The Global Circulation of the Atmosphere 9780691236919

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The Global Circulation of the Atmosphere

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THE GLOBAL CIRCULATION OF THE ATMOSPHERE

Edited by

Tapio Schneider and Adam H. Sobel

Foreword by

Edward N. Lorenz PRINCETON UNIVERSITY PRESS | PRINCETON AND OXFORD

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c 2007 by Princeton University Press Copyright
Published by Princeton University Press, 41 William Street, Princeton, New Jersey 08540 In the United Kingdom: Princeton University Press, 3 Market Place, Woodstock, Oxfordshire OX20 1SY All Rights Reserved Library of Congress Cataloging-in-Publication Data The Global circulation of the atmosphere/edited by Tapio Schneider and Adam H. Sobel; foreword by Edward Lorenz. p.cm. Includes bibliographical references and index. ISBN-13: 978-0-691-12181-9 (cloth: alk.paper) 1. Atmospheric circulation. I. Schneider, Tapio, 1972–II. Sobel, Adam H., 1967– QC880.4.A8G572 2007 551.51
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2006049293

British Library Cataloging-in-Publication Data is available This book has been composed in Minion and Univers Printed on acid-free paper. pup.princeton.edu Printed in the United States of America 10

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Contents

Foreword by Edward N. Lorenz Preface

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xi

Chapter 1

Progress and Problems in Large-Scale Atmospheric Dynamics Isaac M. Held

1

Chapter 2

Theories of Baroclinic Adjustment and Eddy Equilibration Pablo Zurita-Gotor and Richard S. Lindzen

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Chapter 3

The Thermal Stratification of the Extratropical Troposphere Tapio Schneider Chapter 4

Storm Track Dynamics 78 Kyle L. Swanson Chapter 5

Eddy-Mediated Interactions Between Low Latitudes and the Extratropics 104 Walter A. Robinson Chapter 6

On the Relative Humidity of the Atmosphere 143 Raymond T. Pierrehumbert, Hélène Brogniez, and Rémy Roca

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vi | Contents Chapter 7

Quasi-Equilibrium Dynamics of the Tropical Atmosphere Kerry Emanuel

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Chapter 8

Simple Models of Ensemble-Averaged Tropical Precipitation and Surface Wind, Given the Sea Surface Temperature 219 Adam H. Sobel Chapter 9

Dynamical Constraints on Monsoon Circulations R. Alan Plumb

252

Chapter 10

Moist Dynamics of Tropical Convection Zones in Monsoons, Teleconnections, and Global Warming 267 J. David Neelin Chapter 11

Challenges in Numerical Modeling of Tropical Circulations Christopher S. Bretherton Chapter 12

Challenges to Our Understanding of the General Circulation: Abrupt Climate Change 331 Richard Seager and David S. Battisti List of Contributors Index

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Foreword

It is a pleasure to be able to contribute to this volume devoted to the global circulation of the atmosphere, even though I was unable to attend the conference that gave rise to it. I look at the conference as the most recent in an extended series. This has not been a planned series; there have been no First Symposium on the General Circulation of the Atmosphere, Second Symposium . . . , etc. Individual meetings have taken place when the various organizers have felt that the occasion has arisen. What gives continuity to the succession of meetings, and what makes it possible to look at them as constituting a series, is the not-surprising fact that to a considerable extent the participants in any one meeting were those in the previous one, and the ideas that they offered were often extensions of those presented before. Of course, there have generally been a few welcome newcomers, while some contributors of longer standing have retired or acquired new primary interests. Sometimes the organizers have invited specialists in specific related fields. I shall not attempt to enumerate the many meetings that have taken place at many institutions in quite a few nations. Instead I shall mention just two; these seem especially relevant because each one gave rise to a volume not unlike the present one.1,2 Also, having attended each of them, I feel a bit more qualified to describe them. The first of these, which was more specialized than most and was particularly concerned with numerical integration, took place in 1955 at the Institute for Advanced Study in Princeton, New Jersey. Here we were honored by the presence of John von Neumann, possibly the world’s greatest then-living mathematician, who had become a champion of the application of computers to mathematical problems—an activity then frowned upon by many prominent mathematicians—and had identified the weather-forecasting problem as especially amenable to this approach. A highlight of the conference was Norman Phillips’s account of his now famous experiment—the first attempt to model the general circulation numerically. His description, shortly

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viii | Edward N. Lorenz

afterward enlarged and published in the standard literature3, earned him the then recently established Napier Shaw Prize of the Royal Meteorological Society. Nevertheless, many of the papers presented were like what might have appeared at any other general-circulation meeting of that day. Indeed, many participants did not have ready access to computers, and had never contemplated performing numerical integrations. The other meeting took place in 1969 in the Rooms of the Royal Society of London. I had the honor, if it is an honor, of being the first speaker, and I presented what was to me an up-to-date account of the workings of the general circulation, noting a few problems that remained to be solved. I was followed by Joseph Smagorinsky, who described in detail a great many problems that needed to be solved in the still-young field of numerical general-circulation modeling, before results from the models could be considered definitive. Some of the other papers considered the roles of restricted portions of the atmosphere—the lower boundary layer, the stratosphere, and the Tropics—that had generally received less attention in earlier studies, at least partly because suitable observations had not been plentiful. By this time access to computers had become more common, but most of the papers presented did not make much use of computers, other than to speed up some data processing. Both conferences were attended by a large number of those whom one would have expected to encounter at a general-circulation meeting, and one might have supposed that the proceedings would in due time become the works that would be most frequently cited. Possibly they enjoyed this status for a short while, but in preparing this foreword I decided to count the number of references in the present volume to the papers in those proceedings. Out of a total of 752 references (not eliminating duplications), the count came to zero. There are a few references to Phillips’s published paper, which had appeared in shorter form in the Princeton proceedings. How are we to account for this absence? Perhaps many of the same ideas were to be found in more-widely disseminated publications such as journals, which were more conveniently quoted; but, more importantly I believe, our ideas as to what constitutes the general circulation, or what are its relevant aspects, are continually changing over the years, and the last forty or fifty years have been no exception. Almost anyone today would agree that the average or typical tropospheric lapse rate of temperature and the average tropospheric relative humidity, for example, are significant features of the global circulation. Fifty years ago almost anyone, if asked, would probably have agreed, yet these features received little attention then among general-circulation theorists. Possibly their magnitudes were taken for granted. In the present volume they receive some of the recognition that they deserve, in the first, third, and sixth chapters. Likewise, in earlier studies we often treated atmospheric water in its various phases by omitting any explicit reference to it, aside from subsequently acknowledging that it

