure of model 1 in Fig. 9b except that it lacks a lithosphere. The spectra on the right are for the same model with a lithosphere.
49
DYNAMICS OF I H E ICE A G E EARTH
of I marked on the figure, which is for a model with no lithosphere, and the other a three-dimensional plot of the form of the spectral surface, which is for a model which includes a lithosphere. Several important features of these spectra are evident by inspection of the diagram. Most important is the fact that the spectra achieve asymptotic amplitudes both for sufficiently large and for sufficiently small values of the imaginary frequency s. Inco the moduli X(s) and spection of Eq. (3.2) shows that in the limit s p(s) become equal to their elastic values, so that in this limit the Love numbers become asymptotically equal to the Love numbers for an elastic sphere (e.g., Longman, 1963; Farrell, 1972). In this limit, our calculations agree with those of previous authors. In the opposite limit s 0, X(s) K , the elastic bulk modulus, and p ( s ) vs, so that the Maxwell solid becomes like an incompressible viscous fluid. The existence of the small s asymptote is indicative of the existence i n this limit of a new state of viscous gravitational equilibrium. As we will show, this is the state of isostatic equilibrium which obtains in the limit of long time. Also evident by inspection of the spectra shown on this figure (for the model with a lithosphere) is the fact that for sufficiently short wavelength the small s asymptote is reduced to equality with the large s asymptote and this is due to the complete suppression of viscous relaxation at the shortest wavelengths. In order to understand the meaning of the spectral asymptotes at small values of s, we need only consider the inversion of these spectra into the time domain. Since each of the Love numbers h/(a, s). l / ( u , s), k l(a , s) has a spectrum similar to those shown in Fig. 14. they can all be expanded. in the form
-
-
-
-
/?/(a,.s) = /?;/(a,s) + hF(a)
(3.23)
where hF(a) are the elastic asymptotes and /$(a, s) therefore represents the viscous contribution to the response. The Laplace inverse of Eq. (3.23) is just (3.24) Now the integral along the Bromwich path L , in Eq. (3.24) may be evaluated using Cauchy's theorem, which allows us to write
Jf hy(a,
s)eS' ds =
I:
-
I
hy(a, .s)c" ds
+ 2xi c
[
residues at the poles
1
O f t h e integrand (3.25) inside(L, 12)
+
Since the first integral on the right-hand side of Eq. (3.25) goes to zero as the radius of the semicircle tends to infinity, Eq. (3.24) therefore reduces to h(a, t) =
at the poles of the ] + hF(a)6 ( t ) c [.residues rntegrand inside ( L + 6,) I
(3.26)
50
RICHARD PELTIER
Now the poles of the integrand hy(a, s)e“ are located at the zeros of the secular function Dz(s,1 ) in Eq. (3.14) which we determine in the way discussed in the last section. These poles are all on the negative real s axis and we may label them s = - s j where the s j are 20. Furthermore, they are all simple poles, and we may usually write
h,(a, t ) =
c r: e -;‘
+ h ~ ( as(t) )
(3.27a)
J
where the “initial amplitudes” r j can be determined as follows. If h/(a, t ) have time domain forms (3.27a), they then have Laplace transform domain forms h,(a, s) = C r:/(s + s: ) + $(a) (3.27b) J
If solutions of the inhomogeneous problem are obtained for s
=
+s: to give
N
h;V(s:)= h/(a,s j )
-
h:
=
C r:/(s: + s:) I-
(3.28)
1
and if we define the elements of the matrix rn as
m,, = I/(Y; + s:)
(3.29)
then Eq. (3.28) may be written as
hy(s;) =
(3.30)
m,,y:
from which we may compute the r : as r : = In ,,‘h,V(.s;)
(3.31)
This collocation method provides a very efficient means of solving the inhomogeneous problem using the discrete spectrum determined by solving the homogeneous problem. Solutions of the inhomogeneous problem are thereby represented in the form of a superposition of normal modes of viscous gravitational relaxation in which the r: represent the excitation strengths for point forcing. Given the time domain forms (3.27) for the impulse response problem, we may employ them to compute the spectral amplitudes required to describe the response if the load is applied at t = 0 and then allowed to remain on the surface. These amplitudes are obtained by convolving Eq. (3.27) with a Heaviside step function to obtain
hp(a, I )
=
=
h,(a. t ) * H ( t ) (Y;/.s,)(l ,-$if)
c
~
+ 12:
(3.32)
/
where the * denotes convolution in time. Now inspection of Eqs. (3.32) and (3.27) shows that
DYNAMICS OF THE ICE AGE EARTH
lim h,(a, s) 5-0
=
C r ; / s ; + hF = lim @(a, t )
51
(3.33)
1-0
and the meaning of the small-., asymptotes of the impulse response spectra shown in Fig. 14 is thus clear. The difference between the large-s and smalls asymptotes, Z,r;/s;, for each value of I measures the amount of viscous relaxation which would occur if a load of harmonic degree 1 were applied to the surface at t = 0 and left for an infinite time. In Fig. 15 we show a sequence of plots of the viscous parts of the Heaviside Love number time histories hY.V(a, t ) = (r;/s;)(l - e-5;:')
c I
for several values of 1. The time interval covered by these plots is 20 kyr, and adjacent to the right margin of the figure we show the amount of relaxation which has yet to be realized until equilibrium of that harmonic is achieved. Given these time-dependent Love numbers for our radially inhomogeneous viscoelastic earth models we are now in a position to construct the Green's functions which are required to calculate the response to realistic space-time histories of surface loading. Before proceeding to discuss this phase of the analysis, however, we will provide a brief discussion of the method we have developed for calculation of the infinite-time response amplitudes which are shown adjacent to the curves on Fig. 15. These
FIG. 15. Love number temporal history h?."(u, t ) for several values of 1. The parameter h?." is the viscous part of the Heaviside displacement amplitude and the viscosity model is model I shown in Fig. 9b.
52
RICHARD PELTIER
amplitudes cannot in fact be computed b y direct integration of Eq. (3.1 1) at small values of s because in this limit the system of ode's becomes "stiff" and is not directly integrable. The problem is not merely of technical interest. As we will show, the accurate calculation of this long-time response is crucial to the prediction of the free-air gravity anomaly.
3.5. Elastic und Isostatic Asymptotes of the Love Number Spt)c-fru Although the elastic asymptotes of the individual relaxation spectra may be calculated in a straightforward way by setting .Y = n3 in the field equations, inspection of the individual elements of the A matrix listed in Eq. (3.12) shows that the limit s 0 cannot be taken directly. From Eq. (3.2), however. we see that the limit s 0 and the limit I, 0 are equivalent in the sense that in either limit the shear modulus pL:s) vanishes. Under this assumption. the field equations (3.3), (3.4), and (3.5) reduce to the following forms
--
-
0 = POV(P1 0241= 4xGp, P I = -pov
-
-u
PIK06, -
u*
-
VP
(3.34a) (3.34b) (3.34c)
where we have introduced a pressure field (mean normal stress) through the association p = u pogo;r - Kc,,. The static deformation of an inviscid part of the earth has most recently been discussed by Dahlen and Fels (1978), whose analysis agrees with previous conclusions of Smylie and Mansinha ( 1 97 1 ) and others. There it is shown that fluid particles undergoing a quasistatic deformation experience changes neither of pressure nor of density as the deformation proceeds. It therefore follows that the dilatation is zero everywhere. Substituting V u = 0 in Eq. ( 3 . 3 4 ~and ) expanding all variables in spherical harmonics reduces (3.34a,b) to (Wu and Peltier, 1982a)
-
-
(3.35a)
u = -@/go
(3.35b)
In Eq. (3.35b), U is to be interpreted as the displacement ofan equipotential, isobaric, or material surface. Using Eq. (3.10) we may reduce Eq. (3.35a) to a set of two simultaneous ordinary differential equations in the 2-vector Y ' = (a,, (2,)' of the form (3.364 where
53
DYNAMICS OF ‘THE ICE AGE EARTH
In order to calculate the spectral asymptotes at s = 0 we must match solutions of Eq. (3.36a) across the mantle-lithosphere boundary to the elastic solutions which are valid in the lithosphere itself. If (a,M(b-)and Q,M(b-) are solutions of Eq. (3.36a) just below this boundary, then the solution just above the boundary is given by
1 1
1 (3.37)
where b is the radius of the mantle-lithosphere interface and the C, are determined as usual by satisfying boundary conditions (3.17). It will be recognized that this procedure for calculating the infinite-time spectral asymptotes is identical to the procedure which must be followed at arbitrary s for matching solutions in the inviscid core to solutions in the mantle across the core-mantle boundary. The procedure is perfectly stable numerically.
3.6. Green’s Functions fbr the Surface Mass Load Boundarv Value Problem Green’s functions for the gravitational interaction problem may be computed for various signatures of the response by summing infinite series like Eq. (3.7). In general these Green’s functions may be written as the sum of an elastic part and a viscous part due to the fact that the Love numbers themselves can be so expanded. From Eq. (3.7b) we see that the Green’s function for radial displacement may be written in terms of the Love numbers hy(a, s), assuming Heaviside forcing in time, as
54
RICHARD PELTIER
FIG. 16. Radial displacement Green’s function uH.”(O,t ) for viscosity model I shown in Fig. 9b. The Green’s function has been multiplied by a0 to remove the geometric singularity at 0 = 0.
(3.38) Similar forms may be constructed for the perturbation of the gravitational acceleration a(B, t ) (the “gravity anomaly”) and for the perturbation of the gravitational potential d(0, t ) which may be shown (Longman, 1963) to have the forms aH(fl,t)
=
g
-
Me
rxi
C [ I + 2h7
-
(I + l)k?] P,(COSd )
(3.39)
I=O
(3.40) In Eqs. (3.39) and (3.40) the Love number independent terms are due to the direct effects of the surface load, the terms involving hl are due to the displacement of the earth’s surface, and those involving k, are due to the internal redistribution of matter produced by the time-dependent displacement field associated with the adjustment process. Once we have calculated the Love numbers, the Green’s functions are obtained simply by summing the above infinite series, which is a straightforward process except in the elastic limit, where acceleration techniques such as the Euler transformation must be employed (Peltier, 1974).
DYNAMICS OF: THE ICE AGE EARTH
55
One example of a Green's function is shown in Fig. 16, which illustrates the viscous part of the radial displacement response (3.38) for the model with uniform mantle viscosity and 1066B elastic structure. Again we give two presentations of this function. The first is a simple sequence of x-y plots which show uH,'(O, t ) as a function of 0 for several values o f t marked adjacent to each curve in kiloyears. The Green's function has been multiplied by a0 to remove the geometric singularity which would otherwise occur at 0 = 0 for plotting purposes. We also show in this figure a full threedimensional view of the u","(B, t ) surface. Either representation shows that the viscous response is zero at t = 0 as it must be from Eq. (3.32).As time passes, the surface sags under the load while peripheral to the load the local radius is increased. As we shall see in the next section, it is the collapse of this so-called peripheral bulge which explains the submergence of shorelines in the region just outside the ice sheet margin which obtains along the east coast of North America. Also evident from inspection of Fig. 16 is the fact that for the uniform viscosity model the peripheral bulge migrates in time. This effect is absent in models which have high lower mantle viscosity (Peltier, 1974), and so observations of bulge migration can be quite diagnostic of the deep viscosity structure of the earth. Having constructed Green's functions for the radially stratified viscoelastic earth model, such as that shown in Fig. 16, we can proceed to calculate the response of the planet to an arbitrary known history of surface mass loading. This response must in general include the elastic contribution. In the next subsection we will illustrate the characteristic patterns of deformation which are forced by parabolic disk load approximations to the major Pleistocene ice sheets.
3.7. Response to Simple Disk Load Deglaciation Histories Convolution of the Green's functions over simple circular disk loads involves a straightforward exercise in spherical trigonometry, the details of which are provided in Wu and Peltier (1982a). Rather than repeat these details here, we will simply describe a few of the most interesting results which have been derived from such calculations. Figure 17a,b,c shows the radial displacement response forced by surface loading with a circular disk with parabolic thickness profile with disk radius and mass chosen to give reasonable approximations to the Lake Bonneville, Fennoscandia, and Laurentide loads, respectively. All calculations have been performed with the Green's function for viscosity model 1 shown in Fig. 9b and are based upon the assumption that the disk load is applied to the surface at t = 0 and left in place. The computed response is therefore that which would be produced by instantaneous glaciation of the surface, not deglaciation. The disk radii
FIG.17. Radial displacement response forced by a parabolic circular disk load approximation to the Lake Bonneville (a), Fennoscandia (b), and Laurentide (c) loads. Only the viscous contribution is shown, and the viscosity model is model I of Fig. 9b. The response is shown at times Of(-) t = 4 kyr; (- - -) I = 8 kyr; ( - - - ) t = 12 kyr; ( * . .) t = 16 kyr: (- - -) t = IX, kyr.
DYNAMICS 01-
nm
ICE AGE EARTH
57
which approximate the loads at these three sites are shown on the individual plates b y the thin solid line which extends from 0 = 0" at u?." = 0 and ends at the edge of the disk. The radii of the Lake Bonneville, Fennoscandia, and Laurentide disks are respectively 1.2", 8", and 15". Crittenden (1963) estimated the maximum depth of Lake Bonneville to be about 305 m, corresponding to a mass of approximately kg, and we have used his estimates in our disk calculations. For the Fennoscandia and Laurentide loads our parabolic disk models are taken to have central thickness of 2500 m and 3500 m, respectively, and are based upon the ice sheet reconstructions in Peltier and Andrews (1976). Figure I7 shows only the viscous contribution to the total radial displacement response at each of these sites, and this is given for the sequence of times shown in the figure caption. The results of these simple disk load integrations illustrate several important characteristics of the isostatic adjustment process. Inspection of the calculated response for the Lake Bonneville model (Fig. 17a), for example, shows that the maximum amplitude of the response predicted by the model is only about I2 m, compared to the observed maximum of 64 m. Therefore, although viscosity model 1, which has a 120-km-thick lithosphere, predicts a relaxation time which is closc to the observed time (Fig. 13) for the deformation at Lake Bonneville, the predicted amplitude is enormously underestimated. The reason for this is clearly that the lithospheric thickness in this model is excessive for the Hasin and Range region and this leads to a suppression of the viscous relaxation, a fact previously pointed out in reference to the Love number spectra (Fig. 14) and their time domain forms (Fig. 15). In order to fit the observed relaxation at the Lake Bonneville site, the lithospheric thickness can be at most 40 km. But this modification of the model leads to a marked increase of the relaxation time for the dominant wavelengths in the response, which must bc corrected by introducing a lowviscosity zone in the sublithospheric region. This example serves to reinforce the comment made previously i n the discussion of Fig. 13. One cannot obtain an acceptable inference of thc mantle viscosity structure on the basis of the relaxation spectrum alone. Only if response amplitudes and relaxation times are reconciled simultaneously will a reasonable inference result. Comparison of Fig. 17a with Fig. 17b for the Fennoscandia model shows that as the horizontal scale of the load increases the peripheral bulge becomes more substantial. This is reinforced by the result for the model Laurentide load shown in Fig. 17c. The maximum height of the bulge in the region peripheral to Fennoscandia would be only about 20 m, whereas that peripheral to Laurentide would be about 70 m. Also evident as the horizontal scale of the load increases is the fact that the slope of the surface near the edge of the load becomes steeper, an eff'ect which can be understood qualitatively on the basis of the increasing importance of the lithosphere for smaller scale loads.
58
RICHARD PELTIER
8 4 200
3 I
200
- 2
8
12
16
60
0
4
8
12
I6
*OO
4
8
I2
FIG. 18. Uplift remaining at the center of Lake Bonneville, Fennoscandia, and Laurentide depressions on the basis of the calculations shown in Fig. 17. Note the highly nonexponential response of the large-scale Laurentide region when the uniform viscosity model (LI) is employed.
In Figure 18a,b,c we show plots of the uplift remaining at the center of the Lake Bonneville, Fennoscandia, and Laurentide model disk loads as a function of time for each of the viscosity models shown in Fig. 9b. Models 1,2, and 3 in Fig. 9b are labeled L1, L2, and L3 on this figure to emphasize that the models all have lithospheres which are 120 km thick. These response curves are all plotted on a semilogarithmic scale so that if the response were perfectly exponential they would all appear as straight lines. Inspection of these diagrams shows that every model predicts an increase of apparent relaxation time with time, with the effect generally becoming more pronounced as the spatial scale of the load increases and most important for the uniform mantle viscosity model L1. Of greatest interest is the comparison of Fig. 18b and c for Fennoscandia and Laurentide, respectively. Since models L1 and L3 deliver very similar response at the former site, it is clear that for surface loads of this scale both the increase of relaxation time and the decrease of relaxation amplitude produced by high viscosity in the lower mantle may be compensated by a low-viscosity channel. Such is not the case for a Laurentide-scale load, however; at least over the first 8 X lo3 yr of relaxation, models L2 and L3 deliver very similar response, so that the influence of the low-viscosity channel is very small. The most important point to recognize by inspection of Fig. 18c is that the response curve for the uniform mantle viscosity model L1 is strongly nonexponential in shape. The first 8 kyr of the relaxation arf; dominated by a relaxation time near 2 kyr, whereas the response for times in excess of this appears to be dominated by a relaxation time on the order of lo5 yr. This demonstrates very clearly the fundamental property of realistic viscoelastic models of the planet which makes it possible for them to simultaneously explain a short characteristic relaxation time for the initial stages of isostatic adjustment and a large free-air gravity anomaly indicative of a large amount
16
DYNAMICS O F T H E ICE AGE EARTH
59
of uplift remaining. The reason for the transition in the response from one relaxation time to another is clear by inspection of the relaxation diagram for model L1 shown in Fig. 12a. At the dominant angular order n = 6 for the Laurentide load the MO, CO, LO modes all have relaxation times of about 2000 yr, and the initial response will therefore be dominated by a response time of this order. For this model, however, the M1 mode associated with the density jump across the phase transition at 670 km depth is also efficiently excited, and it has a relaxation time of about 2 X lo5 yr. The transition from short to long relaxation time for the Laurentide-scale load and viscosity model L1 is therefore completely explained on the basis of initial MO (CO) and final M 1 dominance of the response. As we will see in the following sections, this property of realistic viscoelastic models is crucial to understanding several different phenomena associated with glacial isostatic adjustment. The prediction of this characteristic of the adjustment mechanism must be considered one of the most important successes of the modern theory of this process.
4. POSTGLACIAL VARIATIONS
OF
RELATIVE SEA LEVEL
Although the radial displacement response curves for simple disk load approximations to actual deglaciation events provide extremely useful insights into the actual histories of isostatic adjustment contained in the Quaternary geological record, they are imperfect approximations to these histories in many important respects. The problem is not simply that the ice sheets are not circular disks with parabolic cross sections which are removed and applied instantaneously. It is more fundamental. Relative sea level data such as those shown in Fig. 7 can be considered as measures of the change in local radius of the earth only to the extent that the local surface of the ocean (the geoid) can be assumed to have maintained a constant local distance from the earth’s center of mass throughout the period of isostatic adjustment. If this were the case, then the flights of beaches cut into continental coastal areas, such as those shown in Fig. 6, would measure the local radial displacement histories exactly. This view turns out to be somewhat naive for the following reason. As the earth deforms in response to the melting of its surface ice sheets there is a discharge of meltwater to the ocean basins, which raises the elevation of the sea surface (geoid) with respect to the center of mass of the earth. If this increase of level were uniform, then it would be possible to simply correct the computed radial response curves for the increase in water level, and the response due to ocean loading could be simply added to the response due to glacial unloading at any point of interest. In fact, the meltwater produced by ice sheet dis-
60
RICHARD PELTIER
integration cannot be added uniformly to the ocean basins since this would violate the equilibrium constraint that the geoid remain an equipotential surface. As the ice sheets melt, water is distributed over the ocean basins in the unique fashion required to ensure that the surface remains equipotential. In order to calculate postglacial variations of RSL accurately we are forced to develop a theory which is capable of predicting global histories of meltwater redistribution. In the next subsection the structure of this theory will be developed and discussed.
4.I . An Integral Equation for Relative Sea Level The sea level equation which we will derive is based upon the Green's function for the perturbation of the gravitational potential defined in Eq. (3.40). Its structure will be most clearly understood if we begin by supposing that all of the ice sheets which were on the surface at glacial maximum melted instantaneously. If L(0,4)is used to denote the ice thickness removed from position (0, 4) at t = 0 and S(0, 4, t ) the amount of water added to the ocean at position (0,@)and time t, then we may compute the net change of potential at any position on the surface by convolution of the surface loads L and S with the Green's function 4Hin Eq. (3.40) to obtain
@(B, 4, t ) = Pr4"
*I L + P W 4 " 0* s
(4.1)
,7
where * and * indicate convolution over the ice and water, respectively, and I
0
and pw are ice and water densities. From Eq. (3.40) it is clear that Eq. (4.1) includes the change of potential due to the vertical displacement of the solid surface of the earth since the Green's function contains the Love number h,. The change of potential given by Eq. (4.1) will force an adjustment of the thickness of the seawater locally in the amount (Farrell and Clark, 1976) pI
where the constant C is chosen such that conservation of mass is ensured. Now Eq. (4.2) is a result of first-order perturbation theory, which is valid for sufficiently small changes of the local bathymetry S. It is important to note that S is, by construction, the local variation of sea level with respect to the deformed surface of the solid earth and is thus precisely the quantity which is recorded in RSL data such as shown in Fig. 7. Substitution of Eq. (4.2) into Eq. (4. I ) leads to the equation
DYNAMICSor; THE ICE AGE EARTH
61
In order to determine the constant C we note that the integral of pwS over the surface of the oceans must equal the instantaneous value of the total mass which has been lost by ice sheet disintegration at time t. Therefore (Pws)O = Pw =
(Pl(4"lS) * L +- P W ( 4 H / s ) * S), + ( C ) , Pw
-M,(f)
0
(4.4)
The minus sign on the right-hand side of Eq. (4.4) is required because M,(t), the mass loss history for all ice sheets combined, is defined as negative for load removal. In Eq. (4.4) the symbol ( ), is used to denote integration over the oceans. Since C is constant at fixed time, therefore (C)o = CA,, where A,, is the area of the oceans. and Eq. (4.4) gives
With Cgiven by Eq. (4.5), Eq. (4.3) is an integral equation for the unknown field S(0, 4, t ) which we call the sea level equation. It is an integral equation since S appears not only on the left-hand side but also in the convolution integral on the right-hand side. Given the deglaciation history L(0, 4, t ) , and thus M,(t),and the potential perturbation Green's function 4" for a specific viscoelastic model of the planet. we may invert this integral equation to find the history of RSL S(0, 4, 1 ) at any point on the earth's surface. This discussion should serve to make clear the basic structure of the isostatic adjustment problem. Two inputs to the theoretical model are required before the model can be employed to make a prediction. The first of these is a viscoelastic model of the planetary interior, whereas the second is a model of ice sheet disintegration. Neither of these functionals of the model is perfectly known u priorr, and we are obliged to proceed iteratively to refine our knowledge of them. We will not provide a description here of the numerical methods required to solve the integral equation (4.3) or the generalization of of it which describes RSL variations produced by realistic deglaciation histories. Detailed descriptions of the numerical methods will be found in Peltier et (11. ( 1978) and Wu and Peltier ( 1982b). Before describing the results which have been obtained through application of Eq. (4.3), we will discuss in the next subsection how one constructs first-guess approximations to the history of surface deglaciation. It is only because a priori knowledge of this functional of the model does exist that we may begin the iterative process which leads eventually to rather precise knowledge of the mantle viscosity profile.
62
RICHARD PELTIER
FIG.19. Disintegration isochrones for the Laurentide ice sheet based upon the map in Bryson ef al. ( 1 969).
4.2. Inputs to the RSL Calculation: The Deglaciution Chronology and Mantle Viscosity Profile Several distinct kinds of information are required to construct reasonable a priori models of the deglaciation histories of the major ice sheets which existed on the earth's surface 20,000 yr ago. Perhaps the most important piece of information is that concerning the location of the ice sheets themselves, and this is provided by ''C-controlled locations of the terminal moraines of these ice masses. In Fig. 19 we show a map of disintegration isochrones based on that in Bryson et al. (1969) which shows the variation in space of the Laurentide ice margin during the deglaciation phase which began at about 18 kyr B.P. Information of the same type is also available for the Fennoscandia region. Besides the ice bound in these major Northern
DYNAMICS 0 1 TIIF IC’E AGE EARTH
63
Hemisphere continental complexcs. thcrc was also considerable mass contained in Alpine complexes in the Rocky Mountains, the Andes. and the European Alps. Indeed, it was on the basis of his observations of evidence of Alpine ice masses that AgassiL first put forward his arguments for the Ice Age itself. As a fraction of the ice bound in the Laurentide and Fennoscandian complexes. however, thcse Alpine contributions are very small indeed. This is not true of the additional mass which was lost from the Antarctic complex. however; evidence from this region suggests that as much as 15-20%>of the total increase in ocean volume produced by deglaciation may have come from a large-scale melting event over the Ross Ice Shelf in West Antarctica. In order to estimate the total volume of ice which was bound in the major complexes we are obliged to rely upon the RSL data themselves. Prior to the new understanding of these data which was realized through the theoretical model embodied in Eq. (4.3),it was assumed by virtually all scientists working in the field of Quaternary geomorphology that the rise of sea level in the global ocean produced by the melting of glacial ice was a constant independent of location. In the literature of the subject this concept is referred to as eirstatic sca levcd, and there has been a great deal of effort expended to measure the eustatic sea level curve. On the basis of our previous discussion, of course, wc know that this concept is of limited utility, since sea level cannot rise uniformly as the ice melts because this would generate a new ocean surface which was not an equipotential surface. In spite of this limitation, however. the variation of the increase of water depth as a function of time between sites at different geographical locations is not extreme if attention is focused upon sites which are sufficiently far removed from the ice sheets themselves. At such sites the oldest beach is inevitably at the greatest depth below present-day sea level, and the maximum submergence is on the order of 100 m . In Fig. 20 we show a typical eustatic sea level curve from Shepard (1963) which is based upon a particularly extensive RSL record from the Gulf of Mexico. These data extend back to I6 kyr B.P. and show an increase of water depth since that time of something in excess of 80 m. Compared to Shepard’s data is the eustatic sea level curve based upon the mass loss history of the deglaciation model ICE- I tabulated in Peltier and Andrews (1976). The ICE- 1 deglaciation model was constructed by combining Shepard’s (1963) eustatic sea level curve with the disintegration isochrone map of Bryson cf a/.( I 969) for the Laurentide ice sheet and equivalent data for Fennoscandia. What we did was simply to partition the total mass loss implied by Shepard’s curve betwecn the Laurentide and Fennoscandian ice sheets roughly in proportion to their surface areas on the basis of the assumption that the thickness protile of each ice sheet had the parabolic profile
64
RICHARD PELTlER
E
1
J W
> W
J
a W
m
0 I-
F
m 3 W
FIG.20. Eustatic sea level curve from Shepard (1963) (dashed) compared to the mass loss history of the ICE-1 model tabulated by Peltier and Andrews (1976) (solid curve).
which obtains under the assumption of perfectly plastic behavior. Three time slices through a slightly modified ICE-2 chronology are shown for both the Laurentide and Fennoscandian complexes in Fig. 2 I . Most rapid disintegration occurs around 12 kyr B.P. as is evident from Shepard’s eustatic curve shown in Fig. 20. The Laurentide history is quite complicated, since the ice center over Hudson Bay collapses first, leaving high stands of ice both to the east over Labrador-Ungava and to the northwest. I t should be recognized, however, from the way in which the ICE-I(2) chronology was constructed, that it is to be considered a first approximation to the actual history of glacial retreat. It will have to be refined as we refine our knowledge of the mantle viscosity profile. The iterative process proceeds by fixing ICE-1 and determining a “best” u(r), then fixing v(r) and refining ICE-1, then refining v(r),etc., until convergence is achieved. This method of attack is feasible only because we have a good a priori estimate of the deglaciation chronology in the form of the ICE-1 model. In the following subsections we will discuss the predictions which are obtained for RSL history when the ICE-1 chronology is inserted into the sea level equation (4.3).By performing such calculations for several different
D Y N A M I C 3 O € I H E ICE AGE EARTH
65
mantle viscosity profiles we will assess the extent to which RSL data may be employed to constrain this property of the earth’s mantle.
4 3 Oiitpzit from the RSL C’ulc iilution Global Sea Level Histories In Fig. 22 we show four time slices through a solution to Eq. (4.3) obtained for earth model L1 and a deglaciation history which is very similar to the ICE- 1 history of Peltier and Andrews ( 1976). This new deglaciation history is called ICE-2 and has been tabulated in Wu and Peltier (1982b), where the slight differences between it and ICE- 1 are also described. It was actually
FIG.2 1. Three time slices through the ICE-2 melting chronology tabulated in Wu and Peltier (1982b). Maps of ice sheet topography are shown for both the Laurentide (a, c, e) and Fennoscandia (b. d. f ) regions.
66
(C)
RICHARD PELTIER
4000 B P
(dl
PRESENT
FIG.22. Four time slices through the solution to the sea level equation for earth model LI and deglaciation history ICE-2. At each point in the ocean basins the RSL variation is shown in meters.
constructed through the iterative adjustment process described in the last subsection. The global maps ofthe RSL rise S(19,4,t ) show several important characteristics. Most obvious is the fact that spatial variations of S(0, 4,r ) are most rapid in the vicinity of the main deglaciation centers, which is hardly surprising since in this region the function varies from positive values corresponding to peripheral submergence of the land (rise of sea level) to large negative values where the land is uplifted. We have not contoured the negative S(0, 4,t ) regions which cover the ice centers, since the amplitudes of emergence are so large that they would completely swamp the sea level variations in the global ocean in a constant-interval contour representation of the field. Inspection of Fig. 22 also shows that as uplift takes place in the once ice covered region a peripheral bulge of water initially propagates toward the ice centers from the far field. This is due to the corresponding migration of the peripheral bulge of the solid part of the planet which is
DYNAMICS 01- THE ICE AGE EARTH
67
visible in the disk load response patterns shown in Fig. 17. Also of interest in Fig. 22 is the fact that the increase of water thickness in the far field of the ice sheets is not a strong function of spatial position. This demonstrates u posteriori the internal self-consistency of our use of far-field RSL data to estimate the integrated mass loss history of the main ice sheets. Based upon such global RSL solutions, we may divide the surface of the world's oceans into a number of different zones, in each of which the RSL signature has a more or less characteristic form. The zone boundaries obtained from the RSL data in Fig. 22 for the ICE-2 deglaciation history and the LI viscoelastic model ofthe interior are shown in Fig. 23. Zone I consists of the deglaciated regions in which the RSL record consists of continuous emergence following disappearance of the ice. Zone I1 is the region peripheral to the ice sheet in which all relict beaches are drowned and in which the record is one of monotonic submergence. The RSL signature at sites on the boundary between zones I and I1 turns out to be quite diagnostic of the viscosity of the deep mantle, as we will show. In this region observed RSL histories are such that, whereas no raised beaches presently exist,'the initial sense of the vertical motion following melting was of emergence. This was later followed by submergence, however, so that the RSL histories in this region are not monotonic. The sea level histories in zone 111 are characterized by delayed emergence following the end of the deglaciation phase.
FIG.23. Relative sea level zone boundaries for earth model LI and deglaciation history ICE2. The characteristic signatures of RSI. within each of these zones are discussed in the text.
68
RICHARD PELTIER
Figure 23 shows a much broader zone I11 than that found by Clark et al. (1978), presumably due to the presence of the lithosphere in our model which was absent in the previous calculations. Zone IV is a region of presentday emergence, which in Fig. 23 is seen to consist only of a fairly narrow region off the west coast of Africa. It is interesting to note that the observed RSL data along the west coast of Africa (Faurk, 1978) do show just this sort of transition from a region of submergence to a region of emergence and then to a region of submergence again in the vicinity of Dakar, Senegal as shown in Fig. 23. The signature of the RSL data in zone V is the appearance of a raised beach immediately after melting ceases. For viscoelastic model L1, Fig. 23 shows a smooth transition between zones 111 and V in the Pacific, with no zone IV separating them. Zone VI consists of all continental shorelines which are sufficiently remote from the main deglaciation centers. The RSL histories at sites in such regions are only very weakly dependent upon the mantle viscosity profile and are characterized by emergence forced by the offshore water load after melting stops. Such continental shorelines are on the bulge peripheral to the water load. It must be kept clearly in mind that although the causes of the characteristic RSL signatures in zones 111, V, and VI are somewhat different, the signatures themselves are really quite similar and it is often difficult i n practice to assign a given site unambiguously to one of these regions. The utility ofthe classification of sites employed by Clark et al. (1978) is therefore not always evident. Nevertheless, the classification does reflect the RSL behavior as a function of distance from the load which is expected on the basis of the disk load calculations discussed in Section 3.7 and so should reflect the interior viscoelastic structure. In the absence of epeirogenic and tectonic processes, the far-field RSL data would probably be quite diagnostic of deep mantle viscosity, but unfortunately these sources of geological noise are important in many localities and the far-field data are not as useful as one might wish. 4 4 RSL Constraints on the Mantle I7r5co5it4’Profile When Initial Iwctuiic Eyzirlihrium I\ 4s ciimed We may obtain global solutions to the sea level equation such as that shown in Fig. 22 for arbitrary profiles of mantle viscosity and from them obtain predictions from the model of the RSL history to be expected at any point on the earth’s surface. We seek to vary the profile in such a way as to obtain a “best fit” to the entire set of RSL data. Although this search procedure may be automated using the methods of Backus and Gilbert (1967, 1968, 1970), we have elected to proceed more cautiously at first in order to convince ourselves that a simple radially stratified model exists
DYNAMICS OF‘ wL- ICE AGE EARTH
69
which is capable of reconciling the majority of the observational data. Just as we have a priori knowledge of the deglaciation history, so we have a priori knowledge of the mantle viscosity profile due to the efforts which have been expended by previous investigators. Among these investigators there evolved a reasonably wcll developed consensus that the viscosity of the upper mantle is near 10” Pa sec and that the thickness of continental lithosphere is near I20 km. The approach which we will adopt for purposes of the discussion in this article is to keep these properties of the viscoelastic model fixed and to focus our attention upon the question of the extent to which the rebound data are able to constrain the viscosity of the mantle beneath 670 km depth; that is, beneath the solid-solid structural phase transition which occurs there. Since we intend to concentrate the discussion upon the question of the viscosity of the lower mantle, we shall restrict our attention to data from sites near the Laurentide ice sheet. The locations of the sites in zone I from which “C-controlled RSL histories are available are shown in Fig. 24, whereas sites from zone I1 are shown in Fig. 25. In Figs. 26, 27. and 28 we show comparisons of observed RSL data (hatched regions) with predictions of the gravitationally self-consistent viscoelastic model for three mantle viscosity profiles at several of these sites. The first two viscosity profiles are identical with models 1 and 2 shown on Fig. 1 1 , whereas the third differs from model 2 only in that the lower mantle viscosity is 5 X loz2Pa sec rather than 10” Pa sec. All three models therefore differ from one another only in the viscosity beneath 670 km depth. these viscosities being 10” Pa sec ( l o z 2 P). l o 2 * Pa sec ( P) and 5 X 10” Pa sec ( 5 X P) for models 1. 2, and 3 , respectively. Comparison of the model predictions with the observations at sites under the ice shown in Fig. 26 demonstrates that the best fit to the majority of the data is obtained with the uniform viscosity model 1. since only this model seems able to fit both the observed amp/ifzidc of cmergence and the relatively low presentday emergence rat”. The data at most sites show the relatively short relaxation time of about 2000 yr, which was stated previously to be characteristic of most Laurentide locations. The model with lower mantle viscosity of 10” Pa sec in general predicts too great a present-day rate of emergence and too long a characteristic relaxation time, whereas the model with lower mantle viscosity of 5 X 10” Pa sec fails to predict the total observed emergence and, where it does predict a reasonable present-day emergence rate, fails even more spectacularly to predict the observed relaxation time. The only exceptions to this general pattern are at a few sites in the southwest of Hudson Bay (e.g., Churchill), wherc some preference for the model with lower mantle viscosity of 10” Pa sec is indicated. The comparisons shown i n t-ig. 27 include sites which are close to the ice margin (N.W. Newfoundland. Prince Edward Island, Boston). and these
4
0
FIG.24. Location map of Laurentide sites in zone I from which radiocarbon-controlled RSL histories are available.
FIG.
25. Location map of Laurentide sites in zone I1 from which radiocarbon-controlled RSL histories are available.
72
RICHARD PELTIER
k y r FROM PRESENT FIG.26. Relative sea level curves from six sites in zone I near the Laurentide ice sheet. The hatched regions denote the radiocarbon-controlled RSL observations. The dashed, long-dashed, and solid curves are theoretical RSL predictions for models in which the lower mantle viscosity P, respectively. The horizontal lines drawn adjacent to the rightis lo2' P. loz3P, and 5 X hand margin on each plate indicate the amount of uplift remaining for each of these viscosity models.
DYNAMICS OF T H E ICE AGE EARTH
73
FIG. 27. Relative sea level curves for six sites along the eastern seaboard of North America. For locations see Figs. 24 and 25. Models and data are represented as in Fig. 26.
strongly reinforce the inference drawn on the basis of RSL comparisons near the ice sheet center. Models with any substantial increase of viscosity in the lower mantle fail to match the observations at such locations, since peripheral bulge migration is strongly inhibited in such models, and this is required in order to explain the nonmonotonic RSL histories at sites nearest the boundary between zones I and I1 (Prince Edward Island, Bay of Fundy, Boston. etc.).
74
RICHARD PELTIER
kyr FROM PRES€NT FIG.28. Relative sea level curves for six sites at increasing distance from the Laurentide ice sheet. Models and data are represented as in Fig. 26.
Figure 28 and the remaining sites on Fig. 27 show comparisons at several of the locations along the East Coast of the United States which are shown on the map in Fig. 25 and at some sites which are considerably more distant from the center of deglaciation. In general, all viscosity models fail to reconcile the data at East Coast sites. since they generally predict excessive submergence (e.g.. Southport, Brigantine, Clinton), although the model with
DYNAMICS OF THE ICE AGE EARTH
75
increased lower mantle viscosity of 10” Pa sec is preferred. Both the uniform viscosity model and that with lower mantle viscosity of 5 X 10” Pa sec are inferior. As discussed in Wu and Peltier (1982b),these misfits could be due to the operation of some epeirogenic process in this region such as that associated with offshore sedimentary loading (Newman et al., 1980), or they could be due to error in the viscoelastic model. As shown in Wu and Peltier ( 1982b), it is not possible to remedy these misfits by inserting a low-viscosity zone into the model. One additional possibility, which turns out to be correct, is that the misfit could be offset by an increase in lithospheric thickness. Farther still from the Laurentide ice center in the Gulf of Mexico (Fig. 28), the uniform viscosity model is again preferred, since both models with increased lower mantle viscosity predict too little submergence. The same tendency is observed at Bermuda. The final plates in Fig. 28 illustrate comparisons at far-field sites in zone VI. These are for Recife, Brazil and South Island, New Zealand and illustrate the point made previously that the ICE-1(2) model predicts raised beaches to appear at such sites at 8 kyr B.P. rather than 6 kyr B.P., at which time they are actually observed. As suggested in Wu and Peltier ( 1982b), however, this misfit may be corrected simply by modifying the deglaciation history slightly to include the tail on the melt curve which is observed on the“eustatic”sea level history of Shepard ( 1963) shown previously in Fig. 20. Inspection of this figure shows that the ICE- l(2) histories are characterized by an abrupt cessation of melting at 5 kyr B.P. If the final disappearance of the Northern Hemisphere sheets were delayed somewhat from that assumed in ICE- l(2) or if some other source of meltwater were still active in the time subsequent to 5 kyr B.P. (such as West Antarctica, for example), thcn this misfit could be simply corrected. With the few exceptions mentioned above, the RSL data from the Laurentide region quite strongly prefer the uniform viscosity model over models which have high lower mantle viscosity. As we will show in the following two sections of this article, this is a conclusion which is further reinforced when other data associated with isostatic adjustment are considered. The first such additional kind of information we will discuss is that contained i n the variation of the surface gravitational acceleration over deglaciation centers.
5 . DEGLACIATION-~NL)II(’ED PERTURBATIONS OF THE GRAVII‘ATIONAL FIELD As mentioned in the introduction to this article, the apparent inability of previously constructed linear viscoelastic models of isostatic adjustment to simultaneously explain both RSL, and free-air gravity data has led to
76
RICHARD PELTIEK
suggestions to the effect that the basic rheological constitutive relation on which such models are predicated could be completely in error. In this section we will show that when the new theory of glacial isostatic adjustment developed in Sections 3 and 4 is employed to predict the free-air gravity anomaly which should be observed over Hudson Bay, then we obtain agreement with the observed anomaly for the same mantle viscosity profile which is required to fit the RSL data. We are therefore able to fully resolve an important question which has remained unanswered in the literature, and at the same time to seriously undercut previous objections to the use of linear viscoelastic models and to the inference obtained from them that the viscosity of the mantle is rather uniform. The ability of the new theoretical model to solve the problem is due to the fact that it includes the complete spectrum of normal modes of viscous gravitational relaxation which is supported by the radial elastic structure of realistic earth models. As discussed in Section 3.6, thc model with uniform mantle viscosity which so well explains the observed record of sea level variations during the past 20,000 yr, a record which is dominated by rather short relaxation times, also supports normal modes with long relaxation times due to the internal density jumps which are associated with the presence of solid-solid phase transitions at the base of the upper mantle. Although the sea level record is dominated by the shortest relaxation times in the complete spectrum, the free-air gravity anomaly depends critically upon the extent to which the modes with long relaxation time are excited. Because the relaxation spectrum of realistic viscoelastic earth models contains modes with relaxation times on the order of 1 O5 yr, and since this time scale is of the same order as the time between successive interglacials (Broecker and Van Donk, 1970), we must also consider the validity of the assumption of initial isostatic equilibrium on the basis of which the previously described calculations of RSL history were performed. Before addressing these questions we will first describe the freeair gravity observations over the main centers of postglacial rebound. 5 . 1 . Satellite and Surface Observations of the Gravity Field over
Deglaciation Centers In Fig. 29 we have reproduced a global map of geoidal heights based upon the GEM 10 data set and the spherical harmonic coefficients of degrees 2-22. This map represents mean sea level over the world’s oceans, while over the continents the geoidal heights are those which would obtain if the continents were cut by a web of thin canals. The anomalies shown on this map are those referred to a reference sphere and are based almost entirely upon satellite observations, although some terrestrial data have also been
DYNAMICS 0 1 : T H E ICE AGE EARTH
77
FIG. 29. Global map of geoidal heights based upon the GEM 10 data set and spherical harmonic coefficients of degrees 2-22
included. Inspection of this map reveals the presence of a global pattern of anomalies with scales on the order of thousands of kilometers and amplitudes varying from -105 m over the Indian Ocean to +70 m over the southwest Pacific. Of particular interest to us for present purposes is the anomaly of -44 m over Hudson Bay. Although this anomaly is very well correlated with the previously shown map of Laurentide ice topography (Fig. 2 I ) , and is therefore most probably associated with deglacial forcing, when we seek a similar feature over Fennoscandia (top left-hand corner of the map) we find no negative anomaly present at all. The difficulty clearly has to do with the fact that the magnitude of the anomalies associated with deglaciation are on the same order as those associated with mantle convective processes. It might well bc, as pointed out in Peltier (1980a) and previous papers, that the isostatic adjustment model can be used as a filter to remove from the global map of geoidal heights the anomalies which are known to have a deglacial cause and thereby to reveal more clearly the convection-related patterns. Before we can carry out this global filtering, however, we have to convince ourselves of the ability of the isostatic adjustment model to correctly predict the amplitude and form ofthe anomalies over the glaciation centers. In order to do this we will have to consider the more accurate representation o f these anomalies which is obtained from surface data. In Fig. 30 we show in parts a and b the free-air gravity anomaly maps constructed by Walcott ( 1 970) and Balling ( I 980) for the Laurentide and Fennoscandia regions, respectively. The map for Laurentide is based on the surface data of Innes et af. (1968). which were averaged on 1 O X 2" grid elements to remove the influence of local topographic variations and near-
78
RICIHARD PELTIER
FIG.30. Free-air gravity anomaly maps for the Laurentide (a) and Fennoscandia (b) regions. The contours are in rnGal. Data sources are discussed in the text.
surface geological structure. Each of the grid elements typically contained about 100 separate observations over land and 40 observations over water (Hudson Bay). Inspection of Fig. 30a shows that the anomalous structure which remains consists of an elongated elliptical trough trending roughly NW with a peak amplitude near -35 mGal. Walcott ( 1 972) has previously cautioned that since the Hudson Bay is itself a Phanerozoic basin, the anomaly actually associated with current glacial disequilibrium could be as much as 10 mGal more negative than shown in Fig. 30a. This must be considered somewhat speculative, however, and for the purposes of the following discussion we will generally assume that the anomaly to be fit by the isostatic adjustment model has its -30 mGal contour surrounding Hudson Bay. Interpretation of the gravity field over Fennoscandia is considerably more complicated than it is for the Laurentide region, because of the combined effects of topography and local near-surface geology on the same spatial scale as that of the ice sheet itself and because the feature related to deglaciation seems also to be biased by the local long-wavelength background. The analysis of this field by Balling (1980) is the most careful which is presently available and is based upon the raw data of Honkasolo (1963). By direct regression analysis, Balling removed from the raw anomalies the spatial part which was linearly related t o the topography to obtain the residual map shown in Fig. 30b, which reveals a peak anomaly somewhat in excess of -10 mGal which is very well correlated with the topography of Fennoscandian ice illustrated previously in Fig. 2 1. The analysis provided b y Balling suggests that this anomaly has been biased by +5 to + 10 mGal due to the large-scale variations, and he concludes that the anomaly rep-
DYNAMICS Of THE ICE AGE EARTH
79
resentative of the current degree of isostatic disequilibrium is between - 15 and -20 mGal. Walcott ( 1972) accepts an estimate of - 17 mGal for the magnitude of the anomaly related to dcglaciation, and we will take this as the observed peak anomaly which our isostatic adjustment model must be expected to reproduce. In the following subsection we will begin to test this model in terms of disk load approximations to the actual deglaciation chronologies.
Just as disk load approximations to the actual deglaciation histories were applied in Section 3.6 to compute approximations to the RSL data in the form of histories of radial displacement, so here we will make use of the same methods to estimate the gravity anomalies to be expected from the more realistic calculations. Free-air gravity signals may be computed in exactly the same way as we previously computed radial displacement, the sole difference being that we employ the Green’s function for the free-air anomaly given in Eq. (3.39). Convolution of this Green’s function over a circular disk with parabolic height profile which approximates the Laurentide ice sheet produces the responsc shown in Fig. 31 when the impulse
FIG. 31. Free-air gravity anomaly as a function of time forced by a parabolic disk load approximation to the Laurentide ice sheet which is applied instantaneously at t = 0. The times are (-) I = 4 kyr; (- - -) 1 = 8 kyr; (- . -) / = 12 kyr; (- - -) t = sc kyr.
80
RICHARD PELTIER
TABLEIVa. Ag FOR LAURENTIDE AND VISCOSITY MODELL1
Time (kYd
Ag : Load added at / = 0 (mCal)
0 -84 -98 - 100 -127 -127
Ag : Equilibrated load removed at I =
0 (mGal) ~
127 -43 -29 -27 -21 0
A g : Nonequilibnum load removed at t = 0 (mGal)
-99 -29 -16 - 14 -14 0
response function used is that for viscosity model L1. This figure shows the time-dependent anomaly which would exist if the planet were initially in equilibrium and the load applied at t = 0; clearly the gravity anomaly reaches a maximum in the limit of infinite time. In order to obtain the anomaly which would be observed if the load were removed instantaneously, afrcr having been resident on the surface for infinite time, we need only subtract from the results shown on Fig. 3 1 the infinite-time anomaly itself. The importance of having accurate calculations of the isostatic asymptotes of the Love number spectra (as discussed in Section 3.4) for the free-air gravity calculations should therefore be clear. Some numerical results for the time dependence of the peak anomaly produced by the Laurentide disk load are tabulated in Tables IVa and IVb for models L1 and L2 respectively. Since the phase of most rapid deglaciation occurs at about 12 kyr B.P. according to Shepard’s eustatic curve (Fig. 20), inspection of this table shows that model L1 predicts the present-day anomaly reasonably well (-27 mGal) when initial isostatic equilibrium of the loaded surface is assumed. Model L2. however, the corresponding results for which are shown in Table IVb, predicts a present-day free-air anomaly of -64 mGal, which is so much larger than the observed anomaly of about -30 mGal (Fig. 30a) that the model must be completely rejected. rlDF TAHIE IVb AK FOR LAURLN
Time (kyr) 0 4 8 12 16 7
Ag Load added at / = 0 (mGal)
0 -33 -5 1 -63 -7 I I27
-
A N D VISCOSITY
MODEI L2
A g Equilibrated load removed at I = 0 (mGal)
A g Nonequilibnum load removed at t = 0 (mGal)
-127 -94 -76 -64 -56 0
-86 -65 -53 - 43 36 0
DYNAMICS OF T H E ICE AGE EARTH
81
FIG. 32. (a) Oxygen isotope stratigraphy from a typical deep-sea sedimentary core showing the ratio “O/”O as a function of time and the implied variation of continental ice volume. (b) 4 sawtooth approximation to this time series which accentuates the 1OS-yrperiodicity in the record of ice volume fluctuations and the large discrepancy between the time scales of accumulation and disintegration.
These initial results serve to tlenionstrate that the free-air gravity anomaly expected for the uniform mantle viscosity model is very much larger than that predicted by previous viscoelastic theories of glacial isostasy. The reason for this was previously explained as being due to the fact that realistic viscoelastic models of the planet support modes of relaxation with much longer time scales than previously recognized. On the basis of the above comparisons of the predictions of models L1 and L2 it also seems quite likely that the observed free-air gravity anomaly over the Laurentide region will prove to be even more diagnostic of the viscosity of the deep mantle than the RSL data themselves. In order to make full use of this datum, however. we are forced to address the question of the validity of the assumption of initial isostatic equilibrium upon which the above-described disk load calculations were based. In order to assess the validity of the assumption, however, we require direct information concerning the actual time scale of the glaciation cycle. It is fortunate for our purposes that such information has been recently forthcoming from studies of deep-sea sedimentary cores taken in the major ocean basins during the course of the Deep Sea Drilling Project (DSDP). The data of interest here are measurements ofthe ratio of the concentrations of the stable isotopes of oxygen (‘*O/“O)as a function of depth in such cores. Although it was originally believed (Emiliani, 1955) that the vari-
82
RICHARD PELTIER
ability of this ratio was a direct reflection of Pleistocene temperatures, it was subsequently established (Imbrie and Kipp, 197 1 ) that the isotopic ratio for the most part reflected the variation of Northern Hemisphere ice volume. Broecker and Van Donk (1970) were among the first to establish on the basis of these data that the Northern Hemisphere glacial-deglacial cycle is very nearly periodic with a time scale of approximately 10’ yr. Kukla et a/. ( 198 1 ) have most recently reviewed the characteristic signature of ice sheet growth and disintegration which defines each cycle. This signature, which is also discussed in somewhat greater detail in Hays el al. (1 976). is characterized by a very slow buildup of the major ice sheets over about 10’ yr followed by an extremely rapid disintegration. This suggests that a reasonably good approximation to the long time scale ice volume fluctuations would be the sawtooth waveform shown in Fig. 32b which is compared to the ‘H0/’60 record in Fig. 32a. Given this information on the previous history of loading and unloading of the Laurentide and Fennoscandia regions (there does not appear to have been any significant geographical migration of successive ice center locations) we may proceed to address the question of the importance of initial disequilibrium upon the inference of deep mantle viscosity from isostatic adjustment data. As shown in Section 3 , all of the viscoelastic impulse response Green’s functions may be written in the form I
G(H, I )
=
GE(H)h ( t )
+c
’If
c rle-‘~‘P,(cos0) /=o
(5.1)
/=I
where I is spherical harmonic degree, 1/s: is the relaxation time for t h e j th mode of the Ith harmonic, and rl is the initial “viscous” amplitude. If we denote by L(B, t ) the load at location 0 and time t then the response at time t is just ~ ( 0 t, ) =
1:
G(t
~
1‘)
* L ( / ’ )dt’
(5.2)
7 ,
where the * denotes spatial convolution over the loaded surface. Suppose that at 1 = 0 the load at location H is ho(H), whereas for t < 0 the “prehistory” of loading is L,(B, t ) . and for 1 > 0 until the present time the loading history may be represented by ho h(0, t ) , i.e.
+
L(0, 1)
with L&H, 0) Eq. (5.2) as
=
= =
&Lo, ho
f),
+ h(H, Z),
t
(5.3)
hO(0)and h(H, 0) = 0. For t > 0 we may therefore expand
+
G(t - 2 ‘ )
* h(”) dr’
(5.4)
DYNAMICS OF 1 H E IC'F AGE EARTH
83
Now the second term on the right-hand side of Eq. (5.4) may be identified with the spatial convolution of' the Heaviside form of the Green's function with the initial surface load h,) * G"'(t). The third term, on the other hand, is the response to the deglaciation phase of the load cycle. This term would be equal to -ho * G"(t) if all the load were removed instantaneously. The first term on the right-hand side of Eq. (5.4) is the response due to the history of loading prior to t = 0. llsing Eq. (5.2) it may be rewritten as (Wu and Peltier, 1982a)
where
Equation (5.6) may be considerably simplified if each location under the load has the same prehistory, since we may then write L,(H, 1 )
=
hdW?,(t)
in which case Eq. (5.5) is replaced by
where
f(=
r'
sie'i'h,(t') dt'
For such a simple model, the functions f; contain all of the information concerning the prehistory of loading. If the ice sheet had remained on the surface for an infinite length of time prior to melting so that the system was in isostatic equilibrium at I = 0. then h,(t) = H(t - t z ) , where H i s the Heaviside step function. With / , = -r; it follows from Eq. (5.8) that f : = 1 , and the total response may be written from Eq. (5.4) as rf * G"(0 + G(t - t ') * h(t ') dt ' (5.9) ti(/) = ho * C C 7c ';'PI + /
1
.$I
Using
Eq. (5.9) may be rewritten as
t ')
* h(t ') dt '
(5.10)
G(t - t ' )
* h(t')dt'
( S 1~1 )
or
ti(2)
=
h()* GH(t
Y )
+
84
RICHARD PELTIER
which shows that the isostatic response to the removal of an equilibrium load at t = 0 may be expressed as the sum of the infinite-time (initial) response and that forced subsequently. If the compensated load were removed instantaneously at t = 0 then the response for t > 0 from Eq. (5.1 1 ) would be ~ ( t =) ho * GH(t= - ho * GH(t) (5.12) XI)
This is the expression which was used to compute the gravity anomalies shown in column 3 of Tables IVa and 1% under the assumption that isostatic equilibrium prevails initially. In order to assess the effect of initial disequilibrium, we may use the general expression (5.4), which gives
u(t) = hO *
c c f I r‘f C ; ‘ P / + ho * GH(t)+ I
/
G(t - t‘) * h(t‘) dt’
(5.13)
sJ
which may be rewritten in the form u(t) = 6 ( l ) - ho * E(t)
(5.14)
where E is the error or correction Green’s function (5.15)
The total response in Eq. (5.14) may therefore be expressed as the sum of the response which would be observed if the load were initially in equilibrium [6(t)]and a correction due to the load prehistory which is expressed as the spatial convolution of the initial load over an “error” Green’s function. In order to determine E(0, t ) we need the f j which are defined in Eq. (5.8) in terms ofthe prehistory of loading h,,(f). For the sawtooth prehistory shown in Fig. 32b this function is given by t li7 -k7 I tI -(k - 1). (5.16) h ( t )=
+
7
where 7 = lo5 yr in the characteristic period of a single ice sheet advance. Substitution of Eq. (5.16) into Eq. (5.8) gives
(5.17)
where N is the number of load cycles in the prehistory. Since the present ice age has continued for 2-3 million yr, a time short compared to the continental drift time scale of lo8 yr on which significant changes of polar
DYNAMICS OF.n IE ICE AGE EARTH
85
FIG. 33. (a) Present-day peak free-air gravity anomaly at the center of the disk model Laurentide load as a function of the viscosity of the mantle beneath 670 km depth. The viscosity of the upper mantle is held fixed at 10" P. The dash-dotted curve is the prediction assuming initial isostatic equilibrium; the dashed curve includes the effect of initial isostatic disequilibrium but the computation has been done for fixed ice sheet radius. The solid curve is the predicted anomaly when the time dependence of ice sheet radius is accounted for under the assumption that the ice cap maintains an equilibrium plastic profile at all times. The hatched region shows the observed peak anomalg over Hudson Bay. (b) Precisely the same analysis for the Fennoscandia region.
continentality can be expected to have occurred, yet long compared to the duration of a given glacial epoch (lo5 yr), we may safely assume A: = 20-30 in evaluating Eq. (5.7). We note from the form of Eq. (5.17) that if S:T is large, which is to say that the characteristic relaxation time is short compared to the time scale of ice sheet advance, then f j N 1 and the mode ( j , 1 ) is very nearly in isostatic equilibrium at t = 0. If s$ N 1 then f; = 0.58, provided N > 2. and the response i n this mode would be just that for a reduced load O . S 8 l 1 , ~ ( 0 which ) was initially in equilibrium. If, however, NS;T-4 1 then f; 'Y h'.viT 12 and by t = 0 the response is but a small fraction of the equilibrium valuc. In column 4 of Tables IVa and I V b we show the peak free-air gravity anomaly at the center of the model Laurentide disk as a function of time for models LI and L2 including sawtooth prehistory with 7 = lo5 yr and R; = 30 ( N = 20 gives almost identical results). Comparing columns 3 and 4 shows that the correction for prehistory at r = 12 kyr is about 13 mGal for L1 and 27 mGal for L2. The calculations therefore demonstrate that the effect of initial isostatic disequilibrium is extremely important insofar as free-air gravity anomaly calculations arc concerned. In Fig. 33a we present the results of a more detailed investigation of the effect of initial disequilibrium on the free-air gravity anomaly to be expected over Hudson Bay.
86
RICHARD PELTIER
This diagram shows the present-day peak free-air gravity anomaly at the center of the Laurentide model disk load as a function of the viscosity of the mantle beneath 670 km depth. The prediction is shown both including and excluding the effect of prehistory and is compared to the observed freeair anomaly of about -35 mGal (hatched region). Inspection of this figure shows that when initial isostatic equilibrium is assumed, the model with uniform mantle viscosity of 10’’ P fits the observed free-air anomaly very well, Under this assumption, as the viscosity of the lower mantle increases the predicted free-air anomaly increases monotonically to approach an asymptotic value near 100 mGal. If the assumption of initial equilibrium were valid, therefore, the viscosity of the lower mantle could not be significantly in excess of the upper mantle value. Otherwise one would predict a free-air anomaly much larger than is observed. Even with a lower mantle viscosity as high as lo2’ P as in model L2, however, characteristic relaxation times are no longer short compared to the time scale of 10’ yr which separates successive interglacials and the assumption of initial isostatic equilibrium is invalid. The second and third curves on Fig. 33a show the predicted present-day free-air gravity anomaly for the model Laurentide load including the influence of initial isostatic disequilibrium. In this case the predicted anomaly is not a monotonica!ly increasing function of the deep mantle viscosity. Rather we may fit the observation for either of two widely spaced values of lower mantle viscosity, one near 3 X 10” P and the other near 5 X lo2’ P. The latter exists as a possible solution because, in the theoretical model, one may trade off the degree of initial disequilibrium against the magnitude of the viscosity in the lower mantle. To the extent that RSL data are relatively unaffected by initial isostatic disequilibrium, however, this solution may be completely ruled out on the basis of the preceding discussion of RSL data in Section 4, and the only acceptable solution is the lower value. It is an extremely important property of the isostatic adjustment data set that RSL data and free-air gravity data complement each other in this way. The explanation of the complementary nature of these two types of data is to be found in the fact that sea level histories are effectively measurements of radial displacement relative to the zero datum established by local present-day mean sea level. These data therefore provide no information concerning the amount of uplift (submergence) which has yet to take place before isostatic equilibrium is restored. The free-air gravity anomaly, on the other hand, is an absolute measurement of the degree of current disequilibrium, and it is for this reason that the two types of observation are influenced to a completely different extent by initial isostatic disequilibrium. This may be shown algebraically by using Eq. (5.14) to compute (5.18)
DYNAMICS OF THE ICE AGE EARTH
87
FIG.34. Radial displacement response at the center of the Laurentide model disk load for viscoelastic models L1 and L2. The solid curves are for calculations done under the assumption of initial isostatic equilibrium, whereas the dashed curves include the degree of initial disequilibrium implied by the oxygen isotope data. Also shown on this figure are data from three sites around Hudson Bay: (A)data from Castle Island; ( 0 )data from Churchill; (m) data from the Ottawa Islands.
where t, is the present time, so that Eq. (5.18) gives the response relative to a zero datum at present and therefore correctly mimics RSL information. The correction of the relative displacement response for the effect of initial disequilibrium is given by the third term on the right-hand side of Eq. (5.18) in the form of a convolution of the initial load over a difference of error Green’s functions. From Eq. ( 5 . IS) the expression in square brackets in Fq. (5.18) is
which is obviously zero for f = I,,. Comparison of Eq. (5.19) with Eq. (5.15) shows that every term in Eq. (5.19) is smaller than the corresponding term in Eq. (5.15) by the factor { 1 - exp[-.r;(t, - t ) ] } .The magnitude of the effect is shown in Fig. 34, where we plot the relative radial displacement response at the center of the Laurentide disk for models L1 and L2 both including and excluding the effect of initial isostatic disequilibrium (solid and dashed lines respectively). Inspection of this figure clearly shows that the effects of initial disequilibrium upon the predictions of either viscosity model are much less than the diff‘erences in response due the viscosity models themselves. This is an extremely imporant point since it assures
88
RICHARD PELTIER
us that viscosity models rejected on the basis of RSL calculations done assuming initial isostatic equilibrium cannot be brought back into contention by invoking this effect. Also shown on Fig. 34 are the RSL data from the Ottawa Islands in Hudson Bay and from two other sites near the center of rebound. These data all lie between the predictions of models LI and L2, implying that the lower mantle viscosity is between that in these two models. This is the same conclusion reached on the basis of the previously discussed free-air gravity data which prefer a value of the lower mantle viscosity near 3 x 102’Pa sec. Figure 33b shows a comparison of disk load predictions of the peak freeair gravity anomaly, with and without the effect of initial disequilibrium, for the model Fennoscandian load. Inspection of this figure confirms the conclusions reached on the basis of the Laurentide analysis. When the effect of initial isostatic disequilibriiim is included, the model with lower mantle viscosity of about 2 X loz2 P predicts the observed present-day free-air anomaly of about - 17 mCal quite accurately. The fact that both the Laurentide and Fennoscandia data require the same contrast of viscosity across the phase transition at 670 km depth in the mantle strongly reinforces the necessity of including this feature in the viscosity profile, establishes the feature as a global property of the real earth, and reinforces our “faith” in the assumptions upon which our realistic viscoelastic models of the earth are based. The model which we have employed here to investigate the influence of initial isostatic disequilibrium upon free-air gravity and RSL predictions assumes, through the expression L&O,I ) = h,(O)h,(t), that the ice sheet radius remains constant while its volume expands and contracts. This is of course not strictly true, since ice sheets expand and contract in their horizontal dimensions as volume increases and decreases. As discussed in Wu and Peltier (1982b), however, when one uses the flow law of ice to fix the variations of ice sheet scale given the characteristic volume fluctuations shown in Fig. 32b, one finds that the assumption of fixed ice sheet radius actually tends to exaggerate somewhat the importance of initial isostatic disequilibrium. This reference should be consulted for a more detailed analysis of this effect. In Fig. 33a,b calculations of the peak free-air anomalies at the center of the Laurentide and Fennoscandia disks are also shown which include the influence of expansion and contraction of the disk radius and which demonstrate this effect.
5.3. Free-Air Anornulie5 from tlw Selflconsistent Model Figure 35a,b shows free-air gravity maps for the Laurentide region predicted from the gravitationally self-consistent theory for viscoelastic models L1 and L2, respectively. All of these calculations have been done under the
DYNAMICS 0 1 I H E IC’E AGE EARTH
89
FIG. 35. Predicted present-day free-air gravity maps for the Laurentide region. Parts (a) and (b) are calculations based upon viscoelastic models LI and L2, respectively. Both calculations assume that isostatic equilibrium prcvails initially.
assumption that isostatic equilibrium prevails initially, and comparison of the peak anomalies on these maps with the corresponding disk load predictions in Fig. 33a shows that the disk load approximations are extremely accurate. Therefore, none of the arguments based upon the disk load analyses will require substantial modification in consequence of application of this more accurate model to calculation of free-air gravity anomalies. As discussed in Wu and Peltier ( I982b), this general conclusion also applies when the effect of initial isostatic disequilibrium is included. The fact that the free-air gravity anomalies predicted by the new theory did fit the observations over the Laurentide region was first demonstrated in Peltier (1981a). 5 4 Gravity Field Construints on [ h e Muntle Viscosity Profile
The few results discussed in the preceding subsections suffice to make the important point that the new theory of isostatic adjustment is able to simultaneously explain both observed RSL histories and free-air gravity anomalies associated with the main centers of Pleistocene deglaciation. No previous analysis of postglacial rebound has achieved this rather important goal. The fact that our realistic linear viscoelastic models are able to reconcile both data sets simultaneously means that there is no evidence in the adjustment data themselves for non-Newtonian or other exotic material behavior. The crucial ingredient which was missing in all previous formulations of the theory is the set of modcs with long relaxation times which is
90
RICHARD PELTIER
supported by the radial inhomogeneity of the elastic structure of realistic earth models. Of particular importance in this respect are the density jumps in the transition region due to the olivine spinel and the spinel pemagnesiowustite phase transitions. With the elastic structure rovskite fixed to that of model 1066B of Gilbert and Dziewonski (1975), which was itself based upon the totality of elastic gravitational free oscillations data, our isostatic adjustment calculations show that the free-air gravity anomaly is a particularly sensitive measure of the viscosity of the lower mantle. When the upper mantle viscosity is fixed to lo2' Pa sec the free-air data require a lower mantle viscosity very near 3 X lo2'Pa sec, with the Laurentide data apparently providing the most sensitive estimate of this number. The RSL calculations discussed in Section 4 show that this viscosity profile is completely acceptable to the RSL data. In the next section we will go on to consider a third set of isostatic adjustment data which are able to provide an extremely useful further corroboration of the validity of this inferred viscosity profile.
+
-
-
6. DEGLACIATION-INDUCED PERTURBATIONS OF PLANETARY ROTATION Given the mass contained in the Laurentian and Fennoscandian ice complexes it should not be too surprising that their melting may have induced very substantial variations in the moment of inertia tensor of the planet. Since the net angular momentum of the solid earth ice water system must be conserved during the internal mass redistributions associated with glaciation and deglaciation, it is clear that the changes of the inertia tensor produced by mass redistribution must be accompanied by changes in the angular velocity vector of the system as a whole. As we will show in the following subsections these changes in the angular velocity vector are astronomical observables which may be invoked to constrain the mantle viscosity profile since the history of surface loading is known.
+
+
6.1. The Historical Records of'Polur Motion and 1.o.d. Variution Since about A.D. 1900 the International Latitude Service (ILS) and more recently the International Polar Motion Service (IPMS) have maintained a set of photo zenith tube (pzt) stations which have provided a more or less continuous record of the monthly mean motion of the rotation pole relative to the conventional international origin (CIO). These polar motion data are shown in Fig. 36, which is based upon the reduction of ILS data by Vincente and Yumi (1969, 1970) as described in Dickman ( 1 977). The upper and lower time series, respectively, show the polar motion along the y and x
DYNAMICS OF THE ICE AGE EARTH
91
FIG.36. Polar motion time series for the motion of the rotation pole in the x and y coordinate directions relative to the CIO. The gcographic orientation of the coordinate system is shown on the inset polar projection which also shows the location of the Wurn-Wisconsin ice sheets (stippled) at the last glacial maximum. The arrow drawn from the CIO shows the direction of the drift of the rotation pole implied by the secular trend in the ILS pole path which is evident in both the x and y coordinate directions. The rate of drift is about 1 "/lo6 yr.
axes of the CIO coordinates, and the geographic location of these axes is shown on the polar projection at the center of the figure. The x axis passes along the Greenwich meridian, and the j' axis along the meridian at long 90" W. Inspection of the polar motion time series shown in this figure shows it to be dominated by a sequence of beats separated by a period of 7 yr. This is precisely the temporal behavior which is expected due to the superposition of the 14-month free Eulerian nutation (Chandler wobble) and the 12-month annual wobble. These oscillatory motions are superimposed,
92
RICHARD PELTIER
however, upon a secular drift of the pole with respect to the coordinate system fixed relative to the surface geography. The direction of the mean motion of the pole is shown by the arrow drawn from the center of the C10 system on the polar projection in the center of Fig. 36. That it is directed toward the centroid of the ancient Laurentide ice sheet should make us suspicious that the drift is deglaciation induced. As we will show, this suspicion turns out to be warranted. Given the rather large difference between the mass of the Laurentian and Fennoscandian ice sheets it should hardly be surprising that the Fennoscandian load, which is of smaller mass and centered at higher latitude, does not significantly influence the response. The observed rate at which the rotation pole is drifting toward Hudson Bay is very near l0/1O6 yr, and this is the observation which we will attempt to fit with our isostatic adjustment model. Of equal interest to the above-described mean motion of the pole for our present purposes are historical observations of variations of the length of day (1.o.d.). It is well known that the rotation of the earth is generally decreasing with time at a rate of ( 1 100 f 100)"/century2 (e.g., Lambeck, 1980), due mostly to the torque exerted on the earth by the moon through the agency of lunar ocean tidal dissipation. In order to ensure conservation of angular momentum of the earth-moon system this decrease in the rate of the earth's rotation is accompanied by an increase of the earth-moon distance. Although it is now understood that the current rate of lunar tidal dissipation is anomalously high and unrepresentative of the past because the ocean basins are currently almost resonant with the tidal forcing (Hansen, 1982), the 1.o.d. variation forced by lunar tidal friction certainly dominates the observed variation of this quantity now. Besides the 1.o.d. change forced by tidal torque, it has been possible to extract from the astronomical record a component of the net 1.o.d. change which is not attributable to this cause. This observation is normally referred to as the nontidal acceleration of rotation and has been measured using various methods to obtain the results listed in Table V. The observation by Currot (1966) is consistent with that of Dicke (1966) and is based upon a n analysis of ancient solar eclipse data. Based upon an assumption of constant lunar tidal acceleration, one may predict from the orbital equations when each of the ancient solar eclipses should have occurred. Knowledge of when they actually occurred provides a measure of the deceleration of rotation which is not attributable to the operation of tidal torques. Muller and Stephenson ( 1975) reanalyzed the ancient eclipse data analyzed by Newton (1972), keeping only the observations corresponding to total eclipses or for which the deviation from totality was explicitly declared. The measurement in the table by Morrison (1973) was based upon observed lunar occultations over the time period
93
DYNAMICS OF T H E ICE A G E E A R T H
TABLE v. MEASUREMENTS OF T H E NONTIDAL COMPONENT
OF T H E
AKFl I RAIION 0 1 RoTA~ION
Source
Value (+2)
Currot ( 1966) Muller and Stephenson ( 19711 Momson (1973) Lambeck (1977)
*
(0.7 0.3) X (1.5 i 0.3) X (2.9 2 0.6) x (0.69 k 0.3) X
10 10 10 10
lo
'(I I" lo
yr-l yr-l yr-l yr~'
1663- 1972. It differs significantly from the others. Lambecks's ( 1977) number was obtained from the diKcrcnce between the value of the net acceleration given by Muller (1975) and the mean value for the tidal acceleration obtained from an ocean model and astronomical and satellite observations. In the following subsections we will show that both the secular drift of the ILS-IPMS pole path and the nontidal acceleration of the earth's rotation are effects due to Pleistocene dcglaciation. We will furthermore demonstrate that these observations may be employed to constrain the viscosity of the earth's mantle and thus to provide valuable confirmation of the validity of the profile of viscosity deduced on the basis of the previously discussed analyses of RSL and free-air gravity data.
If the earth is subject to no external torque, then the principle of angular momentum conservation takes the form of the following Euler equations (e.g., Goldstein, 1980): d + f,,A4W/ =0 (6.1) dt ~
( J l , 9 )
where the angular velocity w is referred to a coordinate system whose axes coincide with the initial direction of the principal axes of inertia of the deformable body with moment of inertia tensor JIJ.In Eq. (6.1) t,,k is the Levi-Civita alternating tensor. If we can determine the J,,(r) which are produced by the Pleistocene glacial cycle. then Eq. (6.1) could be solved for the unknown w , since they would then degenerate to a set of three simultaneous ordinary differential equations. I n fact, the inertia tensor J,, contains contributions from two sources which are of interest to us here due. respectively, to the effect of the dcformation produced by the basic rotation and that associated with the response of the planet to surface loading by the ice sheets. We will proceed to calculate these distinct contributions.
94
RICHARD PELTIER
6.2.1. Pcrltirbations of Inertia Dzir to Variable Rotation. In order to compute the rotational deformation we will employ the formalism of tidal Love numbers in combination with MacCullagh’s formula as described in Jeffreys ( 1970). If the earth is subjected to a disturbing potential of the form
6dr, s)
C %,/(r, S)P,(COS
=
8)
(6.2)
/=0
where s is the Laplace transform variable. r is the distance from the center of mass. and P,is the usual Legendre polynomial, this potential will elicit a response 41(r,.s) such that
61(r.s) = @ d r ,.s)kT(r, .Y)
(6.3)
where kT is the so-called tidal Love number, which differs from our previously defined load Love number k/(r,s) ir? that it is computed for zero normal stress boundary conditions. If the applied potential is the centrifugal potential associated with rotation then
+
+ = ;[wzrl
(6.4)
- (O,X,)2]
which can be split into two terms (e.g., Munk and MacDonald, 1960) as
+ = twZy2 + x where
x
=
1
2
1
+
7
+
:[o~(x: . ~ j 2.~;) ~
* * *
(6.5) ~
~w,w~.vIx~]
(6.6)
is a spherical harmonic of degree two and where the dots denote additional terms obtained by cyclic permutation of the indices. The external gravitational potential V produced by this contribution to the centrifugal potential is, from Eq. (6.3). T’ = (a/r)’x([)* k:(t) (6.7) where the * denotes convolution in time. Now the tidal Love number kT(r) may be obtained from the equivalent time-independent expression for an elastic earth by direct application of the principle of correspondence. For an incompressible, homogeneous earth the elastic tidal Love number (e.g., Munk and MacDonald, 1960) is
where i = 19p/2pga, with p the elastic shear modulus. p the density, a the earth radius, and g the surface gravitational acceleration. For the Maxwell earth the Laplace transform domain expression for the elastic shear modulus (see Section 2) is
DYNAMICS 01 1 H E ICE AGE EARTH
95
The Laplace inverse of Eq. (6.X) is then (6.10)
+
where y = ( p / u ) / ( 1 ,i)is the inverse relaxation time of the 1 = 2 harmonic component of the deformation and where u, as before, is the viscosity of the homogeneous earth model. Introducing the explicit expressions for 4 and k i into MacCullagh's formula we get where it has been assumed that I fi,, + C;, are the elements of the inertia tensor of the rotationally deformed sphere. Invohng the fact that a solid harmonic will produce deformations which leave the trace of the inertia tensor C,, invariant (e.g., Rochester and Smylie, 1974), we may equate like terms on each side of Eq. (6.1 I ), using Eq. (6.6) to obtain
It is a consequence of the incompressibility of the model that the term u 2 r' / 3 in Eq. (6.5) contributes nothing to the response. T o obtain the total inertia of the rotating sphere we have to add to Eq. (6.12) the inertia which the sphere would have in the absence of rotation. This is obtained by assuming that the effect of rotation is to change the moment about the polar axis by an amount 2A/3 and about the two orthogonal equatorial axes by -A/3 (e.g., Burgers, 1955), where A is unknown. Ifwe insist that the resulting principal moments equal the observed values C and A then we get the moment of inertia of the nonrotating sphere as I
=
.,I
+ (C
-
A)/3
(6.13)
The total inertia tensor may then be written as
where the I,, are the contributions due to loading effects. 6.2.2. Prrturbutions of Inerfiu Due to Siirface Muss Loading. The contributions to I,, are due to the ice sheets themselves and to the induced
96
RICHARD PELTIER
deformation. Rather than analyze the response to a realistic unloading event such as that described by the ICE-2 model discussed in Section 4. we will content ourselves here with an analysis of disk load approximations to such histories. Since the actual forcing is dominated by the large-scale Laurentide sheet, whose geometry is well known, and since the polar motion depends only upon the 1 = 2 component of the response to this forcing, the disk load approximation will be a rather accurate one. The strategy which we will adopt to calculate the Ii, will be to take advantage of the symmetry of the circular cap by calculating I,, in a coordinate system which has the cap on its polar axis; if this inertia tensor is called I ; then we can find I , from I : , by multiplying with an appropriate rotation matrix. Now an ice sheet with angular radius a and mass M may be described by the following surface density (Farrell, 1972): u'cE((i) =
M 4aa2 ~
[u,,+ c (21 + /(1 ++ cos 1)(1
1)
/= I
a ) dP/(COS a ) Pl(cos (i)] d COSN
(6.15)
where (i is the angular distance from the center of the cap. We may force our simple disk load to mimic a closed hydrological cycle by assuming that there is a defect of mass in a global ocean outside the ice sheet which is of magnitude -M distributed over the area 2au2(1 cos a). This global ocean has surface density
+
UOC(0)
=
(21 + 1 cos 4 [ -P,, + c 4au/(l+ I ) 1 )(
/=o
-
a ) dP,(cos
(Y)
d cos a
P/(cos ii)]
(6.16)
Conservation of mass is then assured because the surface integral of u((i) = dCL + u(" vanishes. C Z X ~ has the center of the ice cap along In the coordinate system X ~ . X ~ which the x3 axis and distance a from the origin, the perturbations of inertia due to a(e) are U2 L? 2u2 I ; ] = z.2 1 5 2 = - L2 I ; j = - - L2 (6.17a) 3 3 3 where ~
and we have inserted a factor f(s) which depends upon the Laplace transform variable s in order to introduce a time dependence into the forcing function. To the I : , in Eq. (6.17a) we must add the perturbation associated with the deformation of the earth due to loading. The deformation-induced perturbations may be calculated directly from the definition of the moment of inertia tensor I l , ( t )as
97
DYNAMIC3 0 1 . T H E ICE AGE EARTH
(6.18)
where po(r) is the (in general) radially stratified density field, and x denotes the position vector of the mass clement with density po which may be expanded as x = x t- u(x, 2) (6.19) where 2 denotes the initial equilibrium position of the mass element, and u the deformation-related displacement from initial equilibrium. Linearization of Eq. (6.18) in the perturbations from equilibrium gives
I ;,( Y)
=
J
r)(2 2/24h,, - .f,Zl,
-
.2,24,) dV
(6.20)
T o evaluate Eq. (6.20) for disk load forcing we need the displacement vector u = ur6, + zi,], which from Section 3 may be written as
Substituting Eq. (6.21) into Eq. (6.20) and evaluating for I\3(.~)gives
+ [MO. Y,
s) sin H
cos a] sin H dH dq5 dr
The functions [ I , and zi,,. for a load of arbitrary surface density expressed in the form of conbolution integrals as u,(H,
4,r,
s) =
l1
~ ( / jr., c)a(H',$',
(6.22) 0, may
be
.\)a' sin H' dB' dq5'
(6.23a)
8' dH' dq5'
(6.23b)
where GR and G" are Green's functions for radial and tangential displacement which have spherical harmonic decompositions GR(H, Y, s)
=
c G?(Y,
s)P,(cos 0)
(6.244
/ 0
(6.24b) Substitution of Eqs. (6.24) and (6.23) in Eq. (6.22) followed by application
98
RICHARD PELTIER
of the addition theorem for spherical harmonics yields (for the homogeneous sphere)
X
Pz(x’)a2a(x’,G’,fjs dx’ d$’ dr (6.25)
For the incompressible homogeneous sphere the parameters GF and may be determined from the analysis in Wu and Peltier (1982a) as
GI
(6.26a) (6.26b) where t = i)aGp$3, 6 = I 9/a2,d , = 2Gpo/as, d2 = - 1 6Gpo/3u3.Substitution of Eq. (6.26) into Eq. (6.25) then gives the analytic result (6.27) where L2(s)is given by Eq. (6.17b). Now we may make use of the fact (Wu and Peltier, 1982a) that the surface load Love number for the homageneous viscoelastic sDhere is (6.28) to write
I \3(s)
=
-
;U’L,(.Y)k,(s)
(6.29a)
For the incompressible sphere the trace of the inertia tensor is invariant, so that (6.29b) I t l I= 1 5 2 = -1:3/2 The total perturbations of inertia due to surface loading are found simply by adding the contribution from the deformation (6.29) to the direct contribution (6.17a) to obtain
1>2(.7)
=
u2L2 ~
3
[1
+ k*(s)]f(s)
(6.30)
where k,(s) for the homogeneous incompressible model may also be obtained from the general expression for k , ( s )given by Wu and Peltier (1982a)
DYNAMICS OF TI IE ICE AGE EARTH
99
as
(6.3 1)
-
Expression (6.3 1) emphasizes the fact that for the homogeneous incompressible model k ? ( ~ ) -1 as \ 0. In the next subsection we will see that properties of realistic earth models which force this asymptotic value of k2(~y) to differ from - 1 will have important consequences for true polar wandering on the time scale of the Ice Age itself.
-
6.2.3. Solirt ion o f ’ r h e Eider Eyirution.vfOr the nc~~luciation-Induced Polar Motion: The Homogmeous E u r ~ AModel. Given the perturbations of inertia due to rotational forcing C,, defined in Eq. (6.12) and the perturbations I,, = R,,k,Ii, from Eq. (6.30) (where R,, is the matrix of the similarity transformation which rotates I : , into the principal axis system) we have completely specified the J,, in Eq. (6.14) which are required in the Euler equations (6.1). These dynamical equations are clearly highly nonlinear in general and therefore difficult to solve. Sabadini and Peltier (1981) have described a numerical scheme which can be used to solve these nonlinear equations, however, and have employed the exact solution to verify the validity of an approximation scheme proposed by Munk and MacDonald (1960). This approximation scheme is valid as long as the axes of figure and rotation do not wander too far from the reference pole and is based upon linearization of Eq. (6.1) in the small quantities m, = w , / Q (where Q is the initial angular velocity) and I,/C where c‘, as previously. is the principal moment of inertia of the planet. Application of this linearization scheme to Eq. (6.1) leads to the following simple algebraic system in the Laplace transform domain of the imaginary frequency s: (6.32) where (T, = Q(C- A ) / A is the C’handler wobble frequency for the rigid earth and where m(s)= m,(s)+ imz( 5 ) . in which m,(s)and mz(s)are the direction cosines of the rotation axis in the i ,~ ~system . x and ~ and 4 are the follohng “excitation functions” due, respectively, to the rotational deformation and the surface load:
+
(6.33a)
100
RICHARD PELTIER
(6.33b) (6.33~) where 4(s) = +,(s) + i&(s) and where kf = 0.934 is the so-called fluid Love number associated with the centrifugal deformation (Lambeck, 1980). In this scheme the solution to Eq. (6.32) describes the polar motion, whereas the 1.o.d. variations are determined by the following decoupled equation for
*
If in Eq. (6.10) we replace the factor 3/2 by the factor kf = 0.934 for the real earth, then kT(S) =
k
(1
+ ”S)+ Y
(6.35)
and the solution of Eq. (6.32) may be well approximated (Munk and MacDonald, 1960) as
m(\) =
-~ -
y
/go
~-
s
s
+ yiOo iao)dr(s)
(6.36)
-
where a. = arb/(l + f i ) is the Chandler frequency of the homogeneous elastic earth. Inspection of Eq. (6.36) shows that the solution to the polar motion problem upparently consists of the superposition of two normal modes with imaginary frequencies s = 0 and s = iao - y determined by the locations of the poles of m(s)in the complex s plane. This led Sabadini and Peltier (198 I ) to approximate the solution of Eq. (6.36) by neglecting the second term in parentheses on the right-hand side on the basis of the argument that the high-frequency Chandler wobble described by this term should not be efficiently excited by the slowly varying ice sheet loads which contribute to the excitation spectrum $(s). This argument turns out to be incorrect, as has recently been pointed by Peltier and Wu ( 1 982), though Sabadini el al. (1982a,b) have continued to employ it. When the second term in parentheses is neglected, the time domain solutions are radically different from those which include its influence, even when the final solution is subjected to a running average over the period 27r/ao to remove explicit appearance of the Chandler wobble. The reason is clearly (in retrospect!) that this term makes an important nonzero contribution to the mean motion of the pole. The results obtained in the papers of Sabadini et UI. (1982a,b) are therefore completely erroneous. To understand the source of this error from an algebraic point of view,
DYNAMIC’S 0 1 , THE: ICE AGE EARTH
101
we may substitute Eqs. (6.30) and (6.33) directly into Eq. (6.36), assuming that the rotational forcing is produced by a single circular ice cap whose center lies an angular distance fj (say) from the CIO. If the coordinate direction “ 1 ” is assumed to point toward the glaciation center then the explicit form of Eq. (6.36) is
where
$7 where L:
=
=
a2 sin 0 cos (i (C’ - -4)
r.;
u2 sin 0 cos H
4‘2‘ = - L: n(C- A )
Lz/f(s) from Eq. (6. I7b) and the surface load Love number
k2(s)is taken to be given by (6.38) which differs from the exact expression for the homogeneous model (6.28) in the appearance of the small positive parameter I,, which is the so-called isostatic factor discussed in Munk and MacDonald (1960). Its introduction in Eq. (6.38) enables the homogeneous earth model to mimic realistic models of the earth in the sense that lim [ I
+ k,(s)] = I , # 0
(6.39)
\-0
Realistic earth models have I, # 0 because of the presence of the surface lithosphere and of internal density stratification, but the former effect is most important. Although we will not give the algebraic details of the Laplace inversion of Eq. (6.37) here, it can be accomplished analytically. The exact solution for the mean motion of the pole (i.e., Chandler wobble filtered) for an arbitrary time dependence of the ice sheet loading and unloading f ( t ) , is found to be such that
where & , ( t ) is the speed of polar wander in the direction of the centroid of the ice sheet, positive being toward and negative away from this direction. The parameters PI and P2 which appear in Eq. (6.40) are defined as
p P2
=~ n ~ ,1,, u2 -~ - sin 0 cos
i CL
=
P ~
I,
P,
(C-A)
0 L:
(6.41)
(6.42)
102
RICHARD PELTIER
It is useful to consider the solutions (6.40) for the following two different choices of the time history f ( t ) .
Case A In this example we take f(t> =
0,
t i a
=
1,
t 26
(6.43)
From Eq. (6.40) the solution in this case is rn,(t) = 0,
[
+ + y ( l + i)PI
R PI P2 Aao b - a
-~ -
~
t
a5t 5b
t >b
(6.44)
This corresponds to the situation in which a disk load is removed from the surface over the time interval ( b - a) at a uniform rate. The assumption is that the load was initially in isostatic equilibrium. It is clear from Eq. (6.44) that + Iis nonzero fort > b only because 1, # 0 [see Eq. (6.41)]. This model is essentially identical to that of Nakibogliu and Lambeck (1980), although these authors did not employ our simple disk load approximation to the melting history. Both f ( t ) and h(t)for this model are shown in Fig. 37a. Since y = (p/u)/(l ,i depends ) upon the viscosity u and Fzl is an observable ( A , x l0/1O6 yr from Fig. 36), we may invert Eq. (6.44) for t > b to obtain u . This gives u = 6 X 10’’ Pa sec with 1, = 0.006 (Munk and MacDonald, 1960), essentially identical to the value inferred by Nakibogliu and Lambeck (1980) on the basis of a much more complicated model of the deglaciation history. In fact Nakibogliu and Lambeck ( 1980) employed the ICE-1 model of Peltier and Andrews (1976) in their calculations. This serves to demonstrate the adequacy of the disk load disintegration model for the polar wander analysis and reveals the basic physics clearly.
+
Case B Since we know from the oxygen isotope stratigraphy in deep-sea sedimentary cores that the main ice sheets of the Pleistocene have periodically appeared and disappeared with a time scale of about lo5 yr, it is quite clear that the simple unloading model of case A is something of an oversimplification. In order to determine the way in which a load cycle modifies the
103
DYNAMICS OF T H E ICE AGE EARTH
solution (6.44) we will consider the history
=
0,
otherwise
(6.45)
b and a single unloading which consists of a single loading epoch for a 5 t I epoch for b I t I c. Substitution of Eq. (6.45) into Eq. (6.40) gives the explicit solution
=
0,
otherwise
(6.46)
which is plotted along with f ( l ) itself in Fig. 37b. It is quite clear from the form of Eq. (6.40) that no matter how many cycles of the form (6.45) may have preceded the single cycle we have analyzed, the solution within each cycle is completely oblivious of the others since the system has no memory. Therefore, if a large number of cycles of this form have occurred prior to the time t = c and if the ice sheet is not actively accumulating for t > c, then Eq. (6.46) show that &(f) = 0. Since we are presently living in precisely such a time, we see that the homogeneous viscoelastic model of the planet which we have been considering up to now in this section is completely incapable of delivering accord with the observed secular drift of the pole which is revealed by the ILS data shown in Fig. 36. This is completely contrary to the result obtained in Sabadini and Peltier (1981), but their results are invalid because of the neglect of the second term in parentheses on the right-hand side of Eq. (6.36). The above-discussed results lor cases A and B show that the homogeneous viscoelastic model of the earth is completely incapable of delivering accord with the observations. The analysis of Nakibogliu and Lambeck is incorrect because the actual history of loading is cyclic and the case B results show that the predicted speed of polar wander for the present day in such a case is identically zero.This clearly raises the embarrassing question of whether our hypothesis that the secular drift seen in the ILS pole path is in fact due to Pleistocene deglaciation. In the next subsection we will show that this hypothesis is correct but the viscoelastic stratification of the real earth must be taken into account since it contributes in a crucial way to the forced polar motion which is observed in the astronomical data.
FIG.37. (a) Predicted polar wander speed for the Nakibogliu and Lambeck (1980) model of deglaciation. The observed I o / l O h yr secular dnft in the ILS pole path is shown by the cross. (b) Predicted polar wander speed for a sawtooth load cycle. Both calculations are for the homogeneous earth model.
DYNAMICS 0 1 T H E I C L AGE EARTH
I05
6.2.4. Solution ol'the EMICYI~c~iitrtioti.s,fOr the ne~luciation-inducedPolur Motion: Slruf(fkdViscodastic Moddv. As can be shown in a rather straightforward fashion, all of the preceding analysis up to and including Eq. (6.34) will continue to hold for layered viscoelastic models so long as we replace the load and tidal Love numbcrs /
+
With Irf = kT(0) = A-5 Zfi, (/,/\/) required for insertion into Eq. (6.32) from Eq. (6.33a), we may rewrite Eq. (6.47) as (6.48) We may similarly manipulate the general expression for the surface load Love number (6.49) by defining 1
+ kJO)
=
I,
(6.50)
for the layered model, to write (6.5 1 ) Using Eqs. (6.48) and (6.51) i n the Euler equation (6.32) along with the definition of II/ in Eq. (6.33a) and the 4, appropriate for a circular disk load as in the last subsection, the equation which replaces Eq. (6.37) for this general case is (6.52)
(6.53) Since
106
RICHARD PELTIER
we may define
to reduce Eq. (6.53) to the form
Defining
(Y
=
a' sin 8 cos 8 Lily,the real part of Eq. (6.54) is (6.55)
+
Note that for the homogeneous earth N = ( 1 ,ii)P, and s, = y with j = 1 (i.e., 1 mode). In this case Eq. (6.55) reduces exactly to the f i , ( s ) obtained from Eq. (6.37) by approximating -iur/(y - iao) N 1, and similarly replacing the second term in parentheses on the right-hand side of Eq. (6.37) as -iao/(.c y - zag) N 1 (not 0, as assumed in Sabadini and Peltier, 1981). It is this fil(s)which gives the f i l ( t )implied by Eq. (6.40). Now the Laplace transform domain solution (6.55) may be inverted analytically, although the algebra is extraordinarily tedious and will not be given here. The time domain solution requires knowledge of the roots -A, of the degree M - 1 polynomial
+
c [P/ n (s + SJ1 I1
Prr
I(S) =
/
1
n
If,
$1- I
=
I
(s
+ A,)
(6.56)
1
which may be determined numerically with any conventional root-finding algorithm since the s, are known. The exuct inverse of Eq. (6.55) yields a function /Fzl(t) whose time derivative is
(6.57)
DYNAMICS OF T H E ICE AGE EARTH
in which q(s) = s(s
+ A,) -
*
(.s
+ An,
,)
n (.s + A,)
-
, A ~ -I
R,(s) =
1-1
-
107
+ s,)- - - (s + snf) and n (s + s,)
(s
Clearly the general solution (6.57) differs in a n extremely important way from the equivalent solution for the homogeneous model expressed in Eq. (6.40). The important difference is the presence of the third term on the right-hand side, which consists of a sum of terms which are each proportional to the time rate of change of the convolution of f(t) with a decaying exponential. The presence of this term indicates that stratified viscoelastic models will exhibit an instantaneous response which includes a contribution from the forcing applied at all past times. Each of the M - 1 terms in the sum therefore represents a memory of the past state of the system, and it is this history-dependent term. which is clearly absent from the homogeneous solution (6.40), which will allow the layered viscoelastic model to fit the observed polar wander in the ILS pole path even when the load is cyclic. The correct solution [Eq. (6.57)] for the layered viscoelastic model is completely different from that found in Sabadini et al. (1982a,b). Their solution is obtained by neglecting s with respect to s, in the denominator of Eq. (6.54), which amounts to assuming that the earth behaves as a fluid insofar as the rotational response is concerned, which is itself consistent with the assumption in Sabadini and Peltier ( 198 1 ) and the neglect of the second term in parentheses on the right-hand side of Eq. (6.36). That the assumption is physically incorrect follows simply from the recognition that both the tidal and surface load Love numbers, k: and k2 respectively, have precisely the same spectrum of decay times as stated in Eqs. (6.47) and (6.44). The set s, ( j = 1. M ) for k; is the same as the set s, for kZ.Since k: and k2 determine the time dependence of the response to tidal forcing and surface loading, respectively, it is quite clear that there is a basic inconsistency in assuming the fluid limit for one and not for the other. The correct solution to the polar wander equations for models with arbitrary viscoelastic layering is that given by Eq. (6.57), which is valid for arbitrary f(d. As in Section 5 , we determine f ( ( ) by invoking the oxygen isotope data shown previously in Fig. 32a: however, we will employ the slightly more complicated waveform shown in Fig. 38a, which differs from that shown in Fig. 32b in that the deglaciation phase of the load cycle is assumed to take place over a finite rather than an infinitesimal time interval. Although this does not produce any marked effect on the results we have included it as a better approximation to thc actual I8O/I6Odata. Each of the cycles of this wave shape has a mathematical form described by Eq. (6.45). To determine the solution for this excitation we simply substitute f(t) into Eq. (6.57). If there have been N previous load cycles of the form (6.45) then
108
RICHARD PELTIER
FIG.38. (a) The sawtooth approximation to the load history which is employed in all of the calculations in this section. (b) The prediction of polar wander speed as a function of time during the cycle for viscoelastic model I. The dashed line is the speed which is predicted on the basis of the assumption that no glaciation occurs subsequent to the last deglaciation event.
each of the convolution integrals in Eq. (6.57), during and after the current load cycle, is given by expressions of the following form: (i) During the glaciation phase (6.58a) (ii) During the deglaciation phase d dt
-~
( f
* p')
x
[
e ~ a l-
-Pe-Y'
+ (>-Y('-Ao
1
eY(r-A73
at
-
- (>?A1
y AT
]
(6.58b)
(iii) For f = 0 following the latest deglaciation event (i.e., today) (6.58c) The parameter P has the following definition:
DYNAMICS or VrHr ICE AGE EARTH
I09
while At and AT are, respectively, the duration of the deglaciation and glaciation phases ofthe load cycle. The oxygen isotope and other geophysical data suggest A! rn lo4 yr and A ’ / ~N 9 X 10‘ yr. These are the numerical values of the parameters which we have employed in constructing Fig. 38 and in all of the calculations t o be described here. The complete solution for the polar motion of a stratitied viscoelastic model forced by N cycles of glaciation and deglaciation is obtained simply by substituting terms of the form (6.58) into Eq. (6.57).where y is replaced by the appropriate A, for each term. Since the 105-yrcycle has dominated the climate record only through the Pleistocene pa-iod. which began about 2 X lo6 yr B.P., and since it is essentially absent from the record prior to about 3 X 10‘ yr B.P., we have assumed the conservative N = 30 in our calculations. For an earth model with 106hB elastic structure, a constant mantle viscosity of 10” Pa sec, an inviscid core, and a lithosphere of thickness 120 km, the variation of polar wander speed m , is shown in Fig. 38b as a function of time through the load cycle. At a time like the present, of hiatus in the load cycle, the solution follows the dashed rather than the solid curve. The polar wander speed for the inhomogeneous model no longer drops to zero immediately when the load is removed. This is due to the presence of the history terms in Eq. (6.57) which are entirely a consequence of the viscoelastic layering of the earth model which, as we have seen previously, is responsible for supporting a multiplicity of normal modes of viscous gravitational relaxation for each spherical harmonic degree in the deformation. The second point to note from Fig. 38 is that the predicted magnitude of the present-day polar. wander velocity is of the same order as that observed astronomically. Furthermore, the direction of the predicted mean motion is toward the centroid of the disk load (i.e., positive), which is also in accord with the observed apparent motion’s being in the direction of Hudson Bay. It is clear that we may expect to constrain the parameters of the stratified model by fitting it to the rotation data. On the basis of the above discussion it should be clear that the polar motion data will be explicable in terms of glacial forcing only because the real earth is viscoelastically layered. As shown in Section 5 , the layering was also required in order to understand the free-air gravity anomalies observed over present-day centers of postglacial rebound. In that application we showed that the most important features of the layering were the density discontinuities across the phase houndaries located in the mantle transition zone at 420 km and 670 km depth. and that the observed free-air gravity anomaly provided a high-quality constraint on the viscosity of the lower mantle. It might be naively expected that the rotation data would be sensitive only to the mean viscosity of the mantle, since they depend only upon the
110
RICHARD PELTIER
FIG. 39. (a) Prediction of polar wander speed as a function of time for the five realistic viscoelastic models (I-V) discussed in the text. The observed speed of polar wander in the ILS data ( I "/loh yr) is shown on the figure. (b) Predictions of the nontidal acceleration of rotation for the realistic viscoelastic models I-VII. Various of the observational estimates of this parameter are shown on the figure.
degree-two harmonic of the deformation, and that the long-wavelength undulation should sample the mantle throughout its volume. This expectation turns out to be only partly borne out by calculation. Of equal importance to the results, as we shall see, is the magnitude of the isostatic factor I , which is defined for the layered model in Eq. (6.50). Since the magnitude of the isostatic factor is controlled principally by the thickness of the surface lithosphere, the predicted polar motion is very sensitive to this feature of the viscoelastic layering. This may perhaps have been anticipated on the basis of the important role which I , was shown to play for the homogeneous model analyzed in the last subsection. Figure 39a illustrates a samplc of the results obtained with the complete theory for several different stratified viscoelastic models. This plate shows the speed of polar wander as a function of time since the end of the last deglaciation phase for five different layered models (I-V), whose properties are listed in Table VI. Models Iand I1 both have 120-km-thick lithospheres and 1066B elastic structures and differ from one another only in their lower mantle viscosities. which are lo2' Pa sec and Pa sec for models Iand 11, respectively. Since the observed present-day speed of polar wander is 1"/10' yr it is quite clear that the model with high lower mantle viscosity is more strongly rejected by the rotation data than is the uniform viscosity model I. However, it is equally true that the model preferred by the sea level
111
DYNAMICS O F W E ICE AGE EARTH
and gravity data, essentially model I, does not provide an acceptable fit to the rotation data. It predicts a present-day speed of polar wander of only about 0.3”/106 yr, which is only a third of the observed speed. Clearly, reducing the viscosity of the lower mantle somewhat further would further increase the predicted speed. but we would then be unable to satisfy the gravity anomalies observed over Fennoscandia and Laurentia as shown in Section 5 of this paper. Our first recourse must then be to increase I , by increasing the thickness of the lithosphere while maintaining the viscosity of the mantle equal to that of model I. Models I11 and IV have lithospheric thicknesses of 195 km and 24.5 kni, respectively. Inspection of Fig. 39a shows that the model with the thickest lithosphere comes closest to fitting the astronomical observation of present-day polar wander speed (extrapolation suggests that a best fit occurs with L N 300 km). However, this thickness is considerably greater than that suggested by some other lines of evidence, and it is important to inquire whether there may be other geophysical data which could be invoked in support of this number. There are in fact several lines of evidence that suggest a value of L, at least for continental lithosphere, which is as high as this. The most important of these data, insofar as we are concerned here, consists of a subset of the RSL histories which were discussed in Section 4. There it was shown that RSL data from sites along the eastern seaboard of the United States, to the south of the location of the icc margin at 18 kyr B.P. (which was near Boston), all differ systematically from the predictions made on the basis of viscoelastic model I. The theoretical model predicts much more submergence at these sites than is actually observed. It should be fairly obvious intuitively that it is in just this peripheral region that the flexure of the lithosphere is most extreme and therefore in just this region that the theoretical predictions will be most sensitive to lithospheric thickness. When TABLE v1
PROPERTIFS 01 I Hr L A \ F R F D
MODELSEMPLOYFD
IN T H E ROI A I I O N C A L ( lJLATlONS
Model
I II I11 IV V VI VI1
Lithospheric thickness L (km)
IFo\tatlc factoi
Upper mantle
(I)
(Pa sec)
120 120 195 245 120 I20 I20
0 009 I 0 009 I
lo2’ 10” 10” 102’ 2 x lozo 10” 102’
00156
0 0200 0 009 I 0 009 I 0 009 I
viscosity
Lower mantle vlscosltv (Pa sec) 10’I 1 0’* 10’I
2 x 10’” 3 x lo2’ 1023
112
RICHARD PELTIER
RSL predictions are made for models with increasing values of L it is found that the variance between observation and theory is reduced to zero at edge sites with a value of L very near that suggested by the preceding analysis of the polar motion data. The details of this analysis will be presented elsewhere. It should also be clear that this increase of the value of L from that in model 1 will not change any of the results for RSL and free-air gravity at sites inside the ice margin. This expectation is also borne out by direct calculation. We will return to discuss the meaning of this large value for the lithospheric thickness in Section 6.3. On the basis of the above analysis we may take it as established that the polar wander observed in the ILS pole path is a memory of the planet of the glaciation cycle to which the continents of the Northern Hemisphere have been subjected for at least the past 2 X lo6 yr. In fact, there is yet another astronomical observation which may be invoked to check the result obtained from the analysis of the polar motion. This is the so-called nontidal component of the acceleration of rotation. This observation, discussed in Munk and MacDonald (1960) and more recently in Lambeck (1980), may be obtained through an analysis of the historical variations of 1.o.d. by subtracting from the data the variation expected on the basis of the assumption that the lunar tidal torque has not changed significantly. When one does this one obtains a residual which yields an acceleration of rotation corresponding to the values of c&/Q listed in Table V which have been obtained by various authors using different methods of analysis. The theoretical prediction of & follows immediately through Laplace inversion of Eq. (6.34) with thc appropriate 133(s)inserted on the right-hand side. Now 133(,s)for the disk load approximation of the melting history is given by Z33(S)
=
(a'/3)( 1
-
3 cosz d)( 1 + k&:f(s)
(6.59)
with 1
+ k,(s) = I ,
-
c, sr, s +s s/ _____
as in Eq. (6.5 1). The Laplace inverse of 1 3 3 ( ~is) clearly
and from Eq. (6.34) we have
Figure 39b compares the prediction (6.61) to the observations for the same set of stratified viscoelastic models as were discussed in the context of our previous considerations of the polar motion observations. We have
DYNAMICS 0 1 I11L I(‘} AGE EARTI1
1 I3
also used the same cyclic f(r) used to construct Fig. 39a and evaluated the contribution from the history integrals f ( f ) * c on the basis of the assumption that there have been 30 previous cycles in the load history, each of duration l o 5 yr. From the structure of the solution (6.60, 6.61) we see that it will not be sensitive to thc isostatic factor I,, since this appears only in the first term in the square brackets on the right-hand side of Eq. (6.60) where it multiplies !(I). Since J ( I ) = 0 now, I,, will not contribute to the observed nontidal acceleration o f rotation. This explains why models 1. 111. and IV in Figure 39b predict essentially the same present-day nontidal acceleration of rotation (only the curve for model I is shown explicitly). These models essentially differ only in the thickness of their lithospheres and therefore only in their I , values. A more detailed analysis of the solution space for a range of models which all have a lithospheric thickness of 120 km and differ from one anothcr only in the value of the mantle viscosity beneath 670 km depth was presented in Wu and Peltier (in preparation). This shows that there are in fact two values of the deep mantle viscosity which are equally acceptable to this datum. one near 10” Pa sec. which is that which accords with the polar wander requirements, and one much higher near 3 X 10” Pa sec, which is completely rejected by all of the previously discussed information (i.e., RSLs, free-air gravity anomalies. and polar wander speed). The double-root structure of the nontidal 1.o.d. solution was first pointed out in Sabadini and Peltier (198 l ) , whose analysis of this datum does not suffer from the error made in connection with the polar wander analysis. The fact that model I1 is preferred over model I in Fig. 39b is a consequence of the fact that both the Antarctic and Fennoscandian ice sheets have been omitted i n our analysis. When these ice masses are included,the uniform viscosity model is again preferred by the observations. One additional global observable which we can predict reasonably accurately on the basis of the disk load model of the glaciation history which we have employed for all of our previous analyses of rotational dynamics concerns the time dependence ofthe second-degree component in the spherical harmonic expansion of the earth’s gravitational potential field. This is conventionally denoted by .Iz. The Green’s function for the perturbation of gravitational potential for a mass point is given in Eq. (3.40). This determines the potential with respect to a point on the free surface of the niodel since it includes the contribution from the Love number for radial displacement hi. Relative to the earth’s center of mass the potential perturbation due to the point-mass load is ’J‘
(6.62)
1 I4
RICHARD PELTIER
which we will assume is to be evaluated on the earth’s surface. Convolution of this Green’s function over the circular disk load produces a perturbation of potential which has a second-degree harmonic amplitude of
4s ag
M cos
P ~ ( C O@)[ S 1
+ k2(s)]f(s)
(6.63)
This may be simply inverted to the time domain to give AJ2, whose time derivative is
A.J2(l) = u
47T M cos a P I ( C 0 S 0 ) g h Me
Following the last cycle of loading, j ( t ) = 0 and the time derivative of the convolution integral in Eq. (6.64) is given by Eq. (6.58c), so that
Evaluation of Eq. (6.65) for the realistic stratified viscoelastic models discussed previously in this subsection gives the results listed in Table VII. Although J2 has not yet been extracted from the satellite orbital data, the increasingly high accuracy with which J2 itself is being determined promises that this will soon be possible. This observation will then provide us with another means of constraining the viscoelastic layering of the earth and serve as a cross-check on inferences made on the basis of the previously discussed polar wander and 1.o.d. observations. 6.3. Polar Motion and 1.o.d. Construints on the. Earth’s
Viscoelastic Stratification Our previously discussed analyses of RSL and free-air gravity data showed that these data implied a viscosity profile for the planet such that u was essentially infinite in a relatively thin surface “lithosphere” and that this was underlain by an upper mantle in which the viscosity was very near 102’ Pa sec. Across the phase transition at 670 km depth the combined RSL and free-air gravity data required an increase of viscosity but by no more than a factor of perhaps two. The viscosity of the lower mantle according to these data is then about 2 X lo2’ Pa sec, or very nearly the same as that of the upper mantle.
1I5
DYNAMICS 0 1 1 H E ICE AGE EARTH
Model 1
I1 1v V VI v11
J2 (m2 s e ~ ~ )
0.3032 0.1096 0.1777 0.4141 0.6160 0.6035
X
lo-'"
X X lo-'' X X 10 I" X lo-''
The polar motion and 1.o.d. analyses discussed in the last subsection have added somewhat to the further refinement of this picture. In the first instance they are also quite sensitive to the viscosity of the lower mantle and also insist upon a value near that (i.e., low) preferred by the RSL and free-air gravity data. This is useful corroboration. The observation of polar wander speed, in addition to this sensitivity to the deep structure, was also shown to be particularly sensitive to the thickness of the lithosphere. In order to fit the data with a fixed mantle viscosity of lo2' Pa sec we were obliged to employ a lithospheric thickness i n excess of 245 km, which is considerably in excess of that which most would consider reasonable as a measure of the average lithospheric thickness for the entire planet. Although there is very good evidence from RSL data in the region peripheral to the Laurentide ice sheet that the thickness of the continental lithosphere is in fact on this order, it is a number which is quite impossible to accept for oceanic lithosphere which is well constrained seismically and through studies of the flexure and gravity anomalies associated with seamounts and guyots. These data constrain the thickness of the oceanic lithosphere to be less than or equal to about I20 km. The explanation of this apparent inconsistency may be found in the way in which the lateral heterogeneity of lithospheric thickness is sampled by the rotational response. Work on this issue is ongoing.
6 4 Seculur In,tabilitv of the Kotrrtion Pole
As demonstrated in Fig. 38, the Pleistocene glacial cycle excites a true wander of the rotation pole rclative to the surface geography due to the perturbations of inertia associated with the ice load and with the load- and rotation-induced deformations. This polar motion in fact consists of two parts, the first being a slow oscillation of the pole about the initial equilibrium position, and the second a slow unidirectional drift of the equilibrium position itself. In order to demonstrate that the equilibrium position of the
116
RICHARD PELTIER
pole does in fact execute a slow mean drift we may simply average our previously derived solutions over the period of a single glacial cycle. We will consider the homogeneous and stratified cases separately under cases 1 and 2 below.
Case 1. Mean wander speed for the homogeneous model. If we denote by '-' the average over a single glaciation-deglaciation cycle and assume the sawtooth cycle described by Eq. (6.45), averaging of Eq. (6.40) gives the result
-
y
1
+ G, u'
7
P
sin B cos 6 LA -I, C-A 2
(6.66)
which is obviously nonzero only because I , f 0. Since this number is exactly one-half the instantaneous speed of wander which would be observed following removal of an equilibrated load which is given by the expression for t > b in Eq. (6.44), and since this has been plotted for various values of the viscosity ofthe model in Fig. 37a, we see that A , ( [ )x -0. 18"/106 yr, which is near the value of -0.2O/1O6 yr found by Sabadini and Peltier (1981). In fact. Eq. (6.62) is identical to Eq. (57) of Sabadini and Pelticr (1981), so that their calculation of the mean drift speed was not in error even though their calculation of the time-dependent polar motion was completely erroneous. This number of -0.2"/106 yr is sufficiently large that it could conceivably be important to the mechanism of climatic change itself and might in fact be observable in the paleomagnetic record as a residual true polar wander (TPW) after thc data are corrected for the known dnfts of the continents relative to the hot-spot frame. The question is at least sufficiently interesting that we should proceed to examine the magnitude of hi([) for realistic stratified viscoelastic models of the planet. We note further that the mean speed is negative [/Iin Eq. (6.62) is negative], so that on the average the Hudson Bay region IS moving slowly toward the equator at the computed rate.
Case 2. Mean wander speed for stratified viscoelastic models. I'he mean specd of polar wander over the Ice Age cycle for layered models may be determined by direct averaging of Eq. (6.57). Since f = 0, this average may be written
DYNAMICS Of 'I 1-11 ICE AGE EARTH
117
where each of the terms JhJdt has the explicit form
which is clearly nonzero. The factor /3 is the same as in Eq. (6.58d) and accounts for the memory of the system of the past N cycles. Because the two terms in Eq. (6.68) which contain /j do not cancel, it is clear that the average speed of polar wander over each cycle will be a function of time. In Fig. 40 we show a plot of the mean speed of polar drift, and the angular drift itself, as a function of the cycle number in the load history for one of the previously described viscoelastic models. For model I the average speed over 30 cycles of the periodic sawtooth history is -0.005°/106 yr, which is about two orders of magnitude ICSJ than the mean drift speed given by Eq. (6.66) for the homogeneous model. The effect of the viscoelastic layering of the planet upon the predicted drift speed is therefore extremely important. T o understand how this comes about we may simply inspect the dominant term in Eq. (6.67), which is proportional to the first term in brackets on the right-hand side. For the homogeneous model this term is simply -rf(t),whereas for a three-mock layered model it is (6.69) Since the three-mode model is a good approximation to the spectrum of model I, if we associate sI,s2, and .xi with the MO, CO, and MI modes, respectively, inspection of Eq. (6.69) immediately explains the slow' drift speed obtained for the layered model. For model I we have s, = 2.763, .s7 = 0.3746, s 3 = 5.318 X 10 '. A , = 1.438, and A7 = 0.0421. Since[([) = 0.5, evaluation of Eq. (6.69) gives I.' = - 4.59 X lo-' whereas for the layered model -rr(r) = - 4.6. I t is quite clear then that IF1 6 I-rf(t(t)l
118
RICHARD PELTIER
FIG.40. (a) Net angular deflection of the rotation pole relative to the surface geography for viscoelastic model I. (b) Variation of the "mean" speed of polar wander as a function of the number of the glaciation-deglaciation cycle.
because of the long relaxation time s ' '~ associated with the MI mode. Insofar as the mean drift speed is concerned, this slowest decaying mode is therefore the rate-controlling mechanism. This can be understood physically by recognizing that in order to effect a mean drift of the rotation pole relative to the geography, the equatorial bulge associated with the basic rotation must execute the same net drift and this occurs at a rate governed by the spectrum of the degree-two harmonic. As we have seen, this contains at least one important mode with an extremely long relaxation time. This long relaxation time appears to stabili7e the system. That realistic layered viscoelastic models should be rotationally stable to cyclic ice sheet forcing is in contradiction to the recent claims to the contrary which have appeared in Sabadini et UI.( 1982a,b). Their calculations are, however, marred by the mathematical error mentioned previously and are therefore misleading. It does not seem that the analysis of the recent paleomagnetic record by McElhinny ( 1973)and Jurdy and van der Voo ( 1974), as revised by Morgan ( 198 1 ) and Jurdy ( 1981 ), which suggests the existence of as much as 10"-15" of net TPW since the Cretaceous. could then be explained by ice sheet forcing.
DYNAMICS OF THE ICE AGE EARTH
119
In the next section we will focus upon an attempt to understand the mechanism of climatic change which is responsible for the observed IOS-yr glaciation cycle.
7. GLACIAL ISOSTASY
A N D CLIMATIC C H A N G E :
A THEORY OF T H E
ICE, AGE CYCLE There are at least two major unsolved problems connected with variations of global atmospheric climate on the time scale of 104-107 yr. The first of these problems has to do with the question of the origin of ice ages. That is, how and why do ice ages such as the one which has marked the present Pleistocene period originate? Although the geological record shows evidence of several such periods during the past few billion years, they nevertheless appear to be somewhat unusual. Since the current ice age has lasted only about 2 X lo6 yr, and since no substantial change of the degree of polar continentality can have occurred on this time scale, it appears that polar continentality alone cannot provide the explanation for ice age occurrence. The question remains open. In this section we will address a second important question, which concerns the explanation of the almost periodic succession of ice sheet advances and retreats which has characterized the present ice age and which is illustrated so clearly in the record of 'sO/160 variations obtained from deep-sea sedimentary cores. An example of one such oxygen isotope stratigraphy from Shackleton and Opdyke (1973) was reproduced in Fig. 32a and has been employed in our theoretical analysis of the adjustment process in ordcr to provide control on the characteristic time scale of ice sheet advances and retreats required to estimate the importance of deviations from initial isostatic equilibrium. In this section we will argue that the observed quasi-periodic oscillation of the main Northern Hemisphere ice sheets revealed by these data is due to the excitation, by fluctuations in the effective insolation, of a systemic free relaxation oscillation which is supported in crucial part by the process of glacial isostatic adjustment. 7 1 Ox-vgen Isotope Stratigrapiiv und thc Observed Spectrum qj Climatc Fluctuations on tlir rirnc Scule 104-106 Yearv The Milunkovitch Fljpotlzrsr 5 Analysis of oxygen isotopic records such as that shown in Fig. 32a has recently led to an intense revival of interest in the astronomical theory of
120
RICHARD PELTIER
the ice ages which was so strongly advocated by Milankovitch (1941) but which had also been discussed earlier by Adhimar, von Humbolt, and Croll (e.g., see Imbrie and Imbrie, 1978, for an interesting nontechnical discussion). The astronomical theory of long time scale paleoclimatic fluctuations asserts that the observed oscillations of Pleistocene climate are controlled entirely by changes in the effective insolation received b y the earth. Variations in the radiation intensity are governed by the temporal changes in the parameters of the earth's orbit produced by the varying gravitational attraction of other planets in the solar system. Milankovitch's contribution to this idea was to perform the first laborious set of calculations to determine, as a function of latitude and season, the time variations of insolation which would have been produced over the past several hundred thousand years of orbital history. His calculations have been superseded in the more recent literature, however, first by Vernekar (1972, 1974) and more recently by Berger (1978). Figure 41 is redrawn from Birchfield and Weertman (1978) and shows a power spectrum of the insolation time series of Berger (1978) for lat 60" N, which is near the latitude of the maximum thickness of Laurentide ice. The spectrum of insolation variations clearly contains energy at three very well defined periods, these being 19,000, 23,000, and 41,000 yr. The first two peaks are due to the precession of the equinoxes, whereas the third is due to the periodic variation of orbital obliquity. It is crucial for our present purposes to note that there is essentially no variance in the insolation time series at a period of lo5 yr, yet by inspection of Fig. 32a we can see, even visually, that the history of ice volume fluctuations is dominated by a periodic oscillation on this time scale. The ideas which we will develop in this section are concerned with an attempt to explain how the
PERIOD
(lo3yr)
FIG.41. Power spectrum of the insolation time series of Berger (1978) for 60" latitude. Note the absence of energy at periods near lo5 yr.
DYNAMICS O F THE ICE AGE EARTH
121
astronomical forcing on the precession and obliquity time scales might be transformed into a response which is dominantly on the time scale of lo5 yr. To understand this we will clearly have to invoke nonlinear processes. Before discussing the model which we have developed to resolve this problem, however, it is useful to attempt to quantify the extent of the dominance of the I 05-yr oscillation in the stratigraphic record of oxygen isotopic variability. In Fig. 42 we show a sequence of plots of the isotopic ratio '*0/I6Oas a function of depth in centimeters in several Pacific and Atlantic deep-sea cores based upon data in Imbrie ct al. (1973) and Shackleton and Opdyke (1973, 1976) as composited in Oerlemans (1980). In order to transform these isotopic depth series into time series we have to be able to locate at least one time horizon at some depth in the core, and the only method by which it has proved possible to do this is by locating the depth of occurrence
FIG.42. A comparison of oxygen isotope records from four different deep-sea sedimentary cores based upon data from Imbrie ( I / . (1973) and Shackleton and Opdyke (1973, 1976). The heavy vertical bars marked M-R dcnote the depth corresponding to the MatuyamaBrunhes boundary of age 730 (+20) kyr.
122
RICHARD PELTIER
ofthe faunal extinctions which mark the last change in polarity of the earth's magnetic field. This occurred 730,000 yr B.P. (Cox and Dalrymple, 1967; Mankinen and Dalrymple, 1979) with an error which is at most +20,000 yr and is called the Matuyama-Brunhes transition. This horizon has now been located in a reasonable number of sedimentary cores and, subject to the assumption of constant sedimentation rate at each site, leads to a linear mapping of the depth scale to a time scale. For the records shown in Fig. 42, on which the dashed lines join constant time horizons, it is quite clear that the rate of sedimentation varies from site to site. Clearly the cores from sites characterized by high rates of sedimentation will preserve a higher resolution record of the climatic variability than will cores from sites with low sedimentation rates. The sedimentation rate in core V28-238 is about 2 cm/103 yr, whereas that in core V28-239 is closer to 1 cm/103 yr. The most useful representation of the time series obtained by transformation of the data to the time domain is in terms of the quantity 6 ' * 0 , which is simply the variation in the concentration of I8O measured in parts per thousand relative to the ''0 concentration. It is quite generally accepted, as argued in Shackleton (1967) and Shackleton and Opdyke (1973), that this isotopic anomaly (measured in foraminifera tests contained in the sediments) provides a direct measure of the ice bound in continental ice complexes. Time series of this isotopic anomaly from cores V28-238 and V28239 are shown in detail in Birchfield et al. (198 I), and in Fig. 43 we show reproductions of the power spectra of these time series from this paper. These spectra show in a completely unambiguous way that the variance in the ice volume record is dominantly contained in the oscillation with period lo5yr, a fact which was first established by Hays el al. (1976), who performed similar analysis on the data extracted from core RC 1 1 - 120, which was also
2000 Ibl
7
PERIOD 1 1 0 ~ ~ ~ )
FIG. 43. Power spectra of the 6'*O time series from Pacific cores V28-238 (a) and V28-239 (b) reproduced from Birchfield ef al. (1981). Note the dominance of the spectral peak near a period of lo5 yr.
DYNAMICS or: THE ICE AGE EARTH
123
analyzed by Shackleton (1977). Besides the dominant oscillation at 105-yr period, however, the spectra clearly show statistically significant variance at the astronomical periods of -4 1,000 and -23,000 yr. Hays et al. ( 1976) argued on the basis of similar power spectra which they obtained from core RCI 1-120 and one other that the results demonstrated the validity of the astronomical theory of the ice ages. They certainly d o establish that the cryosphere responds to the astronomical forcing, since both the precession and obliquity periods do appear in the power spectra of 6"O. However Hays et al. (1976) were not able to explain why the dominant cryospheric response consisted of a quasi-periodic 1 05-yr oscillation when the astronomical forcing contained no power at this period (Fig. 41). Birchfield et ul. (198 1 ) have recently proposed a model which attempts to explain this observation, and although it achieved very limited success, it is nevertheless instructive since it does contain what appear t o be the main physical ingredients which are required to understand the phenomenon.
7.2. A Preliminary Model of tlie Pleistocene Climatic Oscillation The model of Pleistocene climate proposed by Birchfield et al. (198 1) basically consists of a model for ice sheet flow, which is forced by a particular accumulation function, coupled to a model of glacial isostatic adjustment. The model is used to describe the expansion and contraction of a circumpolar ring of ice whose northern boundary is constrained to the coast of the polar sea (Fig. 44). The model consists of the following simultaneous partial differential equations:
(7.2) In these equations h(0, t ) is the height of the ice sheet above sea level (where 0 is the latitude), H(H, t ) is ice sheet thickness, and h'(0, 1) is the depth to bedrock below sea level. Equations (7. I ) and (7.2) also contain several important parameters which Birchfield rf a/. (198 1) specify as (7.3a) (7.3b)
124
RICHARD PELTIER
FIG.44. Schematic diagram for the paleoclimatic model which consists of an active ice sheet driven by variations of the effective insolation as modified by the process of isostatic adjustment.
In Eq. (7.3a), T is the relaxation time for a harmonic deformation of wave number kH (Haskell, 1937) of a plane half-space with constant viscosity u and density p. The parameter q in Eq. (7.3b) is a time scale which depends upon the density difference Ap between ice and rock, while X in Eq. ( 7 . 3 ~ )
DYNAMIC'S OF T H E ICE AGE EARTH
125
anses from use of the Glen flow law (e.g., Paterson, 1981) to describe the ice flux supported by a given surface slope dhldfl. The final crucial ingredient in the Birchfield et a/ (198 1 ) model is the accumulation function A through which the feedback loop in the model is connected. Birchfield eta/. assume that the accumulation rate A depends upon the height of the ice sheet above sea level and take this dependence to be of the form A
=
a( 1
A
=
u'( 1
-
hh) > 0. hh) < 0.
above the firn line below the firn line
(7.4)
where a > 0 and a' < 0 are accumulation and ablation rates, respectively, at mean sea level. The firn line (e.g., Paterson, 1981) is the intersection of the snow line with the ice sheet and separates the zones of ablation and accumulation. Birchfield el crl ( I98 1) introduce solar forcing into their model by direct variation of the latitudinal location of the snow line by an amount proportional to the insolation anomaly. They compute the shift in latitude 6 % from the expression 6\
=
(7.5)
-C'SQ
where S Q is the insolation anomaly, and determine the constant C from the present day insolation gradient as
Birchfield et al. ( 1 98 1 ) describe several numerical experiments in which the model (7.1, 7.2) is integrated forward in time using Berger's (1978) insolation anomaly time series and the parameter values 7 = 1 x 10-2'sec-' Pa-' C = 43.35 W/m' km
a a'
They have also assumed a constant value for T
=
3000 yr
7
= =
1.2m/yr -2.7 m/yr
(7.7)
in Eq. (7.3a) of (7.8)
based upon the time scale which is observed to dominate the RSL records in Hudson Bay and Fennoscandia. They therefore implicitly employ a constant effective scale for the ice sheet. in spite of the fact that its actual scale is time dependent. Forward integration of this model leads to a prediction of the time variation of ice sheet volume which may be compared directly to the observed time series of 6"O. In Fig. 45 we have reproduced a power spectrum from their paper of the icc volume history predicted by the model. Although the response does contain significant power at the lower frequency end of the spectrum, there does not exist a sharp line at a period of lo5 yr
126
RICHARD PELTIER
FIG.45. Power spectrum of the ice volume time series predicted by the model of Birchfield et al. (1981). Note that although there is some energy in the low-frequency region of the spectrum, it is overwhelmed by that at the period of the astronomical forcing and is rather diffuse rather than concentrated in a well-defined spectral line.
and the response at the input obliquity period strongly dominates, with the response at the precession period also evident and of strength equal to that at low frequency. Birchfield et al. (1981) attempt to reconcile the unsatisfactory result shown in Fig. 43 by arguing that the spectrum should actually have a “red noise” background added to it which would of course produce a relative enhancement of the power at low frequency. This argument is of course entirely ad hoc, and it would be much more satisfactory if it were possible to design a model which could deliver a much closer facsimile of the observed signal of ice volume fluctuations. A suggestion as to how this problem might be resolved is contained in Oerlemans (1980), who employs a model which is virtually identical to that in Birchfield et al. (198 1). The only significant difference is in fact that Oerlemans has treated the constant relaxation time 7 in Eq. (7.3a) to be a variable rather than a fixed parameter. Figure 46 shows a result obtained by Oerlemans (1980) with a model forced at a single period of 20 kyr (approximately equal to the precession period) both excluding and including the effect of isostatic adjustment under the ice load and for various choices of the isostatic adjustment time scale 7 . This figure establishes the adjustment time 7 as a crucial variable in the model. When the adjustment time is short the response is almost entirely on the time scale of the forcing (the result obtained by Birchfield et al., 1981). On the other hand. when the time scale is long, 7 7 10 kyr, the ice volume builds up slowly and then
DYNAMICS OF- THE ICE AGE EARTH
127
collapses after about lo5 yr, in the way which is suggested by the '80/'60 data. Oerlemans (1980) describes a sequence of Milankovitch experiments which are somewhat indecisive since they do seem to indicate a fairly pronounced sensitivity to the choice of 7. We are therefore at something of an impasse. The careful ice volume predictions by Birchfield ct a/ ( 198 l), based upon a value of 7 which is obtained from the sea level data, show that the model fails to predict the observed oscillation. The initial calculations of Oerlemans ( 1980),however, which use a relaxation time very much in excess of that implied by the sea level record, seem to suggest that an oscillation of the observed type is supported under such conditions. These results suggest an explanation in terms of a more accurate description of the isostatic adjustment process in the model. We have shown in the preceding sections of this paper that the sea level record is sensitive only to the shortest relaxation times in the
RIGID EARTH
0
20
40
60
80
I00
120
r=lohyr
.d/"_l
00
20
40
60
80
100
120
2 L r=5 kyr
0
0
20
40
60
80
I00
I20
INTEGRATION TIME ( k y r )
FIG.46. Time series of ice volume fluctuations predicted by the model of Oerlemans (1980) for several different values of the isostatic ad,justnient time scale T. Note that as 7 increases the model seems to sustain an oscillation with a time scale near lo5 yr although there is no forcing at this period.
128
RICHARD PELTIER
relaxation spectrum of a realistic earth model. We have furthermore shown that such relaxation spectra also contain certain modes of relaxation with long characteristic relaxation time which are supported by the internal density jumps in the mantle at the olivine spinel and the spinel postspinel phase boundaries. Just as these modes are required to explain the observed gravity anomalies over the centers of rebound, they may also be necessary to understand the oscillatory nature of the Pleistocene climate cycle. In order to test this hypothesis we are obliged to develop a much more accurate model of the coupling between ice flow and glacial isostatic adjustment than that which is embodied in Eqs. (7.1) and (7.2). This is described in the next subsection.
-
-
7.3. A Spectral Model with Isostatic .4djustment: The Feedback betuwn .4cczimulution Rate and ICCJShcet Topogruphic’ Height The new model which we will develop here is based upon the same equation for the flow of a thin ice sheet which underlies Eq. (7.1) and can be expressed in the form
a.-H at
-
i
a (sin 0 U ) r sin 0 dB
+ A(H, t )
(7.9)
where the ice flux U is given by
which derives from the Glen flow law (Paterson, 1981), where again h is the height of the ice above sea level and H is its thickness. It is useful to expand H = h + h’ as before and to rewrite Eq. (7.9) in the form of a nonlinear diffusion equation as (7.10) where the nonlinear diffusion coefficient which determines the rate of flow (7.1 1 ) Our generalization of the model embodied in Eqs. (7.1) and (7.2) will be to derive a new equation for h ’ to replace Eq. (7.2) which gives a more accurate description of the process of glacial isostatic adjustment. In Section
129
DYNAMICS OF THE ICE AGE EARTH
3 we showed that the radial displacement h '(19,A, t ) produced by an arbitrary ice sheet of thickness H(B, A, t ) could be obtained by direct convolution of the surface load and the Green's function for radial displacement defined in Eq. (3.38). This may be expressed as
h'(8, A, t ) =
s Is dt'
dQ' 24d8/e', X/X, t/t')p,H(e',A',
1')
(7.12)
Use of the Love number expansion for zi, in terms of Legendre polynomials, the spherical harmonic decomposition of H for an axially symmetric load, and the addition theorem and orthogonality properties of spherical harmonics, reduces Eq. (7.12) to the form
(7.13) where 4: have been used here to denote the elastic Love numbers, and the H / are the spherical harmonic amplitudes in the decomposition of ice thickness. Now Eq. (7.13) may be converted to an exact differential equation to replace the approximate equation (7.2) by direct time differentiation to obtain
a 4~ +--mu , ( 2 /4T+ 1 ) C ri / / / ( t ) a 2 p+, q: m, (21 + I ) ~
a2p1
(7.14)
/
A spectral form of Eq. (7.10) may be derived to accompany Eq. (7.14) by expanding H = h h'. K , h, and A in terms of Legendre polynomials. When these expansions are substituted into Eq. (7.10)and this equation multiplied through by P, and integrated from cos 8 = - 1 to cos 0 = + 1 we obtain the following spectral form of the nonlinear diffusion equation:
+
(7.15)
where the interaction matrix
is
130
RICHARD PELTIER
which has elements which depend only upon the basis functions P/ and which may be computed once and for all. The model embodied in Eqs. (7.14) and (7.15) is a spectral model which replaces Eqs. (7.1)and (7.2)and which, although it contains the same physics as the original model, embodies a much more accurate description of the isostatic adjustment process which appears to be crucial to understanding the nature of the 105-yroscillation in the continental ice volume record of the Pleistocene period. Although we will not describe here a set of Milankovitch experiments with this model, we will, in the next subsection, provide a preliminary analytic exploration of its basic properties which leads to a particularly simple expression for the period of the free relaxation oscillation which the model supports. 7.4 An Analysis of the Propertie.\ of u Reduced Form ofthe S p a rul Model
The complexity of the isostatic adjustment equation (7.14) is almost entirely due to the fact that each harmonic amplitude of the deformation has several modes of relaxation accessible to it. In this subsection we will focus on the reduced form of the model which obtains when each harmonic decay may be approximated by a single exponential relaxation. This approximation reduces Eqs. (7.14) and (7.15), after considerable algebra, to
which may be enormously reduced if we neglect the elastic part of the response entirely by taking qy = 0. This approximation was also invoked
131
DYNAMIC'S O F T H E ICE AGE EARTH
in Birchfield ef a/. (1981) in writing Eq. (7.2) and reduces our spectral equations (7.17) and (7.18) to the set
+
( ' / ~ / h / .s/[I
dh,/dt
=
B/,,,,K,,,(t)h,(t)
dhj/d/
=
C / r / h / (-s' t C'/r')h;
-
-
+
C / ( r ' / . ~ ' ) ] h ; A / ( t ) (7.19)
+
(7.20)
in which the constants C, are defined by
u
47r
c, = m, (21+
1)
(7.21)
PlQ2
The nature of the simplified spectral model embodied in Eqs. (7.19) and (7.20) is most clearly revealed by differentiating Eq. (7.19) with respect to t and substituting from Eq. (7.20) to eliminate h' completely. This leads to the following second-order equation for h,(t):
-+d'h/ dl'
.ddh/ d t = Blmn[:[ + .r'C;r']K,,,h,
+ [$ + s'C;r']A,
(7.22)
which is relatively innocuous until one introduces the crucial feedback between ice sheet height and accumulation rate which is described by Eq. (7.4). We may rewrite Eq. (7.4) in the form A
=
~ ( l hh) -
+ E(h)
(7.23)
where E is a term that is nonzero only below the firn line. Now the Legendre decomposition of Eq. (7.23) givcs (for I # 0) ,,1/
=
-uhh/
+ El
(7.24)
which reduces Eq. (7.22) to the form
(7.25) which is now clearly seen to be the equation for a damped simple harmonic oscillator which is forced by a nonlinear term on the right-hand side, which will input energy to the oscillator at frequencies both higher and lower than those contained in the astronomical forcing, and by the term in El, which contains the astronomical forcing itself and some additional but weak hl dependence. The hypothesis which we wish to put forward here is that the Pleistocene ice age cycle is simply the free relaxation oscillation described by the weakly nonlinear damped simple harmonic oscillator equation (7.25).
132
RICHARD PELTIER
With w i the squared free oscillation frequency and X the damping coefficient of the oscillator, Eq. (7.25) may be rewritten as d2h/ at2
~
+ 2X dhl + w;h/ = Bonn + s' dt -
-
C/r']K,,,h,
(7.26) where
ab[s' - c/rq 2h = s' + ab w: =
(7.27a) (7.27b)
All of the parameters a, b, s', r', and C, are reasonably well known for the geophysical system so that we are in a position to inquire as to whether Eq. (7.26) allows a free oscillation and to determine how close this might be to critical damping. We note first that w: > 0 as long as s/ > C / r ' . Now C, is defined in Eq. (7.21) and may be rewritten as (7.28) where pI and pE are the densities of ice and rock respectively. Also, for a homogeneous earth model with density pE, Wu and Peltier (1982a) show that r' 21+ 1 (7.29) s ' + q : = 3
so that w: is positive (with q? small) if pI/pE < 1, which is of course true. The squared eigenfrequency of the oscillation w: is therefore always greater than zero. Using Eqs. (7.28) and (7.29) we may reexpress Eq. (7.27a) as abs'(1
w: =
- PI/PE)
(7.30)
on the basis of which we note that as the relaxation time of isostatic adjustment 7' = ( s ' ) ~ 'decreases, the frequency of the oscillation increases. If we insert into Eq. (7.30) the parameters employed by Birchfield et af.(1981) which give ab = 2.76 X yr-' and 7' = 3 X lo3 yr we predict a period for the oscillation of 27r T/ = N 25,000 yr (7.31) ~
w/
which is far too low to explain the observed ice cover fluctuation. In order to increase the period of the free relaxation oscillation we need to increase the effective relaxation time of isostatic adjustment. Equally important to
133
DYNAMICS 01' THE ICE AGE EARTH
the correct prediction of the period of the oscillation, however, is the extent to which the oscillation is damped. Inspection of Eq. (7.27b) shows that the strength of the damping to which the oscillator is subject decreases as the relaxation time for isostatic adjustment increases, implying that it will be much simpler to sustain a large-amplitude oscillation of low frequency than one of high frequency. Since ah = 2.76 X yr-' from Birchfield et al.'s data and s' = 3.3 X yr-' we can considerably reduce the damping by increasing the isostatic adjustment time scale. T o see how close the system in Birchfield et al. (1981) is to critical damping we simply compare X to wo. For their parameters we find X
3.1 X
' 1
yr
'
wo =
2.5 X
yr-'
(7.32)
so that the system is in fact overdamped and would respond only sluggishly to forcing at the natural oscillation frequency. With the approximation pr/pE6 I , critical damping with X N wOis obtained for s' = ab and subcritical damping for s' < ab. The relaxation time 7, = (ab)-' is about 3600 yr. Unless the isostatic adjustment time scale is much in excess of this value the damping in the system will be too strong to sustain an oscillation. This provides a very nice explanation as to why Oerlemans (1980) required a relaxation time in excess of about 1O4 yr before any relaxation oscillation was excited by the solar forcing. With T / = 104/yr we predict a T/ of approximately 5 X lo4 yr, which increases to lo5 yr for T / = 4 X lo4 yr. Because the parameters a and h are reasonably well known from meteorological observations and the viscoelastic properties of the earth are equally well constrained by geophysical observations, our physical model of the Pleistocene climatic oscillation which is embodied in Eqs. (7.14) and (7.15) has no adjustable parameters. On the basis of the analysis of a simplified version of the general model discussed in this subsection, we have good reason to believe that it will be able to explain the observed oscillation when the required Milankovitch experiments are performed with it. In order to deliver the observed periodic fluctuation, however, we must rely upon the same long relaxation time modes of realistic viscoelastic earth models which were required to explain the free-air gravity data discussed in Section 5. Only from these modes can we obtain the long characteristic relaxation times which are required to support the observed oscillation.
8. CONCLUSIONS In the main body of this paper we have provided a systematic development of the new theoretical model which has been designed to describe the
134
RICHARD PELTIER
phenomena which are associated with glacial isostatic adjustment. This model is based upon a linear viscoelastic constitutive relation between stress and strain, which I have referred to as the generalized Burgers relation, that appears to be uniformly valid in time in the sense that it reconciles not only long time scale adjustment data but also the observations of body wave and free oscillation seismology. In the low-frequency limit which is visible to postglacial rebound, the model behaves like a Maxwell solid so that timedependent processes are eventually governed by a Newtonian viscous response. By fitting the model to the observables of glacial isostatic adjustment we may infer the variation of mantle viscosity with depth. There are three complementary kinds of data which have proved to be most useful for this purpose: ( 1 ) radiocarbon-controlled histories of RSL, (2) surface and satellite observations of the “anomalous” gravitational field related to deglaciation centers, and ( 3 ) certain observed properties of the variation of the earth’s rotation. The principal success of the new theoretical model lies in its ability to explain simultaneously the RSL and free-air gravity data. No previous model of glacial isostasy has ever been successful in this regard. The success of the new theory with respect to these two sets of data is based upon the fact that realistic viscoelastic models of the earth’s interior support an entire spectrum of normal modes of viscous gravitational relaxation for each deformation wave number, rather than the single mode which is found for homogeneous earth models. In order to reveal this property of the viscoelastic models clearly, we have been obliged to cast the analysis in terms of a normal-mode formalism and we have discussed the intimate connection between this formalism and that for the normal modes of elastic gravitational free oscillation which is so familiar to seismologists. The totality of normal modes, consisting of those which are essentially elastic and oscillatory and those which are essentially viscous and exponentially decaying, are represented by points in the complex plane of the Laplace transform variable s. Normal modes of viscous gravitational relaxation are located on the negative real s axis. The spectrum of such modes spans a wide range of relaxation times, and those with the longest relaxation times, which exist in models which have essentially uniform mantle viscosity, turn out to be crucial to understanding the ability of realistic models to simultaneously explain RSL and free-air gravity data. These modes are supported by the density jumps in the mantle associated with the olivine spinel and the spinel perovskite + magnesiowustite transitions at 420 km and 670 km depth, respectively. Because of the relatively efficient excitation of these modes in models with weak viscosity stratification, the response to a loading event of large spatial scale is initially dominated by a relatively rapid re-
-
-
DYNAMICS o~- n i r ICE AGE EARTH
135
laxation with characteristic time scale near 2 X lo3 yr, which is followed after about 8 X lo3 yr by a very sluggish approach to the isostatic state on a time scale of about lo5 yr. This is precisely the behavior which is required to reconcile the RSL data from the Laurentide region, which reveal the short initial time scale only, and the free-air gravity data, which indicate well over 100 m of uplift remaining in the central depression. Models with any substantial increase of viscosity with depth are completely ruled out by both the RSL and the free-air gravity observations. Because of the crucial importance of the modes with long relaxation times in the new theory, we have had to pay particular attention to the question of the influence of the assumption of initial isostatic equilibrium upon the theoretical predictions of RSL and free-air gravity data. We introduced the novel idea in the context of isostatic adjustment studies that the oxygen isotope stratigraphy from deep-sea sedimentary cores could be employed to constrain the previous history of loading and unloading and thus provide us with the essential knowledge necessary to obtain a quantitative estimate of the importance of initial disequilibrium. These data have very clearly established that at least the last 2 X lo6 yr of the Pleistocene period have been characterized by a continuous series of ice sheet advances and retreats with successive interglacials separated by a regular time interval near lo5 yr. Our analysis of the extent to which RSL and free-air gravity data are influenced by initial disequilibrium established that the former measurements are relatively insensitive to disequilibrium effects whereas the latter are influenced to a nonnegligible degrce. This demonstration of the complementary nature of RSL and free-air gravity information is very important to understanding the quality of the constraint which these observations provide upon the mantle viscosity profile. When the two sets of data are combined and the influence of initial disequilibrium is taken into account, the data require a mantle viscosity profile in which the viscosity ofthe upper mantle is near 10” Pa sec and that of the lower mantle is near 3 X 10” Pa sec, so that viscosity increu.sc).sby a factor of about three across the phase transition at 670 km depth. This inference is entirely based upon a simple two-layer representation of the variation of viscosity in the sublithospheric region. If some modest increase of viscosity were introduced in the upper mantle (say across the 420-km boundary), then that allowed at the 670-km boundary would be reduced. I n terms of the new theoretical model, the observed free-air gravity anomaly over Hudson Bay is a very sensitive discriminant between viscosity models. As we discussed in Section 6 ofthis paper, it is possible to test the validity of the viscosity profile inferred from the adjustment data through analysis of certain characteristic properties of the earth’s rotation. We showed that
136
RICHARD PELTIER
both the secular drift of the rotation pole evident in the ILS-IPMS pole path and the observed nontidal acceleration of the earth’s rotation are explicable in terms of glacial-deglacial forcing. When the load cycle inferred from the oxygen isotope record is employed through the Euler equations to predict these observations, we find, in the case of the nontidal acceleration, that two quite widely separated values of the mean mantle viscosity are compatible with the observations. The allowed values of V are near 1.0 X lo2’ Pa sec and near 3 X 10’’ Pa sec. This basic ambiguity in the interpretation of this datum can be removed only by invoking the RSL and gravity observations, which strongly reject the larger of the two possible roots. The smaller root is beautifully compatible with the isostatic adjustment data, however, so that the observed history of the earth’s rotation provides information which enables us to verify the validity of the mantle viscosity profile inferred from them. Our analysis of the speed of polar wander observed in the ILS pole path showed that this datum had a markedly different dependence upon the parameters of the viscoelastic model than the nontidal acceleration. It was shown to be sensitive not only to mean mantle viscosity but also to lithospheric thickness. Although a low value of the mean mantle viscosity is again preferred by this datum, at the time of writing the trade-off between the effect of lithospheric thickness and that of the viscosity stratification has been insufficiently investigated to allow any unequivocal statement to be made. The last section of this article was devoted to an application of the new theory of glacial isostasy to an important problem in paleoclimatology which has to do with the explanation of the observed oscillation of ice cover on the I 05-yr time scale which is so apparent in the oxygen-isotope stratigraphy of sedimentary cores taken from the deep ocean basins. We discussed two recent attempts by Oerlemans ( 1980) and Birchfield et af. ( 198 1) to explain this oscillation using a theory which involved coupling of a model of ice sheet flow subject to insolation forcing with a model of glacial isostatic adjustment. Although the model employed in both these papers was essentially the same, the authors came to diametrically opposite conclusions concerning the plausibility of the proposed mechanism. Oerlemans ( 1980) found that oscillation was possible on the required time scale but only if the relaxation time for isostatic adjustment of a Laurentide-scale load was taken to be greater than lo4yr. Birchfield et af.( 198 1) used a short relaxation time for isostatic adjustment of 3 X lo3 yr which was chosen midway between those apparent from the sea level records in Hudson Bay and the Gulf of Bothnia. Their model did not produce the observed oscillation, though it did deliver energy to the low-frequency range due to nonlinear
DYNAMICS 01. T H E ICE AGE EARTH
I37
processes connected with ice flow that were described through the Glen flow law. In Section 7 we derived a theory which incorporated the same basic physical ingredients but which included a correct description of the isostatic adjustment mechanism. In both Oerlemans (1980) and Birchfield et ul. (198 I ) the isostatic adjustment component of the model was only crudely approximated though it is crucial to the oscillation. The general form of the spectral model which we dcrived contained the complete spectrum of modes of relaxation supported by realistic earth models which includes those with long relaxation times which are required to understand the gravity data over Wisconsin Laurentia. These modes could also provide the long relaxation times which are required to support the Pleistocene climatic oscillation. A reduced form of the spectral model was also derived and analyzed to obtain a damped simple harmonic oscillator equation for each component of the ice height spectrum. This equation shows that the period of the relaxation oscillation supported by the feedback between the accumulation rate and ice sheet topographic height may be expressed analytically in terms of the time scale of isostatic adjustment and that determined by the change of the accumulation rate per meter increase of topographic height. Our analysis showed that it would be easiest to oscillate those harmonics of the system with longest characteristic decay time since these had the smallest damping coefficients. The basic ideas in this new theory are the following:
( I ) The longest relaxation times necessary to support an underdamped oscillation are supported by the radial structure of realistic viscoelastic earth models. (2) Energy is forced from the high-frequency solar input to the low natural frequency of the oscillator by the action of nonlinearity due to ice sheet flow. ( 3 ) The energy appears as a sharp peak in the ice volume record because the system is resonant at the natural frequency of the free relaxation oscillation. Our analysis of the reduced form of the equations of the spectral model shows that this scenario is quite realistic. The Milankovitch experiments which are required to provide a detailed demonstration of the plausibility of this idea will be reported elsewhere. One idea which we have not developed at all in this article. but which is nevertheless extremely important, concerns the implications of the observed dynamic response of the earth to ice age forcing to our understanding of the mantle convection process. This point has been discussed by Peltier ( 1980b, I98 1b). Of most concern is the question as to whether or not the
138
RICHARD PELTIER
magnitude of the viscosity inferred from the rebound data is compatible with theories of the thermal convection process which is responsible for plate creation and destruction. The answer to this question is an unambiguous yes; in fact, one may argue as in the above-cited references that if the mantle viscosity were much different from that inferred from the rebound data then it would be very difficult indeed to produce a theory of the mantle convective circulation. Of equal importance to models of the convective circulation is the inference that there is an increase, by about a factor of about three, of the viscosity across the phase transition at 670 km depth. Although this increase is completely inadequate to confine convection to the upper mantle it could be sufficient to explain the compressive nature of deep seismic focal mechanisms (Isacks and Molnar, 1971). If the upper and lower mantles are filled with separate convective circulations, as some geochemical evidence has been taken to suggest (e.g., De Paolo, 198 I), then there should exist a very sharp thermal boundary layer at 670 km depth, since heat could be transported across the boundary only by conduction and the thermal conductivity is low. If the creep activation energy does not change significantly from one side of the phase change to the other (Sammis et af.,1977), then the expected sharp increase in temperature (Jeanloz and Richter, 1979, suggest 500°C) would be accompanied by a decrease of viscosity by several orders of magnitude, and this is not observed. Therefore, either there is no thermal boundary layer at 670 km depth and the mantle convects throughout its volume or there is a sharp increase of creep activation energy at 670 km depth which just offsets the decrease of viscosity which would otherwise be produced by the temperature increase. The only systematic analysis of the expected variation in the creep activation energy is that by Sammis et af. (1977) which would strongly suggest that the former possibility is correct. This may not be definitive, however, so that direct experimental measurement of the creep activation energy of the perovskite magnesiowustite phase would be invaluable. Peltier (1980b, 1981b) has given a series of arguments based upon dynamical considerations which also support the idea of whole-mantle convection. Of crucial importance among these is that connected with the expected dynamical effects of phase transitions. Richter (1973) has shown that convection through a phase boundary with negative Clapeyron slope such as that at 670 km depth is not significantly impeded by the phase transition. In subsequent analyses the advocates of separate upper and lower mantle circulations have therefore been obliged to invoke the idea that the 670-km discontinuity was a chemical boundary across which there was a significant change ofthe mean atomic weight of mantle material. A chemical layering would of course prove enormously efficient at preventing convective mixing across the boundary (Richter and Johnson, 1974). Until the
+
DYNAMICS OF T H E ICE AGE EARTH
139
new high-pressure diamond anvil data became available from the group at the Carnegie Institute (Yagi et al., 1979), it was quite possible that the 670km boundary could have been a chemical boundary since no direct petrological data were available at such high pressures. The new data clearly suggest that this boundary is an equilibrium phase boundary, however, and furthermore show that all of the seismically observed density increase is explicable in terms of the transformation from the less dense to the more dense phase. It would therefore appear that there is no dynamical mechanism available to explain how the separation between separate upper mantle and lower mantle circulations could possibly occur. The only possibility which remains open here, as far as 1 can see, is that previous calculations of the effect of phase boundaries upon convection are seriously in error. One must either give up the idea of separate upper mantle and lower mantle circulations, and face the geochemical consequences, or demonstrate that convection cannot penetrate a phase boundary with adverse Clapeyron slope. On the basis of this line of argument we can see that the observation from postglacial rebound that the viscosity of the mantle is essentially constant has forced a rather radical rethinking of ideas concerning the mantle convective circulation since it has demonstrated that there is no purely mechanical bamer to the penetration of convection at 670 km depth.
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Hilaire-Marcel. C., and Fairbridge. R. W. (1978). Isostasy and eustasy in Hudson Bay. Geo/o,qy, Jan. pp. 4-12. Honkasolo. T. ( I 963). On the use of gravity measurements for investigation of the land upheaval in Fennoscandia. Fennia 89, 2 1-23, Imbne. J.. and Kipp. N. G. (1971). A new micropaleontological method for quantitative paleoclimatology: Application to a late Pleistocene Caribbean core. I n “Late Cenozoic Glacial Ages” (K. K. Turekian, ed.). Yale Univ. Press. New Haven, Connecticut. Imbrie. J., and Imbrie. C. P. (1978). “Ice Ages: Solving the Mystery.” McMillan. New York. Imbrie. J., Van Donk. J., and Kipp. N. G. (1973). Paleoclimatic investigatior. of a late Pleistocene Caribbean deep-sea core: Comparison of isotopic and faunal methods. Quat. Re.5 (N. Y.1 3, 10-38. Innes. M. J. S., Goodacre, A. K., Weston, A.. and Weber, J. R. (1968). Gravity and isostasy in the Hudson Bay region. Pt. V. Publ. Dom Ohs. (Ottawa). Isacks. B., and Molnar, P. (1971). Distribution of stresses in the descending lithosphere from ~ a global survey of focal mechanism solutions of mantle earthquakes. Rev G ‘ e o p h . ~Spuce Phy.7. 9, 103-1 74. Jackson, J. D. (1962). “Classical Electrodynamics.” Wiley, New York. Jacobs. J. A. (1976). “The Earth’s Core.” Academic Press. New York. Jarvis, G. T., and Peltier, W. R. (1982). Mantle convection as a boundary layer phenomenon. Geophys J . R. Astron. Soc. 68, 389-427. Jeanloz, R. (1981). High pressure chemistry of the earth’s deep interior. A m . Chcm. Soc., I 7 r h State-of-the-ArtSymp. (in press). Jeanloz. R., and Richter, F. M. (1979). Convection, composition and the thermal state of the lower mantle. JG‘R. J. Geophy~s.Res. 84, 5497-5504. Jeffreys. H. (1972). Creep in the earth and planets. Tecfonophysics 13, 569-581. Jeffreys, H . (1973). Developments in geophysics. Annu. Rev. Earth Planet. Sci. 1, 1-13. Jurdy, D. M. ( I98 I). True polar wander. Tectonophysics 74, 1 - 17. Jurdy, D. M . , and van der Voo, R. (1974). A new method for the separation of true polar wander and continental drift, including results for the past 55 M.Y. JGR. J . Geophy.~. Res. 19, 2945-2952. Kohlstedt, D. L., and Goetze, C. (1974). Low stress and high temperature creep in olivine single crystals. JGR. J . Geophys. Res. 79, 2045-205 1. Kukla. G., Berger. A.. Lotti. R., and Brown, J. (1981). Orbital signature of interglacials. Nufure (London) 290, 295-298. Lambeck, K. ( 1 977). Tidal dissipation in the oceans: Astronomical, geophysical and oceanographic consequences. Philos. Trans, R. Soc. 1,ondon. Ser A 287, 545-594. Lambeck, J. ( 1980). “The Earth’s Variable Rotation: Geophysical Causes and Consequences.” Cambridge Univ. Press, London and New York. Lerch. F. J., Klosko. S. M.. Laubscher, R. E., and Wagner. C. A. (1979). Gravity model improvement using GEOS 3 (GEM 9 and 10). JGR, J . Geophj’s. Re5 84. 3897-3916. Libby. W. F. (1952). “Radiocarbon Dating.” Univ. of Chicago Press, Chicago, Illinois. Liden, R. ( 1938). Den senkvartara strandforskjutningens forloop och kronologi i Angermanland. Geol. Foercw Stokholm Foerh. 60, 397-404. Litherland, A. E. ( 1980). Ultrasensitive mass spectrometry with accelerators. .4nnu. Rev. Nucl. Part. Sci.30, 437-473. Liu, H. P., Anderson. D. L., and Kanamori, H. (1976). Velocity dispersion due to anelasticity: Implications for seismology and mantle composition. Geophy.P. J. R. .I.crron. Soc. 47, 4 I58. Lliboutry. L. A. (197 I ) . Rheological properties of the asthenosphere from Fennoscandian data. JGR, J . Geoph.v.5. Rex 16, 1433-1446.
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PLANETARY SOLITARY WAVES IN GEOPHYSICAL FLOWS P. MALANOTTE RIZZOLI Depurtmenl of Mc?c.r)IoCigy und Physical Occunogruph) Massudiriwtls lristrtulr of Technology Curnhrrdge, hfassuchusc~~ts
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Introduction Why Solitary Waves Ma) Be Important in Large-Scale Geophysical Motions Solitary Waves in One Dimension A Short Synopsis The Existing Models for Large-Scale Permanent Structures 3 I A Unified Approach Length Scales of the Order of the External Deformation Radius 3 2 Length Scales Smaller Than the External Deformation Radius Evolution of Coherent Structures The Initial Value Problem 4 I The Single Solitary Eddy in the Wedk- and Strong-Wave Limits 4 2 Collision Expenments Stability 5. I Perturbations in the Initial Conditions Numerical Expenments 5 2 Overlapping Resonances Further Investigations on Coherent Structures 6 I Numencal ACCurdCy 6 2 Dissipation Conclusions References
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LARGE-SCALE GEOPHYSICAL MOTIONS In this last decade a resurgent interest has been focused on finding solitary wave solutions to a very wide class of models suitable to describe geophysical fluid motions. Solitary waves are not a new concept. I shall not quote again the by now famous description given by Scott Russell (1845) of his discovery of the solitary wave in 1834. The interested reader can find it in almost any recent review paper on the topic. I will point out instead that the interest in this kind of phenomenon, given such a beautiful and exotic name, has proceeded in “outbursts” since Russell’s discovery more than a century ago. The first outburst originated from Russell’s observations (1 838, 1845) and their conflict with Airy’s shallow water theory (see Lamb, 1932, for a review of it). This originated the series of papers by Boussinesq ( 1 87 la,b, 1872), Rayleigh ( 1876), and Korteweg and de Vries (1 895), in which the apparent contradiction was explained and Russell’s solitary elevation was given its mathematical formulation. The field was quiescent until interest I47 ADVANCES IN GEOPHYSICS,VOLUME 24
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in surface gravity waves of the solitary type was revived by Keulegan and Patterson ( 1940). Keller ( 1948), and Munk ( 1949). This produced a series of papers in the late 1940s to early 1950s concentrating on surface gravity and internal solitary waves (see Miles, 1980, for a comprehensive review). Successive papers were irregularly produced in the 1960s and early 1970s. Among others, there were the works by Benjamin ( 1962, 1966, 1972), Benney (1966), Clarke (197 l ) , Davis and Acrivos (1967), Grimshaw (1970, 1971). Johnson (1972, 1973). Larsen (1965), Long (1956, 1964a.b), and Long and Morton (1966). Again, most of these works concentrated on surface gravity and internal solitary waves. The last, and present, outburst of interest on the topic was primarily originated by another fundamental piece of work by Zabuski and Kruskal (1965) and Zabuski (1967). They in fact provided what is now the most widely accepted explanation for the Fermi-Pasta-Ulam (FPU) problem. This concerns the lack of thermalization of a one-dimensional lattice of mass-point oscillators coupled by nonlinear springs (Fermi et a!., 1955). Zabuski and Kruskal explained the FPU recurrences and lack of approach toward a final state of energy equipartition with the capability of the anharmonic lattice of supporting nonlinear, permanent-form solutions, the solitary waves. Specifically, they related the discretized lattice with quadratic nonlinearities to a continuum model, the Korteweg-de Vries (KdV) equation, the asymptotic solutions of which are dominated by solitary waves. They also coined the name “solitons,” which solitary waves are now often called in the literature, capitalizing upon their property of being able to collide with one another and reemerge unaltered in shape and speed (amplitude) from their collisions. They found that in the FPU system solitons freely streaming through one another emerged from quite general initial conditions. Their fundamental work again excited intensive research on solitary waves in all branches of physics. As far as large-scale (planetary) motions of geophysical flows are concerned, after the already mentioned isolated papers by Long (1 964b), Larsen ( 1965). Benney ( 1966), and Clarke ( 1 97 1), the flourishing of papers has started during the second half of the 1970s. It was then realized that a new approach was possible toward the understanding of the mesoscale atmospheric and oceanic motions, together with other approaches, such as the linearized theories for infinitesimal amplitude motions or the turbulence theories for the highly nonlinear ones. This approach was just the possibility of using nonlinear permanent-form solitary wave solutions of finite amplitude to explain and model the long-lived structures clearly recognizable in the ocean and atmosphere. Thus, papers dealing with solitary waves are now appearing at an increased rate in every category and class of geophysical fluid motions. The task of embodying all the current
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research into a single review article is impossible and perhaps even useless, because there are too many aspects of solitary wave theory upon which emphasis can be focused. 1 shall mention, for the interested reader, recent review articles or books covering a variety of sectors and exploring the field from different points of view. Scott et al. ( 1973) provide a most comprehensive review of solitary wave models in many different fields (water waves, nonlinear optics, elementary particle theory. plasma physics, etc.) and of their properties. Miles (1980) summarizes a vast number of important works in fluid mechanics, particularly for surface gravity water waves and internal solitary waves. Whitham’s “Linear and Nonlinear Waves” (1974) also furnishes a thorough review of soliton model equations and properties (see also Miles, 1981). In this article we shall not deal with surface gravity or internal solitary waves, as the above-mentioned review paper by Miles (1980) and many other recent works already cover this particular sector quite extensively. Further references for it can be found in the lectures delivered by Redekopp and Keller in the WHO1 Report of the 1980 Geophysical Fluid Dynamics Seminar Study Program. Other types of solitary waves (nonlinear Kelvin and shelf waves) have been studied by Smith ( 1972), Grimshaw ( 1 977), and Boyd (1980a). In this article we shall concentrate instead on the large-scale motions of geophysical fluids and the related solitary wave models. The attention to large-scale motions both in the ocean and the atmosphere, even other planets’ atmospheres, has been exponentially growing in the last decade, thanks to the most advanced technology, the increase in. the basic worldwide data set, and the organization of important experiments such as MODE, POLYGON, POLYMODE, and GARP. As previously remarked, in the context of these mesoscale systems, geophysical fluid dynamicists have resorted either to linearized analytical theories or, given the impossibility of ignoring nonlinearities, to turbulence theories and related numerical studies. Turbulent systems with many degrees of freedom exhibit the usually postulated mixing of phases leading eventually to statistical equilibrium. In weather prediction, this randomness postulate leads to the socalled predictability problem, for which any forecast of how the flow evolves becomes practically useless after a finite time. This time cannot be extended indefinitely by any improvement of the initial data short of absolute perfection. This randomness postulate of turbulence theories is violated by solitary waves. For them, nonlinearities play the opposite role of preserving phase correlations against the effects of dispersion. In a dissipationless system, solitary waves are therefore, in principle, completely predictable motions. To better express the possibility of the soliton’s bearing upon the pre-
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dictability problem, I quote Leith (1978): Is such a new approach in the offing for weather prediction? Perhaps not, but if it were, then conjecture about its possible form would be of obvious interest. My own candidate for such an approach is the recognition, based on nonlinear wave theory, that many onedimensional nonlinear systems with linear dispersion can have remarkably stable and completely predictable solutions called solitons. . . . The earth’s atmosphere is far more complicated than the examples studied so far. but certain intriguing questions arise. Is it possible that at times relatively stable structures can exist in the earth’s atmosphere whose predictability is far greater than that generally estimated? Do these arise from a balance between linear dispersion and nonlinear interactions leading to phase locking of waves of different wavelengths?
This kind of concept has been the primary motivation of the interest in these nonlinear permanent-form models of large-scale oceanic and atmospheric motions-namely, solitary Rossby waves. Solitary Rossby waves will be the focal point of this article. To partially counteract an excessive optimism about the importance of the soliton concept in the real world, I must point out that unambiguous experimental evidence has not yet been produced for the existence of solitary Rossby waves, either in the ocean or in the atmosphere. The work done thus far is essentially theoretical, both analytical and numerical. I also personally think that the suggestions, made in recent works, about how, where, and under which circumstances one might observe and detect solitary Rossby waves in the real world are rather optimistic (see, for instance, Boyd, 1980b). Apart from any difficulty in isolating experimentally a solitary Rossby wave, the model equations used thus far to get solitary solutions are indeed rather simple. In the context of a more complex two- or three-dimensional model, numerical experiments remain still the most direct and useful tool. Nor is it straightforward to obtain coherent structures emerging from general (random) initial conditions or to force them through suitable driving mechanisms. Even so, it is not straightforward to prove that the coherent structures thus obtained really are the solitary wave solutions of our simplified models. However, Charney and Flier1 (1981) speculate that in the category of mesoscale motions there exists a scale range in which the dynamics may be dominated by orderly, phase-coherent structures, much more than predicted by any turbulence theory. Also, this scale range, in which nonlinearities could play this organizing role, is much broader for the oceans than for the atmosphere. The oceans would then constitute the right geophysical environment in which to look for them. Notwithstanding these strong limitations in the experimental evidence for solitary Rossby waves, the field has deepened enough and the completed research is of sufficient dimensions for a review on the topic to be appropriate.
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Thus, Section 2 presents a short synopsis of the relevant one-dimensional solitary wave models which are derived in most instances from the two- or three-dimensional geophysical systems of interest. The most important ones are the KdV equation, its modified form, the Boussinesq equation, and the nonlinear Schrodinger equation. Section 3 first offers some examples of coherent structures observed in geophysical flows which have been related to nonlinear, permanent-form models-Jupiter’s Great Red Spot, blocking phenomena in the atmosphere, and Gulf Stream rings. Successively,the variety of existing planetary solitary wave solutions are investigated in the context of a unified approach. All permanent solutions of this kind can be derived and classified along this line. In Section 3.1 length scales of the order of the external deformation radius are examined. Section 3.2 is devoted to length scales smaller than the external deformation radius. A final table summarizes the existing models for coherent structures, classified according to the type of functional F (analytical or multivalued), expressing the dependence of the potential vorticity upon the flow stream function. Other classifying features are the strength of the solution ( U c, where U is the particle speed and c the wave phase speed) and its symmetry (closed or open streamlines in the reference frame moving with phase speed c). In Section 4 the same permanent-form structures are examined in the context of the initial value problem posed by the considered model, that is, allowing the slow time variation of the flow field. To do this, a specific geophysical model is chosen, namely, the quasi-geostrophic, barotropic potential vorticity equation over variable relief. In Section 4.l the initial value problem posed by the above model is solved explicitly. Numerical evidence is presented for the slow time evolution of these coherent structures in both the weak ( U < c) and strong ( C i > c) wave limits. Section 4.2 contains new results. The basic question to be answered is whether two-dimensional solitary waves survive collisions in a suitable parameter range and how extended this is. The problem is equivalent to exploring, analytically and numerically, the width of the range in which a coupled KdV dynamics holds for n interacting solutions. The case for two interacting permanent waves is examined in detail. A series of new numerical experiments is presented in which one wave amplitude is held fixed and the second wave amplitude is gradually increased. The range in which the two waves behave like solitons, maintaining their identity in the twodimensional collision, is indeed rather wide. The coupled KdV dynamics holds until the varying-wave amplitude is more than one order of magnitude stronger than that of the weaker wave. After this limit, the stronger wave still obeys a KdV model, whereas the weaker one is destroyed, finally evolving into a turbulent field. Thus, three subranges can be defined for the
*
152
P. MALANOTTE RIZZOLI
interaction of permanent solutions to the chosen model: ( 1 ) A KdV subrange in which the model is well approximated by a set of coupled KdV equations, one for each interacting solution. (2) A KdV-to-linear subrange for the weak-amplitude limit. The stronger wave is a KdV solitary eddy; the weaker one is reduced to a packet of dispersive Rossby waves. ( 3 ) A KdV-to-turbulent subrangc The stronger wave is a KdV eddy. The weaker is destroyed in the interaction to evolve as two-dimensional turbulence over topography, in the model chosen in this section.
Transition from ( 1 ) to ( 3 ) is achieved when the stronger wave amplitude becomes at least one order of magnitude higher than the weaker one. Section 5 is devoted to examining the stability properties of the permanent-form solutions discussed in Section 4. General considerations are first made through the use of integral theorems. In Section 5.1 attention is focused on stability to perturbations in the initial conditions, considered as “errors.” If a threshold is reached in the stability of the permanent structure, the perturbation’s energy will grow, destroying the structure itself. This stability threshold can be shown to depend both on the perturbation’s intensity and on its scale content (average length scale). For perturbations with a scale content concentrated at length scales smaller than the average diameter of the coherent structure, a qualitative stability limit can be established on the basis of numerical experiments. This limit is reached when the perturbation’s intensity is of the same order as the basic structure intensity, defining the intensity of the two fields through their root-meansquare velocity u, and vorticity ems. The crossing of this stability limit signals the onset of turbulent behavior and can be visualized through the sudden decorrelation of the locked Fourier phases of the coherent structure. In this context, the collision experiments of Section 4 can be interpreted as stability experiments, with the varying-amplitude eddy being the “perturbation.” Observation of locked Fourier phases and single-eddy energy shows complete similarity of behavior. In Section 5.2 a new interpretation is offered for the stability properties of all permanent structures, be they analytic or modon-like (multivalued). This new interpretation is based on the theory of overlapping resonances, particularly suitable to explain and predict the onset of widespread chaotic motion in systems constituted, in general, by a set of coupled nonlinear oscillators forced by external perturbations. Applying the consequent formalism to the permanent solutions of Sections 4 and 5.1 (though the application could be easily repeated in the same straightforward way for any permanent solution, in particular for multivalued modons), an approximate criterion can be formulated to predict the location of a “stochasticity” bor-
PLANETARY SOLITARY WAVES IN GEOPHYSICAL FIOWS
153
der, separating a region of nonlinear deterministic, wavelike behavior from a region of stochastic, turbulent behavior. This criterion furnishes an approximate relationship for the division of phase space into “order” and “disorder” regions. The relationship relates the perturbation’s intensity, as measured by its Crms, to the perturbation’s scale content, as measured by x = Kso,/Kpert, where K,,, is the average wave number of the basic state (solitary eddy diameter) and Kpenis the average wave number of the perturbation. All the stability expenments performed by different authors upon different kinds of permanent solutions (survival or destruction of the solution itself) can be shown to fall unambiguously into the “order” and “disorder” regions defined by this relationship. This one (Chirikov’s criterion) marks the border beyond which a KdV, or in general a permanentwave, dynamics does not hold any longer, the coherent structures are randomized, and the fields rather obey a turbulence dynamics. The transition from order to disorder seems to occur in a restricted parameter range, and the border region seems to be rather sharp and well defined. Section 6 is devoted to the investigation of further properties of coherent structures. In Section 6.1 the question of numerical accuracy is examined, and the available numerical algorithms are found to be quite adequate to simulate and reproduce the properties of coherent structures. In Section 6.2 the problem of dissipation is studied. A new, general approach is given to investigate the influence of different kinds of dissipation on the permanent eddies. The latter are distinguished according to the usual criterion of being analytic or multivalued (modons). The considered dissipation forms include a linear bottom-drag term as well as higher frictional forms (Newtonian, biharmonic). The general behavior inferred, common to all coherent structures acted upon by friction, is that as t 0, at the beginning of the process, the solution behaves like a permanent-form solution. As t w, dissipation reduces the energy, that is, both the amplitude and the speed. The solutions will slow down and finally stop. With decreasing amplitude they will enter the range of linear dynamics, with an asymptotic behavior given by a decay into linear Rossby waves. The details of the decay, however, are rather different for structures characterized by an analytic, general functional F (relating the potential vorticity to the stream function in the reference frame moving with phase speed cj, and for those of a modon-type nature with a multivalued functional. In the latter case, for those modon structures characterized by a linear interior functional F, all friction laws can be shown to have an identical functional dependence (linear in $) in the interior region. This will lead to the same observed decay of the modon and to an identical time dependence of the modon phase speed c(tj for every type of friction law (linear drag, Newtonian, biharmonic) if the various friction coefficients are numerically chosen to give the same
-
-
154
P. MALANOTTE RIZZOLI
e-folding decay time for the modon intensity. This explains apparently surprising results of numerical experiments exploring dissipative effects upon modon structures (McWilliams et ul., 1982). More complex is the decay under dissipation of coherent structures characterized by a general analytic functional F, like those studied in Sections 4 and 5. New numerical experiments are shown to illustrate their behavior under the previously considered friction laws. As a conclusion to Sections 4, 5, and 6, one may state that for model equations like those examined, suitable to describe large-scale geophysical motions, a well-defined parameter range seems to exist in which they obey a nonlinear dynamics leading to phase-locked, coherent structures. Under dissipation, through progressive reduction of their amplitude, the coherent structures reach the linear dynamics range, with asymptotic dispersion into a packet of linear Rossby waves. The transition to the linear dynamics range is gradual and smooth. Under the forcing produced by external perturbations, a stochasticity border is reached at a critical perturbation amplitude and scale content. This border seems to be well localized. Upon crossing it, phase locking is suddenly lost, and the transition to turbulent behavior occurs rather quickly. Section 7, the conclusion, offers a general discussion of the examined body of research. In its final part, the basic major questions still not answered are put forth, as well as the fundamental doubts and uncertainties still related to these nonlinear permanent models.
2. SOLITARY WAVES IN
ONE
DIMENSION: A SHORT SYNOPSIS
The purpose of this section is to give a brief overview of the important one-dimensional nonlinear equations obtained in most instances during the derivation of solitary wave solutions from complex two- or three-dimensional geophysical models. Again, for a more detailed exposition, the reader is referred to Scott el al. (1973) and Miles (1980). The most important of such equations are the Korteweg-de Vries (KdV) equution: 24
+ auu, + u,,,
=
0
(2.1)
0
(2.2a)
the generalized form of the KdV equation:
ut
+ aZPu, + u,,
=
with p a positive integer. The most important in the class of equations (2.2a) is with p = 2, namely
PLANETARY SOLITARY WAVES IN GEOPHYSICAL K O W S
155
the modified Korteweg-de l’rics (mKdv) equation: u,
+
+ u,,,
0
(2.2b)
+ 111, + al24l2u = 0
(2.4)
( Y U h ,
=
the Boussinesq equation:
the nonlinear Sdiriidinger cqiiation:
u,,
Other equations, such as the sine-Gordon equation, the Born-Infeld equation, nonlinear lattice equations-all admitting solitary wave solutionsare more important in branches of physics other than geophysics. A general equation is the Hirota equation (see Scott et al., 1973), allowing for Nsoliton solutions, which reduce to a linear equation, the mKdV equation, or the nonlinear Schrodinger equation in different parameter limits. All previous equations (2.1-2.4) are normalized. The constant a can assume different values according to the chosen scale and measures the relative strength of nonlinearity versus dispersion. It is to be pointed out that the KdV equation (2.1)-as well as its generalized forms Eq. (2.2a) and mKdV equation (2.2b)-in the form in which it is here written and is commonly used, is in reality a simplification of the complete KdV equation: u,
+ 21, + N U U , + u,
=
0
(2.5)
Eq. (2.1) can be obtained from Eq. (2.5) passing to a frame moving with the linear speed co, here normalized to be unity: co = 1; namely, the frame defined by the change of coordinates s = x - t; 7 = 1. Thus, for future derivation from more complex systems, it must be borne in mind that Eq. (2.1) is the KdV equation after subtraction ofthe linear, dispersionless speed. As a final point I shall make a distinction between the terms “solitary wave” and “soliton.” A solitary wave is a nonlinear wave solution to the considered model. “Soliton” has a more cogent meaning, namely, “a solitary wave which preserves asymptotically its own identity upon collision with other solitons” (Scott et al., 1973). As such, the soliton is not destroyed by a superimposed structure of any intensity whatsoever. As we shall see, the solitary wave solutions to the geophysical models examined in the following sections behave like solitons only in limited parameter subranges. The KdV-mKdV equations are the most common in the set (2.1-2.4) for the geophysical applications that follow. Therefore, I shall proceed in reverse order in this short review, starting from Eq. (2.4), and discussing in more detail the KdV-mKdV models.
156
P. MALANOTTE RIZZOLI
The nonlinear Schrodinger equation (2.4) admits as solution: u
=
A s e c h [ m ( x - 24.41 exp[i(ue/2)(x - u,t)]
(2.6)
in which u, is the “envelope” velocity and u, is the “carrier” velocity. Thus, Eq. (2.6) describes solitary waves which are envelopes of dispersive packets. The “envelope” speed u, is the group velocity of the packet; the “carrier” speed u, is the phase velocity of individual fronts inside the group. Envelope solitary waves are also envelope solitons, in the sense that they behave as such in collision events. Envelope solitons of sea-surface gravity waves have been discussed by Whitman (1974). Envelope solitary waves obeying the nonlinear Schrodinger equation are also discussed by Boyd (1980b), for waves of the Rossby-gravity mixed type, in his review of nonlinear equatorial waves and by Yamagata (1980). See also Pedlosky (1 972, 1976) and Hide (1979). The Boussinesq equation (2.3) can be derived to describe shallow-water waves propagating in both (positive and negative) x directions. It does allow for soliton solutions (Scott et al., 1973), and it has been studied as a model for gravity water waves by Whitham (1974) and Miles (1980). It should be pointed out that Eq. (2.3) is not written in the frame of reference moving with the linear dispersionless speed co = 1. In fact, if in Eq. (2.3) we allow for waves propagating only along one characteristic s = x - cot = x - t, namely, if we allow for waves only moving to the right, we reduce the Boussinesq equation to the KdV equation (2.5). The KdV equation (2.1), first derived in its form Eq. (2.5) to describe the dissipationless propagation of shallow-water gravity waves, is the simplest equation in which a simple nonlinearity and a simple dispersion balance to give soliton evolution. The most general solution to Eq. (2. l ) in the form of a traveling wave can be found as follows. Let c be the speed of the traveling wave. In the frame moving with the wave itself, if s = x - ct, Eq. (2.1) becomes an ordinary differential equation:
u,(au - c) + u, This can be integrated twice to give 1 a C - u: = - - u3 - u2 2 6 2
+
=
0
+ M 1 u+ M2
(2.7a)
(2.7b)
where M I ,M2 are integration constants. The general traveling wave solution to Eq. (2.7b) is given by the elliptic integral
JI
with F(u) = -(a/3)u3
du/\/Fo
=
x - ct
(2.7~)
+ cu2 + 2M,u + 2M2 and uo = u[(x - ct) = 01. The
PLANETARY SOLITARY WAVES IN GEOPHYSICAL FLOWS
157
general solution for M , , M2 # 0 is a periodic wave called the “cnoidal wave,” being given by the square of the elliptic function cn. For the actual solitary wave, the requirement is that the solution be localized in space, together with its derivatives. This implies M I = 0 and M2 = 0. Then Eq. ( 2 . 7 ~is ) integrated to give the solitary wave:
From Eq. (2.8) it is evident that: ( 1 ) The amplitude of the KdV soliton increases with its speed; or, conversely, the correction to the linear, dispersionless speed co = I is amplitude dependent. (2) The sign of the solitary wave depends upon ,the sign of a:
sgn(amp1itude) = sgn a Thus, the KdV soliton is rithc)r compressive or rarefactive, not both. In the generalized form of the KdV equation (2.2a) this is true whenever p is odd. The converse is true when p is even, and therefore for the mKdV equation (2.2b) both solutions are possible, with sgn(amp1itude of compressive wave) = sgn a sgn(amp1itude of rarefactive wave) = -sgn a The KdV equation is endowed with remarkable properties and is the first one to which powerful techniques, such as the inverse scattering method, were successfully applied. The most striking property of the KdV equation is that an infinite number of independent conservation laws, and therefore of integral invariants, exist for it. These were first discovered by Miura et al. ( 1968) through a somehow “magic” mathematical transformation. Each of the integral invariants takes the form of a constant-coefficient polynomial in u(x, t ) and its derivatives, integrated over all space. An infinite number of conservation laws have also been found for the modified KdV equation, the sine-Gordon equation, and the nonlinear Schrodinger equation (see Scott et af.,1973). In another pioneering paper, Gardner et af.(1974) applied and solved the KdV equation through the inverse method. The inverse method is a powerful tool which allows one to find solutions to nonlinear initial value problems-such as that constituted by the KdV equationthrough a succession of linear computations. Lax (1968) gives a general formulation of it, extending its application to other equations. The details of the method are beyond the scope of this article; they can be found in the references. It is enough to say that the method relates the KdV initial value problem to the Schrodinger operator for which the KdV solution u(x, t )
158
P. MALANOTTE RIZZOLI
constitutes the potential well. From the knowledge of the solution of the Schrodinger problem for the potential well uo = u(x, t = 0), one determines the potential well itself for all times u(x, t ) , namely the desired solution. The focal point then is that each bound state of the Schrodinger equation corresponds to a soliton, whose speed and amplitude are determined by the energy level of the corresponding bound state. The continuous part of the spectrum in the scattering problem gives a dispersive wave train which is left behind. The number N of bound states is thus the number of solitons asymptotically emerging from the considered initial condition, uo(x,0), which, together with the dispersive packet, constitute the general solution. The inverse method is indeed very powerful, as it provides the solution to the KdV initial value problem at all times and in all the infinite domain, but is also rather complex. If one is satisfied to know mainly the asymptotic solution, namely. the number of solitons emerging from a given initial condition, and their shape and amplitude (speed), then the use of the conservation laws can give a much quicker-and simpler-answer. In fact, it can be shown that the N-soliton solution to the KdV equation is related to the variational problem of the N 1 integral invariant subject to the condition that the first KdV Nintegral invariants are conserved. Thus, imposing the conservation of N integral invariants allows one to find the parameter range, if any, in which the given initial condition to KdV will asymptotically evolve into N separated solitons, as well as the speeds (amplitudes) and shapes of the solitons themselves. Actually, the method also allows one to find the Lagrange equation for the N interacting solitons which, however, most of the time must be solved numerically because of its complexity. Details of the method are given in Scott et al. (1973) and in Jeffrey and Kakutani (1972). This last paper is a very comprehensive discussion of the KdV equation properties. The existence of the infinite number of integral invariants finds its proper perspective when it is shown that KdV is a completely integrable Hamiltonian system with an infinite number of degrees of freedom. In fact, not only a generalized Hamiltonian can be found for it, but it can also be shown that the inverse scattering problem can be regarded as a set of canonical coordinates for KdV (see Scott et ul., 1973). To the previous properties, one must add that the stability of the soliton as solution to KdV has been thoroughly demonstrated by Benjamin ( 1972). (See also Jeffrey and Kakutani, 1970, for the stability to small perturbations in the propagation direction and Oikawa et al., 1974, for the stability of the one-dimensional soliton to small, transverse perturbations.) All this makes KdV and its solution, the soliton, a deterministic, completely predictable nonlinear system. The above obviously applies to the one-dimensional KdV equation and
+
PLANETARY SOLITARY WAVES IN GEOPHYSICAL FLOWS
159
initial value problem. However, the model equations used for large-scale two- or three-dimensional flows constitute quite a different set of initial value problems. At the most, they can be thought of as being dominated by a KdV dynamics in some parameter subrange. The question, then, is to find this parameter subrange, for both the ocean and the atmosphere, to explore its extent, and to see how flexible the KdV dynamics in two or three dimensions is to the variations of the parameters themselves. Finally, it must be pointed out that the previous results hold only for the KdV equation in the unbounded domain. From the point of view of geophysical applications, mathematical complexities usually prevent us from finding the complete analytical solution. This is then analyzed by means of numerical experiments, camed out in finite, closed domains, very often with periodic boundary conditions. In this context, theoretical tools such as the inverse scattering method are of little practical use. This last, in fact, has been applied until now only in the unbounded domain and, even more stringently, only when the initial disturbance (the initial potential well of the Schrodinger problem) is smooth and localized in space. The difference can be intuitively understood by thinking of the Schrodinger equation, the first step in the inverse method for KdV: it is a very different problem to find the bound states (and the continuous spectrum) of a n isolated, localized potential well versus the same potential well “cut” at some boundaries and repeating itself periodically! Another restriction is that, when passing to numerical experiments, one necessarily truncates the infinite number of degrees of freedom of the original continuum model to a finite number. All the analyzed properties should then be reviewed in the context of the discrete model one is actually dealing with. Very often, these properties hold only to a limited extent. A typical example is the conservation of the integral invariants of a continuum model. In its discretized version, sometimes only a few of the original integral invariants (in their discretized versions!) are conserved. Thus, a point usually neglected which must be borne in mind is that our numerical models are not the analytical ones whose properties we have explored, and that some of these properties may be profoundly modified when discretizing a continuum system. Notwithstanding these limitations, the analysis of the KdV continuum model and of its initial value problem in the unbounded domain usually provides a remarkable insight for the same problem in a finite, periodic domain and in a discretized version. Most often, the solution of the latter problem can be constructed close enough to the solution of the former one to be considered the same for all practical purposes. It is enough to quote the famous paper by Zabuski and Kruskal(1965), in which they explained the Fermi-Pasta-Ulam (FPU) recurrences of the discretized anharmonic lattice by substituting for this last the continuum KdV, thus actually doing the inverse procedure of the usual
160
P. MALANOTTE RlZZOLl
numerical simulations. Let us further remark that properties which in the infinite domain can be looked at in physical space, in the bounded, periodic domain can be equally well observed in Fourier space. Again, as an example let us take the KdV equation. The emergence of N solitons from a given initial condition and their separation from one another in physical space in the unbounded domain are equivalent to the FPU recurrences in the finite, periodic box. Here, because of periodicity at the FPU recurrence period, the solitons produced focus together, reproducing a condition identical to the initial one. If the physical-space picture should not be revealing (for instance, solitons prevented from separation because of shortness of the finite box), observation of the spectral properties provides equally illuminating information. In the example considered, if the flow energy is initially concentrated in a few Fourier modes, and if other modes are excited during the flow evolution, at the FPU recurrence time all the energy will be back only in the initial modes, evidence of the focusing of the solitons. With this background of theoretical knowledge on the behavior of onedimensional solitary wave models, systems will be considered below in two and three dimensions for geophysical flow evolutions which can be analyzed in terms of solitary waves.
3. THEEXISTING MODELSFOR LARGE-SCALE PERMANENT STRUCTURES As remarked in the introduction, coherent structures do exist and are clearly recognizable amidst the variety of background turbulence present in the ocean and the atmosphere. In both fluids, geophysical models for large-scale motions are very often those used for two- and three-dimensional turbulence. Two-dimensional turbulence theories successfully account for a variety of observational aspects of atmospheric and oceanic motions. For example, Leith ( 1 97 1 ) shows that the eddy kinetic energy spectrum in the atmosphere, as measured from the observations of several authors, has a tendency toward a K - 3 dependence at the wave numbers between 10 and 20. This tendency is typical of the inertial range of two-dimensional turbulence. However. no turbulence theory, bearing upon the random phase approximation, can explain the orderly, phase-coherent structures which are observed in the ocean and atmosphere and do persist for times much longer than the expected predictability time. I shall now give some examples of these coherent structures, above all examples recently related to nonlinear, permanent-form models by different authors. The first example refers to the Jupiter atmosphere. The problem is that of an oval structure superimposed on a basic, zonal shear flow. This is the
PLANETARY SOLITARY WAVES IN GEOPHYSICAL FLOWS
161
famous Great Red Spot, persisting for centuries. Observation of motions around the Red Spot suggests very interesting evidence. For instance, for many decades in this century a disturbance existed which, starting at the southern edge, interacted several times with the Red Spot itself. During the interaction, observers noted a rapid acceleration of the disturbance and its reemerging and reforming without a change of shape. These properties stimulated Maxworthy and Redekopp (1976a,b) t o model the Red Spot as a solitary wave eddy. Specifically. Redekopp ( 1 977) formulates the theory of a solitary eddy superimposed on a mean shear flow, both in a barotropic and in a baroclinic atmosphere. For a barotropic atmosphere, the final equation is the KdV model. In a baroclinic atmosphere, with a constant Brunt-Vaisala frequency, the mKdV equation is the result, which allows for both elevation (E) and depression (D) solitary wave solutions. A typical stream function pattern of Maxworthy and Redekopp’s solution together with the assumed mean shear flow is shown in Fig. 1 (adapted from Maxworthy and Redekopp, 1976a). However, it must be pointed out that these solutions are weak-amplitude waves, while the amplitude implied by Fig. 1 may be beyond the range of the small-amplitude theory. Ingersoll and Cuong (1981) model the long-lived Jovian vortices as a different type of permanent structure, the modon (see below). Here a point must be made. It is the writer’s opinion that it may be a rather strong statement to assert that the Red Spot is a solitary wave. Nevertheless, one statement can be made on safe grounds. The Red Spot is the most exotic example of a large-scale, coherent structure characterized by remarkable permanence and stability. For these properties, it is strongly suggestive that there may exist a parameter subrange in which delicate balances between nonlinearity and other physical effects such as dispersion are
FIG. 1 . Theoretical mean shear flow profile and theoretical stream function composed by the superposition of depression (D) and elevation (E) solitary waves. (Adapted from Maxworthy and Redekopp, I976a.)
162
P. MALANOTTE RIZZOLI
FIG.2. Modon solution to the equivalent barotropic vorticity conservation equation on the infinite @-plane.(Adapted from Flierl ef al., 1980.)
achieved. Because of them, these low wave number scales result in being much better correlated than what is predicted by any turbulence theory. These balances are those characterizing the simple solitary wave model. A second experimental configuration which persists for long times is that of the blocking ridge phenomenon of the atmosphere. Recently, blocking activity has been compared with a permanent structure called a “modon,” to which I shall return later in more detail, and consistency has been found between the experimental situation and the theoretical model (McWilliams, 1980a). The modon structure which was compared with a blocking ridge is that of a simple dipole, as shown in Fig. 2, the simplest case of a modon (adapted from Flierl et al., 1980). The third example of coherent structures endowed with remarkable per-
PLANETARY SOLITARY WAVES IN GEOPHYSICAL FLOWS
163
sistence and stability refers to the cyclonic (anticyclonic) rings which are observed to the south (north) of the Gulf Stream. Flierl ( 1979a) has proposed a solitary eddy model for these rings as a solution to the quasi-geostrophic, &plane equations in the continuously stratified form, with a basic shear flow resembling the Gulf Stream pattern. This baroclinic eddy, shown in Fig. 3 (adapted from Flierl, 1979a) is characterized by radial symmetry and fluid speeds exceeding the phase speed of the ring itself. Recently, moreover, during the POLY MODE Local Dynamics Experiment (LDE) “small” mesoscale features were discovered, with radii smaller than the Rossby deformation radius. These coherent structures are also endowed with very long lifetimes, and no theory has yet been formulated for them (McWilliams et ul., 1982). The previous examples of models for coherent structures have been shown because they have been specifically related to observed large-scale eddy configurations. Now the variety of existing solitary wave solutions for mesoscale motions will be investigated in the context of a unified approach. Along its line all existing permanent solutions can be derived and classified. This classifying approach was first used by Flierl (1 979b), and the technique was employed by Ingersoll (1973), Stern (1975), Larichev and Reznik (1976a), Flierl (1979a,b), Flierl ct ul. (1980), Malanotte-Rizzoli (1980b, 1982a), and Charney and Flierl (1981).
- 20
I1
9
3
t
*
-
15
8
-
10
5
0
0
10
-4
20
-7
30
-8
FIG. 3 . Baroclinic solitary eddy with radial symmetry, solution of the quasi-geostrophic potential vorticity equation on the P-planc with a mean shear flow. (Adapted from Flied, 1979a.)
164
P. MALANOTTE RIZZOLI
3.1. A Unlfred Approach: Length Scales of the Order o f t h e External Deformation Radius
-
We consider a fluid which is incompressible, hydrostatic, and non-Bouscc is the sound sinesq in the @plane approximation. For such a fluid, c, speed limit (incompressibility condition); X = H / L 4 1 is the fluid aspect ratio, where H is the fluid depth and L is the horizontal length scale of the considered motions: and ( L / r o )tan Oo 4 1, where ro is the radius of the earth and Oo is the mean latitude. For such a fluid, the equations of motion are, in dimensional form
Db,/dt
+
Duldt - f v = -( l/p)p, Dvldt + f u = -( 1 /p)p,, ( 1IF)(1 + b,/g)pz = b, u, v , + wZ= 0 (continuity) (buoyancy equation) (N’/g)wb, + I?’w = 0
+
(3.1)
where subscripts x, y, z indicate partial derivatives and D/dt = d / d t + u(d/dx) v(d/dy) w(d/dz) is the substantial derivative; f = f o + ~ J isJ the Coriolis parameter in the 0-plane approximation; p = ptotal - @(z)is the pressure after the subtraction of the average z-dependent contribution given by j Z= -gj(z); b, = g ( j - p ) / p is the potential buoyancy for an incompressible fluid, with g the gravity acceleration, j(z) the average vertical density distribution, and p ( x , y , z , t ) the density anomaly: and N z ( z ) = [ - g / j ( z ) ] ( d j / d z )is the Brunt-Vaisala frequency. We shall scale the set of equations (3.1) as given in the tabulation:
+
+
scale for ( u , v ) velocity components horizontal scale of motion vertical scale of motion average Coriolis parameter pressure scale buoyancy scale advective time scale scale for w-velocity component, from continuity With the above scaling, the following dimensionless parameters are defined: fi = (L/ro)cot t = UJfoL (Rossby number) (Froude number, or divergence parameter) A = f;L’/gH S = N’(z)H ‘/f$L2
165
PLANETARY SOLITARY WAVES IN GEOPHYSICAL FLOWS
and w = A will be assumed. With the above, set (3.1) becomes the dimensionless set of equations
+ 2 4 U , + U24,.) + (cA)M)21z ( 1 + by>v = -( 1 + tAb,)px t(v, + uv, + vv,.) + (tA)MT- + ( 1 + py)u = -( 1 + tAh,)p,,
t ( ~ ,
-
(
h,,
I
+ u b , + vhp,,+ A d > , ,
+ cAh,)(p= Asp) = 6, zit + V , + Aw, = 0
(3.2)
-
t~A2SfihP
+ S- A w = 0 -
t
and f = 1 + by. We shall look for steadily translating solutions to equations (3.2); therefore we shall work in the frame of reference moving with the wave phase speed c and defined by s
=
,I--
ct
7 =
I
In this frame, the motion is steady. In this frame, therefore, and for a constant Brunt-Vaisala frequency, set (3.2) will admit Bernoulli’s integral of motion which can be derived for a steady, dissipationless, stratified system. Bernoulli’s integral of motion can be used to obtain the steady translating, permanent-form solutions we are looking for. We shall mostly work, however, with another equation derivable from set (3.2), namely, the potential vorticity equation, and examine its various forms in the various subranges defined by the variations of the two dimensionless parameters A, j.In these subranges, and under expansion in the small parameter characterizing the system, a generalized potential vorticity is conserved. Then, the Bernoulli integral and the vorticity integral can be related to a unique functional whose asymptotic shape is unequivocally determined by the flow assumptions at infinity. The Rossby number t = [ i / f , , L = U/c, if c = f o L is the scale for the solution phase speed, is also the ratio between particle speed and phase speed. As such. it can be used to define the intensity of the solution itself. The wave will be classified as
weak : strong:
li$c
U 2c
--
t < l t 2 I
The general form of the vorticity equation derivable from set (3.2) is
where jX
(a/&)
=
v, - ( u - c), is the flow relative vorticity and D/dt
= (I(
-
c)
+ v(d/dy) + (Aw)(d/dz) is the substantial derivative. For a constant
vertical average profile N 2 ( z ) ,we can also write the Bernoulli integral, in the frame moving with the wave:
P. MALANOTTE RIZZOLI
166
(3.4) The various parameter subranges will now be examined as defined by the variations of A, S. The first subrange to be examined IS the case:
h,-O
A=l
S=o
namely, the planetary, barotropic motions in which divergence plays an essential role, with length scales L of the order of the external deformation x 3000 km for an ocean with depth H 4000 m at middle radius latitudes. Under these assumptions, set (3.2) becomes the usual set of shallow-water equations:
-
m/fo t(u -
t ( u - c)(u - c), + tv(z1 - c), - ( 1 + 0y)v = - t v r c)v,+ tvu, + ( 1 + &)(u - c) + c( 1 + j y ) = -tv, Kt?)+ h)(u - c)l, + [(t? + hN1, = 0
(3.5a)
where ~ ( s y, ) is the surface elevation and h(s, v) the depth. From the continuity equation, a generalized stream function can be defined: (c?)
+ h)(u
-
-*,
c) =
(t?)
+ h)v = +Q,
In terms of Q, the vorticity equation ( 3 . 3 ) and the Bernoulli integral (3.4) can be written, respectively, as
0
(3.Sb)
J(B, *) = 0
(3.k)
J ( V , Q)
with
=
VQ tq
+h
the potential vorticity, and with
the Bernoulli functional. For a similar treatment of the barotropic shallowwater equations see also Charney and Flier1 (1 98 1). Equations ( 3 3 ) and ( ~ S C )expressing , the conservation of potential vorticity and Bernoulli functional, imply
167
PLANETARY S O L I T A R Y WAVES IN GEOPHYSICAL FLOWS
cV.-
V* €7 t I?
( t / 2 ) [ 9 f+ \k5]
+f
-1- ( t q
=
(q
+ h)’
+ h)E,(\k)
-
(c?)
(3.5d)
+ h)2/I(y)
where F1(q), F2(\k)are a prrcm arbitrary functionals. One can now make use of the fact that we are looking for permanentform solutions which decay asymptotically in space. We require, therefore, that the wave contribution to the total field satisfy
This procedure was first used by Flier1 (1979a,b). Different conditions can be imposed upon the flow at -tw. For instance, we can require the flow to have a mean shear U(y);we can require a complex topography. Let us note, however, that terms like (tq /I)2 and (€7 /I)3as well as the right-hand side of Eq. (3.5d), namely, (q h)Fl(\k) and (q + h)’F2(q)-are nonlinear, the nonlinearity being provided by selfinteractions of the wave field functions. This nonlinearity disappears in the quasi-geostrophic approximations of Eq. (3.5d). Thus, quasi-geostrophic permanent models need external features-a mean shear flow, a variable topography, etc.-to provide the necessary steepening effects balancing dispersion. In the shallow-water limit examined here, no external features are required. We shall then assume no basic shear and a constant topography h = H = 1. Then, the requirement that the wave field vanishes asymptotically means
+
+
+
Then, the asymptotic forms of the functionals FI and F2are
+
F17 ( c y )= F , , , ( Z ) = (b/c)Z 1 F 2 . 7 ( ~ .= ~ )F2., ( L )= (b/2c)Z2+Z
+ (t/2)c2
(3.5e)
From Eq. (3.5e) the general property that F , , F2are not independent functionals is evident: F, = (F2)’. Equations (3.5d) with their asymptotic formulation (3.5e) allow us, in principle, to solve the system in the two functions \k, 7. Two possibilities can now be considered. First, the asymptotic functional shapes (3.5e) are valid in the interior of the fluid only along those isolines q = (constant) which extend continuously from -ato +a. Closed isolines can, however, exist. Here, we encounter the fundamental distinction leading to the two basic categories of solitary wave models.
168
P. MALANOTTE RIZZOLI
(1) F,, F2 are analytic, single-valued functionals. The same definitions given by Eq. (3.5e) can be used in the interior of the fluid, on open or closed streamlines. (2) F,, F2 are multivalued functionals. Different definitions will be used in the exterior region on those isolines extending continuously to +-a (for which Eq. 3.5e holds) and on closed isolines in the interior region. The fact that F , , F2 are multivalued implies that there is a “limiting” closed streamline 9 = (constant) upon which F , , F2 will pass from the functional form of the interior region to that of the exterior region. Solutions belonging to the first class of analytic functionals were first found by Clarke ( I97 1) in a channel, for the weak-wave case. Clarke also examined the behavior of these solutions when a mean shear flow U(y)and a variable topography h(y)are included. Charney and Flierl (1 98 1 ) extended the derivation to the strong-wave case. Solutions belonging to the second class of multivalued functionals were first found by Stern (1975) and Larichev and Reznik (1976b) for the onelayer barotropic case in the quasi-geostrophic approximation (see Section 3.2)-not, therefore, starting from the complete set (3.5a). Later, they were reviewed also in the context of a two-layer model by Flierl et al. (1980), always in the quasi-geostrophic approximation. These solutions are strong ( U 2 c) and are also called modons. A typical example for the simplest onelayer case is the dipole of Fig. 2. Strong permanent-form solutions of the two-dimensional vorticity equation (not on the P-plane) characterized by closed streamlines, are also the V-states of Deem and Zabusky ( 1978). These are uniformly rotating and uniformly translating finite-area vortex regions of constant vorticity density, solutions to the two-dimensional barotropic vorticity equation. The dynamics of modon-like solitary vortices is also discussed by Shen ( I98 1 ).
3.2. Length Sculcs Smaller Than the External Deformation Radius The second subrange to be considered is for motions which are very large and baroclinic: A < l ,$+l or L i = N 2 ( z ) H 2 / f ;< L2 < g H / f ; namely, motions smaller than the external deformation radius K H / f o but still much greater than the baroclinic Rossby radius LR. For these motions we take A N O(t) ,$ N O(t2)
PLANETARY SOLITARY WAVES IN GEOPHYSICAL FLOWS
169
Now the starting equation I S the full set (3.2) under the above assumptions, always in the frame moving with the wave speed c. We now split the velocity field according to (U
-
C) =
v
=
-$, +$,
+ A$, + A$,
+ + t$)
=
-$)
=
$s
t$s
to include both the rotational and irrotational parts of the flow. We then + expand all field functions in power series of t , in particular $ = t$l - ,$I = do t$l . Then, the vorticity and Bernoulli equations (3.3), (3.4) are, at lowest order:
+ -
+
+---
+
and 0; = d2/ds2 d'/d$ is the horizontal Laplacian. In the absence of a mean shear flow, repeating the previous derivation, we again find that the two functionals, for the vorticity and for the Bernoulli equation, are expressed by identical functional shapes as in Eq. (3.5e):
In this case, expressions (3.6b) hold for whatever topographic relief h(y) is assigned. Now, however, the various nonlinear terms present in Eq. (3.5d) have disappeared from their equivalent counterpart obtainable from Eq. (3.6a). This means that either we consider modon solutions-namely, multivalued functionals-or, in the analytic case, a variable topography, an external mean shear, or baroclinicity is necessary to provide the required wave steepening. Therefore, as soon as the considered scales of motion become smaller than the external deformation radius for which a barotropic shallow-water dynamics may be appropriate, external steepening of the wave field becomes necessary. This must be provided by external features such as topography or a mean shear, as divergence is too weak on these reduced wavelengths, even when allowing for a free surface. The final case considered here is for length scales of the order of the baroclinic Rossby radius, for which the divergence parameter is small: A < 1 N O(t) (Boussinesq approximation) and baroclinicity is important: $ e O( 1 ) or L = LR, the Rossby radius. We shall work in the quasi-geostrophic approximation, obtainable from set (3.2) when 6 N O ( E and ) upon expansion in power series o f t . Then, from the zero-order geostrophic bal-
170
P. MALANOTTE RIZZOLI
ance, we define the zero-order stream function as
( u - c)
=
-$,
2,
= +$s
and the vorticity equation, which will now be used alone, is
+ ( 1 + BY) + (~/A)(a/az)(+z/.91= 0
(aI/dt”2,+
(3.7a)
with 7 2 -
DH
- --
dt
a2/as2 + a2/a.v2 a ( u c) a + -
-
as
2,
~ J J
(the horizontal Laplacian) (the horizontal substantial derivative)
For a detailed derivation of the quasi-geostrophic approximation see Pedlosky (1979) and Charney and Flierl (1981). It is in the context of Eq. (3.7a) that most ofthe permanent-form solutions in the literature have been obtained. Equation (3.7a) must be coupled with proper boundary conditions in z. In dimensionless form, and in the frame moving with the wave, these are J($-.
c$
+ cv) = 0
at
z
=
0
4$;,4+ cy) = -[(Iadl/H)~]~2(-1)J($,h ) at
z
= -1
+ (lAdl/H)h(.v)
(3.7b)
where lAdl is the fractional variation, assumed small, around the mean depth H . Equation (3.7a) with the boundary conditions (3.7b) has been used by various authors under different conditions. The above-mentioned solutions obtained by Maxworthy and Redekopp (1976a,b) and Redekopp (1977), and of which Fig. 1 is an example, are solutions of Eq. (3.7a) in the zonal channel, superimposed on a basic shear flow U(y). The barotropic version of Eq. (3.7a) leads to the KdV model; Eq. (3.7a) itself with a constant BriintVaisala frequency N(z)leads to the mKdV model, with elevation and depression permanent waves, whereas a variable N ( z ) leads again to the KdV model (Redekopp, 1977). All of these solutions are weak-amplitude waves. Always with steepening effects provided by interaction with a mean shear, in the zonal channel, but with strong intensity, are the solutions found by Long ( 1 964b). Flierl (1979a) studied the completely baroclinic system (3.7a,b), over constant topography, for which the bottom boundary condition is identical to the surface one. Allowing for an external shear flow either barotropic or baroclinic, he has found very interesting solutions in the form of isolated, radially symmetric eddies in the infinite 0-plane. These solutions are strong ( U Z c) and exist only in the baroclinic case. The reader is referred to Flierl
171
PLANETARY SOLITARY WAVES IN GEOPHYSICAL FLOWS
( 1 979a) for a detailed treatment of the fully stratified problem in the case
of analytic functionals both for the vorticity integral (3.7a) and the surfacebottom boundary conditions (3.7b). Malanotte-Rizzoli found solutions to the barotropic form of Eq. (3.7a) in the zonal channel over variable topography. These solutions can be both weak ( U 4 c; Malanotte-Rizzoli and Hendershott, 1980) and strong (Malanotte-Rizzoli, 1980b, 1982a). They have been derived for the barotropic model, but baroclinicity can be easily included by adding a divergent term to obtain an equivalent barotropic model or by starting directly from Eqs. (3.7a,b). In this latter case, the solution upon expansion in the small parameter of the system, at zero order, is separable in s and ( y , z). Further separation in y and z is possible only when the topography h(y) is linear in y. Then, at zero order, the solutions are identical to the linear, topographic Rossby waves with an arbitrary profile " z ( z ) as treated by Charney and Flier1 (1981). However, if the topography is linear in y , the constant topographic slope is equivalent to a @-term.As previously remarked, for length scales smaller than the external deformation radius, the beta effect is not a sufficiently varying index of refraction to provide the necessary steepening. This can be achieved only with a nonconstant topographic slope. Topographic solitary waves with radial symmetry have also been discussed by Henrotay ( 1981 ). A specific example will now be given about how permanent solutions are obtained through the previous treatment. For this, we shall choose a very simple model, namely, the barotropic, quasi-geostrophic vorticity equation over variable topography:
o&+,+ / I , * , + J(+. 02+)= 0
(3.8a)
Passing to the reference frame moving with the wave speed c: Oh+
-
+ M y ) = F(+/c + y )
(3.8b)
+
-
0 The flow is constituted only by the required wave solution. Thus, as x +a. This means F , 0')= h(J4 (3.8~) the asymptotic form of the functional is the topography itself. Excluding all modon-like solutions, we take F ( Z ) to be analytic. Thus F ( Z ) = h ( Z ) everywhere. Then Eq. (3.8b) can be written as (3.8d) F($/c + .I)) - F,(.I,) h(+/c + y ) - h(y) Let us now notice the following. The required permanent-form solutions are both weakly dispersive and weakly nonlinear, and nonlinearity and dispersion are of the same order. At zero order, the solution is a superpo-
%$
1
172
P. MALANOTTE RIZZOLI
sition of linear, dispersionless waves. There are only two ways in which nonlinearity can be made a higher order effect:
Case I. The wave intensity is weak: C’ 6 c or $/c 6 y . Case 11. The waves are strong: LT+ c or $/c y , but the functional F = (topography) is “weak,” namely, quasi-linear.
+
We shall derive permanent-form solutions for cases I and 11, and specifically solutions which are asymmetric in the zonal channel. For case I, asymmetry means d2/dx2 h2 #/ax2, where a2 is the square aspect ratio of north-south to east-west length scales. Weakness of the solution means $ t$ and balance of nonlinearity versus dispersion requires a2 = c. Then Eq. (3.8d) becomes
-
-
t(rCJV
+
4 Y. Y)
=
4 4 / c +Y)- IW
(3.9a)
If we now Taylor-expand the right-hand side of Eq. (3.9a), simultaneously expanding $, c in powers of t, we obtain
Order zero: $on
-
( 1/co)h ’ ( U ) $ O
=
0
g(s)KY) $(O) = $(1) in
(3.9b)
$0 =
Order
01y1
1
t:
$Il.,.
-
(l/co)h ’(Y)$I = -(cl/c;)h’(Y)$o - $OH + [h”(.v)/2c6l$; in 0 5 y I 1 $](s, 0) = $](s, I )
The usual solvability condition gives the KdV equation for &):
(c,/c;)u,g+ (1/2c:,)afl2
+
a3g.,y =
0
(3.9c)
These are the solutions originally found by Malanotte-Rizzoli and Hendershott (1980). The nonlinearity vanishes if a2 = -I: h”(y)& d v = 0, namely, if the topography is linear in y . For case 11, we consider again asymmetric solutions in the zonal channel: d2/dx2 6* d 2 / d x 2 .The aspect ratio is now the small parameter, as the required solutions are strong: $/c 2 O( 1). Let us take a quasi-linear topographic relief, specifically a quadratic relief:
-
h ( Z ) = nzZ
+ (S2/2)Z2
Then Eq. (3.8d) becomes $,u
+ S2$.,
=
m($/c)+ S2[( 1/2c2)$2 + (Y/C)$l
(3.1Oa)
173
PLANETARY SOLITARY WAVES IN GEOPHYSICAL FLOWS
Again expanding both $ and c‘ i n powers of J2:
Order zero: $on
-
(m/co)$o= 0 $0 = , r ( . C ) W 3 $(0)=$(1)
(3.10b) in
O
~
y
~
l
Order 6’:
GI)’?- ( W d $ I = [-(mc.,/d + (J’/cb)l$o Gl(s, 0) = $,(s, 1 )
in
- $O.H
+ (1/2C&G
0 I y I 1.
The solvability condition for the q J ’ ) equation provides again the KdV model for g(s). These solutions are strong and asymmetric. The zero-order problem gives sine and cosine modes in the channel; thus, permanent structures exist only for odd-n modes. The solutions are therefore the equivalent of Long’s waves (1964b) over a mean shear flow. For a detailed treatment of the possible topographic shapes allowing for permanent solutions see MalanotteRizzoli ( 1982a). To summarize, Table I shows the existing models for coherent structures, classified according to the type of functional F of the potential vorticity conservation equation.
4. EVOLUTION 01. COIIERENT STRUCTURES: THE INITIAL VALUEPROBLEM
In the previous section a review has been given of the existing permanentform solutions found by different authors to the model equations suitable for mesoscale oceanic and atmospheric motions. Examples have also been shown of specific model solutions which have been compared to coherent structures actually observed in nature, such as Gulf Stream rings or bloclung configurations. In this section we shall examine the same permanent-form structures in the context of the initial value problem posed by the chosen model, namely, allowing for slow time variation of the flow field. We shall thereafter generalize and extend the procedure to treat the interaction of coherent structures, that is, to study their properties under collisions. To do this, we first choose a specific model of geophysical significance, which allows for permanent solutions, and which will be used in all the following. The chosen model is the quasi-geostrophic, barotropic potential
174
P. MALANOTTE RIZZOLI TABLEI. EXISTINGMODELSOF COHERENT STRUCTURES
Analytic functional F: Quasi-geostrophic Strong: U t c Channel solutions (asymmetric) Barotropic and baroclinic Steepening: mean shear U ( y ) Non-quasi-geostrophic Weak: U < c Channel solutions (asymmetric) Barotropic Only P-plane Non-quasi-geostrophic Weak: U < c Weakly divergent Channel solutions (asymmetric) Barotropic Steepening: mean shear and/or topography Quasi-geostrophic Weak: U < c Channel solutions (asymmetric) Barotropic (KdV) and baroclinic (mKdV) Steepening: mean shear, barotropic or baroclinic Quasi-geostrophic Weak: U < c and strong: U 2 c Channel solutions (asymmetric) Barotropic and baroclinic Steepening: topography h(y) Non-quasi-geostrophic Weak: U < c Asymmetric Barotropic Equatorial waves Quasi-geostrophic Strong: U k c Radially symmetric Only baroclinic Steepening: mean shear or topography Non-quasi-geostrophic Weak: U < c and strong: U k c Radially symmetric Barotropic and baroclinic Multivalued F (modons strong): Quasi-geostrophic Barotropic and baroclinic &Plane Moving and stationary (Stern) Non-quasi-geostrophic Barotropic-no {3
Long (1964b): Larsen (1965): Benney ( 1966); Benney ( I 979)
Clarke ( 197 1)
Clarke (1971)
Maxworthy and Redekopp ( 1967a.b); Redekopp ( 1 977); Husuda ( 1 979); Miles ( 1979)
Malanotte-Rizzoli (1978, 1980b, 1982a)
Boyd (1977, 1980b)
Flierl ( 1979a): Henrotay ( 1 98 I )
Flierl (1979b); Charney and Flierl (1981)
Ingersoll (1973); Stern (1975): Larichev and Reznik ( 1976a); Berestov ( I 979): Flierl et al. (1980): Shen (1981)
Deem and Zabusky (1978)
PLANETARY SOLITARY WAVES IN GEOPHYSICAL FLOWS
175
vorticity conservation equation over variable relief: V*21c,,5+ J(1c,*, V*21c,*+ fo(lAdlH)h)= 0
(4.1)
+
where subscripts indicate partial derivatives, V*2 = d2/&P2 d 2 / d p 2 is the horizontal Laplacian, J(u, 6) = (du/dx*j(db/dy*) - (da/dy*)(db/dx*) is the Jacobian of ( ( I , h), and all starred quantities are dimensional. Equation (4.1) has been chosen because it allows for very different flow behavior in different parameter ranges. Specifically: ( 1 ) For infinitesimal-amplitude motions, Eq. (4. I ) obeys a linear dynamics. The solutions are linear superpositions of topographic Rossby waves. (2) For high-amplitude motions, Eq. (4.1) is a model for two-dimensional turbulence over topography (Holloway, 1978). In the turbulent flow evolutions of Eq. (4.1) no phase correlations between different modes persist beyond a typical turbulent time scale T,, the eddy turnover time. T, - @ ( L / U ) , where L is a typical eddy diameter in the turbulent field and U is a typical particle speed. T, is the time required for the flow to reach essentially a statistically steady state (Holloway and Hendershott, 1977). It can also be considered as a predictability time for the flow, namely, the time over which the total energy of an initial “error” superimposed upon the basic field grows to its theoretical limit of twice the energy of the basic field itself. ( 3 ) For small but finite amplitude, Eq. (4.1) allows for nonlinear, coherent solutions. Their persistence can therefore be a measure of the predictability of these structures when compared with turbulent flow evolutions of the same model. 4.1. The Single Solitary Eddy in rhc Weuk- and Strong- Wave Limits As shown in the previous section, Eq. (4.1) allows for both weak ( U < c) and strong ( U 2 c) nonlinear solutions. T o study their time evolution, we shall explicitly solve the initial value problem posed by Eq. (4. I ) in the weak-wave limit. The analytical treatment of Eq. (4.1j in the strongwave limit is much more complex (even though it can be done and shown to reduce to the same KdV equation as in the weak-wave limit). As the mathematical derivation would add nothing to the physical insight in the wave properties, we shall leave the demonstration of the latter case to the numerical evidence. Equation (4.1) is scaled as in Malanotte-Rizzoli and Hendershott (1980) and is reduced to
(4.2)
176
P. MALANOTTE RIZZOLI
Notice again that h = h(y) only, as is evident from Eqs. (3.8b,c), in order for the solutions to exist. In Eq. (4.2) U U - t =
[fo(lA4/ff>IL
-
c
is the Rossby number, namely, the ratio of the particle to the phase speed, the small-amplitude parameter;
is the aspect ratio of the wave field, namely, the ratio of north-south (crosschannel) to east-west (along-channel) length scales. In the limit 62 = t, Eq. (4.2) allows for solutions obeying the KdV equation. In the solitary limit, these solutions are
I),,= g,(s = x - cn t 1dJA.Y 1 c,
= Con
+ tCln + - - -
with @,, solution of the Sturm-Liouville problem @nv,
- (l/cdhy$ri =
&(O)
=
0 @,,(I)
in
0 Iy I 1
of which conare the eigenvalues and
+ (l/c&)a2nggn, + a3ngnrc.s = 0
(cln/c&)algny
with
Details of the derivation can be found in Malanotte-Rizzoli and Hendershott ( 1980). For the initial value problem posed by Eq. (4.2) we shall follow the procedure originally employed by Benney (1966) and later generalized by Redekopp and Weidman (1978) for the interaction of solitary structures in a zonal shear flow. We introduce the operator
L
=
+
&(d,>”) h,d,
in terms of which Eq. (4.2) is written as
Slli =
--s2&(&x)$
-
4$$,,
Expand:
JL
= $1
-
I).”&)(a,,.)lt - ts2($xdl,
+ 4 2 + 62& +
* *
’
- $dX)(&JI)
(4.3)
177
PLANETARY SOLITARY WAVES IN GEOPHYSICAL FLOWS
The zero-order problem is then
h$,= o(t, S 2 )
(4.4a)
Choosing $, to be given by a single mode: =
A @ , t)4(y),
4yJ- (1/c)hJ4 = 0
with
(4.4b)
the zero-order problem becomes A , + ( A , = O(t, S2)
(4.4c)
Following Benney (1966) we put A,
+ cA, = t N ( A ) + S2a)(A)
(4.4d)
in which N and 3 are, respectively, nonlinear and linear operators acting on A(x, t). Inserting the expansion for into Eq. (4.3)-taking into account Eqs. (4.4b) and (4.4c), equating equal powers o f t and S2-the equations for G2 and q3 are obtained:
+
a
2 =
-($1xd,
a
3 =
-4(&x)$I
+lLdJ(d",)+l
~
-
-
W4d,,4
(4.5a) (4.5b)
B(A)d,,4
Choosing
D(A) = s,4\,\
"(A)
=
2rA,
(4.5c)
and = f(l
$2
)'4'
=
$3
(4.5d)
g(.V)A,,
Eqs. (4.5a) and (4.5b) give the equations for f(y),g(y):
4,f
-
d,,g
-
( l / c ) h f = (h,,/2C2M2 + r(hy/c2)4 (l/c)h,g = [C(h,/C2) - 114
The solvability condition required for the boundedness of +2, Eqs. (4.6a) and (4.6b) gives
(4.6a) (4.6b) applied to
$j
(4.6~) and Eq. (4.4d) becomes the evolution equation for A(x, t): A,
+ cA, = t2rAA, + S2sA,,
namely, the KdV equation. The procedure can be generalked to $1
=
Ln '%LA>
?I
interacting solutions taking
WLV)
(4.7)
178
P. MALANOTTE RIZZOLI
with 4nl.y -
( 1 /G,)4dh = 0
The derivation is only mathematically lengthy. For two interacting solutions $1
=
A141 + A242
one obtains two coupled KdV equations: A,/ + c41, A 2 + C2-42,
= 62SlAI.ux
+ t[2rlA,A,x + VlA2’4Ix + XIAlA2xl
= 62s2A2.,xx
+ 42r2A2A2x + U2’4IA2.X + X 2 & 4 1 x I
(4.8)
The procedure is analogous to that followed by Redekopp and Weidman (1978), to which the reader is referred for details. Asymptotically, when the two waves are well separated spatially, system (4.8) reduces to two independent KdV equations, one for each individual wave. As previously remarked, the high-amplitude initial value problem can be shown to reduce to an evolution equation identical to Eq. (4.7), the KdV model. In fact, in the strong-wave limit the cross-channel eigenmodes are simple Fourier modes (see previous section). For them, both .I(+ $J, = O(1) and .I(+ GxaY) , = O(S2) can be shown to vanish identically, at zero order and 0(6’), respectively. In Fig. 4a and 4b (from MalanotteRizzoli, 1980a) we show two typical permanent-wave evolutions, for the weak- and strong-wave case, respectively. Table I1 summarizes the properties of the two fields. The strong solution has a Rossby number-that is, a wave steepness-of the same order as those observed in Gulf Stream eddies. Also, in Table I1 eddy turnover times and traversal times are compared using the dimensionless scaling appropriate for the numerical experiments. As can be seen, both fields last in their evolution without appreciable changes of shape for many eddy turnover times, over which any different flow would have become completely randomized. Thus, these nonlinear waves, solutions of the time-dependent KdV equation (4.7), constitute highly predictable flow realizations also in two (and three) dimensions. The experiment of Fig. 4b, in which the wave amplitude was changed and the corresponding change in the wave properties observed (MalanotteRizzoli, 1980a), is very interesting. It actually shows the evolution to a solitary wave from an initial state which is not the actual solution. In fact, the weak-amplitude analytical solution was given as initial condition to Eq. (4.1) in the high-amplitude case. With a Rossby number t 2 1 , the dimensionless form of Eq. (4.1) can be written as $,w
+ h.V$.X + J($>l J l J
+ fi2[$,,,
+ J($, $ . d l
=
0
(4.9)
for the asymmetric solutions in the zonal channel, derived in the previous section. There, we saw that the cross-channel eigenmodes for a quasi-linear relief are Fourier modes in the channel, and that permanent strong waves do exist only for odd-n modes. A topography like the one used in experi-
PLANETARY SOLITARY WAVES I N GEOPHYSICAL FLOWS
179
+
FIG.4a. Evolution of = -0.02 sech'(R\-)&(y) over the relief h(y) = psin(2.y) in the model equation (4.l),respectively. at t = 0 (a) and after one-quarter (b), one (c). and two (d) basin traversals. The field is scaled by lo4. (Adapted from Malanotte-Rizzoli, 1980a.)
ments (4b)-namely, h(y) = -sin J' sin 2y-cannot be approximated to a quasi-linear relief, except near J' = 0. However, the lowest eigenfunction of the weak-amplitude problem is almost sin(y), and it can be written as -
4L(~v) = sin J) + fi'q(y) for which J(&, 4Lvy) = 0 at zero order. We can think of q(y) as composed of the higher harmonics in the channel, in a Fourier decomposition of the lowest eigenmode 4L(y).Then, as already pointed out, it can be shown that Eq. (4.9) reduces again to the KdV time-dependent equation (4.7). Being dominated by KdV dynamics, the flow spontaneously evolves into the appropriate high-amplitude solitary wave plus a residual dispersive wave packet. This is recognizable in the amplitude oscillation of the flow field of Fig. 4b around a mean value, the excursion of which is much wider than for the weak wave of Fig. 4a, where only numerical noise is present.
180
P. MALANOTTE RIZZOLI
PLANETARY SOLITARY WAVES IN GEOPHYSICAL FLOWS TARII 11. PROPIR rim
W*
=
C'*
=
c,X
=
(Traversal (Eddy turnover
5
=
OF
181
SOLITARY EDDIES
(L'/T)$(, (L/T)$,I (L/T)c,,,
(q*,Lp, c: are dimensional quantities; $0, co are the corresponding dimensionless quantities)
(width/co*)(l / T )
( L / L ' * ) (l / T )
=
=
~ K L / C J=- 27r/c.0
l/$o
Weak wave
Strong wave
t = L,*/co+ = $O/CO x 0.15 (Traversal time)d~lCsP = 47 (Eddy turnover time)d.lesr= 50
t = $"/CO = 4 (Traversal time)d.le\r4 10 (Eddy turnover time)d.ler,2 1
To my knowledge, the only other works in which the evolution of (isolated) coherent structures has been simulated, numerically solving the initial value problem constituted by a conservation equation similar to Eq. (4. I), are those by McWilliams et a/. (198l), for barotropic and baroclinic modons, and Ingersoll and Cuong ( 198 1). Most authors, in fact, have limited themselves to finding the analytical solutions and showing patterns of them. Initial value problems like that posed by Eq. (4.1) are usually mathematically troublesome. It is natural then to resort to numerical experiments to actually observe the pattern evolution. Even where (as in Redekopp and Weidman, 1978) the initial value problem can be analytically dealt with, the pattern evolutions thereafter computed are in reality solutions of the approximate set of equations analytically dcrived from the original complete problem. This means that the solutions are actually valid only in the parameter subrange in which the starting model can be well approximated and substituted for by the equations finally derivcd. (In the Redekopp and Weidman case, the two coupled KdV equations in place of the starting quasi-geostrophic potential vorticity equation, for the study of the interaction of solitary vortices in a zonal shear flow). However, the equation of geophysical significance is the starting one. Solving numerically its initial value problemnamely, not eliminating a prior/ any of its terms-allows us to observe how well the total problem obeys the dynamics of the approximate analytical FIG.4b. Evolution of $ = -2.0 sech'(Ru)rb,(y) over the relief h(y) = -sin y sin(2y) in the model equation (4.1). at: (a) t = 0: (b) t = I : (c) t = 3; (d) after one-half basin traversal; (e) after one basin traversal; ( f ) after four basin traversals. Dimensionless time units I = ( I/fa)(lAdl/H).(Adapted from Malanotte-Rinoli, 1980a.) -
182
P. MALANOTTE RIZZOLI
model in the proper parameter range, how and when the extra terms affect the analytical solution’s behavior, and how wide is the parameter range itself (which often turns out to be much wider than what is analytically predictable; see the discussion on the stability properties of Section 5). 4.2. Collision Experiments The second property to be examined in this section is how coherent structures in two dimensions behave upon collision. The basic question is whether two-dimensional solitary waves are also solitons, namely, whether they survive collisions as in the one-dimensional case. The first investigation of collision properties in two and three dimensions is the already mentioned work by Redekopp and Weidman (1978), for the interaction of weak-amplitude waves propagating in a zonal shear flow. In their paper, they examine the modal interactions in a barotropic atmosphere, specifically the interaction of the two lowest modes in the zonal channel. For their model, they analytically approximate the solutions of the initial value problem constituted by the coupled KdV equations of set (4.8). They present results for collisions in which the large wave overtakes and swallows the smaller, showing only a single peak or both wave peaks at the interaction center, according to their relative amplitudes. However, the set of coupled KdV equations (4.8) is a good approximation to the actual and complete problem [equivalent to our model (4.1) in dimensional form] only in the limited range in which the two waves have comparable amplitude. At the most, in fact, in their results one wave amplitude is the double of the other. This does not answer the previous question about how wide the parameter range is in which the initial value problem (4.1) for interacting waves is well approximated by set (4.8). Indeed, if one were to repeat Redekopp and Weidman’s computations allowing for one wave amplitude to be significantly bigger than the other, starting from two separate waves at t -a, one could show that the two coupled evolution equations d o not constitute any longer set (4.8), but become, at the interaction,
-
-
In this, A , ( x , t ) is the stronger wave [of 0(1)]and A2 = O(6). Then, at t +a, when the two waves are again well separated spatially along their different characteristics [, 7,the equivalents of Redekopp and Weidman’s evolution equations (2.4a,b) become
183
PLANETARY SOLITARY WAVES IN GEOPHYSICAL FLOWS 11,
+ RllluE + v,
(4. lob)
-1
Equations (4. lob) show that the stronger wave asymptotically still evolves as a KdV soliton, while the weaker is destroyed to a packet of dispersive waves. This is true if we always remain in the weak-amplitude limit, for which from the KdV subrange we shift to the infinitesimal amplitude subrange dominated by linear, dispersive dynamics. But if, on the contrary, we shift toward the high-amplitude subrange, the approximations leading to results (4.8)or (4.IOa,b) break down. Then, we must solve the initial value problem (4.1) for coupled evolutions in total generality or resort to numerical experiments. We shall now present a series of numerical experiments performed upon our model (4.1) for the interaction of two permanent-form solutions of the model itself. As our numerical box is periodic, our eddies are actually cnoidal waves of modulus 1. Except in the first collision experiment, we study the interaction of the first and second lowest eigenmodes in the zonal channel over the topography h(y) = -sin y - sin 2y. The eddies fill up the numerical box. Thus, we adopt the following procedure commonly used in stability and predictability experiments. Let and ~,b~ be the two lowest permanent structures in the zonal channel over topography, numerical solutions to the numerical approximation of Eq. (4.1). Experiment l is the evolution in Eq. (4.1) of the total field +b = +2. Experiment 2 is the evolution of the stronger wave 1c/, by itself, with varying, increasing amplitude (and correspondingly varying inverse width B so as to remain the correct numerical permanent solution). Then the evolution of in the collision is given by [Experiment 1 - Experiment 21. This is compared with the evolution of +b2 (whose amplitude is fixed) by itself, to observe the preservation of the soliton’s properties upon collision. The first experiment to be shown is a proper collision experiment. Over the topographic relief h(y) = sin(2y), both eastward- and westward-going waves are allowed. Cross-channel eigenfunctions corresponding to the same absolute phase speed lcol are phase-shifted with respect to each other. The experiment is camed out in the weak-wave limit with A , = A2 = -0.02. Initially, the two waves (second lowest modes 42and 4-2)are superimposed at the (periodic) channel borders. Figures 5a and 5b show subsequent evolution. The two eddies emerge and begin to propagate toward the center of the box, where they superimpose to reemerge unaltered in shape and speed. Our model is indeed obeying the coupled KdV dynamics given by set (4.8). The phase shift is observable in a sudden movement of the high (low) of the eddies at the moment in which the interaction is weakest (the
+
+*
~
t = 11 t = 15 FIG.5a. Collision experiment of $, = -0.02 sech’(Bx)&(y) and $* = -0.02 sech’(Bx)&2(y) over h(y) = psin(2.v). at the labeled dimensionless times. The field is scaled by lo4.
I84
t = 33
t = 38
FIG.5b. As in (5a), at the labeled dimensionless times. 185
186
P. M P LANOTTE RlZZOLl
TABLE 111. O V E R T A K I N G EXPERIMENTS Fixed wave: $.
=
4 2 sech2(B2.u)@2(J~)
& (y) = second cross-channel eigenmode A.
=
-0.02
over h ( y ) = -sin
,I’ -
sin 2y
Colliding with:
GI
=
illsech’ ( B , x ) ~ , ( y )
@,(y)= first cross-channel eigenmode
(a) A ,
=
-0.02
(b) A ,
=
-0.05
(c) A ,
=
-0.1
(d) A ,
=
-0.5
(e) A ,
=
-1
(f) A ,
=
-2
two centers are farthest away from each other). However, immediately after, the periodic box brings the waves to interact again, and the highs (lows) shift back to the previous position. Overtaking experiments were successively carried out, keeping fixed the amplitude of +2 and gradually increasing the amplitude of so as to reach the high-amplitude subrange. Table 111 summarizes the series of experiments and Fig. 6 shows the initial eddies (+, is shown with A , = -0.5). Figure 7a shows the coupled fields at t = 0 and after one complete basin traversal. Figure 7b shows the difference field +2 at the same time. Here A , = A2 = -0.02; the second mode is clearly recognizable in the total field (+I &), giving rise to four peaks in cross-channel direction. The difference field shows that there is no difference in evolution from an isolated solitary eddy. Figures 8a and 8b show the analogous patterns for the total field and the difference G2 = $ - + I , at the same times ( t = 0 and after one basin turnover). Now the amplitude is A , = -0.05, and the stronger eddy has “swallowed” the weaker. Only the stronger wave peak remains evident in the coupled evolution. Still, the weaker wave preserves its permanent properties. Figures 9a and 9b show now the evolution of only the difference field at the same times, respectively, under collision with A l = -0.1 (Fig. 9a) and
+
PLANETARY SOLITARY WAVES IN GEOPHYSICAL FLOWS
187
FIG.6 . Overtaking experiments of l’ablc Ill. Initial condition for the second cross-channel eigenmode, scaled by lo4 (upper panel) and the first cross-channel eigenmode, with absolute amplitude 0.5 (lower panel). Topographic relief h(v) = -sin y - sin(2y).
188
P. MALANOTTE RIZZOLI
FIG. 7a. Evolution of the coupled fields of Fig. 6, for the overtaking experiment (a) of Table 111 at t = 0 (left panel) and after one basin traversal (right panel). The field is scaled by lo4.
A , = -0.5 (Fig. 9b). In Fig. 9a the beginning of instabilities is evident after one basin turnover. In Fig. 9b (shown also at the intermediate times t = 10; t = 25) the stronger eddy succeeds now in shearing and twisting the weaker wave, and we are no longer in the parameter range in which our model (4.1) can be approximated by a coupled KdV dynamics. Finally, Fig. 10 shows how the weaker wave is destroyed when interacting with
FIG. 7b. Evolution of the difference field G2 = -0.02 sech2(B2x)&(y)from the overtaking experiment (a) of Table 111 at the same times as in Fig. 7a. The field is scaled by lo4.
PLANETARY SOLITARY WAVES IN GEOPHYSICAL FLOWS
189
FIG.8a. Evolution of the coupled fields of Fig. 6. for the overtaking experiment (b) of Table 111 at t = 0 (left panel) and after one basin traversal (right panel). The field is scaled by lo4.
A , = -2. Its final evolution is that of a turbulent field, while the stronger eddy maintains its solitary wave character (not shown). Thus, three subranges can be defined for the interaction of permanent solutions of model equations like Eq. (4.1): ( 1 ) A KdV subrange, in which Eq. (4.1) is well approximated by a set of coupled KdV equations, such as (4.8). This both for the weak-amplitude
FIG. 8b. Evolution of the difference ficld
$2
from the overtaking experiment (b) of Table
111 at the Same times as in Fig. 8a. The field is scaled by lo4.
190
P. MALANOTTE RIZZOLI
FIG.9a. Evolution of the difference field $z from the overtaking experiment (c)of Table 111. = 0 (left panel) and after one basin traversal (right panel). The field is scaled by lo4.
at t
and the strong-amplitude waves. The two eddies have amplitudes of the same order and they survive collisions like one-dimensional solitons. (2) A KdV-to-linear subrange. We are in the weak-amplitude limit. The stronger wave is a KdV solitary eddy; the weaker is reduced to a packet of dispersive Rossby waves. (3) A KdV-to-turbulent subrange. The stronger wave is a KdV eddy. The weaker is destroyed to evolve as two-dimensional turbulence over topography. Collision experiments for asymmetric KdV eddies have also been carried out by R. Watanabe (unpublished report, 1980). For the strong, multivalued solitary vortices (modons) described in the previous section, collision experiments have been carried out by Makino et al. ( 198 1 ), McWilliams et a/. (198 I), and McWilliams and Zabusky ( 198 1). with similar results. Headon collisions and overtaking experiments both show that the vortices recover their initial shapes after the interaction, even though with changes in the propagation speeds. In the head-on collision the two vortices of the smaller modon separate from each other. Each of them couples with the corresponding vortex of the bigger modon, and then rotates around it. After the rotation, the two vortices join again together to reproduce the smaller modon and continue the evolution. In the overtaking experiment, the bigger dipole (faster) swallows the smaller, overcomes it, and leaves it behind unaltered. The experiments are very interesting, and the reader is referred to the original papers for details. In the next section we shall examine the stability of coherent structures
PLANETARY SOLITARY WAVES IN GEOPHYSICAL FLOWS
191
under superimposed perturbations and shall include the present results in a more comprehensive picture. 5 . STABILITY This section will be devoted to examining the stability properties of the permanent-form solutions discussed thus far. Stability is one of the important questions to be thoroughly investigated to realize the possible bearing of KdV dynamics on modeling coherent structures.
FIG.9b. Evolution of the difference field $? from the overtaking experiment (d) of Table 111, at the labeled dimensionless times ([ = 40 corresponds to one basin traversal). The field is scaled by lo4.
i 92
P. MALANOTTE RlZZOLl
FIG. 10. Evolution of the difference field $2 from the overtaking experiment ( f ) of Table 111, a t the labeled dimensionless times. T h e field is scaled by lo4.
For permanent analytic solutions like those derivable from Eq. (4. l), the KdV dynamics is only an approximation to the complete model. In fact, the procedure capitalizes upon expansion in a small characteristic parameter: t , the small amplitude parameter in the weak-wave case; h2, the small aspect ratio in the strong-wave case. The KdV equation is obtained at some given order in the expansion. As such, the complete model will behave accordingly only until the remaining terms are effectively negligible. Thus, the investigation of the stability of the two-dimensional permanent solu-
I93
PLANETARY SOLITARY WAVES IN GEOPHYSICAL FLOWS
tions will elucidate the extension of the parameter range in which a KdV dynamics holds. Special attention must be given to modons, because modons are exact, multivalued solutions of conservation equations such as (4. I >.For them, again, a stability analysis will elucidate the extent to which the large-scale atmospheric and oceanic motions may embody these orderly structures. General considerations about stability can be made through the use of integral theorems. The procedure adopted has already been used by Blumen ( 1968) and extended by Charney and Flier1 ( 1 98 1). Equation (4. I), our basic model (but the procedure can be repeated for any model), can be written as
v*++ H ( y ) = F(+ + cy)
+
(5.1)
Here H ( y ) = f $ h ( y ) = f o ( l A d ( H ) h ( y )as , compared to Eq. (4.1); is the permanent-wave basic state, steadily translating with speed c over the variable topography H(y). Let us work in the reference frame moving with the basic-state speed c. In it, let us consider a perturbation superimposed to +: 1C/total
++Y
=
Linearizing in the perturbation’s amplitude, supposed small (Y be easily shown that the perturbation satisfies the equation (d/dt)r
+ LJ
Vr
F’U * V$’
< +), it can
0
(5.2) where r’ = 0’1)’ is the perturbation’s relative vorticity; U = ( ( u - c), v } are the basic-state velocity components in the moving frame; V = ( d / d x , dldy) is the gradient operator; and F‘ = dF.’(Z)/dZ, as given by Eq. (5.1). To study the stability, we put 1c/‘ = #(.I- cr, J+? *
-
=
-
obtaining an eigenvalue problem for the growth rate a:
ar’ = -u .V(V’
-
F’)\L’
(5.3)
with the periodic or zero boundary conditions in the considered volume. In Eq. (5.3) we examine only the cases for which F’ # 0, and we consider Eq. (5.3) together with its complex conjugate. Then, multiplying Eq. (5.3) through -Y* and its conjugate through transforming, wherever possible, all internal products into flux-form operators V ( ); remembering that V U = 0; adding the two complex conjugate equations; and integrating over the considered volume where the flux-form integrals vanish, we obtain the energy equation
-+;
-
e
e
..
194
P. MALANOTTE RIZZOLI
r*
Multiplying now Eq. (5.3) through and its complex conjugate through and repeating the previous procedure, one finds, after integration over the considered volume, the enstrophy equation:
If we now add Eqs. (5.4) and ( 5 . 5 ) we obtain (u
+ a*) JJ { lvq12+
(5.6)
BOX
Whenever F' > 0. the integrand is a positive definite quantity. Then Eq. (5.6) implies (5.7) Re(u) = 0 that is, the basic state is neutrally stable to the superimposed infinitesimal perturbations. Being a dissipationless system, neutral stability is the only kind of stability we can expect for these permanent solutions. F' > 0 (and the analysis is easily extended to F' = 0) is thus a sufficient condition for neutral stability. However, the analysis fails for functionals F ( Z ) which change sign over the considered domain. For the analytic permanent-form solutions of Eq. (5.1), the functional is the topography itself, as shown in Section 3: F ( Z ) = H ( Z ) . In the numerical experiments carried out by Malanotte-Rizzoli (l980a), sinusoidal shapes were assumed for h(y);for them the above analysis does not hold. Through an expansion procedure in the small dimensionless parameter of the system, Malanotte-Rizzoli and Hendershott ( 1 980) have formulated a linearized stability theory which proves the neutral stability of the permanent solutions of Eq. (5.1) to infinitesimal, two-dimensional perturbations. To our knowledge, this is the only analytical stability theory available for two-dimensional coherent structures. This stability analysis generalizes to whatever topography the condition F' > 0 above derived. In fact, for our model (5.1), F' > 0 means h'(y) > 0. For the weak-amplitude solutions of Eq. (5.1) the zero-order problem is given by set (3.9b). If lz'(y) > 0 in the definition interval 0 5 y I1, then the Sturm-Liouville theory ensures the existence and reality of an infinite denumerable set of eigenvalues A, = l/con, all of the same sign (negative or positive) and corresponding orthogonal eigenfunctions d,, (Ince, 1956). These solutions, all eastward- or westward-going, are then ensured to be neutrally stable by h ' ( y ) > 0. The stability analysis of Malanotte-Rizzoli and Hendershott is then an equivalent proof. Conversely, if F ' = h ' ( y ) < 0 in 0 Iy I 1, in the problem (3.9b) we can change co to
PLANETARY SOLITARY WAVFS IN GEOPHYSICAL FLOWS
195
(-co). For these opposite-going waves, h‘(y) > 0 again, and we are back to the previous case. Thus, for weak-amplitude permanent solutions of the model (5. l), F ( Z ) = h ’ ( Z )> 0 is a sufficient condition for neutral stability, equivalent to the more complex analysis of the reference mentioned, valid also when the topographic shape is not a monotonic function. 5 . I. Perturbations in the Initial Conditions: Numerical Experiments
Stability studies can be performed through numerical experiments, which can complement and extend the results of any linearized analytical theory far beyond its range of validity. For asymmetric coherent structures in the zonal channel, the only stability studies are those carried out by Malanotte-Rizzoli (1980a, 1982b). The most important of such experiments are those concerning stability with respect to perturbations in the initial conditions. We also mention that other kinds of stability have been investigated, namely, stability to: (1) random perturbation in the initial phases of the soliton Fourier components; (2) random, small perturbation in the topography field. In both cases, the numerical permanent-form solutions exhibit neutral stability behavior (Malanotte-Rizzoli, 1982b). Let us now make the following consideration. We consider a purely turbulent flow and two initial states differing from one another by an “error” energy. Then their representative points in phase space will be separated by a well-defined distance, very small if the error is small. In the cases in which the error energy is observed to grow in time, the two point trajectories diverge in phase space until the two original states become as widely separated as two randomly chosen ones. The error energy itself will then approach a saturation limit given by the sum of the energies of the two initial states. This behavior we can expect I f and when we reach the threshold in the stability of the permanent structure. Upon crossing it, the perturbation’s energy will grow, destroying the structure itself. The series of numerical experiments carried out in Malanotte-Rizzoli ( 1980a) allows definition of a qualitative stability limit for perturbation fields similar to those considered, namely, with a scale content concentrated at length scales smaller than the average diameter of the coherent structure. The stability limit is crossed when the perturbation reaches an intensity comparable to the intensity of the basic state itself, taking quantities such as the root-mean-square velocity zi,,, and vorticity Crms as measures of the intensity. Figure 1 la (adapted from Malanotte-Rizzoli, 1980a) shows the behavior of the perturbation’s energy in a stable case (upper panel) versus an unstable one (lower panel). In both cases the disturbance is constituted by a random field with an isotropic energy spectrum proportional to K 3 , namely, concentrated at high wave
196
P. MALANOTTE RlZZOLl
TIME
FIG. 1 la. Time evolution of the total energy of a random perturbation superimposed to the asymmetnc solitary eddy of Eq. (5. I). Dimensionless time units as in Fig. 4b. The perturbation’s energy is normalized by the basic-state energy. Upper panel: stable case: energy units in lo-’. Lower panel: unstable case. (Adapted from Malanotte-Rizzoli, l980a.)
numbers. In the stable case, the perturbation has an intensity given by
u,,,
N
0.006
CrmS = 0.07
In the unstable case, its intensity is Zlrms
0.03
Crms
0.35
The basic-state intensity, in the same units, is (Unns)hs
0.02
(Crms)bs
0.07
The upper panel of Fig. 1 la, the stable case, shows that the perturbation’s intensity initially grows until reaching a saturation limit, at which it stays
PLANETARY SOLITARY WAVES IN GEOPHYSICAL FLOWS
197
for the remainder of the numerical experiment. Were it to be continued for more basin traversals, probably the perturbation’s energy would exhibit FPU recurrences, decreasing again toward a value almost equal to the initial value (apart from the energy of a dispersive packet). In fact, in this case we are in the range in which the model equation (4.1) indeed obeys a KdV dynamics. A properly scaled initial field must organize itself into a superposition of nonlinear KdV solutions plus a dispersive packet. As it is impossible to distinguish them in the limited physical space constituted by the numerical box, the only signs of their occurrence are the FPU recurrences, namely, the oscillating, periodic behavior in the random field energy. This is what we believe is evident in the experiment reported in Fig. 10 of Malanotte-Rizzoli ( 1980a). Here, the perturbation is a more organized structure (one single mode initially in both x and y directions). The decrease of its energy toward a level almost equal to its initial value is the indication of the first FPU recurrence. The lower panel of Fig. 1 la, on the other side, the
FIG. 1 Ib. Time evolution of the Fourier phases of modes u i . ~ at (- -), and a , 6 (---) of the solitary eddy. Dimensionless time units as in Fig. 4b. Vertical scale in degrees. (a). (b) stable cases (KdV range): (c) unstable case. (Adapted from Malanotte-Rizzoli. I980a.) (-)?
198
P. MALANOTTE RIZZOLI
TIME
FIG.12a. Time evolution of the total energy of the solitary eddy with amplitude A? = -0.02 in the collision experiments of Section 4.2. (Colliding eddy with successive amplitude A , = -0.1: -0.5; - 1 as indicated.) The total energy is normalized to 1 by its initial value. Dimensionless time units as in Fig. 4b.
unstable case, shows that the stability threshold has been overcome: the perturbation’s energy is shooting up toward its theoretical limit, the permanent wave itself is destroyed, and the field evolves into turbulent flow. This passage from a range dominated by a KdV dynamics to a higher amplitude range dominated by turbulent behavior is clear in the sudden decorrelation of the locked Fourier phases of the coherent structure. Figure 1 1b (adapted from Malanotte-Rizzoli, 1980a) shows the phases /11.2, c?,.~, of the three most energetic Fourier modes u ~ ,~ 1~. 4, , a,.h in a Fourier decomposition of the basic state. The first two cases correspond to the perturbed basic state in the stability (KdV) range; in case (c), after the phase locking of the least energetic mode a1,6 is lost, all phases are completely randomized, signaling the onset of chaotic motion. The collision experiments shown in the previous section can also be interpreted as stability experiments. Here the basic state, the weak-amplitude four-eddy solution, is perturbed by a larger eddy, the lowest mode in the
PLANETARY SOLITARY WAVES IN GEOPHYSICAL FLOWS
199
channel, namely a two-eddy structure with progressively increasing amplitude. From the point of view of stability, the perturbation’s scale content is now at larger length scales than the basic field. This will be destroyed when the model (4.1) will not obey any longer a coupled-KdV dynamics, as given by set (4.8). Figure 12a shows the evolution of the total energy of the basic state with constant amplitude A Z = -0.02, when interacting with the larger structure having, successively, an amplitude of A I = -0.1 ; A = -0.5; A , = - 1. The normalization value is the basic-state energy at the initial time. Here ( A , = -0.1) the energy is rigorously constant. Big oscillations, with increasing amplitude, signaling the onset of chaotic motion, occur when A , = -0.5. When A , = -1, the basic-state energy explodes after less than one quarter of a basin traversal time. Figure 12b shows that the initially locked Fourier phases of the eddy undergo randomization in the interaction with the lowest mode at A , = -0.5. Phases are randomized in time sequence according to the energy
,
TIME
of the Founer phases of modes (-), (-.-), (---) of the solitary eddy with amplitude A 2 = -0.02 during the collision experiment with the eddy having amplitude A , = -0.5. Dimensionless time units as in Fig. 4b. Vertical scale in degrees. FIG. 12b. Time evolution
200
P. MALANOTTE RIZZOLI
of the corresponding mode. Thus a1,6 the least energetic, is the first to be randomized; u ~ ,the ~ , most energetic, is the last. As far as modons are concerned, stability experiments to investigate their resistance to perturbations have been carried out by McWilliams et al. (1981). Specifically, they performed a series of studies varying both the perturbation’s amplitude and its scale content. Their results are very interesting, and in agreement with the results of the collision experiments of the previous section. Their conclusion is that modons are destroyed by perturbations having moderate amplitudes. The critical amplitude for modon stability is dependent upon the scale content of the perturbation, for which large-scale eddies are much more efficient in destroying the modon than small-scale eddies. They also identify the destroying mechanism in the advective shearing out of the modon contours by the perturbation field.
5.2. Overlapping Resonances The behavior of all the considered coherent structures when interacting with superimposed perturbations, and their consequent stability properties, can be put into a common frame of rationalization. This rationalizing basis is what is known as the theory of overlapping resonances. This theory has been constructed originally for quasi-Hamiltonian systems, and applied successfully to a two-degrees-of-freedom system by Walker and Ford ( 1969), who give a very clear explanation of it (see also Zakharov, 1974; Zaslavskii and Chirikov, 1972; Tabor, 1979). A Hamiltonian, integrable system has a closed, periodic phase-point trajectory in phase space. The ensemble of all possible orbits for the given integrable system, submitted to possible integral constraints, such as conservation of total energy, constitutes in phase space what is called a torus. We now make the system quasi-Hamiltonian, forcing it with an external, nonintegrable perturbation in resonance with its basic frequencies. The effect of this extra term is to cause a gross distortion of the torus in the vicinity of the resonance. Beyond a certain threshold in the perturbation’s intensity and composition, the torus is not only deformed but finally destroyed, the point trajectory wanders throughout the allowed phase space and chaotic motion (= turbulence) ensues. For quasi-Hamiltonian systems with a finite number of degrees of freedom and, in general, for a set of N one-dimensional nonlinear oscillators forced by an external perturbation, the theory of overlapping resonances can be used to explain and predict the destruction of tori and the onset of widespread chaotic motion. More specifically, let us consider a nonlinear oscillator, whose frequency is therefore amplitude dependent: w = w ( A ) . Let us force it with a perturbation whose frequency is in resonance with W : po + q62 = 0, with p , q integer numbers at the initial time. In the linear
PL.ANETARY SOLITARY WAVES IN GEOPHYSICAL FLOWS
20 1
case, it is well known that the motion “blows up.” In the nonlinear case, the resonance begins to increase the oscillator’s amplitude. However, as this increases, the frequency, now amplitude dependent, varies and the system goes out of resonance. Thus nonlinearity stabilizes an isolated resonance. Let us now consider a system composed of many nonlinear oscillators w, = @ , ( A , with ) a forcing perturbation also composed of many frequencies Q,, so the set of resonances is everywhere dense. Then, if at t = 0 the system is in resonance with Q n = p/qw,,, it will afterward go out of resonance but, as the resonances are dense, it will enter into resonance with another frequency a, = r/sw,(t = 1 ) = p/yw,,+,. Then, as the resonances “overlap” the system will never stabilize itself and transition will occur to chaotic motion. These simple concepts can be used to explain the observed behavior of permanent structures in stability experiments. In the case of asymmetric solutions in the channel they obey a KdV dynamics. Now, the KdV model has been shown to be a completely integrable Hamiltonian system with an infinite number of degrees of freedom (see Scott et a/., 1973). In the numerical experiments, moreover, we actually deal with the discretized version of the continuum model, namely. with a system having a finite number N of degrees of freedom. When forcing the system with external perturbations, we make it quasi-Hamiltonian. and in principle, for the KdV permanent structures, we could apply the theory of overlapping resonances, writing the model (4.1) in its Hamiltonian form in the range in which the KdV dynamics holds. In general, every permanent structure can be looked upon as composed of many nonlinear oscillators. The theory briefly outlined in the following has been constructed for the asymmetric, permanent-form solutions of our model (4.1), but it can be very easily repeated for any permanent solution. Let us consider the discretized version, composed of N degrees of freedom, of the dimensionless model equation (4.1):
The procedure would be the same for Eq. (4.1) scaled for the strong-wave solutions in the zonal channel. We consider a field given by
(5.9) where p is the perturbation’s amplitude relative t o the basic-state amplitude. In cross-channel direction, the natural set of eigenfunctions is the set of modes &, solutions of the zero-order Sturm-Liouville problem (3.9b). In axial direction, both in the infinitc channel and in the numerical, periodic box. a proper orthogonal set o f cigenmodes is simply given by Fourier modes. As the basic state in our case is given b y one single cross-channel
202 eigenmode $,
P. MALANOTTE RIZZOLI
we decompose
(5.9b) for a most general perturbation. For a modon solution, the proper orthogonal sets would be Fourier modes in both x and y directions. 2K + 1 = N is the total number of degrees of freedom considered. From a different point of view, we are simply repeating, analytically, the decomposition used in numerical algorithms bearing upon Fourier decomposition! (see Orszag, 197 1). Substituting Eqs. (5.9a,b) into (5.8) and carrying out the lengthy, but straightforward, mathematical manipulation, we amve at the following set of equations for the nonlinear oscillators h,(t) composing the permanent solution:
Here (5.11a) is the exact linear frequency of the nth oscillator, as derivable by substituting Eq. (5.9b) into the linearized version of Eq. (5.8).
(5.1 lb)
-
as, at zero order. the / I , are simple sinusoidal modes h, e-’n‘omr . Au,,,,is the amplitude-dependent nonlinear frequency width. The right-hand side gives the forcing of the external perturbation upon the basic state, composed of many resonant frequencies. Considerable manipulation gives
(5.1 lc)
PLANFTARY SOLITARY WAVF S IN GEOPHYSICAL FLOWS
203
The effect of the forcing perturbation is therefore to broaden the width of the nonlinear frequency shift A u , ~ , , , ~ ( ~for ) , which Eq. (5.10) can be written as dbnldt + l { ~ n +m Aw:,rn(b)}bti= 0 (5.12a) with +K
aw?,m
=
C
+ CP
~ u n , t > i
~/.ua(,i-/).L,
/.,,=-K
= Aw,,.m
+ 4Jun.tn)p
(5.12b)
For systems composed of many nonlinear oscillators, an approximate method to predict the onset of chaotic motion, namely, the overlapping of resonances, is a criterion proposed by Chirikov (see Zaslavskii and Chirikov, 1972; Tabor, 1979). Chirikov’s criterion states that resonances will overlap when
s
=
(5.13)
Au,l,,,f/AQ 2 1
Here Awn,,n is the nonlinear frequency width produced by the forcing perturbation, and AQ is the distance between two adjacent resonances. Chirikov’s criterion (5.13) will thus provide an approximate way to predict the location of a stochasticity border, separating a region of deterministic, wavelike behavior from a region of stochastic, turbulent behavior. is the nonlinear width In our case the important contribution to produced by the perturbation, ( AU,~,,,,),,. The separation Ail between two adjacent resonances can be evaluated according to
where w ~ , , , ,N (comlnis the permanent-wave dispersion relationship in its zero-order approximation [see Malanotte-Rizzoli, 1980a, or Eq. (5.1 la)] and An = I . Thus, Eq. (5.13) becomes i K
B (aWn,m)p -
J = ___
AQ
~
/L
2
14/,~4n-/),v
K
2 1
(5.13a)
kOml
Various approximations can be found to Eq. (5.1 Ic) for perturbations having different scale contents. and specifically composed of: ( 1 ) Scales larger than the scale of the basic state. (2) Scales smaller than the scale of thc basic state. (3) All scales, but the perturbation’s energy peak is concentrated at the same length scale as the basic state.
204
P. MALANOTTE RIZZOLI
In the various cases:
c an.”[f s,’
m- I
1
~ (* m)(2n n
~
1)
$i($vhv)v
i
s < (2)
K
P(n * m)(K - n )
1)3)
c
Q,l,,,
v=mtl
I
P(n * m )
an,”
$?nhyy
.‘v- J” 0
*=I
s,’ 412m$”L~?.r.
$;n$v\,hy
4’1
dv
dY
” I
(5.13b) The criterion (5.13b), in the three approximations, is of little practical utility, as it is not expressed in any quantity directly comparable with the numerical experiments. Chirikov’s criterion is by itself an approximate tool. We shall therefore try to relate Eq. (5.13b) to quantities directly obtainable from the numerics. even though to do this we can only proceed in a rather qualitative way. Thus let us notice:
Where Fn,+is given respectively by the three expressions (5.13b) in the three approximations. The expansion (5.9b) is performed upon the perturbation stream function. The value Kp is the perturbation’s dominant wave number; qrm5,fms its root-mean-square values of the stream function and vorticity. But then, in Eq. (5.13b), if n, m are the dominant wave numbers of the basic state, then ( ~ * w z ) K:
-
where K, is the total wave number of the basic state (the eddy diameter). Thus-even though in a very qualitative way-we can write rather generally the criterion (5.13b) for the stability border, namely, for the transition from order to disorder in the stability experiments of a coherent structure s
(frms)pcrXZIK:
2
1
or (5.14) where a is a proportionality constant. Equation (5.14) is, as we emphasize. a very approximate criterion. However, its form is general enough to be directly compared with all the stability experiments available in the literature-namely, those by Malanotte-Rizzoli ( 1980a), previously discussed: the collision experiments reported in Section 4 and considered as stability
PLANETARY SOLITARY W A V E S IN GEOPHYSICAL FLOWS
205
experiments; the stability experiments performed upon modons by McWilliams et ul. ( 198 1 ). For this comparison, we adopt the scaling used by the authors of this last paper, in which the modon permanent-form solutions have amplitudes in the stream function and u,, of O( I ) . Thus, in our stability experiments we first normalize the perturbation’s energy and enstrophy with respect to the total energy of the permanent basic state, which therefore has a u, = 1. Second, we renormalize again the perturbation field so that, when the perturbation has an amplitude equal to I , also its Crms = 1. All the computed quantities for all the stability experiments mentioned are inserted in Eq. (5.14). This relationship is shown in Fig. 13. In view of the considerable approximations and of the qualitative nature of the criterion (5.14), it is indeed remarkable that all the experiments carried out thus far fall rather well on either side of the shaded region, limited by the two values u = 2; u = 4. All the experiments in the region with u i 2
FIG.13. Chirikov’s criterion for the stochasticity border (shaded region) separating wavelike behavior (order) from chaotic motion (disorder). Vertical coordinate: c = of the perturbation. Horizontal coordinate: x = K,,,/K,,,, where K,,, is the average wave number of the basic state (eddy diameter): K,, is the average wave number of the perturbation. Only exception: experiment R with c > 10 is a case of disorder, even though located just below the limits of the border region (curve with a = 2).
rr,,,,
206
P. MALANOTTE RIZZOLI
correspond to the survival of the coherent structure. All the experiments with u 2 4 correspond to its destruction. The only exception is the experiment R with {ms > 10, which is a case of destruction of the basic state, even though located just below the limits of the border region (curve with a = 2). Obviously, many more experiments would be necessary to determine with greater confidence the width of this stochasticity border, namely, the range of values allowed for a. Thus, coherent structures, permanent-form solutions to large-scale model equations, do behave like one-dimensional solitons in a well-defined parameter range, the range in which the considered models obey a KdV dynamics. When made to interact with random-phase features, according to the intensity and scale content of these latter ones, a border can be crossed beyond which the KdV dynamics does not hold any longer, the coherent structures themselves are randomized, and the fields rather obey a turbulence dynamics. The transition from order to disorder seems to occur in a restricted parameter range. Even though approximate, the criterion (5.14), defining the border region, might be used to predict the onset of chaotic motion in future experiments, similar to those discussed, but under more general-and more realistic-conditions.
6. FURTHER INVESTIGATIONS
ON
COHERENT STRUCTURES
6 .I . Numerical Accuracy
A fundamental question which naturally arises in the study of coherent structures concerns the reproducibility of these analytical solutions through suitable numerical algorithms. These necessarily truncate to a finite number the infinite number of degrees of freedom of any continuum model. The flexibility and accuracy of numerical techniques in this respect must be carefully checked if we want our simulations to reproduce without too big an error the features for which the chosen analytical model is important. Experiments to check the accuracy of numerical techniques have been carried out by McWilliams et af. ( 1 98 1). In these, the degree of accuracy has been tested as a function of the resolution, simulating barotropic modons through second-order finite difference schemes. Even with a very small number of degrees of freedom, they succeed in simulating the numerical analog of the analytical modon. The accuracy, however, is obviously too low. With a very fine resolution (200 degrees of freedom for a box with L = 10 modon radii) the accuracy attained is good, but not what might be expected from the scheme used (see references for details). Thus they conclude that a clear difference remains between analytical modons and their numerical counterpart.
PLANETARY SOLITARY WAVES IN GEOPHYSICAL FLOWS
207
Modons are, however, “troublesome” solutions, as-at least for the chosen cases-they present a finite discontinuity in the radial gradient of the vorticity. For analytical solutions in a zonal channel, numerical techniques should provide better approximations, at least for numerical schemes based upon spectral truncation (Orszag, 1971) such as those used in MalanotteRizzoli (1980a) and in the collision experiments reported in the Section 4.2. It has been found that spectral methods are more efficient, even though marginally, than finite-difference ones (C. E. Leith, personal communication). They are able not only to reproduce the form-preserving solutions of Eq. (5.1) but also to distinguish between the separate effects of a (weak) nonlinearity and a (weak) dispersion (Malanotte-Rizzoli, 1980a). With 32 X 32 degrees of freedom, and at least for the gravest modes in the zonal channel, the numerical analogs of the analytical solutions show amplitude oscillation of 5 1% over long numerical simulation (many traversals of the periodic box of 2 a X 2 ~ )The . numerical eddy speeds differ from the predicted ones by 2-39’0, but it must be pointed out that the “theoretically” predicted speeds also must be evaluated numerically and are therefore approximate by themselves. Thus, the numerical accuracy of the spectral schemes seems to give results comparable with finite difference schemes, only slightly better for the experiments carried out thus far. The numerical techniques used seem therefore to be quite adequate, at least for the experiments carried out thus far, of which the most complex are the collision experiments. However, the problem of how many degrees of freedom are necessary-and sufficient-to obtain the desired model behavior under more complex and general circumstances is still an open and very important question.
6.2. Dissipalion
The models, and consequently the solutions, explored thus far are unforced and dissipationless. Apart from forcing-a totally open, still unexplored line of research-another important question is the behavior of coherent structures under the influence of dissipation. Dissipation is always present and intense, in the atmosphere as well as in the ocean. Dissipation must model, in our case, energy loss from large-scale structures. Energy loss can occur through a linear drag or through different mechanisms, more scale selective and affecting the higher wave number range in the energy spectrum. Thus it is meaningful to explore the effects of different forms of dissipation on the permanent structures examined. These last ones can be of two natures, analytic or multivalued modons. For modons, the physical space consists of two different regions, the interior
208
P. MALANOTTE RIZZOLl
and the exterior, matched together at the modon circumference. In other words, in the exterior region fluid particles belong to a continuous stream and can move freely from one side of the considered domain to the other: in the interior region, fluid particles are trapped inside the “core” of the eddy and can preserve their own identity for long times (as observable for instance in T-S anomalous relationships inferred to be preserved for times as long as 1-2 yr; see McDowell and Rossby, 1978; see also McWilliams ez al., 1982, for the “small” mesoscale coherent structures of the LDE experiment). Therefore, we shall divide the discussion of the effects of dissipation into the two cases of multivalued and analytic solitary eddies. We shall also examine different kinds of dissipation, modeling a linear drag term or higher dissipation forms. In general, they can be wntten as
Dn = (-I)”K,V’(n-I)
<
(6. la)
with [ = V2$ the flow relative vorticity. Thus, Eq. (6.la) can assume the different form:
D ,= - K
1 {,
D2 = K202{, Di = -K3V4{,
for for for
n n n
= =
=
(linear friction-bottom (Newtonian friction) (biharmonic friction)
1 2 3
drag) (6.1b)
and so forth. We want to explore dissipation effects in the context of the simplest analytical model. Thus, for modons we shall treat barotropic modons. namely, the model used by McWilliams el af. (198 1): j; + D#x
+ J($, 0 = Dn
(6.2)
The modon solution is obtained in the dissipationless case passing, as usual, to the frame moving with the modon speed c:
V2$
+ DJ,
=
F($
+ cy)
(6.3)
-
with p = c = 1. Eq. (6.2) is Eq. 1 of McWilliams et a/. (1981). In the exterior rcjgion, $ 0 as r cn: 3
(6.4a) and (6.4b) ( { - $ i f p = c = I). In the interior rexion, let us choose the simplest case of barotropic modon, namely, again a linear functional: F , ( Z )= -mZ
+n
(6.5a)
209
PLANETARY SOLJTARY W A V l - S IN GEOPHYSICAL FI OWS
with m > 0, from which
< = v2$
=
-tn#
-
(nic
+ p)y + n
(6.5b)
From Eq. (6.4b), Eq. (6.2) assumes the following forms in the exterior region: -K,< = -K,(P/c)$ (6.6a) l l + P*, + J($> .r) = +K2V2t = +K2(P/C)*$ -K3V4{ = - K @/c ) ~ $
II
and so forth. Thus, for every barotropic modon obeying Eq. (6.2), in the exterior region all friction laws will be identical respectively for n odd and n even, and the decay rate (negative if n is even) will only depend on the chosen values of K,,, 8, c. From (6.5b) Eq. (6.2) assumes the following forms in the interior region:
+ (mc + p)y n] K,m[m# + (mc + P)y n] K1[m$
-
=
D,
K2 K1 K3tn’[m$ + (mc + p)y - n] = K3 - m2DI Kl (6.6b) -
= - mD,
and so forth. Thus, all friction laws will have an identical functional shape (linear in $) in the interior region for those modons whose interior functional F is linear, as in Eq. (6.5~1).T o see more profound effects of different friction laws, the interior functional must not be linear. The constant m of Eq. (6.5a) is related to the modon wave number K(c) in the interior region. In fact, for /j = c = 1, and for the solutions used in McWilliams et a/. (1981): i n = K’ (6.6~) We can obtain more insight into the dynamics ofthese barotropic modons characterized by an interior linear functional by analyzing the general form of their interior equation, as derivable from Eq. (6.6b):
V’$l
+ p$,
i
J ( # , V2, $) = -K*V2$
(6.7a)
with K* = K , , mK2, m2K3, and so forth, for successively higher friction laws. The above K* values are thosc valid for the interior solution. Equation (6.7a) is also the equation for thc exterior region if we allow K* to,be negative for even friction laws. I t must be pointed out that the K,,’s of Eq. (6.2) are “overall” friction coefficients, as the observed modon behavior is of actual decay in all three cases.
210
P. MALANOTTE RIZZOLI
Let us perform the substitution: )I
e A*[4(x,y , I )
=
upon which Eq. (6.7a) becomes:
v?$, + /34,+ C A * ' J ( @ ,v24)= 0
(6.7b)
-
From Eq. (6.7b), the modon behavior under dissipation is evident. As t 0, at the beginning of the process, the solution is still a modon. As t m, dissipation will reduce the energy, i.e., both the amplitude and the speed. The modon will slow down and finally stop; with decreasing amplitude the solution will enter the range of linear dynamics, and the asymptotic behavior of Eq. (6.7b) is given by a decay into linear Rossby waves. Let us pass, as usual, to the system defined by
-
.s
=
x
-
c(t)t
7 = 1
where now, however, because of the dissipation effects, c system, Eq. (6.7b) becomes:
=
c(r). In this
(6.7~) with c' = dc/dt. For a slow time variation and not too long an evolution, Eq. ( 6 . 7 ~ is ) approximately the exact modon equation if we put 1
e-A*' ~- - -
c+
y
c"7
This has the solution
c=-
:( I - - -
,.-K*r
uK*
where a is an integration constant. Equation (6.8) satisfies lim c = O 7-t7
To satisfy the boundedness condition for lim c
=
T
-
0, namely
c(,=~
7-0
the integration constant is chosen to be a
=
y / K * . Then
y
=
CI,=~
lim
c' =
aK*
=
r-0
Thus Eq. (6.8) has the expected behavior and the same functional decay in time for every type of friction law, if K* is the same. That is, if the
PLANETARY SOLITARY WAVES I N GEOPHYSICAL FLOWS
21 1
various coefficients K , , K2, K3 arc chosen to give the same e-folding decay time in the maximum vorticity value, with /3 = c = in = I , the same K* is obtained; then the nature of the modon speed decrease will be identical in all the dissipation experiments considered (as in McWilliams et al., 1981). The friction laws (6.6b) for the Newtonian and biharmonic friction are now scale selective in the interior region. This is immediate, recalling that m = K*, where K(c) defines the modon dispersion relationship. The decay under the linear friction, given by the first of Eqs. (6.6b), is shape preserving; for higher friction laws, the high-wave-number region of the modon spectrum will be more damped. This is immediate if we look at equations (6.6b) in wave-number space, that is, decomposing the stream functions according to: +N
$
=
2
n= ,v ,?I=
+N
c
.K
(6.9)
a,l,,n(f)&t”nY)
Under this decomposition, the cases of linear and Newtonian frictional decays become, respectively: an,,
- ian.rnan.rn
+
1
~ r , \ a r . \ a c n r),(tn-s) =
2
Ar,\a,,\a(,j-r).(m-.s) = - U n 2
pKlan,m
r.r+n.m
bn.m -
iw,.,nan.rn+
(6. I Oa)
+ m*)an,rn
r , +n,m
with =
+ m2) nr + rns 1-2,-+n- + in*
n/(n*
+ s * ] (ns n* + in* r*
-
mr)
The first of Eqs. (6.10a) shows that, in the linear friction case, each mode has the same decay rate; the second of Eqs. (6.1Oa), for the Newtonian friction, shows that mode an,rndecays proportionally to exp[-K2(n2 + rn2)t], with the highest damping for the highest modes. The Fourier space representation of Eq. (6.6b) allows us to explain the effect, noticed in the above reference, that high-order dissipation processes are not as efficient as one estimates on the basis of a shape-preserving argument. In fact, the estimates of K2, K 3 thus obtained were less than the numerical values actually required to give the same bulk decay as in the linear friction case. We can express Eqs. (6.10a) in the Fourier polar plane as $
= e-K~r
s s
1c/
Clkrcos(8-(y)K&y
(6. lob)
for the linear friction, with a the angle between the wave vector K and the
212
P. MALANOTTE RIZZOLI
space vector r; and (6.1Oc) for the Newtonian friction, with K' = n2 + m2, in relationship to Eqs. (6.10a). $K,B, a& obey different equations, namely, the Fourier representations of Eq. (6.7b) in the two respective cases. Then one can put ug,@N GK,#only in the two asymptotic ranges: t 0 (pure modon dynamics); t co (linear wave dynamics). The range we wish to explore is for 1 0, when both cases slowly decay away from the pure modon dynamics. In it we can write, Taylor-expanding for t 0:
-
-
-
=
e-K''A(K,0)
N
A(K, 0)
-
( K , t ) A ( K ,0)
-
(6.1 la)
with
and $2
= A ( K , 0) - K2t
ss d0
(az," N $K,B)e'Krcos(8-n)K3 dK
K2A(K,0)
A(K, 0) d ( K 2 ) ]
-
(6.1 lb)
$2 to have the same time decay in the early stages of the We want process, which means
(6.12a)
K2 = K I / K 2is the value for K2 one obtains under the hypothesis of shape preservation and exponential decay. The actual value is larger, however, as indicated by Eq. (6.12a) and in agreement with the numerical results, and depends upon the integral properties of the modon solution. Analogously, for the biharmonic dissipation law (6.12b)
..
A
Thus, K 1 , K2, K 3 are integral, rather than local, dissipation coefficients, depending upon the modon's behavior in both the interior and exterior regions. This had already been conjectured in the above reference. The analysis carried out thus far is valid only for multivalued solutions of Eq. (6.3). Even more specifically. the simplest modon case has been
PLANETARY SOLITARY WAVES IN GEOPHYSICAL FLOWS
213
+
treated in detail, namely, that for which F($ cy) is a linear functional also in the interior region. The actual value of the decay coefficient will depend only on the speed of the structure, apart from the basic features of the chosen model. This is true in general for the behavior under dissipation laws given by Eq. (6.la) of all the permanent solutions of Eq. (6.3) (or similar model equations), whenever the functional F is linear. More complex is the behavior when F is not linear, as it is, for instance, for the solutions of the model equation (5.1). For it I.'(%)
=
h(Z)
(6.13)
Now, different friction laws will produce very different patterns. To be precise, a sinusoidal topographic relief h(j9 = -sin(2y), like that used in the weak-wave solutions shown in Section 4, gives for the relative vorticity
< = v'$
=
sin(2y) .- sin[2($
+ cy)]
while 02{,V4j- will have a rather complex shape. To show the difference from the dissipation experiments carried out by J. McWilliams upon the previous modons, analogous experiments have been camed out for the weak-wave solutions over the above topography, in the two cases of linear and Newtonian friction, with respective decay coefficients K , = 0.2; Kz = -0.014. Figure 14 shows the shape-preserving decay under linear friction. The solution is considerably slowed down with respect to the dissipationless case. At T = 50, over more than one basin-traversal time, the decaying eddy has crossed over only half of the basin. At this time the amplitude has been reduced by more than two orders of magnitude. Figure 15 shows the analogous evolution, at the same times, under Newtonian friction. The behavior is now totally different, at T = 50 the eddy has crossed over less than one-quarter of the basin, implying a much more reduced speed than in the previous experiment. At T = 50, however, the amplitude is now reduced by only one order of magnitude, correctly indicating that dissipation is more concentrated in the high-wave-number part of the spectrum. As a conclusion to this and the previous sections, we may state that, for model equations like those examined, suitable for describing large-scale geophysical motions, a well-defined parameter range seems to exist in which they obey a KdV dynamics or, more generally, a nonlinear dynamics leading to phase-locked, coherent structures. This is true both for the weak-wave case ( U < c) and the strong one ( b' 2 c):both for analytic and for multivalued solutions. Under dissipation, through progressive reduction of their amplitude, the solitary eddies reach the linear dynamics range, with asymptotic dispersion
214
P. MALANOTTE RIZZOLI
t = 35
t
=
50
FIG. 14. Time evolution of )I = -0.02 sech'(B.r)+2(j>)over the relief h(y) = -sin(2y) under = -K,.C Dimensionless time units as in Fig. 4b. The field is scaled
the linear friction law 11, by lo4 at t = 0.
into a packet of linear Rossby waves. The transition to the linear dynamics range is gradual and smooth. If the wave is weak, its amplitude can be reduced to a very low value, essentially negligible, before beginning to irradiate Rossby waves. Under the forcing produced by external perturbations, a stochasticity border is reached at a critical perturbation amplitude. The border seems to be rather well localized, depending not only on the perturbation's intensity but also on its scale content. Upon crossing it, phase locking is suddenly lost, and the transition to chaotic behavior seems to occur rather quickly. Then, we enter the turbulence dynamics range. Figure 16 summarizes these conclusions.
PLANETARY SOLITARY WAVFS IN GEOPHYSICAL FLOWS
215
7. CONCLUSIONS In the previous sections a review has been given of the body of research carried out thus far on mesoscale coherent structures observable in both the ocean and the atmosphere. The observation of these structures has been the primary motivation for a renewed interest in nonlinear, permanent-form waves, solutions of one-dimensional model equations endowed with remarkable properties. Specifically, general initial conditions allowed to evolve in these model equations asymptotically break into a number of well-defined, separated pulses each of which is one of the nonlinear wave solutions.
t = 50 t = 35 FIG. 15. Time evolution of J. = -0.02 sech*(Bx)@&) over the relief h ( y ) = -sin(2y) under Dimensionless time units as in Fig. 4b. The field I S scaled the Newtonian friction 112 K2Vz{. by lo4 at t = 0.
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FIG. 16. Existing parameter ranges and their transitions for model equations of large-scale geophysical flows.
Thus, these can be considered the nonlinear normal modes of the chosen model. In one dimension, these nonlinear modes have been found to exist for quite a variety of equations. This remarkable property-by which they maintain their own identity in mutual collisions-has suggested the name “soliton” in analogy to the elementary particles of quantum physics. Thus, in Section 2 a short synopsis has been given of the one-dimensional equations which have been found to be relevant for the dynamics of oceanic and atmospheric motions. and of their solutions. Among these equations, the most important and most studied is the Korteweg-de Vries (KdV) equation. This is the nonlinear model we have mostly encountered and dealt with in the investigation of our geophysical systems. The oceanic and atmospheric motions we have concentrated upon are those characterized by length scales which can reach the dimensions of oceanic basins, that is, of the order of lo3 km. In general, the length scales we are interested in are greater than the first baroclinic Rossby radius L , = N ( Z ) H / f , ,and can reach the ordcr of the external deformation radius Lz = Typical values of L , for the ocean and the atmosphere are about 50 km and 1000 km, respectively; the external deformation radius, in both cases, is of the order of 3,500 km at a latitude Bo il 30”. To study the dynamics of these mesoscale motions, one has to deal with two- and three-dimensional systems. much more complex than the one-dimensional equations of Section 2. For them, typical linear solutions are the Rossby waves. If nonlinear solutions, analogous to the permanent waves of Section 2, do exist, they will be nonlinear Rossby waves.
mZ/fo.
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Thus, the first question that naturally arises is whether there exists a parameter subrange in which our two- and three-dimensional models of mesoscale motions d o admit similar permanent solutions. Or, in other words, in which their dynamics can be approximated by these equations. Thus, Section 3 has been devoted to reviewing all the existing nonlinear planetary-wave solutions found by different authors in the context of a unified approach. Examples havc also been given of coherent structures commonly observed in nature, the dynamics and interactions of which no linear theory can adequately explain. Jupiter’s Red Spot, the blocking ridge phenomenon of the atmosphere, and the by now widely observed Gulf Stream rings and eddies constitute the most striking examples of these coherent structures. The findings of Section 2 allow us to state that, for scales as large as the external deformation radius, allowing for a free surface on a P-plane is sufficient to induce important steepening effects in the wave (divergence is important). Thus, for these longest, essentially barotropic motions (- 3,000 km), permanent waves can exist without any required interaction with external properties of the fluid environment. However, for shorter length scales, when divergence effects become less important or even negligible, interaction with external features is necessary to provide the required steepening counteracting dispersion. Then. one must allow for a mean shear flow and/or a variable topography and/or baroclinicity to provide a sufficiently-but always smoothly-varying environment. The solutions can be weak ( U -4 c) or strong ( U >, c); symmetric or asymmetric; analytic or discontinuous. For most of them, the final model approximating our initial system is the KdV equation. Thus, the answer to our first question is positive and we can assert that, in a suitable parameter subrange, mesoscale oceanic and atmospheric motions indeed obey a KdV dynamics. The second question is then whether we can reproduce them through numerical techniques, those commonly used to simulate and predict geophysical motions. This question in reality embodies a series of questions. In fact, the analytical tools at our disposal allow us to study the evolution of these waves-namely, the initial value problem constituted by their model equation-only under very particular circumstances. Let us first simplify our systems. considering them in the proper parameter range in which we can substitute a one-dimensional, nonlinear dynamics to the more complex two- or three-dimensional one. Then, even under this approximation, we can usually determine the system’s evolution starting from a given initial condition only in the infinite domain. What if we truncate the system, substituting a discrete set of niodes to the infinite number of degrees of freedom of the continuum model (as it is done when dealing with numerical experiments)? Which numerical accuracy is required to reproduce the analytical features with satisfactory approximation? What if we put our system
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in a finite domain, imposing meridional boundaries with zero cross fluxes, as in the ocean? What if we consider periodic boundary conditions in our finite domain, as more appropriate for the atmosphere? The one-dimensional models we arrive at, even the most studied, such as KdV, can behave quite differently under these different boundary conditions. KdV has been related through inverse theory to the study of the potential-well problem in the Schrodinger equation. Then it is enough to remember that a potential well decaying to ~ C allows L for a set of bound states (= the solitons) different from the set of the same potential well cut at some meridional boundaries x = k u and repeating itself periodically. Most of these questions cannot be answered analytically, even in the context of the one-dimensional models. The answer is usually found through numerical experiments. Section 4 is thus devoted to the study of the initial value problem posed by the evolution of coherent structures. The first part of Section 6 is devoted to the required numerical accuracy needed to reproduce them. There, our successive answers are again positive. Our numerical algorithms are quite adequate to reproduce the coherent structures, even in their simplest versions (finite differences) and for the most complex solutions (the discontinuous, multivalued modons). We can reproduce both the weak and the strong waves through long numerical simulations of the complete model equations, thus allowing for sufficient time for the effects given by terms not present in a pure KdV dynamics to manifest themselves-and they do not. In the proper parameter subrange we are even able to reproduce in two and three dimensions the analog of the one-dimensional collisions. The waves preserve their identity in mutual interactions. Thus, the numerical experiments allow us to state with confidence that, in the above parameter subrange, these coherent structures behave like one-dimensional solitons. However, the KdV dynamics is only approximate for our initial models and valid only in this limited subrange. The starting model, unlike KdV, may not be a Hamiltonian system. And ifwe were able to find a Hamiltonian for it, this does not mean that the system is integrable-namely, in the continuum case, that it allows for an infinite number of conservation laws, like KdV. Suppose. furthermore, that the starting system were special enough to allow for an infinite number of conservation laws. Its numerical discretized version with N degrees of freedom would most probably not conserve the discretized versions of the corresponding N conservation laws. This is finally what matters for the practical purposes of numerically simulating and predicting geophysical flows. Thus, we can anticipate that the parameter range, in which a KdV-or coupled KdVs-dynamics is obeyed will be limited. Upon crossing its borders one will enter the range of linear dynamics on the infinitesimal-amplitude side or of turbulence dynamics on
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the high-amplitude side. Then the important question concerns the width of this parameter subrange and of its sensitivity to a slow, gradual variation in the parameters themselves: whether, in other words, the transition to a region of different dynamics is gradual and smooth or rapidly occurring. The way to answer these questions is through numerical stability-or predictability-experiments of the inviscid solutions to superimposed perturbations. Thus, in Section 5 we explore the transition from the KdV dynamics to a turbulence dynamics. The answer emerging from numerical experiments is that permanent structures are indeed rather robust, be they analytic or multivalued. They can coexist with a turbulent background having urms, [,.,,s of the same intensity without being destroyed. The threshold to different behavior constitutes a rather narrow stochasticity border, which can be found through the theory of overlapping resonances. On crossing it, locked phases are suddenly-and rapid1y-decorrelated, nonlinear cascades spread energy to the whole spectrum. and turbulent behavior and dynamics dominate. Large-scale features larger than-or as large as-the coherent structures themselves are much more efficient in inducing this transition to turbulence than small-scale random features. Section 6.2 examines the transition to the range of linear dispersive dynamics through the effects of dissipation. Various functional laws are considered, of successively higher order and, therefore, more scale selective. For solutions characterized by a linear functional dependence of the potential vorticity upon the stream function, all dissipation forms have the same functional shape (linear in.))I Only the decay rate varies as the high region of the wave number spectrum is more and more affected. For nonlinear functional forms, as in the barotropic quasi-geostrophic vorticity equation over variable relief, only the linear friction provides an approximately shapepreserving decay. Higher frictions induce a much more complex behavior, not amenable to simple interpretation. However, in both cases the asymptotic regime is the same, namely, radiation of a packet of dispersive Rossby waves. The transition to the range of linear dynamics, contrary to what occurs for the transition to turbulence, is smooth and progressive. The structure can even become so damped as to be of negligible amplitude before the final Rossby waves are beginning to disperse. What next? There is a great deal to do, because the major questions of whether or not these models are relevant for mesoscale geophysical flows have not yet been really answered. The answer, and with it the answer to enhanced flow predictability in a certain parameter subrange, lies in the possibility of obtaining these coherent structures emerging from general initial conditions-from chaos-or to force them. Again, the path lies in numerical experiments more than in anything else, as analytical tools are most probably insufficient for these complex problems. In nature, and above
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all in the ocean, evidence has been accumulated in recent years for the production of coherent, long-lived structures from a given environment. It is enough to think of the Gulf Stream rings and eddies, whose formation, evolution, and interaction with the mean stream or other rings, has been more and more documented after experiments like MODE and POLYMODE (see, for instance, Richardson. 1980; McWilliams et al., 1982). But the mechanisms of forcing and production are far from being understood. So, how relevant are these models in the context of this oceanographicor equivalent atmospheric-evidence'? In the one-dimensional wave tank it is very difficult not to produce solitons when forcing through a piston at one end of the tank. What about our two- (or three-) dimensional numerical tanks? Can we force these structures in an equivalent way through boundary forcing or through more complex ways (for instance, topographic forcing)? This obviously implies that we have already put our model equations in the parameter subrange in which the proper nonlinear dynamics is obeyed. If so, supposing our forcing is general enough, how can we recognize them, in the middle of possible background radiation of a turbulent nature? Can we make our numerical tanks long enough for them to emerge asymptotically out of these initial conditions? If this is unfeasible, can we devise intelligent criteria for unambiguous identification, for instance, through the conservation of admitted integral invariants, Fermi-Pasta-Ulam recurrences or similar properties? The model equations treated thus far have been mostly stripped of unnecessary complexities, to better illustrate these solutions. Only dissipation has been studied. But for more realistic systems, what can be the behaviornamely, energetics and predictability-namely, if we want, stability-under more general situations, for instance, allowing for external sources and sinks of energy? These are a few of the urgent questions to be explored next. On our capacity to give answers to them will depend the emphasis given to the study of such nonlinear models. If we are able to force them from general background conditions through appropriate mechanisms; if we are able to show that this behavior is indeed representative of a well-defined parameter range. and not of a set of zero-measure in phase space; if, in other words, the KdV dynamics will actually be shown to emerge from different types of dynamics and in significantly wide and robust circumstances: then will the soliton become an important geophysical concept, and not go back into the realm of mathematical curiosities. ACKNOWLEDGMENTS This research was partially carried out with the support of the National Science Foundation (IDOE) as part of the POLYMODE project, Grant OCE 76-80409. I am gratcful t o Glenn R.
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Flierl of Massachusetts Institute of Technology, Cambridge; Myrl C. Hendershott of Scripps Institution of Oceanography, La Jolla, California: Greg Holloway of the Institute of Ocean Sciences, Patricia Bay. British Columbia: and James McWilliams of the National Center for Atmospheric Research, Boulder, Colorado. for careful reading of the manuscript, useful suggestions, and pointing out o f imperfections. Finally. it is a pleasure t o thank Joel Sloman, who typed the successive versions of the manuscript with care, ability, and patient endurance
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ORGANIZATION AND STRUCTURE OF PRECIPITATING CLOUD SYSTEMS ROBERTA.
HOUZE,
JR. A N D PETER v. HOB=
Departmcni c?f Airnospheric Sciences L1nivmsityof Washington
Sc.arrk cc'oshinglon
1
2
3
4
5
Introduction Extratropical Cyclones Introductory Comments and Histoncal Perspective 2 I 22 Classification of Rainbands in Cyclones 2 3 Warm-Frontal Rainbands 24 Warm-Sector Rainbands 25 Wide Cold-Frontal Rainbands 26 The Narrow Cold-Frontal Rainband 27 Prefrontal Cold Surge 28 Postfrontal Rainbands 29 Some Interactions between Rainband$ 2 10 Orographic Effects 2 1 I Vortices in Polar Air Masses Midlatitude Convective Systems 3I Thunderstorms 32 Multicell Storms 33 Supercell Storms 34 Midlatitude Mesoscale Convecti! e Complexes 35 Midlatitude Squall Lines 36 Effects of Downdraft Spreading Tropical Cloud Systems The Spectrum of Clouds in the Tropics 4I Types of Cloud Clusters 42 43 Squall-Line Cloud Clusters 44 Nonyuall Cloud Clusters 45 Generalized Cloud Cluster Structure 46 Hurncanes Conclusions References
225 229 229 2 32 234 2 37 238 239 243 243 244 246 247 247 247 249 256 275 278 284 287 287 289 290 293 296 300 303 305
1, INTRODUCTION
Precipitating clouds are important in the global circulation and climate because they interact strongly with large-scale motions through latent-heat release, cloud-scale vertical air motions, and in-cloud radiative transfer. From a local perspective, precipitation is important because it is often depended upon as a source of water and, at the same time, poses a forecasting problem because it may arrive in storms, which are sporadic, difficult to 225 ADVAYCES IN GtOPHYSlCS VOLUME 24
Copynghi Q 1982 bv Academic Press Inc All nghts of reproduction in any form reserved ISBN 0-12-0 18824-4
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FIG.1. Globally averaged annual precipitation. Types of cloud systems associated with peaks are indicated. Adapted from Sellers ( 1965).
predict, and sometimes violent. An understanding of the storms producing precipitation is therefore desired both for facilitating detailed local weather forecasts, on time scales of 1-10 hr, and for improving longer term regional and global climatic predictions through realistic parameterizations of the effects of storm clouds in large-scale models. Improvements in both types of forecasts should lead to better management of water as a resource. Basic understanding of atmospheric precipitation processes has been elusive because the scales of phenomena involved in precipitation development cover a wide range, extending well below the minimum temporal and spatial scales resolvable with standard meteorological observations. This fact necessitates the use of numerical models of clouds and the mounting of special field experiments in which meteorological radars and instrumented aircraft are deployed to observe the smaller scale processes. In the past 35 years, beginning with the Thunderstorm Project (Byers and Braham, 1949), many field experiments involving radar and aircraft have been conducted over diverse parts of the earth. During recent years, numerical modeling of clouds has also become quite sophisticated. As a result, considerable knowledge has been accumulated on the organization and structure of storms with which precipitation is associated.
'
' Hobbs ( 1 98 la) points out that the phenomena involved in the development of precipitation range from the nucleation of cloud particles to the scale of baroclinic waves. In terms o f spatial scales this range involves fifteen orders of magnitude. which is the same as the range of scales involved in comparing the linear dimensions of the earth with that o f t h e Milky Way!
FIG. 2. Idealized vertical cross section through a midlatitude cyclone, according to the Norwegian model. (Note that the vertical scale is stretched by a factor of about thirty compared to the horizontal scale).
FIG.3. Satellite photographs of cloud patterns associated with extratropical cyclones. (a) “Open wave” stage when the warm front (-), the cold front (A), and the warm sector (cloudless region between the warm and cold fronts) are distinct. (b) The larger cloud system to the right is typical of that associated with a cyclone in its occluded stage of development. The smaller cloud system, located just to the left of the center of the photograph, is a “polar vortex.” (c) The spiral cloud system in the upper part of this photograph is typical of a cyclone in its mature stage of development.
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In this article, we review this knowledge, emphasizing precipitating cloud “systems,” which contribute the bulk of precipitation over the earth. Most precipitation occurs in storms that are 20-2000 km in horizontal dimension (the meso-/3 through meso-oc scale ranges of Orlanski, 1975, and Hobbs, 198 1 a). Small showers may greatly outnumber larger precipitating cloud systems, but these do not contribute significantly to total precipitation (Lbpez, 1978; Simpson el a/., 1980; Houze and Betts, 1981). Therefore, the cloud systems that we consider in this article are all in the mesoscale size range. Besides implying large size, the term svsrern is fitting because, as we shall see. the mesoscale storms accounting for most precipitation have complex internal structures. For example, meso-oc frontal clouds contain mesop rainbands, which in turn contain smaller structures, whereas meso-/3thunderstorms group together to drive meso-cu circulations, and so on. Precipitation occurs over thc globe in three major latitude belts (Fig. 1). The midlatitude maxima of the Northern and Southern Hemispheres are essentially mirror images of cach other and are accounted for by cloud systems associated with extratropical frontal cyclones and midlatitude thunderstorms. The equatorial maximum is accounted for mainly by rainfall from tropical cloud clusters and, to some extent, by hurricanes and smaller scale convection. The remainder of this article is organized around the major precipitating cloud systems of each latitude belt. Extratropical cyclones are described in Section 2, midlatitude convective systems in Section 3 . and tropical cloud systems in Section 4.
2. EXTRATROPICAI. CYCLONES
2.1. Introductory Commenfs and flistorrcal Perspective As noted in Section 1, many of the distinctive cloud patterns seen in satellite photographs of the earth are associated with extratropical cyclones (“cyclones,” for short). Cyclones dominate the weather in midlatitudes and are the familiar systems followed on daily weather maps. They are characterized by “fronts” that curve outward for thousands of kilometers from the low-pressure centers of the storms. Upward air motions associated with these fronts (Fig. 2). and the clouds and precipitation that form in response to these air motions, coincide in broad outline with the patterns of the fronts (Fig. 3 ) . The regular occurrence of ditlerent types of clouds, precipitation. and other weather in various regions of cyclones has been recognized since the earliest days of synoptic meteorology (see, for example, the work of Abercromby, 1887). The classical picture of the large-scale structure and life
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cycle of cyclones was developed by the Bergen school in the early 1900s (e.g., Godske et a/.. 1957), and their models are still widely accepted. The Bergen school introduced the concepts of “warm,” “cold,” and “occluded” fronts. In this model (Fig. 2) the clouds and precipitation associated with warm fronts are depicted as being essentially uniform and produced by the slow, widespread uplifting of the warm-sector air as it rides up over denser, colder air. Precipitation diminishes appreciably, and may be absent, in the warm sector which follows the passage of the warm front. At the cold front, the undercutting of the warm-sector air by denser air of polar origin can produce heavy, convective precipitation. Behind the cold front, the weather is bright with scattered convective showers. The classical model depicts an occluded front as resulting from the cold front’s “catching up” and merging with parts of the warm front. The reader is referred to P a l m h and Newton ( 1 969) for detailed descriptions of the larger scale aspects of cyclones. Ttie organization of clouds and precipitation on the mesoscale, and the meso- and microphenomena that lead to precipitation in cyclones, are not discussed in detail in the classical Bergen model. This is not surprising, since at the time this model was being developed observational facilities suitable for detailed studies of subsynoptic features were not available. Before turning to a description of our current understanding of the mesoscale and microscale structure of cyclones and the mechanisms leading to the production of precipitation in these storms. we will give a brief review of earlier work that has provided the foundation for recent progress in understanding. An important advance in understanding the microphysical processes leading to the formation of precipitation in cyclones was made by Bergeron ( 1 935). who proposed that most (if not all) precipitation particles in these storms originate as ice crystals in clouds. According to this hypothesis, ice crystals are nucleated in a much larger population of supercooled droplets at temperatures below about - 10°C. Since the saturation vapor pressure in a mixed cloud which is dominated by supercooled droplets is significantly in excess of ice saturation, the ice crystals grow rapidly by deposition from the vapor phase and may reach sufficient size to fall out as precipitation. Another major advance in the study of clouds and precipitation processes in cyclones occurred in the early 1950s as a result of the increasing use of radars in meteorological studies. The common occurrence in cyclones of radar “bright bands” just below the melting level (produced by snow melting to rain) confirmed the importance of ice particles in the production of precipitation. As the resolving powers of radars were improved, increasingly finer details became apparent in the precipitation patterns. First, the precipitation was observed to be nonuniform, particularly aloft, even ahead of warm fronts. Second, “mare’s tails,” or streamers of snow, produced by the continuous formation of snow crystals in “generating cells” aloft, were
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often observed in radar displays. The reader is referred to the work of Marshall and Gordon (1957) for a review of these early radar studies. One of the earliest coordinated aircraft-radar studies of cyclones was carried out by Cunningham ( 195 1 ), who made several flights through a deep cyclone in Massachusetts. The structure of the clouds was found to be quite heterogeneous, even in the warm-frontal region where, according to the Norwegian model, the slow, uniform ascent of air should have produced fairly uniform cloud layers. In addition to the growth of ice particles in upper-level generating cells, growth of ice particles by coagulation at lower levels and the growth of raindrops by coalescence below the melting level were found to be important in different regions of the cyclone. In the 1960s and early 1970s. attention was focused on studies (mainly involving radar and/or serial rawinsondes) of the organization of precipitation on the mesoscale in cyclones. Nagle and Serebreny ( 1962) identified the basic pattern of rainbands in frontal systems approaching the West Coast of the United States. Elliott and Hovind (1964), in studies of fronts off the coast of California, identified organized bands of convective precipitation, some 35-70 km wide and 55- I00 km apart, embedded within general frontal precipitation. From rawinsonde data, Elliott and Hovind (1965) also deduced that the frontal structure included several important subsynoptic features. The warm-frontal region consisted of alternating tongues of warm, moist air and cold, dry air. The wavelength of this alternating pattern was about 200-300 km, and the bands of prefrontal convection generally occurred within the warm, moist tongue closest to the occluded front. Potential instability in the frontal lifting zone was maintained by cold advection aloft. Kreitzberg ( 1964) studied Pacific cyclones entering Washington State by combining the use of serial rawinsondes and a vertically pointing radar. He concluded that the vertical motions are much more complex than is suggested by the Bergen model, that certain recurring mesoscale features are observable, and that on the mesoscale the frontal zones (particularly the warm front) are composed of multiple subzones. Kreitzberg noted that in occlusions’ air of lower moist static energy amves in a series of pulses (called “prefrontal surges”) ahead of the occlusion. Associated with each prefrontal surge is a region of vertical air motions and associated precipitation. Kreitz-
’
Frontal systems entering the West Coast of the United States often are presented on analyzed weather maps as occlusions, and their structures bear some resemblance to classical occluded fronts. Often. however, these frontal systems do not evolve by the classical occlusion process postulated by the Bergen school. Instead. they may form discontinuously as so-called instant occlusions. This process is not well understood. In this paper we are not concerned with the process by which an “occlusion” forms, but rather with the structure and organization of precipitation processes once the larger storm has developed. We use the term occlusion loosely, to refer to these frontal systems regardless of their origin.
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berg and Brown (1970) found that most of the widespread precipitation associated with cyclones in New England occurs in mesoscale bands and groups of showers. A subsynoptic core of cold, dry air in the middle troposphere, ahead of a surface occlusion, was found to suppress widespread cloudiness in the upper regions of the cyclone but to furnish potential instability lower down. Nozumi and Arakawa (1968) made an extensive study, using radar, of cyclones in Japan. In 82% of the cyclones studied, one or more mesoscale bands of precipitation were detected in the warm sector. Austin and Houze (1972) used quantitative radar measurements and a network of precipitation gauges to study the organization of precipitation in cyclones in New England. They found that the precipitation patterns consistently displayed a hierarchical organization, in which small (- 10-1 O2 km2) and large (- 10’- I O4 km’) mesoscale precipitation areas, each containing convective cells, were embedded in the general cyclonic rain shield. The studies outlined above showed that considerable substructure exists within the basic frontal precipitation patterns associated with cyclones, and that “mesoscale rainbands” are a major feature of this substructure. Detailed studies of rainbands in cyclones have been carried out in the British Isles by K. A. Browning and his co-workers and in the Pacific Northwest by P. V. Hobbs and his co-workers. The results of these studies, which we will now summarize, have synthesized much of the early work on the mesostructure of cyclones and provided new insights into cloud and precipitation processes.
2.2. Classification of Rainhnndc in C y c h e s The principal types of rainbands (or “bands,” for short) observed in cyclones, and their positions in relations to fronts, are shown in Fig. 4. The types are: Type 1. ”mn-frontul DnndJ. These bands occur within the leading portion of the frontal system, where warm advection occurs through a deep layer, and they have orientations similar to that of the warm front. These bands are typically about 50 km wide. They may be located ahead of the warm front (Type la), coincide with a surface warm front (Type lb), or simply have an orientation similar to that of a warm front even though no well-defined warm front can be seen in standard thermodynamic or wind data. Type 2. Warm-wctor handy. These bands are in the warm sector and are oriented parallel to the surface cold front. They are typically up to about 50 km wide.
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FIG.4. Schematic depiction of the types of rainbands (numbers 1-6) observed in extratropical cyclones. From Hobbs ( I 9 8 1b).
Type 3. Wide co1d:fiontal brrnds. These bands are oriented parallel to the cold front. They are about 50 km wide and either straddle or are behind the surface cold front. In the case of occlusions, they are associated with the cold front aloft. Type 4. The nurroM1 cold,froritul hand. This type of band differs markedly from the other types of bands. 11 is very narrow ( - 5 km) and coincides with the position of the cold front at the surface. Type 5. Prefrontal, cold-surge hunch. These bands are associated with the surges of cold air ahead of the cold front. of the type described by Krejtzberg ( 1 964). Otherwise these bands are essentially the same type of feature as the wide cold-frontal bands. Type 6. Postfronral hand.y. ' I hcse bands are lines of convective clouds that form well behind and generally parallel to the cold front. In addition to the above are smaller, \.t*uvelikPrainbands that occasionally
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are superimposed with the other rainbands (Houze et al., 1976b; Matejka et al., 1980; Parsons and Hobbs, 19821). Unbanded mesoscale patterns of convective clouds, sometimes organized into roughly hexagonal-shaped cells, occur in the maritime polar air well behind the cold front. Although the full classification of rainbands described above is based on observations in the Pacific Northwest (Houze of al., 1976b; Hobbs, 1978; Matejka 01 al., 1980), it is consistent with observations of rainbands in the United Kingdom (Browning and Harrold, 1969, 1970; Browning ct al., 1973, 1974; Harrold, 1973: Browning and Pardoe, 1973; Harrold and Austin, 1974), and in the northeastern United States (Cunningham, 1951; Boucher, 1959: Austin and Houze, 1972). It is also consistent with observations made in subtropical oceanic cyclones near Japan (Nozumi and Arakawa, 1968). There is, therefore, good reason to believe that the picture in Fig. 4 is representative of the inherent mesoscale organization of precipitation in extratropical cyclcnes. It should be noted that not all of the rainbands shown in Fig. 4 are necessarily present in any one cyclone. Note also that the observed sequence of bands will be dependent on the location of the observer with respect to the large-scale features of the cyclone. In the following sections we describe, in more detail, what is known about the mesoscale features depicted in Fig. 4. In each case we present information, insofar as it is available, on the air motions within the rainband, the substructure of the rainband, the precipitation-producing mechanisms, and the dynamic processes responsible for the formation of the band.
2.3. Warm-Frontal Ruinhands Warm-frontal rainbands arise when precipitation becomes enhanced in mesoscale regions embedded within the large area of cloudiness and stratiform precipitation produced by the widespread lifting associated with warm advection in the leading portion of the cyclonic system. Shown in Fig. 5 is a schematic which summarizes information on the structure of warmfrontal rainbands and the processes by which precipitation is formed in these rainbands. There is considerable observational evidence that the precipitation within these rainbands involves “seeding” from above by ice particles (Cunningham, 1951; Plank et al., 1955; Browning and Harrold, 1969; Houze et a/., 1976a, 198 lb: Hobbs and Locatelli, 1978; Herzegh and Hobbs, 1980; Matcjka et a/., 1980). The ice particles are nucleated in groups of generating cells aloft, where they grow to precipitable size. probably b y a combination of vapor deposition and riming (i.e., collecting supercooled water droplets). They then fall through stratiform cloud below, where they continue to increase in mass by deposition, aggregation, and riming. In this
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FIG. 5. Model of a warm-frontal rainband shown in vertical cross section. The structure of the clouds and the predominant mechanisms lor precipitation growth are indicated. Vertical hatching below cloud bases represents prccipitation: the density of the hatching corresponds qualitatively to the precipitation rate. The heavy broken line branching out from the front is a warm-frontal zone with convective ascent in the generating cells. Ice particle concentrations . motion (ipc) are given in numbers per liter; cloud liquid water contents (Iwc) are in g m ~ 3 The of the rainband In the figure is from left to right. From Hobbs (1978) and Matejka ef a/. ( 1980).
type of rainband, riming growth is usually small, whereas growth by deposition can be very important. It should be noted that whereas aggregation (which may be particularly important near the melting level) can have an appreciable effect on particle fall speeds, it can not change precipitation rates. The streamers of ice originating from individual generating cells give rise to enhanced precipitation rates over small mesoscale regions within each warm-frontal rainband. These regions account for the small mesoscale areas and cells identified in these rainbands by Austin and Houze (1972) and Hobbs and Locatelli (1978). Although seeding from above appears to play a crucial role in enhancing precipitation in warm-frontal rainbands, typically only 20-35% of the total mass of precipitation reaching the ground originates from the “seeder” zone. The remaining 65-80% originates in the “feeder” clouds below, although the ice particles from above are needed to collect this mass (Cunningham, 195 1 ;Herzegh and Hobbs, 1980: Houze c’t al., 198 1b). There is also evidence that in some cases the feeder clouds are enhanced by nonconvective mesoscale lifting and that on occasion this lifting is sufficient to increase precipitation rates through liquid-phase processes alone (Herzegh and Hobbs, 1980; Houze et a/., 198 lb). The “seeder-feeder’’ process in warm-frontal rainbands has been quantitatively reproduced in a numerical model described by Rutledge and Hobbs ( 1982).
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FIG.6 . Model of two warm-sector rainbands shown in vertical cross section. The structure of the clouds and the predominant mechanisms for precipitation growth are indicated. Vertical hatching below cloud bases represents precipitation: the density of the hatching corresponds qualitatively to the precipitation rate. Open arrows depict airflow relative to the rainbands; Ow is wet-bulb potential temperature. Ice particle concentrations (ipc) are given in numbers per liter: cloud liquid water contents (Iwc) are in g m-3. The motion of the rainbands in the figure is from left to right. From Hobbs (1978) and Matejka rt a/. (1980).
Since the generating cells associated with warm-frontal rainbands are often located in potentially unstable layers, it is evident that they are produced by the lifting of these layers to release their instability. Potentially unstable air above warm fronts tends to arrive behind tongues of warm, moist air that branch out from the warm front (Kreitzberg, 1964), and it has been noted (Kreitzberg and Brown, 1970; Matejka et a/., 1980) that warm-frontal rainbands are associated with such branches (Fig. 5). Questions remain about the source of the potential instability and the mechanism for its release. Air-trajectory analysis indicates that layers of potential instability above warm fronts in extratropical cyclones may sometimes originate in the potential instability of the subtropical air mass (Houze et al., 1976a). Alternatively, the potential instability may be generated by differential advection in the middle troposphere (Elliott and Hovind, 1964: Harrold, 1973). A third possibility that has been suggested is that the generating cells form when a layer of moist tropospheric air overrun by dry stratospheric air is lifted and becomes saturated (Wexler and Atlas, 1959). Lindzen and Tung ( 1976) have investigated the ducting of gravity waves in a statically stable layer that is bounded above by an unstable or neutral layer. They find that mesoscale gravity waves can propagate under these conditions, and they propose that vertical motions associated with these waves may initiate warm-frontal rainbands. These vertical motions could
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produce rainbands either by promoting the release of instability in the potentially unstable layer above the duct (i.e.,by producing a mesoscale seeder zone with a high concentration of gcnerating cells) or by creating a denser feeder cloud at lower levels as a result of the increased lifting and condensation on the mesoscale. This mechanism has appeal because the stratification required by the theory is similar to that observed and the theoretical wavelengths and phase speeds are similar to those measured for warmfrontal rainbands (Parsons and Hobbs, 1982b). To prevent undue dispersion of the energy of the waves, it is necessary that the wind at some altitude within the unstable or neutral region be about equal to the velocity of the wave. The latter requirement implies a “steering level” for rainbands, at an altitude which also agrees with observations (Hobbs and Locatelli, 1978). Symmetric instability is another possible mechanism for lifting in baroclinic zones where rainbands occur (Bennetts and Hoskins, 1979). The alignment of rainbands along the direction of the thermal wind, their spacing. and the stable lifting needed to release potential instability, indicate that symmetric instability, modified by moisture, could be responsible for the generation of warm-frontal rainbands.
2.4 W’arm-Sector Rainhands The vigor and intensity of warm-sector rainbands varies considerably. In their most vigorous form, they may take the form of prefrontal squall lines. The less vigorous warm-sector rainbands often show similarities to squall lines; more work is needed to distinguish clearly the weaker and stronger cases. In the present discussion we will describe the weaker warm-sector bands. Midlatitude squall lines are treated in Section 3.5. Figure 6 summarizes some o f the dominant features of the weaker type of warm-sector rainbands. In contrast to the warm-frontal bands, which are primarily stratiform. with shallow embedded convective cells aloft, the warm-sector bands can contain deep convective cells, extending vertically through the full depth of the rainband. These bands are fed by boundarylayer convergence concentrated at a surface gust front, similar to those of squall lines. As depicted in this figure, rainbands often occur in series, with the younger, more vigorous bands preceding the older bands. The clouds of younger rainbands are strongly convective, containing relatively high concentrations of supercooled cloud water, but low concentrations of ice. Growth of ice particles by riming is an important feature of these bands. However, the older, less intense bands are dominated by ice particles, which continue to
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grow by aggregation. The heavy riming (including graupel formation and growth) that occurs in the young active warm-sector bands is in marked contrast to the slight riming and predominance of vapor deposition and aggregation in the more stratiform warm-frontal bands. In some warm-sector bands, seeder-feeder processes can be involved in the growth of the precipitation. In one case, 10-20% of the mass of precipitation from the rainband originated in the seeder zone, and 80-90’70 in distinct regions below the seeder zone. One of these regions was a zone of deep, vigorous convection in which 50-60% of the mass of precipitation developed. The other region consisted of stratiform cloud, which served as a feeder zone, in which 30-40% of the total mass of precipitation developed. The precipitation efficiencies in the convective and stratiform regions were -40 and -80%. respectively (Hobbs et a/., 1980). More work is needed to understand the difference between this type of warm-sector band and the deep convective type. Warm-sector rainbands may be associated with internal gravity waves that propagate away from the cold front. During their initial stages of formation, positive feedbacks involving moisture may be important (Parsons and Hobbs, 1982b). Possible mechanisms for formation of the gravity waves are geostrophic adjustment (Ley and Peltier, 1978) and frontal convection (Ross and Orlanski, 1978). At large distances from the cold front, ducting of internal gravity waves (Lindzen and Tung, 1976) is possible and may explain the maintenance of warm-sector rainbands. Other possible mechanisms for maintaining warm-sector rainbands include wave-CISK (LindZen, 1974; Raymond, 1975), forced symmetric instability (Bennetts and Hoskins, 1979), and the release of potential instability (Kreitzberg and Perkey, 1976. 1977).
2.5. Wide Cold-Frontal Ruinbands Wide cold-frontal rainbands occur when lifting over the cold-frontal surface is enhanced by several tens of centimeters per second over horizontal distances of several tens of kilometers. Structurally they resemble warmfrontal rainbands. Release of potential instability in the form of generating cells occurs aloft. Ice crystals that form in these cells grow as they fall through lower cloud layers to give rise to rainbands (Hobbs, 1978; Matejka et a]., 1980; Hobbs el ul., 1980), as depicted schematically in Fig. 7. Since the steering level of these rainbands is located at the height of the generating cells, they can move faster than the surface cold front. As i n the case of warm-sector rainbands, those wide cold-frontal rainbands in which a seeder-feeder mechanism operates have a high (- 100%) precipitation efficiency (Hobbs and Matejka, 1980).
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FIG.7. Model of the clouds associated with a cold front showing narrow and wide coldfrontal rainbands in vertical cross section. ’The structure of the clouds and the predominant mechanisms for precipitation growth arc indicated. Vertical hatching below cloud bases represents precipitation: the density of thc hatching corresponds qualitatively to the precipitation rate. Open arrows depict airflow relative to the front: a strong convective updraft and downdraft above the surface front and pressure trough. and broader ascent over the cold front aloft. Ice particle concentrations (ipc) are given in numbers per liter; cloud liquid water contents (Iwc) are in g m-3. The motion of the rainhand in the figure is from left to right. Horizontal and vertical scales are approximate, but typical of aircraft and radar observations in specific cases. From Hobbs (1978) and Matejka el (11. (1980).
Symmetric instability (Bennetts and Hoskins, 1979) is a likely mechanism for the formation of wide cold-frontal rainbands. The predictions of this theory, with respect to the location, movement, and spacing of rainbands, are in good agreement with observations (Parsons and Hobbs, 1982b).
2.6. The Narrow Cold-Frontal Rainburxi The narrow cold-frontal rainband occurs at the leading edge of a cold front, where converging air produces a narrow ( - 5 km wide) updraft. Air may ascend in this updraft at a velocity of a few meters per second directly above the windshift at the surface when the cold front reaches the ground, or above the cold-frontal passage aloft in a warm-type occlusion (see Figs. 4 and 7). The cloud towers associated with the updraft may penetraJe the larger cloud shield associated with the cold front (as shown in Fig. 7), but more commonly they do not. The source of moisture in the updraft is a low-level, southerly jet situated just ahead of the cold front, onto which the easterly moving cold front continually encroaches (Browning and Harrold,
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1970: Hobbs et ul., 1980). The updraft typically forms a coupled system with a downdraft, the latter coinciding with the heavy precipitation (- 100 mm hr-’) associated with the narrow cold-frontal rainband (Fig. 7). The cloud band associated with the updraft in the narrow cold-frontal rainband contains large amounts of liquid water, but relatively low iceparticle concentrations. Consequently. the ice particles grow primarily by riming. The particle concentrations are high (- 100 liter-’) in the downdraft. Precipitation aloft consists of rimed aggregates of ice particles. Graupel and hail may form in these bands. The heaviest precipitation in the narrow cold-frontal rainbands is organized, on the smali mesoscale, into ellipsoidal areas oriented at angles of 30-35’ to the synoptic-scale cold front (Hobbs. 1978; James and Browning, 1979: Hobbs and Biswas, 1979: Hobbs and Persson, 1982). Hobbs and Biswas refei, to these areas as “precipitation cores.” and we will use this term here. A schematic ofthe structure of a cold front on the small mesoscale is shown in Fig. 8. where it can be seen that the precipitation cores are located in regions of high surface convergence. The precipitation cores are separated by “gap” regions where the mesoscale cold front “kinks,” resulting in reduced convergence and therefore lower precipitation rates. The relationship between the passage of a precipitation core and variations in wind, pressure, temperature, and rainfall on the surface are indicated schematically in Fig. 8. Both windshifts and pressure checks occur -5 min before a peak in rainfall rate, and temperature drops occur at the time of, or shortly after, the heavy rain associated with the downdraft of a precipitation core (James and Browning, 1979; Hobbs and Persson, 1982). The sequence of events on the surface during the passage of a gap region depends on whether or not small. convective precipitation areas within the windshift zone pass over the ground station. If they do not, the pressure check and windshift occur slightly before or at the same time as the fall in temperature, and the rainfall rate does not peak (James and Browning, 1979). If a convective precipitation area does pass over the station, a distinct peak occurs in the rain rate, followed by a pressure check, windshift, and temperature drop. The last three parameters may change simultaneously, or the pressure check and windshift may occur just prior to the temperature drop (Hobbs and Persson, 1982). In many respects, the passage of a precipitation core associated with a narrow cold-frontal rainband resembles a squall-line gust front. Although the outflow of cold air from a squall line is generally not accompanied by precipitation, a pressure j u m p and a shift in the wind generally occur 5-10 min ahead of a drop in temperature. The circulation of the air at low levels in the vicinity of the northeastern tip of a precipitation core (i.e.. near one of the “kinks” in the temperature
FIG.8. Schematic summarizing the small-mesoscale structure of a cold front. The areas of the precipitation cores (PC) on the ground are outlined by the elongated ovals along the temperature front (- A - A -) and they are separated by gap regions (GR) where the temperature front jags. The large arrows represent the airflow relative to the motion of the precipitation cores. The hatched areas are the areas of strong updrafts, and the stippled areas are the areas of weak downdrafts. Adapted from Hobbs and Persson (1982).
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FIG. 9. Schematics of: (a) the “bow” echo that Fujita (1981) associates with mesoscale downbursts and tornadoes; (b) the “boomerang” echoes that Hobbs and Biswas ( 1979) associate with narrow cold-frontal rainbands.
front) is such as to induce cyclonic rotation of the winds; this forms a “mesolow” in which there is strong wind shear (Hobbs and Persson, 1982). Carbone ( 1982) has documented a case where a moderate tornado developed from such a mesolow in northern California. Fujita (1981) has pointed out that in the central United States mesoscale downbursts (i.e., localized currents of rapidly sinking air, which induce an outward burst of damaging winds on or near the ground-see Section 3.6) and tornadoes are often associated with radar echoes that have a “bow” shape (Fig. 9a). The radar echoes associated with the mesolows of cold fronts have a similar shape (Fig. 9b), called “boomerang” echoes by Hobbs and Biswas (1979). It appears therefore that Fujita’s bow echo represents a n extreme example of the more common boomerang echo associated with cold fronts. Gravity-current models, such as that described by Benjamin ( 1 968), which simulate the dynamics of a high-density fluid overtaking a lower density fluid, provide reasonable predictions for the speeds of propagation of cold fronts (Carbone, 1982; Hobbs and Persson, 1982) as well as gust fronts associated with thunderstorms (Charba, 1974). Moreover, the breakup of the narrow cold-frontal rainband into precipitation cores and gap regions resembles the pattern seen when gravity currents are produced in laboratory tank experiments (Simpson, 1972). Another possible mechanism for the formation of precipitation cores and gap regions is through instabilities produced across the cold front by the strong horizontal shear of the component of the wind parallel to the front (Matejka, 1980; Hobbs and Persson, 1982).
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In an occlusion, the cold air advances over the warm front in a series of pulses. The strongest pulse is generally analyzed as the cold front itselt the weaker pulses are referred to as prefrontal cold surges (Kreitzberg, 1964; Kreitzberg and Brown, 1970; Browning et al., 1973; Matejka et af., 1980). A surge is marked at the surface by a temporary slight rise in pressure or a decreasing fall in pressure (Fig. 10). Behind the prefrontal cold surge aloft is a core of low moist static energy air that tends to suppress upper cloud layers but enhances potential instability below. A deep band of cloud and precipitation precedes, or straddles, the leading edge of the prefrontal cold surge. This prefrontal-cold-surge (or “surge,” for short) rainband is similar in structure to the wide cold-frontal and warm-frontal rainbands. The mechanisms responsible for the formation of surge rainbands are probably the same as those that form wide coldfrontal bands (Parsons and Hobbs, 1982b).
2.8. Postfrontal Rainbands Postfrontal rainbands (Type 6 in Fig. 4) are lines of convection that form in cold air masses behind zones of strong subsidence, immediately following
FIG. 10. Model of rainbands associated with a prefrontal surge of cold air aloft, ahead of an occluded front. The broken cold-front symbol indicates the leading edge of the surge. (The primary cold front is off the figure to the left.) The structure of the clouds and the predominant niechanisms for precipitation growth are indicated. Vertical hatching below cloud bases represents precipitation: the density of the hatching corresponds qualitatively to the precipitation rate. Open arrows depict airflow relative to the cold surge and convective ascent. Ice particle concentrations (ipc) are given in numbers per liter. The motion of the cold surge and the rainbands in the figure is from left to right. From Hobbs (1978) and Matejka ef a/. (1980).
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the passage of a cold front. Since postfrontal bands are not usually obscured by upper-level clouds or embedded in widespread layer clouds, they may often be observed visually from the ground and in satellite photographs. On the small mesoscale, postfrontal rainbands comprise groups of convective clouds occupying horizontal areas ranging from -50 to 10’ km2. Sometimes new lines of convection form immediately ahead of an existing line of decaying cells, suggesting that in these areas they behave as organized convective systems such as squall lines (Houze r t ul., 1976b). In this respect they resemble warm-sector rainbands. As in warm-sector bands, the microphysical structures of postfrontal rainbands depend strongly on the age of the convective cells being sampled. In young cells containing relatively large quantities of supercooled water, ice crystals grow by riming; showers of graupel are common in such situations. In older cells, which tend to be glaciated, particle growth is primarily by aggregation. The unstable conditions associated with postfrontal rainbands suggest that a wave-CISK mechanism, incorporating horizontal temperature gradients and vertical shear, is a possible mechanism for their formation (Parsons and Hobbs, 1982b). Following the passage of postfrontal rainbands, large areas behind the cyclone are often covered by regions of convective clouds. In many respects those clouds are similar to those of postfrontal rainbands, in that they originate in unstable layers and are convective in nature. However, they are organized into hexagonal cells -40 km across and are separated by clear regions -20-50 km across. The centers of cells are often cloud-free, in which case they are referred to as “open hexagonal cells” (Krishnamurti, 1975a). The heights of the radar echo tops of these cells are similar to those of the postfrontal bands, and they move with the wind at about the level of free convection. The convection model proposed by Krishnamurti ( 1975a,b,c) provides a reasonable explanation for open hexagonal cells. However, despite the fact that this theory provides accurate predictions of the location of hexagonal cells and some of their characteristics (e.g., movement, cloudy or cloud-free centers), it often underestimates the spacing of cells by up to an order of magnitude (Krishnamurti, 197%: Parsons and Hobbs, 1982b).
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2.9. Sornc~Inteructions hctwccw Ruinhands Narrow and wide cold-frontal rainbands exhibit the most obvious interactions (Parsons and Hobbs, 1981). The observed interactions can be divided into three categories (Fig. l l ) . In the first, a wide cold-frontal band moves over and ahead of a narrow cold-frontal band. This modifies the
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FIG. I 1 . Schematic showing three modes of interaction of a wide cold-frontal rainband with a narrow cold-frontal rainband. (a) The w8ide cold-frontal rainband overtakes the narrow coldfrontal rainband. The narrow cold-frontal rainband is disturbed but re-forms after the wide cold-frontal rainband moves on. (b) Thc wide cold-frontal rainband overtakes the narrow coldfrontal rainband and the latter dissipates. ( c ) The wide cold-frontal rainband reaches, but does not move ahead of. the narrow cold-frontal rainband. From Parsons and Hobbs (1981).
narrow cold-frontal band, though it continues to exist (Fig. 1 la). The modification to the narrow cold-frontal band begins when the wide cold-frontal band is located over the surface cold front. At this stage, it may be difficult to locate the narrow cold-frontal band. The passage of the wide band over the narrow may cause some decrease in the frontal convergence in the boundary layer, which is necessary for the maintenance of the narrow coldfrontal band (Hobbs and Persson. 1982). As the wide cold-frontal band moves ahead of the surface front, the narrow cold-frontal band begins to re-form as an irregular line containing precipitation cores, although the latter may be without distinct alignment. Later the cores align at their usual angle of 30-35" to the front. The time scale for the re-formation of the precipitation cores ranges from 10 to 75 min. In the second category of interaction, a wide cold-frontal band moves
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over and then ahead of the narrow cold-frontal band but the latter does not re-form (Fig. 1 lb). After the wide cold-frontal band moves ahead of the surface front, the narrow cold-frontal band dissipates and the frontal windshift is greatly weakened. The third category of interaction that has been observed occurs when the wide cold-frontal band moves over the surface front but then dissipates (Fig. 1 lc). In this case, the narrow cold-frontal band and the precipitation cores are particularly well defined, while the wide cold-frontal band aloft is rather weak. 2.10. Orogruphic Efects
Orography can have a profound influence on rainbands. In the case of small (- 50 m high), isolated hills, precipitation may be increased on the windward slopes as enhanced condensation produces a feeder cloud that is scavenged by precipitation falling from higher level seeder clouds (Bergeron, 1935). The seeder-feeder process leads to the strong correlation between elevation and precipitation amounts reaching the ground. This situation has been modeled quite well by Storebei ( 1 976), Bader and Roach (1977), and Gocho ( 1 978). Lifting of the air by topographic features can produce convection in the mid-troposphere and perhaps initiate rainbands (Browning c’t a/., 1974). However, high terrain (e.g., mountain ranges b 1 km in height) may lead to downward motion on the lee slopes (Fraser et a/., 1973) or it may, on the windward slopes, block the low-level flows necessary for the maintenance of rainbands (Hobbs el a/., 1975). In a recent study, Parsons and Hobbs ( 1982a) observed that warm-frontal and wide cold-frontal rainbands are generally only interrupted by descent in the lee of large orographic features. Since precipitation is produced mainly by the seeder-feeder mechanism in these two types of rainbands, any condensate produced by orographic lifting enhances the precipitation from the rainbands. Similarly, orographic lifting enhances the precipitation from those warm-sector rainbands in which the seeder-feeder mechanism plays an important role in the production of precipitation. There is a tendency for rainbands to be generated over hills when the atmosphere is unstable either in the lower layers (e.g., in postfrontal conditions) or aloft (e.g., in the warm sector). When the airflow in the lower layers is stable and parallel to a range of mountains, channeling of the flow by the mountains can cause dissipation of rainbands. The precipitation associated with narrow coldfrontal rainbands is generally unaffected by orography, although high mountains can change the orientation of the bands.
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2.11. Vortices in Polar Air MU,V,W.Y
In Fig. 3b the clouds just to the left of the center of the picture, and to the west of the trailing front, are associated with a small vortex in the polar air mass. In Europe such vortices are referred to as “polar lows,” while in North America they are called “comma clouds” (Reed, 1979). These vortices exhibit many similarities to cyclones, although they are smaller in scale, and they can give rise to significant weather. For example, in California they account for a fairly large proportion (20-50%) of the precipitation (Monteverdi. 1976). Locatelli et al. (1982) documented three case studies of vortices that contained mesoscale rainbands of the types described above, and features which, in the case of cyclones, would have been analyzed as fronts. Locatelli et al. also point out that in the cases they studied, the vortices played key roles in forming so-called instant occlusions.
3. MIDLATITUDE CONVECTIVE SYSTEMS 3.1. Thitnderstorms
During the warmer half of the year, precipitation over midlatitude land masses is dominated by deep convective events, collectively referred to as thunderstorms, which stand in contrast to the large-scale cyclonic storms described in the preceding section. The “Glossary of Meteorology” (Huschke, 1959) defines a thunderstorm as “. . . a local storm invariably produced by a cumulonimbus cloud, and always accompanied by lightning and thunder. usually with strong gusts ofwind, heavy rain, and sometimes with hail.” In addition, it defines a tornado as “a violently rotating column of air, pendant from a cumulonimbus cloud and nearly always observable as a funnel cloud. . . . On a local scale it is the most destructive of all atmospheric phenomena.” The term “severe thunderstorm” is usually reserved to describe thunderstorms that are accompanied by tornadoes, very large damaging hail, especially strong nontornadic winds associated with storm downdrafts (including “gust fronts” and the type of sudden downdraft referred to by Fujita, 198 I , as a “downburst”), or some combination of these phenomena. Thunderstorms may occur either in isolation or grouped together in mesoscale complexes or squall lines. These groups of storms are quite significant and the subject of much recent research. In Sections 3.2 and 3.3, we consider the structure of individual thunderstorms. Then in Sections 3.4 and 3.5 we discuss mesoscale complexes of storms and squall lines.
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As elsewhere in this article. we focus our discussion on the internal airmotion and precipitation structures of storms. We do not treat thunderstorm forecasting or the electrical activity of the storms. 3.2. Mult icell Storms The internal structure of thunderstorms was first investigated as a specific observational objective in the Thunderstorm Project (Byers and Braham, 1949). This project was the first coordinated use of instrumented aircraft and radars together with intensive soundings and surface observations to explore the structure of a particular type of storm. The storms investigated were the common summertime thunderstorms of Florida and Ohio. These thunderstorms typically occur in widespread convectively unstable air masses characterized by low-level warm, humid air and little vertical wind shear. This type of storm has come to be referred to as the “air mass” thunderstorm. In the Thunderstorm Project, the internal structure of the storm was found to consist of a generally random pattern of “cells” (Fig. 12). The term “thunderstorm” is used to refer to the overall aggregate of cells, and its lifetime (several hours in the case of air mass storms) considerably exceeds that of an individual cell (- I hr). Thus, the pattern of cells within the air mass thunderstorm is continually changing. A special case occurs when only a single cell develops, matures, and dissipates, but no adjacent or subsequent cells develop to form a larger storm complex. Chisholm and Renick ( 1972) assert that such “single-cell storms” are the most common type of thunderstorm. This is probably true, if every towering cumulus that reaches considerable height and precipitates is considered to be a thunderstorm. However, the significance of single-cell storms in terms of precipitation (Simpson et al., 1980) or storm damage (see Chisholm and Renick’s Fig. I ) is practically negligible. Hence, we focus here on thunderstorms consisting of more than one cell. In the Thunderstorm Pro.iect, it was deduced that each cell within a thunderstorm undergoes a life cycle with characteristic stages. The life cycle of a single cell within a thunderstorm has since been simulated quantitatively in numerous cloud models. For example, Ogura and Takahashi (197 1 ) used a simple model that computes the areal averages of in-cloud properties at
FIG. I ? . (a) Plan view of an example o f an air mass thunderstorm observed in Ohio during the Thunderstorm Project. Developing cells contained updrafts (U); mature cells, both updrafts and downdrafts (D); and dissipating cells. only downdrafts. In (b) and (c) vertical cross sections along B-B and A-A’ are shown. Adapted fi-om Ryers (1959).
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FIG. 13. Time-height cross sections of (a) vertical velocity (m/sec), (b) excess temperature (“C).(c) liquid and solid water content (g/kg). (d) content of cloud droplets (g/kg). (e) content of raindrops (g/kg). and ( f ) content of ice crystals (g/kg) for a thunderstorm cell simulated by a one-dimensional, time-dependent cloud model. From Ogura and Takahasi ( 197 I ),
a series of heights in a cylindrical cell. Their results illustrate the stages in the life cycle of a thunderstorm cell (Fig. 13). The developing stage (called the “cumulus” stage by Byers and Braham, 1949) is characterized by a growing cloud pushing its way up toward its maximum height (Fig. 13d). The interior of the cell at this stage is filled with buoyant air (Fig. 13b) moving upward (Fig. 13a). Precipitation particles begin developing early and near cloud base but d o not yet reach the ground (Fig. I3e). The tendency of the precipitation particles to fall is offset in this early stage by the strong updraft.
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Eventually, the weight of the precipitation particles becomes considerable and their drag on the air initiates a negatively buoyant downdraft in the lower portion of the cell. The appearance of this downdraft (at 40 min in Fig. 13) marks the beginning of the “mature” stage of the cell, in which updraft and downdraft coexist. Losing their supporting upward motion, the precipitation particles begin reaching the ground (50 min in Fig. 13). The disappearance of the updraft (after 60 min) defines the beginning of the “dissipating” stage of the cell. During this stage, a weak downdraft persists until the remainder of the precipitation falls out as light rain. The cells making up the air mass thunderstorm depicted in Fig. 12 were in various stages of their life cycles. Those with updraft only were in their developing stages, those with both updraft and downdraft were in their mature stages, while those with downdraft only were in their dissipating stages. The formation of new cells in such storms is favored where the cold downdraft spreading out at low levels from an older cell helps to lift ambient air to its level of free convection (e.g., the new cell near A‘ in the cross section of Fig. 12b). New cells are thus formed in the vicinity of old cells, especially between cells, where two downdraft outflows collide (Byers and Braham, 1949; Simpson et al., 1980). It is largely by this process that the multicellular cluster comprising the thunderstorm is developed and maintained. While emphasizing the importance of this regeneration mechanism for cells, Byers (1959; see also Byers and Braham, 1949, pp. 77-79) also noted that: In many cases the time interval between the beginning of the outflow and the appearance of the new cell on the radarscope is too short to permit explanation of the new one as a result of the underrunning cold air or a similar time-consuming process. There are cases, as indicated by the radar echoes, in which one new cell or a cluster comes into existence almost simultaneously with the initial or parent cell; this suggests that a preferred region of convergence and ascent favors the development of several cells.
Such a region of convergence can occur along an “arc cloud line,” which marks the boundary of a downdraft outflow emanating from a distant older thunderstorm or storm complex (see Section 3.6.2 for further discussion), or be produced by some other highly localized forcing (e.g., sea-breeze convergence; see Simpson e/ ui., 1980). The air mass thunderstomi, described by the Thunderstorm Project, is a member of a broader class of storms referred to generally as “multicell” thunderstorms. Another type of multicell thunderstorm, which we shall call the “organized’ multicell storm, differs from the air mass storm in that it occurs in an environment of substantial wind shear and, as a result, the cells form and move through the storm in a systematic rather than a random fashion. This process is illustrated in Fig. 14, which is a vertical cross section along the direction of motion of a multicell thunderstorm that produced
FIG. 14. Schematic model of a multicell hailstorm observed near Raymer, Colorado. It shows a vertical section along the storm’s north to south (N-S) direction of travel, through a series of evolving cells. The solid lines are streamlines of flow relative to the moving system; they are shown broken on the left side of the figure to represent flow into and out of the plane and on the right side of the figure to represent flow remaining within a plane a few kilometers closer to the reader. The chain of open circles represents the trajectory of a hailstone during its growth from a small particle at cloud base. Lightly stippled shading represents the extent of cloud and the three darker grades of stippled shading represent radar reflectivities of 35, 45, and 50 dBZ. Environmental winds (m/sec, deg) relative to the storm are shown on the left-hand side of the figure. From Browning et al. (1976).
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hail in eastern Colorado. The figure can be thought of either as an instantaneous picture of the storm. with cells in various stages of development, or as a sequence of stages in the life of one cell as it moved, in a relative sense, through the storm. New cells (at n + l in Fig. 14) formed on or just ahead of the leading edge of the storm. As cells moved through the storm, they underwent life cycles vcry similar to the life cycle of a cell in an air mass thunderstorm. At n + l and 17, the cells were in the developing stage, with updraft air filling the cells and precipitation particles developing aloft but not yet falling to the ground. Precipitation particles were initiated near cloud base n f 1 and grew by collection of supercooled cloud water. Above the 0°C level, the collectors were primarily ice particles, whose growth, after their formative stages, was dominated by the accumulation of rime ice, which formed as cloud liquid water was accreted. Continuation of this riming built up graupel particles and hailstones, which eventually became big enough to fall relative to thc ground (Dye et a/.,1974). The schematic hail trajectory in Fig. 14 was based on an assumption that the particle, once initiated, remained within the same cell throughout its lifetime. Heymsfield ef al. ( 1 980) present evidence that optimal hail production in a multicell storm occurs by the initiation of graupel particles and hailstones in smaller cells and their subsequent advection into the updraft of the most intense cell of the storm. Other studies note still finer scale patterns of vertical velocity (Battan, 1975. 1980). radar reflectivity (Barge c’t ul., 1976), and surface hailfall (Coyer, 1977). which indicate that further variability in hail growth is superimposed on the basic cellular pattern of the storm. The cell at n-1 was the most intense in the storm depicted in Fig. 14. It had the characteristics of a mature thunderstorm cell: its maximum height had been attained: the updraft in its upper regions coexisted with a strong downdraft at lower levels; and heavy precipitation, including hail, was reaching the ground. By n-2, the cell had the characteristics of a dissipating cell: the updraft had disappeared; weak downdraft existed throughout the cell: and precipitation. though still falling, was considerably weakened. Chalon et ul. (1976) showed that the motion of the organized multicell thunderstorm in Fig. 14 was the result of two components: one was the result of the movement of individual cells (V, in Fig. 1%) along the direction of the middle-level winds and slightly to the left of the overall storm movement, while another (V,) was the result of the periodic and discrete propagation by new cell formation on the right forward flank of the storm. Thus, the new cell formation always occurred on the storm’s leading edge. Other organized multicell thunderstorms frequently exhibit new cell formation on their right rear flanks, with cell dissipation on the left forward flanks (Brooks, 1946; Browning, 1962; Chisholm and Renick. 1972; Newton and Fankhauser, 1975). In such cases. the discrete propagation retards the motion
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FIG. 15. Organized multicell thunderstorm motion (V,) as the sum of individual cell motion (V,) and discrete propagation by new cell formation (V,) for (a) forward-moving, (b) rightmoving, and (c) left-moving storms. Case (a) is from Chalon el a/. (1976). Cases (b) and (c) are described by Newton and Fankhauser (1975).
of the storm and makes it deviate to the right of the individual cell motion (Fig. 15b). Since individual cell motions are typically within +30° of the mean wind in the cloud layer,3 the overall storm motion is also typically to the right of the mean wind. These rightward-deviating storms are quite common; however, organized multicell storms also occasionally deviate to the left of their individual cell motion (and hence to the left of the mean wind in the cloud layer) as a result of systematic cell formation on the left flank of the storm and dissipation on the right (Fig. 15c; see also Hammond, 1967; Newton and Fankhauser, 1975). The ambient wind shear determines whether a multicell thunderstorm takes on the characteristics of an air mass storm, with random cell regeneration, or of an organized storm, with an orderly pattern of cell formation on a favored side of the storm. As in the air mass storm, new cell formation in the organized multicell storm tends to be triggered at the edge of downdraft outflow from an older cell. However, since the organized storm moves through an environment of substantial vertical wind shear, there is strong flow relative to the storm at various levels, including low levels. Thus the boundary layer becomes an inflow layer feeding into a particular side of the thunderstorm, and new cell generation is highly favored where this strong low-level inflow meets the downdraft outflow of the mature cell. At this point (the zero horizontal coordinate in Fig. 14), maximum and highly These departures from the mean wind are to the left for smaller cells, to the right for larger cells, and are greater when the environmental winds are strongly veered (Newton and Fankhauser. 1964, 1975).
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concentrated velocity convergence leads to new cell formation. As long as the relative flow pattern remains intact, the tendency to form cells on this side of the overall storm is maintained. The orientation of the ambient shear vector, and hence the low-level flow vector, determines on which flank of the storm cells regenerate. In the case of air mass thunderstorms with little shear in the environment, there is little flow relative to the storm at any level. A horizontal inflow layer cannot become firmly established, and no side of the storm is strongly favored for cell development. However, even in air mass storms with slight shear, Byers and Braham (1949) noted a tendency for new development on the low-level upwind.side of downdraft outflows (see their pp. 77-79). When the relative flow in the organized multicell storm is examined at all levels, it is seen that the cells constituting the storm are either superimposed on a circulation pattern on the scale of the thunderstorm itself, or the cells together constitute such a pattern. It can be seen from Fig. 14 that the circulation pattern is characterized by general ingestion and upward flow of warm moist air entering the storm in the low-level inflow layer. The rising air encompasses the updrafts of the developing and mature cells. It exits in an anvil on the upper-level downwind side of the storm. Dry midtropospheric air enters on the middle-level upwind side, is cooled as precipitation particles evaporate into it and sinks in a broad downdraft, which occupies nearly the whole lower portion of the storm and contains the downdrafts of all the mature and dissipating cells. As the downdraft air generally spreads out at the surface, part of it goes against the environmental low-level flow. This portion of the overall storm downdraft region is associated with the downdraft of the mature cell, the leading edge of which runs under the inflowing updraft air in the region of new cell formation. Another portion of the downdraft air flows out of the storm on its low-level downwind side, Whether the larger storm-scale updraft-downdraft pair of the organized multicell thunderstorm really constitutes a circulation physically distinct from the individual cells, or is simply an agglomeration of the air motions of the cells closely spaced in order of their successive stages of development, is not clear. Whichever is the case, the continued low-level inflow of warm moist air and midlevel inflow of dry air maintained by the storm-scale flow allows the storm to last for a long time. In contrast, the lifetime of an air mass storm, which lacks an organized storm-scale flow, is limited, since it can draw only upon the boundary-layer air in its near environment. For this reason, the organized multicell storms are generally longer lived and more severe than air mass storms (Weickmann, 1953; Newton and Fankhauser, 1975).
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3.3. Sirpcwell Storms The most severe thunderstorm is the so-called supercell thunderstorm. The name supercell (coined by Browning, 1964) refers to the fact that although this type of storm is about the same size as a multicell thunderstorm, its cloud structure, air motions, and precipitation processes are dominated by a single storm-scale circulation consisting of one giant updraft-downdraft pair. Smaller scale features in supercells have been noted by Barge et ul. (1976), and Battan (1980) suggests that further structural details might be discernible from very high resolution radar observations. However, these superimposed finer scale structures do not appear to be separate thunderstorm cells, and the major aspects of supercell structure can be understood in terms of the storm-scale circulation alone. Advances in understanding the growth of large damaging hail and the formation of tornadoes have followed from recent numerical modeling and detailed observational documentation of the storm-scale circulation. It has been recognized for a long time that supercell storms occur in environments of great potential instability and strong vertical wind shear (Newton, 1963). Recent numerical modeling studies confirm this fact and further show that multicell and supercell storms comprise two distinct classes of thunderstorms (Weisman and Klemp, 198 1, 1982). The multicell storms occur in weak to moderate shear. Weaker shear allows the downdraft gust front at low levels to move ahead of its parent cell; the warm inflow to the original updraft is cut off, and a new cell is triggered along the outflow boundary (Thorpe and Miller, 1978; Weisman and Klemp, 198 1 : Wilhelmson and Chen, 1982). Supercell storms occur in moderate to strong shear, which allows the updraft and downdraft to adopt a configuration in which they propagate together. A detailed discussion of supercell organization and structure is given in the following subsections.
3.3.1 Visuul Appearance of the Supercell Storm. The visual appearance of a supercell thunderstorm viewed from the side is shown in Fig. I6 in the form that is usually taught to ground-based tornado spotters. The tornado vortex is visible as a funnel-shaped cloud pendant from a rotating wall cloud extending downward from cloud base. Usually, the rotation in the tornado and wall cloud is cyclonic and is also suggested by striations of the primary cumulonimbus cloud base. A tail cloud is sometimes seen streaming cyclonically into the west side of the wall cloud from the region of cool air and heavy precipitation. The tornado usually occurs near the peak of a wedge of low-level warm air entering the region of the storm typically from the east or southeast. This warm air rises over the gust front to form the updraft of the storm-scale circulation. Cold downdraft air deposited by the
FIG. 16. Schematic visual appearance of a supercell thunderstorm. Based on National Severe Storms Laboratory publications and an unpublished manuscript of H . Bluestein.
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FIG. 17. Plan view of an idealized supercell thunderstorm as it would appear in a satellite picture and in the low-level precipitation pattern that would be detected by a horizontally scanning land-based radar. Cloud features seen by satellite include the flanking line, the edge of the anvil cloud, and the overshooting cloud top. Position of gust front (given by frontal symbols) and tornado are also shown. Based on National Severe Storms Laboratory publications.
storm at the surface spreads out behind the gust front. In Fig. 16, precipitation reaching the ground behind the gust front forms a curved backdrop for the tornado. Weaker tornadoes can occur along the southwest (or rearflank) gust front (Bates, 1 968; Davies-Jones and Kessler, 1974; Forbes and Wakimoto, 1982) or along the lateral boundaries of “downbursts” (see Section 3.3.7). We will concern ourselves here with the predominant type of tornado found at the peak of the wedge of warm air near the storm center. Warm air rising along the rear-flank gust front or confluence line results in a “flanking line” of towering cumulus (Lemon, 1976). Lifting is most intense near the peak of the gust front, where the visibly active cumuliform growth is seen to extend up through the tropopause to form an overshooting cloud top. Divergence at the tropopause level gives rise to the anvil, which extends downwind to the east or northeast at upper levels. Mammatus structures are commonly seen at the base of the anvil; explanations for their occurrence have been suggested by Scorer (1972) and Emanuel (198 1). An orderly pattern is seen in the precipitation to the northeast of the tornado in Fig. 16. Closest to the tornado, large hail occurs, then, progressing northeastward, small hail, heavy rain, light rain, and virga. As will be shown in Section 3.3.4, this sorting is a result of the intense internal storm-scale air motions. Interestingly, two of the attributes of the storm that classify it as severe, namely, the large hail and the tornado (cf. Section 3. I), are found in close proximity near the center of the storm. 3.3.2. Appearance of the Storm in Satellite and Radar Imagery. An idealized horizontal projection of the cloud-top topography of a supercell,
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as it would appear in a satellite picture, and the low-level precipitation pattern that would be detected by a horizontally scanning ground-based radar are shown superimposed in Fig. 17. The near coincidence of the tornado, the peak of the wedge of warm air, the overshooting cloud top, and the indentation in the horizontal precipitation area are evident. The horizontal distribution of the size-sorted precipitation particles produces a distinctive radar reflectivity pattern, since light rain, heavy rain, small hail, and large hail produce increasingly greater echo intensities. The large hail produces an extremely intense echo surrounding the notch in the precipitation pattern where the tornado is located. This radar reflectivity pattern is generally referred to as a “hook echo.” The radar reflectivity patterns vary significantly with height in the storm (Fig. 18). The notch in low-level horizontal echo pattern ( 1 km in Fig. 18a)
FIG. 18. Schematic illustration of supercell structure typically observed by radar in Alberta. Horizontal sections of reflectivity ( Z e , in units of dBZ) at various altitudes are shown in (a). Vertical sections are shown in (b) and (c). Cloud boundaries are sketched; BWER refers to the “bounded weak-echo region.” From Chisholm and Renick (1972).
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FIG. 19. Perspective view of an Alberta supercell storm. Storm-relative airflow, radar reflectivity (solid lines labeled in dBZ), cloud boundaries (sketched and dashed), and environmental wind profiles are indicated. Wide arrow indicates storm motion. Adapted from Chisholm and Renick (1972).
is associated with a “bounded weak-echo region” (BWER) or “echo-free vault” that extends upward toward the overshooting top of the storm (Fig. I8a, 4 and 7 km; Fig. 18b,c).
3.3.3 Environrnental Wind Shear and Updraft Structure. The BWER seen in Fig. 18b,c is associated with the intense updraft of the storm-scale circulation (Fig. 19). At low levels, the wedge of warm air flowing toward the center of the storm rises over the gust front. It rises so rapidly that hydrometeors in the updraft do not grow to radar-detectable size until they reach great altitudes. Thus, the BWER coincides with the core of the updraft (see the vertical cross sections on the back and side walls of Fig. 19). Air reaching the top of the cloud turns and exits toward the east or northeast, in a direction consistent with the upper-level winds. The inpouring of air at low levels from the southeast is also consistent with the environmental winds. As will be further shown in Section 3.3.5, the strong wind shear in the environment (see back and side walls of Fig. 19) is crucial to the formation and maintenance of the storm-scale circulation of the supercell. 3.3.4. EIailfall Pattern of the Supercell. The extremely strong updraft in the supercell storm (- 10-40 m/sec; Barnes, 1970; Manvitz, 1972; DaviesJones, 1974; Klemp ef af., 1981) makes possible the growth of very large
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hailstones. The growth process has been hypothesized to occur more or less as shown in Fig. 20. Various particle trajectories can ensue depending on the size and location of hail embryos when they first appear in the main updraft. Embryos may be initiated within the main updraft or introduced into the main updraft from feeder clouds located along the flanking line (Barge el a/., 1976; English et a/., 1982). Particles initially located in the core of the updraft and following trajectory 0 in Fig. 20b are carried aloft and into the storm’s anvil cloud before becoming large enough to fall out. Particle 1, beginning near the front edge of the updraft, falls back into the updraft for another cycle of growth by accretion of cloud liquid water before it is carried over to the rear of the storm, where the largest hailstones fall at the front edge of the precipitation area (trajectory 3 ) , while smaller particles are carried farther back into the precipitation region (dotted trajectory). This sorting ofparticles by size together with horizontal airflow normal to the plane of the cross section in Fig. 20 (further details in Section 3 . 3 . 5 ) accounts in a general way for the distribution of precipitation shown in Figs. 16 and 17. Further variations of hailstone structure, size, and trajectory can occur within this basic pattern, depending on such factors as whether the particles are initiated in the main updraft or in feeder clouds, hailstone density (Pflaum et al., 1982), or inhomogeneity in the structure of the main updraft (Battan, 1980; Nelson and Knight, 1982). The fallout of precipitation illustrated in Fig. 20a drives the downdraft of the supercell by precipitation drag and evaporation into and cooling of entrained midlevel environmental air. In Fig. 20, the downdraft occupies the region between 0 and 20 km on the horizontal axis. As in the multicell storm (cf. Fig. 14), the air from the storm-scale downdraft spreads out at low levels and sustains the storm-scale updraft. To understand the relationship between the updraft and downdraft of the supercell more fully, the storm’s circulation pattern is examined in three dimensions and in time in the next subsection.
3.3 5. Storm Splitting; Lest- and Right-Moving Storms. Supercell thunderstorms occur in environments in which the vertical shear of the horizontal wind is very strong. The shear is usually especially strong in one direction (e.g., back wall of Fig. 19), with lesser shear in the orthogonal direction (side wall of Fig. 19). Basic aspects of supercell formation and structure can be understood by considering first the direction in which the shear is particularly strong. Strong vertical shear of the horizontal wind is associated with vorticity about a horizontal axis normal to the shear. If an updraft is formed in this environment, the initially horizontal vortex tubes are bent upward (Fig. 2 la). The updraft then contains two counterrotating
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FIG. 20. (a) Vertical section showing features of the visual cloud boundaries superimposed on the radar echo pattern of a supercell thunderstorm in northeastern Colorado. The section is oriented along the direction of travel ofthe storm, through the center of the main updraft. Two levels of radar reflectivity are represented by different densities of hatched shading. The locations of four instrumented aircraft are indicated. namely, D- 130, QA (Queen Air), DC-6, and B (Buffalo). Bold arrows denote wind vectors in the plane of the diagram as measured by two of the aircraft (scale is only half that of winds plotted on right side of diagram). Short thin arrows skirting the boundary of the vault represent a hailstone trajectory. The thin lines are streamlines of airflow relative to the storm drawn to be consistent with other observations. To the right of the diagram is a profile of the wind component along the storm’s direction of travel, derived from a sounding 50 km south of the storm. (b) Vertical section corresponding to (a). The echo distribution and cloud boundaries are as before. Trajectories I , 2, and 3 represent the three stages in the growth of large hailstones. The transition from stage 2 to stage 3 corresponds to the recntry of a hailstone embryo into the main updraft prior to a final upand-down trajectory during which the hailstone may grow large, especially if it grows close to the boundary of the vault as in the case of the indicated trajectory 3. Other, less favored hailstones will grow a little farther from the edge of the vault and will follow the dotted trajectory. Cloud particles growing within the updraft core are carried rapidly up and out into the anvil along trajector) 0 before they can attain precipitation size. From Browning and Foote (1976).
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centers of vorticity about a vertical axis. This vortex-couplet structure is seen in the results of three-dimensional numerical cloud models when an initial perturbation in a thermodynamically unstable environment is allowed to grow in an environment of wind shear similar to that indicated in Fig. I9 (Klemp and Wilhelmson, 1978a,b; Wilhelmson and Klemp, 1978,
FIG. 2 I . Schematic of storm-splitting process leading to left- and right-moving supercell thunderstorms. Cloud boundary is sketched. Prccipitation is hatched. White tube represents a vortex tube. Heavy arrows are updrafts and downdrafts. GF indicates gust front. Stornis move over a horizontal surface shown in perspcctive. Point 0 is fixed to the surface and is directly under the center of the cloud i n (a). Storms move away from 0 in time. Storm cross sections arc in vertical planes outlined hy dashed lines. In (b) the split occurs. and subsequently gust front propagation leads to a component of storm motion away from the center line of thc horizontal plane in (c) and (d). In ( c ) divergence (DIV) and convergence (CONV) are indicated, and dashed vortices indicatc their being weakened by divergence.
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198 1; Schlesinger, 1978, 1980, 1982a,b; Clark, 1979; Blechman, 198 1). The counterrotating vortices entrain potentially cool midlevel air on the downwind side of the cloud. This introduction of cool air together with the downward drag of developing precipitation starts a downdraft in the center of the cloud, and the previously upward-bent vortex tube is bent downward in its middle (Fig. 2 I b). The downdraft then contains counterrotating vortices of its own, and the updraft is split into two parts, one on each side of the downdraft. The split of the original updraft by the downdraft is seen in the model simulations to be followed by one updraft-downdraft pair’s moving to the left of the mean wind while the other moves to the right of the mean wind as a result of gust-front propagation. This motion makes the split complete, leading to two separate storms, which continue to propagate away from each other (Fig. 2 lc,d). The storm moving to the right in the figure is referred to as the “right-moving” storm. The other is referred to as the “left-moving’’ storm. We shall temporarily focus attention on the right-moving storm. If Coriolis effects are ignored, the left-moving storm may be regarded as the mirror image of the right-moving storm. Rotunno ( 1 98 1) has analyzed the development of rotation in the right-moving storm that forms in Wilhelmson and Klemp’s (1978) model experiments. He shows that, at midlevels (more specifically, the level of nondivergence), the centers of cyclonic and anticyclonic rotation occur at first to the right of the updraft and downdraft cores, respectively, as shown in the right half of Fig. 2 1b. However, as the storms continue to move apart, the centers of rotation migrate to the draft cores, resulting in a cyclonically rotating updraft and an anticyclonically rotating downdraft (Fig. 2 Ic). At low levels, Rotunno finds (in accordance with work of Brandes, 1978, 1981; Heymsfield, 1978; and Bluestein and Sohl, 1979) that the vorticity structure is strongly modified by convergence, which strengthens the rotation in the updraft, and divergence, which weakens the counterrotation in the downdraft (the weakened downdraft rotation is indicated by the dashed lines in Fig. 2 1 c). Rotunno further finds that the center of cyclonic rotation at low levels migrates from its initial position to the right of the updraft core (Fig. 2 1b) past the center of updraft (Fig. 21c), finally coming to rest on the gust-front boundary separating the downdraft and updraft air (Fig. 2 ld).4At this stage, the center of cyclonic rotation extends vertically from the gust-front boundary at low levels to the updraft core at midlevels. This vertically continuous region of cyclonic rotation in the right-moving storm (dotted line in Fig. 2 1 c) is referred to as the “mesocyclone” (Fujita. 1965; Burgess, 1976; Lemon et ul., 1978: Burgess et ul., How the downdraft air with its weakened anticyclonic rotation actually develops cyclonically curved streamlines near the gust front (as indicated in Fig. 16) is a question under active investigation. See comments by both Brandes (1981) and Rotunno (1981).
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1982), and it is recognized as the likely origin of most tornadoes formation (see Section 3.3.7 for further details). The counterpart to the mesocyclone in the left-moving storm is referred to as the “ me s o a n ti~ y c lo n e ” ~ (Burgess, 198 1). Examples of the three-dimensional structures of the left- and right-moving storms formed by splitting in unidirectional shear are shown in Fig. 22. We shall again restrict discussion to the right-moving storm, except where indicated otherwise. Qualitatively. the right-moving storm has the characteristics of the supercell storm described in preceding sections. At low levels (Fig. 22a), the precipitation field resembles that of a supercell, except for the absence of hail, which does not occur because the ice phase is excluded from these calculations. The horizontal rainfall pattern, however, exhibits a hook-echo configuration, with a notch at the core of maximum updraft intensity. South of this maximum, the updraft region is elongated along an apparent flanking line. Warm air streams in toward the flanking line and updraft core region from the east. Air is seen diverging from the downdraft, which is centered in the precipitation area. This downdraft air flows into the region behind the flanking line and forms a gust front, which meets the air flowing in from the east. At middle levels (Fig. 22b,c), the cyclonic rotation in the updraft core is clearly seen. The counterrotation in the downdraft core is not as evident in the plotted winds; however, it can be seen clearly in analyses of the vorticity field (e.g., Figs. 14 and 15 of Wilhelmson and Klemp, 1978). Midlevel ambient flow is deflected around the rotating updraft, especially around the south side, as though the updraft were an obstacle to the flow. After passing around the updraft, this air is entrained into the precipitation area on the east or forward side of the storm. The midlevel air is dry, and the precipitation particles readily evaporate into it. The cooling from the evaporation together with precipitation drag induces the air to subside after it is entrained. Passage of midlevel air around the updraft core prior to its entrainment into the precipitation region (Fig. 22b,c) and the exit of downdraft air and outflow at low levels, behind the flanking-line gust front (Fig. 22a) were postulated on the basis of indirect observational evidence by Browning (1964). Comparing the locations of the updraft centers at different levels in Fig. 22a-d, we see further that the updraft core slopes over the downdraft with height (also postulated by Browning) and that divergent anvil flow is centered on the updraft summit6 (Fig. 22d).
’
The mesoanticyclone in the left-moving storm should not be confused with the thunderstorm high-pressure area typically observed behind the gust front. Note that the cloud-top level in this model simulation is lower than that observed in nature as a result of modifying the ambient sounding to reduce the model domain for computational simplicity.
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FIG. 22. Three-dimensional cloud model results showing three-dimensional structures of left-moving ( y > 0) and right-moving ( y < 0) thunderstorms formed by splitting in unidirectional shear. Horizontal relative wind vectors are shown with vertical velocity (m/sec) superimposed. (a) z = 0.25 km, max vector = 14 m/sec; (b) z = 2.25 krn, max vector = 13.1 m/ sec (c) z = 3.75 krn, max vector = 13 m/sec; (d) z = 5.75 km, max vector = 13.0 m/sec. The heavy dashed line marks the outer boundary of the rainwater field, except in (d), where it encloses the cloud water field. From Klemp and Wilhelmson (1978b).
The left-moving storm in Fig. 22 is very nearly the mirror image of the right-moving storm. The slight departure from exact symmetry resulted from inclusion in the model of Coriolis terms, which had somewhat different effects on the left- and right-moving storms. The occurrence of storm splitting, in which both left- and right-moving storms are well defined, is ob-
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served only infrequently in nature; however, it seems to occur, as expected, when the ambient shear is nearly unidirectional (e.g.. see Charba and Sasaki 197 1, and Wilhelmson and Klemp’s, 198 I , discussion of observed soundings and hodographs). Klemp and Wilhelmson ( 197%) showed, however, that slight departures from unidirectional shear in the environment will favor either the left- or right-moving storm. For example, the shear depicted on the side wall of Fig. 19 modifies the unidirectional shear shown on the back wall to favor right-moving storm development. Hodographs favoring the development of symmetrically splittipg, left-moving and right-moving model storms are shown in Fig. 23. In model simulations with hodographs (b) and (c) in Fig. 23, the basic storm-splitting process seen in the case of unidirectional shear occurs, but
(cl
FIG. 23. Hodographs favoring (a) symmetrically splitting, (b) right-moving, and ( c ) leftmoving supercell thunderstorms. From Kleinp and Wilhelmson ( 197%).
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the disfavored storm is greatly weakened (Klemp and Wilhelmson, 1978b). If this process occurs in nature, the disfavored storm probably never develops sufficiently to be readily observable. Since severe storm situations in North America are usually characterized by the type of shear shown in hodograph (b) of Fig. 23, the right-moving storm is the type of supercell usually observed. The low-level flow pattern of a supercell simulated using the wind and thermodynamic soundings for a specific case of right-moving supercell development (shear in the environment like that of Fig. 23c) is compared in Fig. 24 with the flow pattern observed at two times in the actual storm by
FIG. 24. Flow pattern and radar reflectivity at the I-km level in the Del City, Oklahoma, tornadic thunderstorms observed by multicell Doppler radar at (a) 1833 LST and (b) 1847 LST and simulated by a three-dimensional numerical model (c). Updraft velocities (solid lines) and downdraft velocities (dashed lines) are contoured at 5 m/sec intervals in (a) and (c) and 10 m/sec in (b). Shaded regions designate areas of negative vertical velocity ( -i I m/sec). The heavy solid line outlines the rainwater field enclosed by the 0.5 g/kg contour in (c) and by the 30 dBZ contour in (a) and (b). Wind vectors are scaled such that one grid interval represents 20 m/sec. From Klemp C! uI. ( I 98 I).
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FIG.25. Idealized version of radar reflectivity and dual-Doppler observed flow at the 0.4km level in the Del City, Oklahoma. tornadic thunderstorm. Low- and high-density shading is for reflectivity thresholds of 30 and 40 dBZ, respectively. Solid (dashed) arrows show horizontal flow in downdraft (updraft) region. Heavy solid line is boundary between downdraft and updraft regions. Large dot shows tornado location. Adapted from Fig. 5 of Brandes ( 1 9s I).
multiple-Doppler radar techniques. Several model and Doppler radar comparisons such as this one have now been made, and the similarity of major features is striking. The success of these comparisons lends confidence to both the modeling and radar methodologies.
3.3.6. Genesis of the Tornarlo. The Doppler radar observations depicted in Fig. 24a,b show a prominent center of cyclonic rotation, which is the low-level manifestation of the mesocyclone discussed in Section 3.3.5 and illustrated in Fig. 21d. The mesocyclone is the favored region for tornadogenesis, though, as we noted in Section 3.3.1, tornadoes may also occur along gust-front or “downburst” boundaries. These gust front and downburst tornadoes, though quite damaging, are not as severe or as well documented as the mesocyclonic tornado. We will restrict the present comments to the mesocyclonic tornado. The vortex comprising the tornado is much more intense and occurs on a much smaller scale than the parent mesocyclone. The tornado vortex is in fact too small to resolve i n multiple-Doppler radar analyses; however, it can often be detected in single-Doppler data as a location of extreme wind shear from one data sampling volume to the next (Brown et ul., 1978) or in the velocity spectrum for a single sampling volume (Zrnic et ul., 1977). This so-called tomado-vortex signature (TVS) is observed aloft, at the
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FIG.26. Conceptual model of mecocyclone core evolution. Thick lines are low-level wind discontinuities. Tornado tracks are shaded. Inset shows the tracks of the tornado family. and the small square is the region expanded in the figure. From Burgess et a/. (1982).
2-5-km level within the mesocyclone, some 20-30 min before it appears within the low-level mesocyclone and reaches the ground as a visible tornado funnel. At low levels, thc tornado appears after the mesocyclone center moves from the updraft core to the boundary between the downdraft and updraft (cf. Fig. 21c,d). In the Doppler radar data in Fig. 24, the circulation center was in the updraft at the earlier time (Fig. 24a), but by the later time (Fig. 24b) the updraft region had become distorted into the shape of a horseshoe by an intrusion of cyclonically rotating downdraft air. Klemp ct al. (198 1 ) have shown that this downdraft air began its sinking motion in the main precipitation region to the north of the mesocyclone and that it subsequently spiraled downward, around the mesocyclone center, where it split the updraft core as it sank to low levels. Brandes ( 198 1 ) has related this intrusion to the formation of a tornado in the location indicated in Fig. 25. Brandes ( 1 978: see his Fig. 18) and Golden and Purcell ( I 978a; see their Fig. 8) have shown similar downdraft intrusions in relation to the appearance of tornadoes at low levels. Brandes ( 198 1 ) finds that once the downdraft intrusion has become established, both tilting and stretching of vortex tubes on a highly localized scale become active (stretching was of greater magnitude in this case) in forming the tornado at a location along the boundary of the updraft and downdraft air circulating around the mesocyclone (e.g., Fig. 25). Brandes argues further that, soon after the situation illustrated in Fig. 25, the updraft in the vicinity of the mesocyclone center was completely cut off from the inflow of warm air by the downdraft intrusion, and the
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tornado and mesocyclone dissipated. A new supercell updraft and mesocyclone center then formed ahead of the rear-flank gust front. Cutting off of the mesocyclone center in this way, while the rear-flank gust front propagates ahead and generates a new main updraft and mesocyclone, which in time spawns another tornado, accounts for the association of multiple tornado paths with a single supercell storm (Fig. 26). Lemon and Doswell (1979) noted that this sequence of secondary mesocyclone development bears an intriguing resemblance in miniature to synoptic-scale cyclone evolution. Burgess et al. ( 1982) have verified the sequence of events depicted in Fig. 26 statistically from Doppler radar observations. They note that, although long-track tornadoes have been observed, they are extremely rare and that “many tornadoes initially appearing long are in reality a family of segments, each associated with a different mesocyclone core.” High-resolution three-dimensional model simulations confirm the tendency of the mesocyclone center to be cut offwith new main updraft and vorticity centers developing along the gust front to the east (Klemp and Rotunno, 1982), and recent photographic documentation of multiple-tornado storms by tornado chase teams is consistent with the sequential mesocyclonic and tornadic evolution observed on radar (Rasmussen e/ al., 1982; Marshall and Rasmussen, 1982).
FIG.27. Damage path of the Union City, Oklahoma, tornado with sketches of the funnel and associated debns cloud as seen from the south. Letters A-H indicate damaged farmsteads. From Golden and Purcell ( I978a).
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FIG.28. Tornado structure from photogrammetry and damage surveys. (a) Outline of Union City, Oklahoma, tornado funnel and upper wall cloud showing features tracked on movie loops. Sequential outline of cloud tags and streamers in wall cloud and typical composited trajectory and displacement of debris aggregate are superimposed. (b) Schematic plan view of horizontal streamlines (solid lines) and low-level vertical motion patterns (dashed lines) around decaying tornado. From Golden and Purcell ( 197%).
3.3.7. Life Cycle and Structure of the Tornado. An example of an individual tornado funnel that formed within a mesocyclone is described in detail by Golden and Purcell(1978a); for other examples, see Fujita (1960), Hoecker (1960), and Davies-Jones ( 1982a). The structure of the funnel observed by Golden and Purcell is illustrated schematically in Fig. 27 for several stages in its life cycle. The “organizing stage” was characterized by a visible funnel touching the ground intermittently, though the damage path was continuous. In the “mature stage,” the tornado was largest. In the “shrinking stage,” the entire funnel decreased to a thin column. The “decaying stage” was characterized by a fragmented and contorted but still destructive funnel. Air motions within and near the tornado (Fig. 28) were determined by tracing debris particles and identifiable cloud elements
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(“cloud tags”) in motion pictures (Golden and Purcell, 1978b) and by surveying surface damage patterns (Davies-Jones et al., 1978). In the mature stage, tangential velocities at a radius of 200 m and heights 60-120 m above ground were 50-80 m/sec, in agreement with deductions from various earlier but less intensively documented cases. The flow pattern was notably asymmetric. At the wall-cloud level, there was strong downward motion on the southwest side of the funnel with upward motion on the northeast side (Fig. 28b). The tornado funnel is sometimes observed to contain one to six smaller subvortices, 0.5-50 m in diameter (Fujita, 1970, 1971, 1981; Fujita d al., 1970; Agee et al., 1975, 1976, 1977). These “suction vortices” may be stationary or orbit around the tornado center (Fig. 29). They contain some of the strongest winds of the tornado and leave a complex pattern of narrow trails of debris and extreme damage within the general path of the tornado. The observation of multiple funnel clouds [e.g., the 1965 Elkhart, Indiana, twin tornado (Fig. 46 ofFujita c’f d., 1970) or the 1979 Wichita Falls, Texas, tornado, which exhibited six simultaneous vortices (Fig. 17 of Fujita, 198l)] is explained by the breakdown of the parent funnel into suction vortices.
FIG.29. A model of a tornado with multiple suction vortices proposed by Fujita i n 197 I . From Fujita (1951).
50 N
30 N
100 w
9ow
FIG. 30. Infrared satellite picture of a mesoscale convective complex over Oklahoma at 0330 GMT on 23 May 1976. Shading levels are for -32°C (medium gray), -42.2"C (light gray), -53.2"C (dark gray), -59.2"C (black), and -71.2"C (white).
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This breakdown has been simulated in laboratory models (e.g., Ward, 1972; Leslie. 1977; Church ef a/., 1977) and is related theoretically to the “swirl ratio,” which is a nondimensional parameter expressing the relative strengths of circulation and forced convection in the parent vortex (DaviesJones, 1976, I982b). The breakdown of the mesocyclone into tornadoes, the tornadoes into suction vortices, and even the latter into twin suction vortices twisting around each other (FuJita, 198 1 ) brings to mind the famous rhyme of L. F. Richardson: Big whirls have little whirls that feed upon their velocity And little whirls have lesser whirls and so o n to viscosity.
3.4 Midlatitude Mesoscale C‘onvccfiw Complexes It is quite common for several thunderstorms to be grouped together within a mesoscale complex, which covers an area one or more orders of magnitude greater than that covered by an individual thunderstorm. The storms comprising such a complex typically share a common upper-level cloud shield, which appears very prominently in satellite imagery when the system matures (e.g., the cloud centered over Oklahoma in Fig. 30). Maddox (1980b) has used this fact to dcfine a midlatitude “mesoscale convective complex” (MCC) in terms of the time and space scales of its cimform cloud top. as it appears in satellite data (Table I). Fritsch et al. (1981) have presented evidence that 50-60% of the summer rainfall in the Great Plains and midwestern United States is accounted for by MCCs identified by Maddox’s criteria. Their work indicates. moreover, that the precipitation falling from an MCC at a given time typically covers a continuous area of mesoscale TABLE1 CRITERIA U S t D TO I D € N I IrY MIDI A rITUDE MESOSCALE CONVECTlVt I N I N I R A R C D SATELLITE DATA‘ COMPLEXES Phy.iical charactenstics
Size:
Initiate: Duration: Maximum extent: Shape: Terminate: a
(A) Cloud shield wilh continuously low infrared temperature s -32°C must have an area 2 100.000 km2 (B) Interior cold cloud region with temperature s -52°C must have an area t 50.000 km2 Size definitions ( A ) and ( B ) are first satisfied Size definitions ( A ) and (B) must be met for a period 2 6 hr Contiguous cold cloud shield (infrared temperature s -32°C) reaches maximum size Eccentricity (minor axis/major axis) 2 0.7 at time of maximum extent Size definitions ( A ) and (B) no longer satisfied
From Maddox (1980b).
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FIG.3 I . “Mass-flow’’ stream function (at intervals of lo7 mbar m’ sec ~ ’ for ) the azimuthally averaged flow about an origin at the centroid ( R = 0) of a rainstorm with the characteristics of a midlatitude MCC. The sense of the flow is indicated by the arrowheads. From Bosart and Sanders ( 198 I ),
dimensions; that is, the individual thunderstorms are embedded in a larger, mesoscale region of precipitation falling from the cloud shield. The occurrence of a continuous area of precipitation on a scale greater than that of the individual thunderstorms indicates that an organized circulation occurs on the mesoscale in conjunction with the MCC. Bosart and Sanders ( 198 1 ) deduced the circulation associated with an MCC that produced severe flooding in Johnstown, Pennsylvania, in 1977 (Fig. 3 1). Maddox (198 I ) has found a similar circulation to be characteristic of the mature midlatitude MCCs he has identified. This circulation should not be confused with the air motions of an individual thunderstorm. The MCC circulation is a much larger scale overturning in which the thunderstorms are embedded. The mesoscale circulation of the mature MCC, shown in Fig. 31, was determined for a circularly symmetric region centered on the MCC. A radius-height cross section of the symmetric pattern is shown, with the origin of the graph corresponding to the center of the disturbance. The circulation was characterized by mean ascent over a region several hundred kilometers wide extending vertically through the entire troposphere, except for the boundary layer (below 950 mbar), where weak divergence and descent oc-
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curred in the mean. This low-level net divergence reflects the influence of a mesohigh resulting from the spreading of convective downdraft issued by the various embedded thunderstorms. This mesohigh was of the type described in early work of Fujita (1955) and others. Although convergence and new thunderstorm development undoubtedly occurred along the edge of the mesohigh (or “cold dome”), the mesohigh dominated the boundary layer flow to an extent that the divergence and sinking prevailed in the mean pattern. The mean upward motion above the boundary layer connected a region of strong convergence at 900-700 mbar with a region of strong divergence centered at 200 mbar. The convergence was associated with a mesolow in the low to middle troposphere, while the divergence aloft was linked to a pronounced mesohigh. The upper-level mesohighs associated with MCCs have been described by Ninomiya ( 197 1a,b), Maddox ( 1980a, 1981), Fritsch and Maddox (1981a,b), and Maddox et al. (1981). The mesoscale circulation of the mature MCC develops by progressing through a life cycle described qualitatively by Maddox (1980b, 1981). In its early stages, the MCC is highly convective, being dominated by the individual thunderstorms it contains. As the system evolves, however, lifting on a larger scale becomes more pronounced. A stratiform upper cloud shield develops and precipitation becomes continuous, evidently consisting of a mixture of stratiform and convective components. This life cycle is similar to that typifying tropical cloud clusters (see Sections 4.2-4.5). Detailed radar studies of tropical cloud clusters have clearly distinguished convective and stratiform processes and hence have led to better understanding of the mesoscale organization and dynamics of the clusters. Similar work is needed to distinguish the convective and stratiform processes in midlatitude MCCs. The dynamics of the development of the mesoscale circulation of the MCC (Fig. 31) are not yet well understood; however, some of their aspects may be surmised from existing observations, as well as from mesoscale models and by comparing MCCs to tropical cloud clusters. The initial development is apparently driven by convective heating (which is dominated by release of latent heat in the convective updrafts; Houze, 1982) associated with the embedded thunderstorms. Mesoscale models with parameterized (sub-grid-scale) convection (Brown, 1974, 1979; Kreitzberg and Perkey, 1977; Fritsch and Chappell, 1980; Fritsch and Maddox, 1981b; Maddox ef al., 1981) show that hydrostatic adjustment to the convective heating initiates a mid to low tropospheric mesolow and an upper tropospheric mesohigh of the sort evident at 900-700 mbar and 200 mbar in Fig. 31. Ninomiya ( 197 l a,b) amved at a similar conclusion from observational evidence. As the mesolow and mesohigh intensify, mesoscale lifting strengthens and widespread cloud and precipitation develop. At this stage, condensation, evaporation, melting, and radiative transfer can become important in the
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stratiform cloud and precipitation areas and combine with the heat release by the embedded thunderstorms to provide the total thermal driving force
for the mesoscale circulation. If the convection embedded in the MCC weakens or dies out after this stage, the MCC should be able to continue to exist since the mean circulation can continue, driven by the stratiform and/or radiative processes. Thus, the MCC becomes less convective and more stratiform as it ages. This sequence of thermodynamical and dynamical events is consistent with Maddox’s (1980b, 198 1) descriptive life-cycle observations and with tropical cloud clusters. Houze (1982) has examined the relative magnitudes of convective, stratiform, and radiative heating in tropical cloud clusters and has found them all to be significant, with the stratiform and radiative processes becoming more prominent in the later stages of a cluster. Further work is needed to distinguish these processes more clearly in midlatitude MCCs.
3.5. Midlatitude Squall Lines A type of mesoscale convective system that has been recognized for a long time is the midlatitude squall line. Some confusion exists, however, regarding the definition of a squall line. The “Glossary of Meteorology” (Huschke, 1959) defines a squall line as “any non-frontal line or narrow band of active thunderstorms.” This definition appears to be inadequate in at least two respects: The nonfrontal requirement is too restrictive, since thunderstorms associated with a front (or even frontal rainbands without thunderstorms) can occasionally take on squall-line characteristics (e.g., Fujita, 1955; Sanders and Paine, 1975; Sanders and Emanuel, 1977; Hobbs and Person, 1982), whereas to say that “any” line of thunderstorms qualifies as a squall line is too general, since most observers recognize a squall line as a special class of convective line characterized by rapid propagation and certain mesoscale pressure, wind, and precipitation patterns. Fujita (1955) clearly identified these features. He showed that a squall line starting from a point, probably as one or two individual thunderstorms, successively expands its area of influence (Fig. 32). The leading edge of the system, commonly called the gust front (but referred to by Fujita as the “pressuresurge line”), has the characteristics of an intense cold front. As it propagates, it continually lengthens, and an expanding pool of downdraft air is left in its wake (Fig. 33). A narrow band of intense thunderstorms is found along the leading edge, with a broader region of light precipitation falling from an extensive cloud region to the rear. A mesohigh is found at the surface in the trailing region of lighter precipitation. A “wake-depression,” or mesolow, induced by subsidence warming, is found in dry air to the rear of the precipitation zone. The horizontal extent of the system (-400 km), its widespread area of precipitation, and its oval shape make it comparable in
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FIG.32. Spread of area influenced 11) an advancing squall line. Isobars of excess pressure are drawn for two times (0100 and 0100) with hourly isochrones and envelopes of the various pressure values From Fujita (1955)
size and shape to an MCC. A single squall-line system of this type develops an upper-cloud shield, which can attain the characteristics that qualify it as an MCC according to Maddox’s (1980b) criteria ( e g , Fig. 30). Thus, Fujita’s ideal midlatitude squall-line system can be considered as a special case of an MCC. However, the midlatitude squall line ISidentified as an MCC by Maddox’s criteria (Table I) only when it occurs in isolation. When several squall-line systems break out along a larger scale front or windshift line, leading edges may intersect and form a long chain of squall systems (e.g., see Figs. 3743 of Fujita, 1955; or cf. Figs. 13.8 and 13.9 of PalmCn and Newton, 1969). In this situation, upper cloud shields merge and the individual squall systems become impossible to distinguish as separate MCCs by strict application of Maddox’s criteria, since his rules are designed to identify only isolated (i.e., round or oval) cloud shields. Therefore, while the isolated squall-line system in Fig. 30 is readily identified as an MCC, a squall-line system that is one of several along or ahead of a front cannot be separately identified in satellite data unless the time continuity of the cloud pattern is closely followed and subjective allowance is made for mergers. As shown by Fujita’s (1955) work, radar and other detailed surface observations are needed to distinguish unambiguously the separate mesoscale squall systems forming parallel to a front. Such a distinction seems necessary, however, to understand both the isolated squall system and families of squall systems occurring parallel to fronts. Squall lines occur in a variety of large-scale environments. Palmen and
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FIG.33. Fujita’s model of a squall line, in vertical cross section (upper) and in surface plan view (lower). Lines on surface map are isobars. Wind vectors are indicated by small arrows; the motion of the system at upper levels relative to the moving system is represented by the large open arrows on the surface map. UPD and DWD indicate updrafts and downdrafts, respectively. From Fujita (1955).
Newton (1969) note that, “Although squall lines most commonly occur in the warm sectors of cyclones (or in tropical air well removed from frontal systems), they often extend far north of warm fronts and occasionally form some distance behind cold fronts: in both cases the thunderstorms are in the warm air above the front.” Thus, squall lines form in different situations of environmental wind shear, and, not surprisingly, squall lines take on various configurations. Often the winds in the environment are strongly veering, usually ahead of an approaching upper trough (Fig. 2 of Newton, 1963), and a line of severe organized multicell or supercell storms develops and may exhibit characteristics of the squall-line model of Fujita (1955) (Figs. 32 and 3 3 ) . Newton and Fankhauser (1964) and Newton ( 1966) investigated severe convective lines of this type (Fig. 34). They consist of aligned thunderstorms, with wind shear favoring new development on the southern ends of individual squall lines. By this process, the southern end of a line to the north may join the northern end of a line to the south, thus forming a longer squall line. The ambient shear favors anvil blowoff to the north-northeast, along and ahead of the line. The older thunderstorms on the northern ends of lines evolve into “extensive stratified cloud masses”;
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FIG. 34. Simplified horizontal map of principal features of one type of midlatitude squall line (north at top of page). Solid lines show precipitating cloud (regions of heavier rain cores hatched); dashed lines, the general anvil outline (which may be smaller or more extensive depending on the age of the squall line). New cell formation is most likely inside dotted boundaries, but may occur elsewhere. Inset shows most typical environmental winds. veering between lower (V, ) and upper (V(,) levels. and characteristic movements of storms of different From Newton and Fankhauser ( 1964). sizes relatives to vector mean wind
(v).
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H O R I Z O N T A L DISTANCE (KM)
FIG. 35. Conceptual model of the 22 May 1976 Oklahoma squall-line system. Outside contour outlines radar echo. Heavy lines denote regions of more intense echo. Light shading indicates regions of system-relative horizontal wind component directed from right to left relative to the system. Heavy shading shows jet of maximum horizontal wind from left to right. System was propagating from right to left. These streamlines show two-dimensional relative flow consistent with observed wind and echo structure. Hypothesized ice particle trajectories are denoted by asterisks and broad white arrows. Environmental wind relative to the system is indicated just ahead of the leading anvil echo. From Houze and Smull (1982).
however, an extensive stratiform precipitation area is not typically found in this type of squall line. In other situations, when the environment is characterized by a different type of wind shear, midlatitude squall lines exhibit an extended region of cloud and lighter precipitation behind the leading line of deep convection (Newton, 1950; Fujita, 1955; Chisholm, 1973; Sanders and Paine, 1975; Sanders and Emanuel, 1977; Ogura and Liou, 1980; Matejka and Srivastava 1981; Zipser and Matejka, 1982; Houze and Smull, 1982). This extended trailing precipitation enlarges the overall precipitation area of the system and gives it a continuous character of the type Fritsch et a/. (1981) have associated with MCCs (cf. Section 3.4). Radar analyses of the trailing precipitation (Matejka and Srivastava, 198 1 ; Zipser and Matejka, 1982; Houze and Smull, 1982) reveal that this trailing precipitation is similar to that which occurs to the rear of tropical squall lines (Houze, 1977: Leary and Houze, 1979b) in that it is highly stratiform, with an echo bright band in the melting layer. The isolated squall-line system, whose upper cloud is shown in Fig. 30, was of the type that exhibited trailing stratiform precipitation. Its structure is illustrated schematically in Fig. 35 in a vertical cross section along the direction of propagation of the system. The structure depicted is inferred from studies of composited sounding data (Ogura and Liou, 1980) and detailed radar reflectivity and single-Doppler radar air motion fields (Houze and Smull, 1982). The airflow into the front of the system is depicted as rising first above a gust front associated with downdraft outflow and then progressing into a region of convective cells. A new cell, identified by a first echo aloft, is shown at 60 km above the core of rising air, which was directed primarily
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into a weak-echo indentation on the leading side of a mature cell echo at 70 km. As the core of rising air approached the mature cell, it developed a strong front-to-rear component of horizontal motion between 2 and 5 km. The core of the updraft was thus tilted substantially. Upon reaching the tops of convective cells, the updraft air split into distinct components directed to the front and rear of the system. The part directed forward carried ice particles into the leading anvil. As the particles fell into the inflow stream in mid levels, they were either evaporated or swept back into the region of active cells. Mature cells, such as the one at 70 km in Fig. 35, were followed by dissipating cells, such as the one at 90 km, which is shown as having lost its core of updraft air. The horizontal outflow of downdraft air was strongest under mature cells, but was also seen under the dissipating cells. Consequently, the forward-rushing downdraft air was seen at low levels throughout the region of forming, mature, and dying cells (between 50 and 100 km in Fig. 35). The sequence of forming, mature, and dissipating cells at the leading edge of the system is rather similar to that of the fonvard-propagating multicell thunderstorm depicted in Fig. 14. The thunderstorm structure along the leading edge of the squall system may vary. Occasionally-particularly, early in the lifetime of a squall system-individual storms along the line may develop rotation and take on the character of a supercell storm rather than a multicell storm. In the case illustrated in Fig. 35, Doppler observations showed that the relative airflow in the stratiform region was everywhere directed from front to rear, except at the very back edge of the system, where some inflow from the rear occurred. The airflow through the squall system was particularly interesting at mid levels, where a jet of maximum front-to-rear flow occurred (heavily shaded region). Hourc and Smull (1982) found that the largest front-to-rear velocity components within this jet occurred near the front of the system at the locations ofthe convective cells. The jet extended through the trailing stratiform region, with magnitudes of velocity tapering off toward the back edge of the system, where the jet met the midlevel inflow from the rear. A rather similar horizontal velocity component field was reported by Sanders and Painc ( 1 975) and Sanders and Emanuel (1977). The latter authors attributed the generation of the maximum horizontal velocities primarily to horizontal pressure-gradient accelerations directed into a mid-tropospheric mesolow, apparently of the type known to be characteristic of MCCs. Houze and Smull(l982) noted that this jet was located just above the melting layer and appears to have played a role in spreading the trailing precipitation region out into a stratiform pattern. Houze ( 198 I ) pointed out that a melting layer 100-km wide, to the rear of a convective
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line that is propagating by the successive development of new cells at its leading edge (as in the Oklahoma squall system depicted in Fig. 35), can be explained by the relative horizontal motions of ice particles falling from the tops of cells. The particles move in a relative sense toward the rear of the system as a result of the continuing redefinition of the position of the leading edge of the system by the formation of new cells. This effect was exaggerated in the Oklahoma squall line by the jet of particularly strong relative wind just above the melting layer. Hypothesized particle trajectories are indicated in Fig. 35. The stratiform precipitation occasionally seen trailing midlatitude squall lines thus appears to develop partly by the successive incorporation of old cells into the trailing region (as noted by Sanders and Emanuel 1977) and partly by rearward spreading of ice particles by the midlevel jet. In addition, mesoscale lifting above the melting layer, indicated by Ogura and Liou’s (1980) results and depicted schematically in Fig. 35, may contribute to the development of the stratiform precipitation. Ogura and Liou’s results also indicate that the air below the melting layer subsides in a mesoscale downdraft. Matejka and Srivastava ( 198 I ) have confirmed this mesoscale updraft-downdraft structure with Doppler radar data obtained in the trailing stratiform region of an Illinois squall line. The mid- to upper-tropospheric mesoscale updraft and mid- to lowertropospheric mesoscale downdraft associated with a broad continuous area of precipitation are consistent with the divergence and vertical motion profiles obtained for MCCs by Bosart and Sanders (1981) and Maddox (198 1). Distinguishing between the stratiform and convective portions of the squall systems, however, has helped resolve and distinguish the different roles of convective (cellular) and mesoscale components of motions. Further radar studies should improve the understanding of this type of squall line and of MCCs in general.
3.6. Efcws
Qf
Downdra$ Spreading
The downdrafts from a single thunderstorm, or from a group of thunderstorms occurring in an MCC or squall line, spread horizontally in the boundary layer. In Sections 3.2 and 3.3.5, it was seen that downdraft spreading serves as a mechanism for the discrete regeneration of multicell storms, where downdrafts from older cells trigger new cells, and as a mechanism of continuous regeneration for left- and right-moving supercell storms in which the downdraft air continually spreads under the inflowing updraft air. Below MCCs and squall-line systems, larger regions of cold air are formed by the merger of the various downdrafts emanating from the nu-
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FIG. 36. Schematic drawings showing the airflow patterns associated with cold front, gust front, downburst, and burst swath. From Fujita (1981).
merous thunderstorms embedded i n these systems. These large pools of cool air in the boundary layer account for the low-level mesohighs characterizing these systems. Some effects of spreading downdrafts are discussed in the following subsections. First, violent downdraft outflows characterized by severe low-level winds are discussed in Section 3.6.1. Then the role of old downdraft outflow boundaries from past or decaying storm systems in triggering new storms is described in Section 3.6.2. 3.6.1. Nontornadic Severe W’inds. As noted in Section 3.1, strong winds that can classify a thunderstorm as “severe” can be associated with downdrafts as well as tornadoes. Fii.jita (1981) has classified downdraft winds according to horizontal scale (Fig. 36b-d). His classification has been determined largely from studying patterns of surface damage. Debris and fall patterns of certain flora left on the ground by downdraft winds are distinguishable from those left by tornadoes since the downdraft patterns reflect divergent flow, while tornadic damage reflects convergent flow. The largest thunderstorm wind pattern is the “straight-line wind” asso-
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ciated with the gust front marking the leading edge of the mesohigh of a squall-line system (cf. Figs. 32 and 36). Fujita (1981) reports, however, that “against our expectations, the damage by straight-line winds behind these gust fronts turned out to be minimal.” Instead, damage is concentrated in “downbursts” (referred to earlier as “pressure noses” by Byers and Braham, 1949) which are -10 km or less in horizontal dimension and occur sporadically at points along the gust front. Smaller scale downbursts are referred to as “microbursts.” Emanuel (198 1 ) has suggested that downbursts are a manifestation of unsaturated downdraft instability of the type that he has described theoretically. Within a downburst, a concentrated “burst swath,” < 1 km in dimension, can occur. Fujita ( 198 1 ) notes further that a downburst can occur along a supercell gust front as well as along a squall-line and that downbursts along supercell gust fronts, by virtue of the rotational flow of the supercell, have a “twisting airflow” at the surface rather than the purely divergent flow depicted in Fig. 36. Further damage can occur when tornadoes are generated at the lateral boundaries of downbursts (Fig. 9a). When a downburst occurs along a squall-line gust front, as in Fig. 36, the radar echo of the leading line of thunderstorms juts out ahead of the main squall line to form a “bow echo.” The relationship of the bow echo and downburst airflow is discussed by Fujita (1981). As pointed out in Section 2.6, the bow echoes associated with downbursts appears to have a counterpart in the “boomerang echoes” associated with strong cold fronts (Fig. 9b).
3.6.2. Arc Cloud Lines and Triggering ofNew1 Storms. Satellite studies (Purdom, 1973, 1979; Purdom and Marcus, 1982) show that the boundaries of downdraft outflows from thunderstorms or thunderstorm complexes can maintain their identities as arc-shaped lines of cumulus clouds for several hours after the storms that produced the downdrafts have dissipated. These lines can trigger new deep convective development up to 200 km from the location of the original storm. A great proportion of new deep convective developments occur where propagating arc lines intersect each other or where an arc line encounters preexisting convection (Purdom and Marcus, 1982).Aircraft data show that the arc lines contain some lightly precipitating cumulus (Sinclair and Purdom, 1982). However, these lines are mostly invisible to radar. Thus new deep convection triggered by an arc line may seem to appear at a random location on radar. However, in satellite data the location of development can often be anticipated by observing the history of the arc lines. Perhaps some of the simultaneous cell-formation events in the Thunderstorm Project, noted by Byers (1959) to be unexplainable in terms of locally preexisting cells (see Section 3.2), were occurring along outflow boundaries from previous, distant thunderstorms. The ability of the arc lines to maintain themselves for several hours, while
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traveling great distances from the sites of their originating storms, is not readily explained. Sinclair and Purdom ( I 982) suggest that subcloud evaporation of the light rain from some of the cumulus along the line plays a role. Observations of arc cloud lines emanating from cumulonimbus, possibly being perpetuated by precipitation showers along the line, and contributing to the development of new storms, have also been made in the tropics (Warner et a/., 1979; see also review of Houze and Betts, 1981). 4. TROPICAL CLOUDSYSTEMS 4 1. The Spectrum of Clouds IFI the Tropics
Clouds in the tropics occur in a spectrum of sizes ranging from small isolated cumulus to large “cloud clusters.” The cloud clusters are identified in satellite pictures by their mesoscale cirrus shields, each shield being 100-1000 km in dimension (Fig. 37). Statistical studies indicate that the tropical cloud spectrum, whether measured in terms of heights, areas, durations, or rainfall rates, tends to be distributed log-normally (see review of Houze and Betts, I98 1). That is, smaller, isolated cumulus and cumulonimbus greatly outnumber cloud clusters. Nevertheless, the cloud clusters, owing to their size, dominate the mean cloudiness and total precipitation of the tropics. The cloud clusters resemble midlatitude MCCs in being identified by their large cirrus canopies. They contain continuous rain areas covering up to 5 X lo4 km2. The cirrus shields of cloud clusters are typically not as cold as required by Maddox’s (1980b) criteria for midlatitude MCCs (Table I)’; however, they are of the same scale and qualitatively rather similar. The tropical systems could be called tropical MCCs. However, the term “cloud cluster,” which emerged from early satellite studies of tropical cloudiness (e.g., Frank, 1970; Martin and Suomi, 1972), has become traditional, and for the present we retain its usage. Cloud clusters generally have lifetimes of a day or less (Martin, 1975; Martin and Schreiner, 1981) and are confined to a very low latitudes. Occasionally, however, a cluster evolves into a longer lived tropical storm or hurricane,’ which can move out of the tropics into midlatitudes. Since large,
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The somewhat weaker cloud shields in the tropics are probably a result of the convection over tropical oceans being generally weaker than extratropical continental convection (see comments of Zipser and LeMone, 1980; Simpson and van Helvoirt, 1980; and Simpson e/ ul., 1982). According to the “Glossary of Meteorology’’ (Huschke, 1959, p. 593), a “hurricane” (in the Atlantic, Caribbean, and Gulf of Mexico) or “typhoon” (in the Pacific) is a cyclone that originates over tropical oceans and attains a maximum wind of 65 knots or higher. In extreme cases such storms attain maximum winds of 175 knots o r more. In this paper, we use the term hurricane to include the typhoon and other local names for the same phenomenon
FIG.37. Visible satellite photograph showing tropical clouds ranging in size from small cumulus to large cloud clusters. Cloud clusters are evident by the large cirrus shields at 17"N42"W, 9"N24"W, 9"N21"W, 7"N16"W, 8"N 12"W. 7"N2"W, and 14"N13"W. The last one mentioned was a squall cluster with a well-defined arc cloud line on its leading (southwest) side. Photograph from the SMS- 1 satellite is for 1 I30 GMT, 5 September 1974.
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precipitating cloud systems are the subject of the present article, we shall concentrate the remainder of our discussion on cloud clusters and hurricanes. Knowledge of cloud clusters has expanded greatly during the past decade as an outcome of international field experiments conducted during the 1960s and 1970s. The most ambitious of these were the Global Atmospheric Research Program's Atlantic Tropical (GATE) and Monsoon (MONEX) experiments. GATE was carried out over the eastern tropical Atlantic in the summer of 1974, and MONEX was held over the South China Sea near Malaysia and Indonesia in the winter of 1978-1979 and in the regions of the Arabian Sea and Bay of Bengal in the summer of 1979. In addition to this work, the U.S. National Oceanic and Atmospheric Administration (NOAA) conducts ongoing research on hurricanes. Their studies involve both storm modeling and observations obtained by extensive aircraft penetration of hurricanes. In the last several years, the instrumentation on NOAA research aircraft has improved sufficiently to have produced a new generation of hurricane data that promises to revolutionize the understanding of these storms. Characteristics of clouds observed in GATE have been reviewed in detail by Houze and Betts (198 I). Cloud clusters have been further examined by Houze ( 1982). To avoid redundancy with these papers as much as possible, we shall concentrate here on GATE, MONEX, and hurricane studies that have been completed since 1980.
4.2. Types of Cloud Clusters Two types of cloud clusters are generally recognized. Squall clusters are associated with tropical squall lines of the type identified by Hamilton and Archbold (1945) and Zipser (1969). They are notable in geosynchronous satellite imagery by their rapid propagation ( 15 m/sec), explosive growth, high brightness, and distinct convex leading edge (Martin, 1975; Aspliden et al., 1976; Payne and McGarry, 1977; Martin and Schreiner, 198I ). Nonsquall clusters travel more slowly than squall clusters (typically only a few meters per second) and do not possess the distinctive oval cirrus shield or arc-shaped leading edge of squall systems. A squall cluster is seen over Africa at lat 15" N, long 17" W in Fig. 37. The other clusters seen in the figure are nonsquall clusters. The importance of nonsquall clusters lies in their sheer number. They are the predominant type of cloud cluster in the tropics. The more dramatic squall clusters are relatively rare. However, the well-defined structure and motion they possess make them more amenable to study, and much attention has been devoted to squall clusters in the literature. Despite
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differences in motion and appearance in satellite data, squall and nonsquall clusters exhibit strong similarities in other aspects of their structure-both to each other and to midlatitude convective complexes. Thus, much of the understanding gained in studies of both types of clusters is consequently applicable to a general understanding of mesoscale convective systems. In Sections 4.3 and 4.4 below, the structures of squall and nonsquall clusters are described in turn, and their similarities to each other and to midlatitude systems are pointed out in Section 4.5. Hurricanes are treated in Section 4.6.
4.3. Squall-Line Cloud Clusters The tropical squall line (or “disturbance line”) was first described as a distinct meteorological phenomenon by Hamilton and Archbold ( 1945). The first documentation of a tropical squall line observed during an organized field experiment was presented by Zipser (1969). More recent field experiments have revealed further details of these systems (Betts et al., 1976; Zipser, 1977; Houze, 1977; Leary and Houze, 1979b; Fortune, 1980). Houze and Betts ( I98 1) summarized knowledge accumulated from this past work. The squall line is found to be part of a propagating mesoscale disturbance, namely, the “squall-line cloud cluster” or simply the “squall cluster.” As in the midlatitude squall system (Section 3.5, Fig. 35), the squall line is the leading portion of the disturbance and consists of cumulonimbus line elements (or cells), while an extensive, precipitating mid- to upper-level stratiform cloud shield, or “anvil cloud” (as defined by Brown, 19791, trails the squall line (Fig. 38). The convective elements composing the squall line contain buoyant updrafts that carry air of high moist static energy from the boundary layer to the upper troposphere. As in the midlatitude squall line, the tropical squall line tends to travel by a combination oftranslation and discrete propagation, wherein new cells systematically form on the leading edge of the system. As the cells mature, they become the main cells of the squall line. Dissipating cells occur to the rear of the mature cells. Although similar in spatial arrangement and propagation, the cells of the tropical squall line are typically weaker than those of midlatitude squall lines. The cells in the midlatitude case described in Fig. 34 appeared with a first echo well aloft, exhibited weak-echo regions in zones between the first echoes and mature cells, and contained peak reflectivities centered aloft. The cells in the tropical squall line, on the other hand, exhibit peak echo intensities at low levels as a result of their weaker updrafts (Caracena et al., 1979; Zipser and LeMone, 1980; Szoke and Zipser, 1981; Cheng, 1981).
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FIG.38. Schematic ot'a typical cross section through a tropical squall system. Dashed streamlines show convective-scale updraft and downdraft motions associated with the mature squallline element. Wide solid arrows show mesoscale downdraft circulation. Wide dashed arrows show mesoscale updraft circulation. Dark shading shows strong radar echo in the melting band and in the heavy precipitation zone of the mature squall-line element. Light shading shows weaker radar echoes. Scalloped line indicates visible cloud boundary. Adapted from Houze (1977).
Negatively buoyant downdrafts associated with the convective towers of the tropical squall line carry air of low moist static energy downward from the low to middle troposphere into the boundary layer. A portion of this convective downdraft air spreads forward and produces a gust front at the leading edge of the mesoscale system, similar to the gust front at the leading edge of the midlatitude squall system (Figs. 33,35, and 36). Another portion of this convective downdraft air spreads rearward, leaving an extensive wake of cool, stable air in the boundary layer. The trailing anvil region of the squall cluster, in contrast to the leading line of convective cells, has a predominantly stratiform structure. As noted in Fig. 38, the cloud and precipitation in this region tend to be horizontally uniform, with distinct vertical layering. Precipitation particles in the upper portions of the anvil cloud are initially in the form of ice particles, which grow as they drift downward until they melt in a shallow layer and then evaporate partially while falling as rain through unsaturated air below the base of the anvil. Most of the ice particles are probably generated in the tops of convective cells located along the leading squall line. These particles then move. in a relative sense, toward the rear of the system as a result of the continuing redefinition of the position of the leading edge of the system by the formation of new cells (as suggested by Houze, 1981). The airflow at anvil levels apparently aids in this process. Chen and Zipser (1982) have shown the existence of a midlevel jet of front-to-rear flow emanating from the cells just above the height ofthe melting level in a tropical squall system in a position somewhat similar to the jet shown by Houze and Smull ( 1982) and Zipser and Matejka ( 1 982) to be spreading ice particles back into the
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stratiform region of a midlatitude squall system (Fig. 35). The layer in which the ice particles melt after they are spread rearward into the anvil region appears on radar as a bright band of the type normally associated with stratiform precipitation. The bright-band region behind tropical squall lines appears to be wider than that behind midlatitude squall lines. Bright-band regions behind tropical squall lines can be 200 km across (Houze, 1977; Rappaport, 1982), while those behind midlatitude squall lines appear to be I00 km in most cases. The reason for this difference is not yet understood. The vertical air motions in the trailing anvil region of the squall cluster are widespread, generally nonconvective, and on the order of tens of centimeters per second. Downward motion occurs below the base of the anvil, while rising motion occurs within the anvil cloud itself (Fig. 38). Widespread subsidence below the anvil was first shown to exist from observed penetration of low moist static energy to low levels throughout the anvil region (Zipser, 1969). Its magnitude was determined from observed low-level divergence (Zipser, 1977), and Brown ( 1979) successfully simulated the downdraft below the anvil in a mesoscale model. The existence of the mesoscale updraft in the tropical squall-line anvil has been more difficult to establish firmly, although its existence has been indicated indirectly by a variety of evidence (see discussion by Houze and Betts, 198 1). Recently, however, Gamache and Houze (1982) have demonstrated the existence of the mesoscale updraft and computed its magnitude by using time-to-space conversion to group wind soundings in relation to the radar echo pattern of a squall line observed in GATE. The ability of the radar to identify the horizontally uniform precipitation falling from the anvil and distinguish it from the intense vertical cells that characterize the squall line made it possible to locate wind observations with respect to the squall-line (i.e.. convective) and anvil (i.e., stratiform) regions separately. From the wind pattern in the radar-delineated anvil area, the vertical profiles of divergence within and below the anvil were obtained, and from these profiles the vertical velocities in the mesoscale anvil updrafts and downdrafts were determined (solid curve in Fig. 39). The mesoscale updraft above the base of the anvil cloud (-650 mbar) and the downdraft below are both clearly shown. The vertical velocity profile determined from the winds in the squall-line region is also shown (dashed cuve in Fig. 39a). It was the result of convective updrafts and downdrafts as shown in Fig. 39b. The squall-line region was characterized by boundary-layer convergence feeding the deep convective updrafts. The convective downdrafts spreading out in the boundary layer intensified the low-level convergence feeding the convective updrafts. The circulation in the anvil region, on the other hand, was characterized by midlevel convergence supporting both the mesoscale updraft within the anvil cloud and the mesoscale downdraft below. This
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FIG.39. Average vertical velocity o for: (a) the squall-line (dashed), anvil (solid), and combined (dotted) regions of a tropical squall cluster; (b) the squall-line updrafts and downdrafts (dotted). From Garnache and Houze ( 1982).
midlevel convergence is probably akin to that associated with midlatitude MCCs (Section 3.4). However, the mesoscale divergence and vertical velocity pattern determined for the MCCs was not decomposed into separate convective and stratiform components and is therefore probably a mixture of convective and stratiform motion (as is the combined squall-line plus anvil region curve shown for the tropical system by the dotted line in Fig. 39a). Further work using radar observations to delineate the stratiform and convective components of the midlatitude MCCs might shed light on the dynamics of the midlatitude complexes and their similarities to tropical systems.
4 . 4 . Nonsquall Cloud Cluster\ Beneath the large cirrus shield that identifies a cloud cluster in a satellite picture, there is typically found one or more mesoscale rain areas, each attaining a maximum horizontal dimension -100-500 km. Leary and Houze (1979a) extended the conceptual model of a tropical squall-line system (Fig. 38) to describe the structure and behavior of these rain areas. They
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arrived at the more general concept of a “mesoscale precipitation feature” (MPF), of which the rain area of a squall cluster is an example, but which also applies to the rain areas of nonsquall clusters. This model has been elaborated on by Zipser ( 1980), Houze and Betts ( 1 98 I), and Houze (1982). Squall clusters and some nonsquall clusters contain just one MPF, while other clusters contain several MPFs interconnected by a common mid- to upper-level cloud shield. Intersections and mergers of the MPFs can add complexity to the precipitation pattern of the cluster. However, Leary and Houze ( 1979a) found that when the individual MPFs making up the pattern are identified and followed closely in time, they each exhibit a life cycle similar to that of a squall-line MPF. To illustrate this life cycle, we use as an example nonsquall clusters that were observed over the South China Sea during winter MONEX (Houze et al., 198 la; Johnson and Priegnitz, 198 1 ; Johnson, 1982; Churchill, 1982). These clusters formed diurnally off the northern coast of Borneo and typically contained one MPF (Fig. 40), which progressed through the stages of the life cycle identified by Leary and Houze (1979a). The ,formativr stuge of an MPF is initiated with an imposed mesoscale convergence at low levels (Zipser, 1980). This convergence may be associated with a downdraft outflow boundary from a previous cloud cluster (e.g., Houze, 1977; Fortune, 1980), a confluence line in a larger scale flow (e.g., Zipser and Gautier, 1978), or some other feature that intensifies convergence locally. The winter monsoon clusters used as an example here are triggered by the convergence of the nocturnal land breeze from Borneo with the large-scale northeasterly monsoon flow over the South China Sea (Fig. 40a). The triggering of convection by low-level convergence is followed by the growth of several discrete cumulonimbus elements, which may be randomly distributed in a group or arranged in a line. This initial spatial arrangement probably depends on the form of the initiating convergence. The intensifying stage of the MPF is not shown explicitly in Fig. 40. It corresponds to the period of transition between Figs. 40a and b. During this stage. older convective elements grow and merge while newer elements continue to form. Gradually, this process leads to a large continuous rain area composed of convective cells interconnected by stratiform precipitation of moderate intensity falling from a spreading mid- to upper-level stratiform cloud shield. The mature stage of the MPF is reached when the stratiform precipitation between cells becomes quite extensive, covering areas 100-200 km in horizontal dimension (region between cells in Fig. 40b). This stratiform precipitation resembles that associated with the anvil clouds of squall clusters. Associated with the stratiform precipitation region of a nonsquall MPF, moreover, are a mesoscale downdraft below the melting level and a me-
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FIG.40. Schematic of the development of diurnally generated nonsquall cloud cluster off the coast of Borneo. Various arrows indicate airflow. Circumscribed dot indicates northeasterly monsoon flow out of page. Wide opcn arrow indicates the component of the typical eastsoutheasterly upper-level flow in the plane of the cross section. Heavy vertical arrows in (a) and (b) indicate cumulus-scale updratts and downdrafts. Thin arrows in (b) and (c) show a mesoscale updraft developing in a mid- to upper-level stratiform cloud with a mesoscale downdraft in the rain below the middle-level base of the stratiform cloud. Asterisks and small circles indicate ice above the 0°C level melting to form raindrops just below this level. From Houze et al. (1981a).
soscale updraft above, similar to those of squall-line anvils. For evidence of these mesoscale motions in the stratiform regions of nonsquall clusters, see Zipser and Gautier (1978), Leary and Houze (1979a), Zipser et al. (198 I), Houze and Betts (198 I ) , Johnson and Priegnitz (198 l), Johnson (1982), and Churchill (1982).
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In the dissipating stage of a cluster’s MPF, the formation of new convective cells diminishes. However, the feature can persist for several hours as a region of mostly mid- to upper-level cloud, with continuing light precipitation or virga (Fig. 40c). 4.5. Generalized Cloud Cluster Structure The ability of the Leary-Houze conceptual model to describe the evolution of both squall and nonsquall mesoscale precipitation features is supported by studies of the development of precipitation in both types of clusters (Fig. 41). In each case, the rainfall is dominated in the formative stage of the MPF by convective cells. However, as the upper cloud shield develops (during the intensifying stage) the stratiform precipitation begins to account for a large proportion of the total precipitation from the MPF. By the mature stage, the stratiform component can equal or surpass the convective component. The stratiform component continues to be strong into the dissipating stage, although both the convective and stratiform components gradually weaken. In the squall case shown in Fig. 4 la, the integrated stratiform component accounted for 40% of the total rain, whereas in the nonsquall cases in Fig. 41b,c the stratiform rain accounted for 30% and 50% of the totals, respectively. Similar results have been obtained for other squall clusters by Gamache and Houze (1981) and Rappaport (1982). They found stratiform totals of 57% and 42%, respectively. Zipser et al. (1981) found essentially similar results for a nonsquall cluster. The quantitative similarity of the squall and nonsquall precipitation curves, together with the descriptive similarities of squall and nonsquall MPFs noted by Leary and Houze (1979a), allows certain generalizations to be made about the mesoscale structure of tropical cloud clusters. To the extent that midlatitude MCCs are similar to tropical cloud clusters, these generalizations also apply to those systems as well. Consider a cluster containing a single MPF in its mature stage. Whether a squall cluster (Fig. 38) or a nonsquall cluster (Fig. 40b), the MPF consists partly of convective cells and partly of stratiform precipitation falling from a mid- to upper-level cloud shield. The mature midlatitude squall system (Fig. 35) also exhibits this structure, and it is likely that nonsquall midlatitude MCCs do also. However, this latter possibility awaits confirmation by detailed radar studies to distinguish the convective and stratiform components of midlatitude MCCs. From the observed structures of mature MPFs in tropical cloud clusters, Houze (1982) has computed the net sensible heating in a large-scale area (2 X lo5 km2) containing an idealized cluster (Fig. 42). The heating associated with the convective cells is dominated by the latent-heat release in
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FIG. 4 I . Time variation of total rain integrated over areas covered by the convective and nonconvective regions of (a) a squall cluster and (b) and (c) two nonsquall clusters. The squallline case is from Houze (1977). The nonsquall cases are from Leary (1981) and Churchill ( 1 982).
the deep convective updrafts. This convective heating is distributedthrough the full depth of the troposphere (dashed curve in Fig. 42). To this heating is added the heating and cooling associated with the stratiform regions of
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d T / d t (DEG/DAY)
FIG.42. Total heating of large-scale region (2 X lo5 km2 in area) by a mature cloud cluster (solid curve). The total heating by the convective towers in the cluster (dashed curve) is shown for comparison. From Houze (1982).
the cluster. Latent heat released by condensation in the mesoscale updraft in the mid- to upper-tropospheric cloud shield (e.g., Fig. 40b) together with net radiative absorption in the cloud shield (daytime conditions were assumed) increases the total heating aloft (upper part of solid curve in Fig. 42). Cooling associated with melting and evaporation in the mesoscale downdraft region of the stratiform precipitation area (e.g., Fig. 40b) decreased the total heating in the lower troposphere (lower part of solid curve in Fig. 42). From these calculations, it is apparent that in considering the effects of mesoscale convective systems on their environments, the structure of the mesoscale stratiform clouds and precipitation that develop in association with these systems must be considered. Moreover, when they are considered, results such as those indicated by the solid curve in Fig. 42 are obtained, indicating that the net effect of a mature cluster (or MCC) on its environment should be felt most strongly in the mid to upper troposphere, where the convective, stratiform and radiative processes reinforce to produce strong heating. Large-scale vertical velocity profiles (Houze, 1982) and horizontal wind fields (Esbensen et a/., 1982) in the vicinities of GATE cloud clusters conform to this expectation. Strong responses also have been noted in the upper-level wind fields around mature midlatitude MCCs (Nimomiya, 197 1a,b; Maddox, 1980a, 198 1;Fritsch and Maddox, 198 1 a; Maddox et al., 1981; cf. Section 3.4). In generalizing about the structure of cloud clusters and midlatitude MCCs, we have emphasized the development of mesoscale stratiform structures, which accompany the development of deep convection in these systems. While making these generalizations, we must, at the same time, recognize and try to understand the fundamental dierences between squall and nonsquall systems and tropical and midlatitude systems. For example, why do squall systems propagate while nonsquall systems d o not? For some discussion of this problem, see Houze and Betts (1981). Such differences
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FIG.43. Visible satellite image of Hurricane Allen, 2123 GMT, 8 August 1980. Photograph provided by the National Hurricane Research Laboratory.
appear to be more than incidental. For example, Zipser (197 1) has pointed out that tropical squall-line clusters may be relatively ineffective at influencing large-scale development, whereas nonsquall clusters tend to produce positive feedbacks. His speculation, moreover, appears to be borne out by mesoscale humcane modeling results (Rosenthal, 1980). Clearly, much work lies ahead to sort out the various similarities and differences among these mesoscale systems and their effects on larger scales of motion.
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FIG.45. Schematic horizontal radar echo pattern in asymmetric and symmetric humcanes. Shading thresholds are for echo intensities of 30 and 40 dBZ. Humcane symbol is located at the center of the wind circulation. Symmetric storm is patterned after humcanes Anita (1977), David (1979), and Allen (1980). Asymmetric storm is patterned after humcanes Frederick (1979), Floyd (1981), Gert (1981), and Irene (1981). From Jorgensen (1982a).
4.6. Hurricanes A small fraction of tropical cloud clusters are associated with disturbances that develop into hurricanes (Frank, 1970). When hurricane development occurs, upper-level winds take on a high degree of anticyclonic rotation, and the upper cloud shield of the initial cluster becomes circular. In welldefined storms, a clear spot or “eye” is found near the center of the cloud shield (e.g., Fig. 43). The precipitation falling from the hurricane cloud shield is generally concentrated in a mesoscale “eyewall rainband,” which surrounds the eye of the storm, and in several mesoscale “outer rainbands.” Examples of rainbands in several mature hurricanes are shown in Fig. 44. Lighter precipitation occurs throughout much of the area between rainbands. Early papers on rainbands in hurricanes include Maynard (1945), Wexler (1947), Kessler and Atlas (1956), Senn and Hiser (1959), Atlas et af. (1963), Fujita et al. (1967), and Gentry et af. (1970).
FIG. 44. Radar renectivity patterns showing structure of precipitation in four mature hurricanes. Data obtained with radars aboard National Oceanic and Atmospheric Administration research aircraft on (a) 12 September 1979 in Humcane Frederick, (b) 30 August 1979 in Humcane David, (c) 5 August 1980 in Humcane Allen, and (d) 8 August 1980 in Humcane Allen. Photographs provided by the National Research Laboratory.
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That deep convection is important to the development of the hurricane through cooperative interaction between the convection and larger scale flow is well known from the classic work of Ooyama (1964) and Charney and Eliassen ( 1964).9 Accordingly, the precipitation in the rainbands exhibits a high degree of convective character. However, the precipitation can also be partly stratiform, with well-defined melting layers (as evidenced by radar bright bands) occurring over considerable portions of the storm (Atlas et al., 1963; Black et al., 1972; Hawkins and Imbembo, 1976). The latter two studies did not indicate specifically whether the observed bright bands were within rainbands or in the regions of lighter rain between bands. The observations of Atlas et al. (1963), however, were obtained in outer rainbands. These bands were composed of convective cells at one end, with their remaining portions being stratiform. The presence of mesoscale rainbands, partly stratiform in character, suggests similarity between hurricane precipitation processes and those of cloud clusters (Sections 4.1-4.5). Current study of hurricanes is being directed toward better understanding of both the eyewall and outer rainbands, particularly regarding their convective versus stratiform structure, their similarities to cloud-cluster precipitation features, and their dynamics. Radar observations indicate two apparent modes of rainband organization (Jorgensen, 1982a,b). “Symmetric” hurricanes (Fig. 45b) are characterized by a closed circular eyewall rainband. The center of the wind circulation is located in the center of the circle defined by the eyewall rainband, while the outer rainbands take on various forms (convective, stratiform, spiral, concentric). “Asymmetric” humcanes (Fig. 45a) have an eyewall rainband that is not closed. Outer rainbands in these storms tend to be more spiral than concentric, and the center of the wind circulation is not coincident with the geometric center of curvature of the eyewall but is displaced toward the eyewall. An example of asymmetric hurricane structure is shown in Fig. 44a, while symmetric structures may be seen in Fig. 44b-d. The eyewall rainband in a symmetric storm is often observed to contract; that is, it propagates toward the center of the storm and thus shrinks (Marks, 1981; Willoughby et al., 1982). As the eyewall rainband contracts, the central pressure of the storm lowers. After 1-2 days, the radius of the eyewall band reaches its minimum size, and as the band disappears, it is replaced by a new eyewall rainband at a radius of about 50- 150 km and the central pressure of the storm rises (note the old eyewall band near the storm center and the new one farther out shown schematically in Fig. 45b and by actual examples in Fig. 45b,c). The life cycle of the eyewall band is then repeated. Thus, a symmetric storm is often characterized by a succession of shrinkmg eyewall rainbands and a pulsating central pressure. For a comprehensive review of hurricane dynamics, see Anthes (1974, 1982).
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A theoretical explanation for the propagation of the symmetric eyewall rainbands inward has been suggested by Shapiro and Willoughby (1982). In earlier work, Shea and Gray (1 973) showed that the maximum vertical motion (and hence the maximum cloud and precipitation development) associated with eyewalls is consistently located close to the radius of maximum wind. The rising motion in the eyewall region is associated with the convergence at low levels of radial inflow and outwardly directed components of supergradient flow within the eye. Shapiro and Willoughby explain the inward propagation of this eyewall structure in symmetric storms as a secondary-circulation response to a point source of heat placed near the radius of maximum wind in a hurricane-like vortex. In that idealized situation, temporal increases in tangential wind are greatest just inside the radius of maximum wind. This effect leads to contraction of the zone of maximum wind as the vortex intensifies. Willoughby et al. (1982) find observational support for this theory in data from recent humcane flights. The eyewall rainbands of asymmetric storms do not appear to shrink and undergo the life cycle exhibited by the symmetric eyewall rainbands. A satisfactory explanation for this difference between symmetric and asymmetric storms has not yet been obtained. Jorgensen ( 1981, 1982a,b) has compiled radar, wind, thermodynamic, and cloud-physics measurements from flights through eyewall and outer rainbands. A schematic cross section through the eyewall region of a symmetric storm (Fig. 46) summarizes his findings. Air flowing radially inward
RADIAL DISTANCE FROM STORM CENTER (KM)
FIG.46. Schematic vertical cross section through north-northwestem section of humcane Allen (1980). See. Figs. 42 and 43c for horizontal views. From Jorgensen (1982b).
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meets radial outflow from the eye (between 30 and 35 km radius) and rises along an outward-sloping mean streamline pattern. Superimposed on the mean upslope motion are intense convective updraft cores. The sloping tangential wind maximum also shown is accounted for by conservation of absolute angular momentum of the low-level inflow, which turns upward in the eyewall convergent zone. Fallout of precipitation initiated in the upward flow above the tangential wind maximum accounts for the sloping radar echo core below the wind maximum. Convective downdraft cores occur in the core of heavy rain. The maximum echo intensity occurs at low levels, as in the convective echoes of tropical cloud clusters (Houze, 1977; Leary and Houze, 1979a; Caracena et al., 1979; Zipser and LeMone, 1980; Szoke and Zipser, 1981; Cheng, 1981). A region of stratiform precipitation characterized by a well-defined melting layer occurs adjacent to and just outside the convective eyewall zone. Similarities in the patterns of circulation, cloud and precipitation in Fig. 46 to those of the squall-line cloud cluster (Fig. 38) and the midlatitude squall system (Fig. 35) are striking. Jorgensen (1981, 1982a,b) finds the structure depicted in Fig. 46 to be characteristic of the eyewall rainbands of both symmetric and asymmetric storms. Less is known about outer rainbands. Jorgensen (1982a,b) finds that some outer bands resemble eyewall rainbands. However, a variety of structures occur: some are purely stratiform, others convective; whereas some, such as those reported by Atlas et al. (1963) are'a mixture of convective and stratiform structure.
5 . CONCLUSIONS In this article, we have surveyed the major types of cloud systems that contribute to precipitation over the earth. A common attribute of these systems is their tendency to become organized on the mesoscale. In extratropical cyclones, this organization is manifested in mesoscale rainbands. In deep convective cloud systems, mesoscale organization is apparent in the tendency for cumulonimbus elements to occur in groups, which in turn drive mesoscale circulations, with which are associated mid- to upper-level cloud shields, stratiform precipitation, melting layers, mesohighs and -lows, gust fronts, and arc lines. Mesoscale systems driven by deep convection include midlatitude mesoscale convective complexes and squall lines, tropical squall and nonsquall cloud clusters, and humcane rainbands. In the case of frontal rainbands, progress has been made in classifying the various types of bands that occur and in documenting their basic air motions and cloud microphysical processes. In the case of deep convective phenomena, progress has been made in understanding the basic modes of cumulonimbus structure and dynamics, particularly in the identification of
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the splitting mechanism and subsequent development of rotation that occurs in the evolution of supercell thunderstorms, and in describing the evolution of groups of deep convective thunderstorms into mesoscale systems. In these descriptions, intriguing similarities are seen among midlatitude and tropical systems-compare, for example, the midlatitude squall-line system (Fig. 3 9 , the tropical squall cluster (Fig. 38), and the hurricane eyewall rainband (Fig. 46). Further progress can be made in nearly all aspects of the mesoscale organization of frontal and deep convective cloud systems. Certain outstanding problems, however, seem to be particularly wanting, and in closing we take special note of these. The question of the origin of frontal rainbands remains unresolved. That is, the basic reasons for precipitation to become enhanced in mesoscale bands have not yet been identified. As was pointed out in Section 2, several dynamical instability mechanisms can act on the scale of the rainbands; however, the association of specific instabilities with specific types of rainbands has not been satisfactorily accomplished. In the case of mesoscale systems associated with deep convection, problems remain in comparing midlatitude and tropical systems. In the tropics, radar observations have been used to distinguish between convective and stratiform regions of mesoscale systems, and as a result the deep convective and mesoscale stratiform components of the air motions in these systems have been identified. Similar radar work with midlatitude mesoscale convective complexes and squall lines should be camed out to improve the understanding of the midlatitude systems and make possible their comparison with tropical systems. For both the tropical and midlatitude mesoscale convective systems, a better understanding of their life cycles should be sought. Mesoscale models, such as those of Brown (1974, 1979), Kreitzberg and Perkey ( I977), Fritsch and Chappell(l980), and Fritsch and Maddox ( I98 1b), together with the presently available descriptive studies, give glimpses of understanding; but further work, both observational and theoretical, is needed to understand fully the chain of events involved in the development of the mesoscale mid- to upper-level cloud and stratiform precipitation that accompanies deep mesoscale convective systems in their mature stages. We look forward to intensive research on mesoscale cloud systems in upcoming years. Better understanding of the mesoscale phenomena we have described in this article will contribute to the basic understanding of precipitation processes in the atmosphere, and will have benefits to society ranging from improved detailed weather forecasting to better general-circulation models and improved management of water resources on local, regional, and global scales.
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ACKNOWLEDGMENTS The authors’ studies in this subject are supported by the National Science Foundation under grants ATM-8017327 and ATM-8009203 and by the National Oceanic and Atmospheric Administration under grant NA80RAD00025. Helpful comments of J. C. Fankhauser, B. F. Smull, and E. J. Zipser greatly enhanced the quality of Sections 3 and 4. This article is Contribution No. 627, Department of Atmospheric Sciences, University of Washington.
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Wilhelmson, R. B., and Klemp, J. B. (1981). A three-dimensional numerical simulation of splitting severe storms on 3 April 1964. J . Atmos. Sci. 38, 1558-1580. Willoughby, H. E., Clos, J. A., and Shoreibah, M. G. (1982). Concentric eye walls, secondary wind maxima, and the evolution of the humcane vortex. J. Atmos. Sci. 39, 395-41 I . Zipser, E. J. (1969). The role of organized unsaturated convective downdrafts in the structure and rapid decay of an equatorial disturbance. J. Appl. Meteorol. 8, 799-8 14. Zipser, E. J. (1971). Internal structure of cloud clusters. GATE Experiment Design Proposal, Vol. 2, Annex VII. World Meteorol. Organ., Geneva. Zipser, E. J. ( 1977). Mesoscale and convective-scale downdrafts as distinct components of squall-line circulation. Mon. Weather Rev. 105, 1568-1589. Zipser, E. J. (1980). Kinematic and thermodynamic structure of mesoscale systems in GATE. In “Proceedings of the Seminar on the Impact of GATE on Large-Scale Numerical Modeling of the Atmosphere and Ocean,” pp. 9 1-99, Natl. Acad. Sci., Washington, D. C. Zipser, E. J., and Gautier, C. (1978). Mesoscale events within a GATE Lropical depression. Mon. Weather Rev. 106, 789-805. Zipser, E. J., and LeMone, M. A. ( 1980). Cumulonimbus vertical velocity events in GATE. 11. Synthesis and model core structure. J. Atmos. Sci. 37, 2458-2469. Zipser, E. J., and Matejka, T. J. (1982). Comparison of radar and wind cross-sections through a tropical and midwestern squall line. Prepr., Conf Severe Local Storms, 12th, 1982 pp. 342-345.. Zipser, E. J., Meitin, R. J., and LeMone, M. A. (1981). Mesoscale motion fields associated with slowly moving GATE convective band. J. Atmos. Sci.38, 1725- 1750. Zrnic, D. S., Doviak, R. J., and Burgess, D. W. (1977). Probing tornadoes with a pulse Doppler radar. Q. J. R. Meteorol. Soc. 103, 707-120.
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INDEX
A Air mass thunderstorm, 248-249, see also Thunderstorms defined, 251 Alpine ice masses, Ice Age and, 63 Analytic functional F coherent studies, 174 Angular momentum, conservation, 93 Annual precipitation, globally averaged, 226, see also Precipitation Arc cloud lines, new thunderstorms and, 286-287 Asthenosphere, 8 Asymmetric Korteweg-de Vries eddies, collision experiments, 190, see N I S O Korteweg-de Vries eddies
B Baroclinic atmosphere, modified Kortewegde Vries equation and, 161 Baroclinic Rossby radius, 169 Baroclinic solitary eddy, with radial symmetry, 163 Barotropic atmosphere, Korteweg-de Vries model and, 161 Barotropic vorticity conservation equation, modon solution, 162 Bernoulli equation, 169 Bernoulli functional, 166- 167 Bernoulli’s integral of motion, 165- 166 Biharmonic friction, 208-21 1 Boomerang echoes, in narrow cold-frontal rainbands, 242 Born-Infeld equation, 155 Bounded weak-echo region, of supercell thunderstorm, 260 Boussinesq approximation, 169 Boussinesq equation, 151, 155- 156 Bow echoes, in mesoscale downbursts and tornadoes, 242 Bromwich path, integral along, 49 Briint-Vaisala frequency, 164-165 Burgers body complex s plane for, 25
Debye peak, 22 generalized, see Generalized Burgers body homogeneous spherical, 19-27 Maxwell solid and, 34 modal Q predictions, 22 modes on negative real s axis, 25 short time scale viscosity for, 21 spherical, 28-32 Burgers body rheology absorption band part, 19 for radial, spheroidal, and toroidal modes, 24 Burgers body solid equations, 16 versus Maxwell solid, I5 Burgers gravity anomaly, topographic wavelength and, 3 BWER, see Bounded weak-echo region
C Carbon-14 dating, see Radiocarbon dating Cauchy’s theorem, 49 Chandler wobble frequency, 33, 91 for homogeneous elastic earth, 100 for rigid earth, 99-101 Chirikov’s criterion, 153 for overlapping resonances, 203-205 for stochasticity border separating wavelike behavior from chaotic motion, 205 CIO, see Conventional international origin Clapeyron slope, adverse, 138- 139 Climatic change, glacial isostasy and, 119133 Climatic oscillation, Pleistocene, 123- 128 Cloud clusters, types and lifetimes, in tropics, 287-290, see also Cloud systems; Tropical cloud systems Cloud particles, nucleation, 226 Cloud patterns, in extratropical cyclones, 228
317
318
INDEX
Clouds numerical modeling, 226 precipitating, see Precipitating cloud systems tropical, 257-303, see ulso Tropical cloud systems types, 225-226 Cnoidal wave, defined, 156-157 Coherent structures collision experiments, 182- 191 dissipation in, 207-208 evolution, 173- 191 examples, 161-163 existence, 160 existing models, 174 Fourier space representation and, 209212 further investigations, 206-214 large-scale eddy configurations and, 163 locked Fourier phases, 198 numerical techniques, 206 as one-dimensional solitons, 206 reproducibility of solutions through algorithms, 206 small mesoscale, 208 superimposed perturbations and, 200 Collision experiments for asymmetric Korteweg-de Vries eddies, 190 for coherent structures, 182- 191 coupled Korteweg-de Vries dynamics, 183-186 cross-channel eigenfunction phase-shifting, 183-184, 187 difference field evolution, 188 as stability experiments, 198 Comma clouds, 247 Constant lunar tidal acceleration, ancient solar eclipses and, 92 Continental drift, Wegener’s hypothesis, 3 Convective systems, midlatitude, see Midlatitude convection systems Conventional international origin, rotation pole motion relative to, 90 Coriolis parameter, in P-plane approximation, 164 Coupled fields, evolution in collision experiments, 189-190 Coupled Korteweg-de Vries dynamics, in
see also Korteweg-de Vries dynamics Coupled Korteweg-de Vries equations, in collision experiments, 182, see ulso Korteweg-de Vries equations Crust, MohoroviCic discontinuity and, 36 Cumulus stage, in thunderstorm, 250-251 Cyclones aircraft-radar studies, 231 cloud patterns, 228-229 clouds and precipitation, 229-230 cross section, 227 extratropical, 229-247 gravity-current models, 242 Pacific, 231 radar bright bands, 230-231 rainbands, 232-246 warm-frontal rainbands, 234 wavelike rainbands, 232-234
D Debye peak, of Burgers body, 17, 22 Deep mantle, viscocity, 9, see a l s o Mantle Deep Sea Drilling Project, 81 Deep-sea sedimentary cores, oxygen isotope stratigraphy, 81, 102 Deglaciation Fennoscandian, 3-4 Fennoscandian free-air anomaly and, 10 Pleistocene, 9 Deglaciation centers, gravity field, 76-79 Deglaciation chronology, relative sea level calculation and, 62-65 Deglaciation-forced rotational effects theory, 93-114 Deglaciation-induced perturbations of gravitational field, 75-90 of planetary rotation, 90- 118 Deglaciation-induced polar motion, Euler equation solutions, 99- 104 Deglaciation-related free-air anomaly. Laurentide or Fennoscandian depressions and, 9 Difference field, evolution in collision experiments, 188- 192 Disintegration isochrones, for Laurentide ice sheet, 62-63 Disk load approximations. initial isostatic disequilibrium and, 79-88
INDEX
j , predicted, 113 for polar wander analysis, 102 Dissipation, in coherent structure models, 207 - 208 Dissipation forms, on permanent eddies, 153 Doppler radar observations in squall-line sytems, 282-283 in tornado studies, 269-271 Downdraft spreading, from thunderstorms, 284-287 DSDP, see Deep Sea Drilling Project
E Earth angular momentum conservation, 93 Chandler wobble frequency, 99- 101 elastic gravitational free oscillations, 3 elastic properties and density, 3 homogeneous spherical Burgers body model, 14-32 homogeneous viscoelastic model, 103 observed elastic structure, 35-38 planetary mantle viscosity, 3 seismic discontinuity, 36 spherically averaged elastic model, 35-36 viscous half-space model, 3-5 Earth model L1, relative sea level zone boundaries, 67 Earth models homogeneous, 99- 104 Laplace transform domain Love number spectra, 47-49 Laurentide disk load as function of time, 85 Earth rotation, Pleistocene deglaciation forcing, 10 Eddy, solitary, single, sco Single solitary eddy; Solitary eddies Elastic solid, Hookean, 16- 18 Error energy, turbulent flow states and, 195 Error Green’s function, 84, see also Green’s function Euler equations for deglaciation-induced polar motion, 105- 1 I4 solution, 99- 104 Eulerian nutation, free, 14-month, 91 Eustatic sea level curve, 63 External deformation radius, length scales of order of, 164- 168
3 19
Extratropical cyclones, 228-247, see also Cyclones
F Feeder cloud, in warm-frontal rainbands, 235 Fennoscandia deglaciation, 3-8 Fennoscandia ice sheet, eustatic sea level curve and, 63-64 Fennoscandia region free-air gravity anomalies, 10, 11I free-air gravity anomaly maps, 77-78 interpretation of gravity field, 78 loading and unloading history, 72 radial displacement response model, 5558 relative sea level records, 125 Fermi- Pasta- Ulam recurrences of discretized anharmonic lattice, 148, 159-160, 220 stability studies and, 197 Flux-form integrals and operators, energy equation and, 193 Fourier modes, in theory of overlapping resonances, 201-202 Fourier phases locked, 198-199 time evolution, 199 Fourier space representation, 209-212 Free-air gravity anomaly at center of disk model Laurentide load versus mantle viscosity, 85-86 as function of time, 79 Green’s function, 79 for self-consistent model, 88-89 Free-air gravity anomaly maps, for Laurentide and Fennoscandia regions, 7778, 88 Free-air gravity data Hudson Bay, 76 Newtonian viscoelastic model and, 9 relative sea level and, 10, 86, 134-135 Free-air gravity signals, computation, 79 Free Eulerian nutation, 14-month, 91 Free relaxation oscillation, increased period of, 132 Frontal rainbands, origin, 303-304 Functionals analytic, 166-168 changed signs for, 194
320
INDEX
G GARP experiment, 149 GATE experiments, 289-292, 298 Generalized Burgers body, 14-19, see also Burgers body complex s plane for, 25 phenomenological utility, 32-34 Generalized Burgers body rheology , period and Q of mode 5,. 26 Generalized Burgers relation, defined, 134 Generalized cloud cluster structure, in tropics, 296-300, see ulso Clouds; Cloud structure Geodynamic processes, time scales, 14 Geoidal heights, global map of in GEM 10 data set, 76-77 Geophysical flows, planetary solitary flows in, 147-220 Geophysical motions, large-scale, 147- 154 Glacial isostatic adjustment theory, 11 -22 Glacial isostasy climatic change and, 119- 133 synthetic relaxograms, 46 Glacial isostasy theory basic ideas, 137- 139 mantle rheology model and, 12 Glacial isostatic adjustment, phenomena associated with, 134 Glacial isostatic model, 12 Glaciation/deglaciation cycle, in Northern Hemisphere, 2 Glaciation history, disk load model and, 113 Glen flow law, 125 Global Atmosphere Research Program, 289 Gravitational field, deglaciation-induced perturbations and, 75-90 Gravitational interaction problem, Green’s functions, 51, 53-55 Gravitationally self-consistent theory, for viscoelastic models Ll-L3, 88-89 Gravitational potential, Green’s function for perturbation, 113 Gravitational relaxation, viscous, see Viscous gravitational relaxation Gravity anomaly free-air, see Free-air gravity anomaly lack of, 2 Gravity data, Wisconsin Laurentia, 137 Gravity waves, ducting of, 236 Great Red Spot of Jupiter, 151 as solitary wave, 161- 162
Green’s function(s) convolution over simple circular disk loads, 55 error or correction forms, 84 for free-air anomaly, 79 Love number and, 129 for perturbation of gravitational potential for point mass, 113 potential perturbation, 61 for radial and tangential displacement, 97 radial displacement type, for viscosity model 1, 54 for surface mass load boundary value problem, 51, 53-55 for viscoelastic impulse response, 82 for viscosity model 1, 35, 54-55 Gulf Stream rings, 151 as coherent structures, 173
H Hailfall pattern, in supercell thunderstorm, 260-261 Half-space model, Newtonian viscous, 3-7 Hamiltonian integrable system closed periodic phase-point trajectory of. 200 Korteweg-de Vries model as, 201 Haskell half-space model, 3-7 Haskell-Vening-Meinesz model, 5 Heaviside displacement amplitude, Love number temporal history and, 51 Heaviside step function, 18 time domain forms and, 50 Hirota equation, 155 Homogeneous compressible sphere, equation, 40-41 Homogeneous earth model complex eigenspectra, 21 and Euler equations for deglaciationinduced polar motion, 99- 101 mean pole wander speed, 116- 117 polar wander speed, 104 Homogeneous incompressible spherical Burgers body, see ulso Burgers body free oscillations, 19-27 physical properties, 22 viscous gravitational relaxation, 28-52 Hookean elastic solid, 12-13 equation, 16-18
INDEX
Hudson Bay free-air gravity anomaly, 76 in Phanerozoic basin, 78 raised beaches, 31 -32 relative sea level records, 125 relaxation times, 8 sea level adjustments, 137 Humcane Allen cross section, 302-303 satellite image, 299 Hurricanes, 300-303 eye, 300 eyewall rainband, 300-302
I ICE-I deglaciation model, 63-64 melting cessation, 75 ICE-2 deglaciation model melting cessation, 64, 75 time slices through, 65-66 Ice Age Alpine ice masses and, 63 earth dynamics, 1-139 geophysical importance, 2 Ice Age cycle, theory, 119-133 Ice cap, rotational forcingproduced by, 100101
Ice sheet, topographic height versus accumulation rate, 128- 130 Ice volume fluctuations, in Pleistocene, 126-127 ILS-IPMS pole path, secular drift, 93, see also International Latitude Service; International Polar Motion Service Impulsive forcing, Love number spectra, 46-52 Incompressible homogeneous Burgers body, 28-32, see also Burgers body Inertial perturbations surface mass loading as cause, 95-99 variable rotation and, 94-95 Initial isostatic disequilibrium, disk load approximations in, 79-88 Initial value problem, coherent structure evolution and, 173- 191 Insolation gradient, present-day, 125 International Latitude Service, 10, 90 International Polar Motion Service, 10, 90 Isolated squall-line system, 282 Isostasy, concept, 2 Isostatic adjustment process, 2
32 I
characteristics, 57 equation, 129- 130 Isostatic disequilibrium, initial, 79-88
J Johnstown flood, 276 Jupiter atmosphere, example of coherent structure, 160- 163 Jupiter Great Red Spot, 151 as solitary wave, 161-162
K Korteweg-de Vries continuum model, 159 Korteweg-de Vries dynamics coupled, 151 and higher amplitude range of turbulent behavior, 198 in mesoscale oceanic and atmospheric motions, 217-218 overlapping resonances and, 201 transition to turbulence dynamics, 219 Korteweg-de Vries eddies, 152 collision experiments, 190 Korteweg-de Vries equations coupled, 178, 181 generalized form, 154 modified form, 155 as most important equation in geophysical flow investigations, 216 one-dimensional, 158- 159 remarkable properties, 157 strong-wave limit and, 175 time-dependent, 178 in unbounded domain, 159 Korteweg-de Vries initial value problem, solution, 158 Korteweg-de Vries-modified Kortewegde Vries equations, 154-155 Korteweg-de Vries model, as completely integrable Hamiltonian system, 201 Korteweg-de Vries soliton, amplitude versus speed for, 157, see also Solitons Korteweg-de Vries subrange, 152 Korteweg-de Vries-to-linear subrange, in weak-amplitude limit, 190 Korteweg-de Vries-to-turbulent subrange, Korteweg-de Vries eddies and, 190 Kelvin-Voight elements, chain of, 17
322
INDEX
L Lake Agassiz, history, 4 Lake Bonneville disappearance, 5, 8 radial displacement response model, 5558 Laplace transform domain forms, 50 of imaginary frequency s, 99 Love number spectra for earth model, 4749 Large-scale geophysical flows, existing parameter ranges and transitions, 216 Large-scale geophysical motion parameter range, 154 solitary waves in, 147- 154 Large-scale permanent structures, existing models, 160-163 Laurentia, gravity anomalies, 1 1 1 Laurentide complex, Hudson Bay ice center collapse and, 64 Laurentide disk load, time dependence of peak anomaly produced by, 80 Laurentide disk model as function of time for LI and L2 earth models, 85 -86 radial displacement response at center, 87 Laurentide ice sheet deglaciation-induced drift and, 92 in deglaciation phase, 62 disintegration isochrones, 62 disk load approximation, 96 eustatic sea level curve, 63-64 at glacial maximum, 8 Laurentide ice sheet sites, relative sea level constraints in mantle viscosity profile near, 69-74 Laurentide load, radial displacement response model, 55-58 Laurentide rebound, relict beaches near, 31 Laurentide region free-air gravity anomaly maps, 77-78,8889 loading and unloading history, 82 Laurentide-scale load, isostatic adjustment, 136 Layered model properties, in rotation calculations, 111 LDE, see POLYMODE local dynamics experiment
Leary-Houze model, of tropical squall and nonsquall precipitation, 295-2% Length-of-day constraints, on earth’s viscoelastic stratification, 114- 115 Length-of-day variations historical, 112 polar motion and, 92 Length scales, of order of external deformation radius, 164- 166 Levi-Civita alternating tensor, 93 “Linear and Nonlinear Waves” (Whitham), 149 Linear friction in coherent structures, 208-21 1 shape-preserving decay under, 213 time evolution under, 214 Lithosphere, viscosity stratification beneath, 6 Load history, sawtooth approximation, 108 Load Love number, 94, see also Love number Locked Fourier phases decorrelation, 198- 199 randomization, 199 1.o.d. variations, see Length-of-day variations Love number elastic, 129 load, 94 surface load, 98, 101, 105 tidal, 94 Love number expansion, Green’s function and, 129 Love number spectra elastic and isostatic asymptotes, 52-53, 57, 80 for impulsive forcing, 46-52 Lunar tidal function, 1.o.d. variation and, 92
M McConnell model, 5-6 Magnetic field, Matuyama-Brunhes polarity transition, 11 Mantle, see also Earth discontinuity as chemical boundary, 38 homogeneity, 38 Maxwell time and, 14 olivine-to-spinel structure changes, 36
323
INDEX
and phase equilibrium design for system (Mg, Fe),SiO, as function of pressure or depth, 37 polycrystallinity, 13 rheological behavior, 13- 14 seismic discontinuities, 36 Mantle density, jumps, associated with olivine + spinel and spinel + perovskite + magnesiowustite transitions, 134 Mantle mixing, homogeneity and, 38 Mantle rheology generalized Burgers body, 14- 19 linear viscoelastic model, 12-34 Mantle viscosity for earth model with 1066B elastic structure, 109 first quantitative estimate, 3-4 isostatic adjustment data and, 3-4 Laurentide model disk load as function of, 85 - 86 variation of with depth, 134 Mantle viscosity profile gravity field constraints, 89-90 relative sea level constraints in under initial isostatic equilibrium, 68-75 Mantle viscosity studies, early work, 3-4 Matuyama-Brunhes polarity transition, in earth’s magnetic field, 1 1 Maxwell earth, impulse response, 34-59 Maxwell solid, equation, 15- 16 MCCs, see Mesoscale convection complexes Mean pole motions for arbitrary time dependence of ice sheet loading and unloading, 101 case examples, 102- 104 Melting history, disk load approximation and, 112 Mesoscale atmospheric motions, new understanding, 148 Mesoscale cloud systems, future research, 304 Mesoscale convection complex, 275-278 cloud cluster resemblance to, 287 defined, 275 mass-flow stream function in, 276 mesoscale circulation, 277 midlatitude complexes and, 293 nonsquall midlatitude, 296
satellite picture, 274-275 Mesoscale downbursts, bow echo in, 242 Mesoscale motions, 150 Mesoscale precipitation feature dissipating stage, 2% squall clusters, 294-296 Mesoscale updraft latent heat release, 298 in tropical squall lines, 292 Midlatitude convective systems, 247-287 mesoscale, 275-278 midlatitude squall lines and, 278-284 Midlatitude cyclone, vertical cross section, 227, see also Cyclones Midlatitude squall lines, 278-284 as mesoscale convection complex, 279 models, 280-282 principal features, 281 Milankovitch hypothesis, 120- 123 Modal Q, Burgers body and, 22 MODE experiment, 149, 220 Modified Korteweg-de Vries model, for barotropic version of vorticity equation, 170 Modon-like solitary vortices, dynamics, 168 Modons exact multivalued solutions of conservative equations, 193 multivalued, 207-208 simulation through second-order finite difference schemes, 206 stability experiments in perturbation resistance of, 200 troublesome solutions in coherent structure problems, 207 MohoroviEic discontinuity, 36 MONEY experiments, 289 MPF, see Mesoscale precipitation feature Multicell hailstorm, model, 252 Multicell thunderstorms, 249-255, see also Thunderstorms ambient wind shear, 254-255 motion, 253 Multivalued F coherent structures, 174
N Narrow cold-frontal rainbands, 239-242 boomerang echoes, 242
3 24
INDEX
National Oceanic and Atmospheric Administration, U.S., 289 Navier-Stokes equation, for microscopic motions of liquids, 33-34 Newtonian friction, 208-212 time evolution under, 215 Newtonian viscoelastic earth models, relative sea level and free-air gravity data related to, 9 Newtonian viscous fluid, defined, 13 Newtonian viscous half-space model, 3-7 NOAA, see National Oceanic and Atmospheric Administration Nonhydrostatic bulge, existence, 6 Nonlinear Korteweg-de Vries solutions, superposition, 197 Nonlinear Schrodinger equation, 155- 156, see also Schrodinger operator Nontidal acceleration of rotation, 92 North American eastern seaboard, relative sea level curves for sites along, 73 0 Occlusions, in Pacific cyclones, 231 Olivine + spinel phase boundary, internal density jumps of mantle, 128 Open hexagonal cells, in convective clouds, 244 “Origin of Continents and Oceans, The” (Wegener), 3 Orography, influence on rainbands, 246 Ottawa Islands, radiocarbon ages of shells, 45 Overlapping resonances, 200-206 Chirikov’s criterion, 203-205 for quasi-Hamiltonian systems, 200-201 Oxygen isotope stratigraphy climatic fluctuations, 120- 123 from deep-sea sedimentary cores, 81, 201
P Paleoclimatic change theory, 11 Paleoclimate model, ice sheet movement, 123-124 Permanent structures, large-scale, see Large-scale permanent structures Permanent-wave dynamics, 153 Perturbation resistance, of modons, 200 Perturbations of inertia, see also Random perturbation
due to surface mass loading, 95-99 due to variable rotation, 94-95 Phase transitions, expected dynamical effects, 138 Photo zenith tube stations, 90 Planetary mantle, effective viscosity, 3, see also Mantle Planetary rotation, deglaciation-induced perturbations, 90- 118, see also Polar motion; Polar wander Planetary solitary waves, in geophysical flows, 147-220, see also Solitary waves Pleistocene climatic oscillations, 1 1 , 123- 128 deglaciation, 9, see also Deglaciation glacial cycle, rotation pole wander and, 115
main ice sheet appearance and disappearance time scale, 102 Polar air masses, vortices, 247 Polar drift, mean speed versus angle of, 117-118 Polar lows, 247 Polar motion, historical records, 90-93 Polar motion constraints, on earth’s viscoelastic stratification, 114- 115 Polar motion times series, for rotation pole motion relative to conventional international origin, 91 Polar wander, present-day speed, 103 Polar wander analysis, disk load integration model, 102 Polar wander speed function of time during cycle for viscoelastic model, 108- 110 homogeneous earth model, 104 in International Latitude Service pole path, 136 POLYGON experiment, 149 POLYMODE Local Dynamics Experiment, 149, 163, 220 Postfrontal rainbands, 243, wave-CISK mechanism, 244 Precipitating cloud systems, see also Cloud clusters; Clouds in global circulation and climate, 225 latent-heat release, 225 organization and structure, 225-304 Precipitation, see also Rainbands; Thunderstorms
INDEX
convective, 23 I in cyclones, 229-230 globally averaged annual, 226 latitude belts, 229 microphysical processes leading to, 230 prefrontal cold surge and, 243 Precipitation core, passage of, 240-241 Prefrontal cold surge, precipitation and, 243 Pressure noses, 286 pzt stations, see Photo zenith tube stations
Q Quaternary geological record, isostatic adjustment, 59 Quasi-geostrophic potential vorticity equation, 163 Quasi-Hamiltonian systems, overlapping resonance theory and, 200
R Radial displacement response, at Laurentide disk model center, 87 Radiocarbon-controlled relative sea level data, 7, see also Relative sea level Radiocarbon dating, 4 relative sea level histories and, 4-9 Rainbands in cyclones, 232-246 frontal, 303 in hurricanes, 300-302 interactions between, 244-246 narrow cold-frontal, 239-242 orographic effects, 246 postfrontal, 243-244 warm-frontal, 234-237 warm-sector, 236-238 wide cold-frontal, 238-239 Raised beaches, at Richmond Gulf of Hudson Bay, 31-32 Random perturbation, time evolution of total energy of, I%, see also Perturbation resistance Realistic layered viscoelastic models, rotational stability to ice sheet forcing, 118 Red noise background, in Pleistocene climatic oscillation, 126 Relative sea level deglaciation chronology and, 62-65
325
free-air gravity data and, 9-10, 86, 134135 ICE-1 and ICE-2 models related to, 65-67 ice thickness history and, 60-61 initial isostatic disequilibrium and, 86 integral equation, 60 mantle viscosity profile and, 62-65 postglacial variations, 59-75 radiocarbon-controlled histories, 134- 135 Relative sea level calculation global sea level histories and, 65-68 inputs, 62-65 Relative sea level curve, for Richmond Gulf beaches, Hudson Bay, 31-34 Relative sea level histories, carbon-I4 dating and, 4-9 Relaxation time versus angular degree for earth model sequences, 44-45 function of spherical harmonic degree, 42 Richmond Gulf,Hudson Bay raised beaches, 31 -32 relative sea level curve, 31-34 RSL, see Relative sea level Rossby number defined, 165, 176 for permanent-wave evolution, strongwave case, 178 Rossby radius, baroclinic, 169 Rossby waves decay of coherent structures into, 153 for one-dimensional equations, 216 solitary, 150-151 wave amplitude reduction and, 214 Ross Ice Shelf, large-scale melting event, 63 Rotation, see also Polar motion planetary, 10, 90-118 variable, 94-95 Rotation acceleration, nontidal component, 92-93 Rotational effects, deglaciation-forced, 93114 Rotational forcing, by single circular ice cap, 100-I01 Rotation calculations, layered model properties, 1 1 1 Rotation pole, secular instability, 115 Rotation pole motion, conventional international origin and, 90, see also Polar motion
3 26
INDEX
Rotation pole wander, Pleistocene glacial cycle and, 115-118 S
Schrodinger equation, 151 nonlinear, 155- 156 Schrodinger operator, Korteweg-de Vries initial value problem related to, 157 Sedimentary basins, formation, 12 Shallow water theory, 147 Simple Burgers body, versus generalized, 26-27, see also Burgers body Simple disk load degiaciation histories, response to, 55-59 Sine-Gordon equation, 155, 157 Single solitary eddy, in weak- and strongwave limits, 175-182, see also Solitary eddies Solar eclipses, ancient, 92 Solitary eddies under dissipation, 213 on mean shear flow, 161 properties, 181 single, 175-182 Solitary Rossby waves experimental evidence, 150- 151 isolation, 150 Solitary vortices coupled Korteweg-de Vries equations and, 181 dynamics, 168 Solitary waves, see also Solitons defined, 147, 155 distinguished from solitons, 155 in large-scale geophysical motions, 147154 nonlinear permanent-form solutions, 148 in one dimension, 154 perturbations in initial conditions, 195200 stability properties, 191-206 Solitons, see also Solitary waves coherent structures and, 206 defined, 155 distinguished from solitary waves, 155 as important geophysical concept, 220 predictability problem and, 149- 150 solitary waves as, 148, 155 Spectral models with isostatic adjustment, 128- 130
reduced form, 130- 133 Spherical harmonic degree, inverse relaxation time as function of, 42 Spheroidal system, secular function for, 42 Spinel + post-spinel phase boundaries, internal density jumps in mantle, 128, 134 squall clusters, trailing anvil, 290-291 Squall h e , model, 280, see also Midlatitude squall line Squall-line cloud clusters, in tropics, 290293, see also Cloud clusters: Tropical cloud clusters Squall-line system, isolated, 282 Stability properties, for solitary waves in geophysical flows, 191-206 Stability studies numerical experiments, 195 perturbations in initial conditions, 195200 Standard linear solid, equation, 16 Steady-state creep, rheological law, 13 Stochasticity border, Chirikov’s criterion, 205 Storms, organization and structure, 226, see also Thunderstorms Storm splitting, in supercell thunderstorms, 261-269 Stratified viscoelastic models deglaciation-induced polar motion, 105 114 mean wander speed, 116-118 Sturm-Liouville problem, 176, 201-202 Sturm-Liouville theory, 194 Supercell thunderstorms, 256-275, see also Thunderstorms bounded weak-echo region, 260 hailfall pattern, 260-261 left- and right-moving, 263-264, 267 low-level flow pattern and, 268 perspective view, 260 radar imaging, 258-260 right-moving, 264-267 satellite picture, 258-260 storm splitting, 261-269 three-dimensional structure, 266 visual appearance, 256-260 visual cloud boundaries, 262 Surface load Love number, 98-101 modified expression, 105
-
327
INDEX
Surface mass loading Green’s function for boundary problem, 53-55 inertial perturbations due to, 95-99
T Tanks one-dimensional, 220 two- or three-dimensional numerical, 220 Thunderstorm Project, 226, 249-251 Thunderstorms, 247-249 ambient wind shear, 254 arc cloud lines and triggering of new storms following, 286-287 cumulus stage, 250 downdrift spreading, 284-287 internal structure, 249-250 multicell, 251 -252 nontornadic severe winds, 285-286 supercell, see Supercell thunderstorms Tidal Love number, elastic, 94, see ulso Love number Tornadoes bow echo, 242 damage, 271-272 Doppler radar data, 269-271 genesis, 269-275 life cycle and structure, 272-275 mesoscale convective complex and, 274278 Tropical cloud clusters hurricanes, 300-303 natural mesoscale precipitation features, 2% Tropical cloud systems, 287-303 nonsquall cloud clusters, 293-296 squall lines, 290-293 Tropical squall lines, 290-293, see also Squall clusters; Squall line mesoscale updrift, 292 Tropics cloud spectrum, 287-289 disturbance level, 290-293 squall-line cloud clusters, 290-293 Turbulence shears, two-dimensional. 160, 254 Turbulent behavior, at higher amplitude range, 198, 219 Turbulent flow, error energy and, 195
Two-dimensional turbulent theories, in atmospheric and oceanic motions, 160, 219
U Uniform viscosity model, Laurentide relative sea level data influence on preference for. 75
V Variable rotation, inertial perturbations due to, 9-95 Varying-wave amplitude, Korteweg-de Vries dynamics and, 151 Venig-Meinesz model, 3-5 Viscoelastic earth models gravitationally self-consistent theory, 8889 of mantle rheology, 12-34 polar wander speed as function of time for, 108-110 realistic layered, 118 relaxation time versus angular degree, 4445 stratified, 105- 114 Viscoelastic impulse response, Green’s functions, 82 Viscoelastic problem, for models with radial heterogeneity, 38-41 Viscoelastic stratification, 105- 114 length-of-day constraints, 114- 115 polar motion and, 114-115 Viscosity, of deep mantle, 9 Viscosity model 1 , Green’s function, 54-55 Viscosity models, relaxation diagrams, 43 Viscous gravitational relaxation of homogeneous, incompressible, and spherical Burgers body, 28-32 normal modes, 41-46 Vorticity equation, barotropic version, 170 V-states, defined, 168
W Warm-frontal rainbands, gravity waves and, 236-237 Warm-sector rainbands, 23 6- 238
328
INDEX
Wave-CISK mechanism, in postfrontal rainbands, 244 Wave field, aspect ratio, 176 Weak-amplitude level. in collision experiments, 183 Wide cold-frontal rainbands, 238-239
Winds, from thunderstorms, 285-286 Wind shear, ambient, 254
2
Zero-order stream function, defined, 170