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Foreword | ix

might be a modifying influence. At the London meeting, after noting a reidentification of pressure systems as circulation systems, I concluded my talk by speculating that a future generation might be talking about water systems. While the term “water system" has not invaded the present volume, the presence of water plays an essential role in the arguments presented in at least nine of the twelve chapters. Methods of dealing with the general circulation have also changed. At the earlier meetings there were talks devoted to the new or growing field of numerical simulation, and implicitly hailing it as another approach to the problem. Today numerical modeling appears to have become the approach of choice. Much of what we know or believe that we know about the global circulation as it is, as opposed to knowing why, is actually what we have observed in the output of numerical models. Perhaps the most timely change in attitude, however, is our identification of the global circulation with the climate. This might be just a matter of semantics, except for the fact that our view of the climate itself has changed. Richard Pfeffer, who edited the proceedings of the Princeton meeting, was ahead of his time in titling the volume “Dynamics of Climate"; this was still the age when “climatology" was often irreverently defined as adding up thirty numbers and dividing by thirty. Some standard textbooks, including the one that I best recall from my student days,4 bore no suggestion that the climate had ever deviated from its present arrangement. By the time of the meeting we all recognized that the climate during the recurring ice ages must have differed from the present one, and we generally assumed that some day the ice might come back. Harry Wexler offered a paper on the possible causes of climatic change, but there was little mention of climate in the remaining contributions. Today the study of climate seems to be dominated by the problem of climate change, and we are acutely aware of the possibility that a new climate may well appear within our own lifetimes. It therefore seems quite appropriate that this volume should conclude with a chapter on abrupt climate change. Such a phenomenon was unanticipated forty years ago, and, indeed, the proxy observations that revealed its presence were altogether unavailable then. When the observations did appear some twenty years ago, their interpretation was seriously questioned; slow climate changes were easier to accept. The existence of climates as different as those typifying glacial and interglacial periods, following one another by intervals as short as two decades, is now fairly well accepted. Great advances have been made since several generations ago, when experts were still attempting to show, by nonquantitative reasoning from basic physical principles, that the atmosphere must circulate in the particular manner that was then observed. The present volume leaves little doubt that great advances will continue to be realized. Edward N. Lorenz

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x | Edward N. Lorenz

Notes 1. Pfeffer, R.L. (Ed.), 1960: Dynamics of Climate. New York, Pergamon Press, 137 pp. 2. Corby, G.A. (Ed.), 1969: The Global Circulation of the Atmosphere. London, Royal Meteorological Society, 257 pp. 3. Phillips, N.A., 1956: The general circulation of the atmosphere: A numerical experiment. Q. J. Roy. Meteor. Soc., 82, 123–164. 4. Kendrew, W.G., 1942: The Climates of the Continents. New York, Oxford University Press, 473 pp.

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Preface

This book is an attempt to summarize our current understanding of the mechanisms controlling the global circulation of the atmosphere and to define questions for future research. The book is an outgrowth of a three-day conference held at the California Institute of Technology in Pasadena, California, in November 2004. The centerpiece of the conference was a set of ten invited lectures. Each of the invited lecturers was asked to write a chapter of this book based on the lecture material. We contributed another two lectures and two chapters. The global circulation of the atmosphere is a broad topic, and current research uses a variety of approaches. Observations and the phenomena they describe form the first pillar of our knowledge, and observational work employs a large fraction of our field’s material and intellectual resources. Numerical simulations with the most comprehensive and complex climate models form the second pillar, and their importance continues to grow. The models are continually improving, as are the computers used to run them, and in response to this, as well as to the rapidly increasing prominence of climate change in the public eye, the demands on these simulations have never been greater. Theory forms the third pillar of our knowledge. By theory, we mean that stage of scientific investigation in which mechanisms based on general principles are posited as responsible for observed or simulated phenomena. Theory in our field generally involves the construction of idealized mathematical models that (unlike comprehensive climate models) omit many details, rendering them simple enough that the chain of cause and effect in them can be laid bare, but that still retain sufficient complexity to be relevant for understanding the atmosphere. This book focuses on theory. Key aspects of the observations motivating the theoretical issues at hand are presented in many of the chapters, but a comprehensive treatment of the observations is not contained here. Numerical simulations with comprehensive climate models are discussed in detail only in a small subset of the chapters, particularly chapters 11 and 12.

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xii | Sobel and Schneider

The tropical and extratropical atmospheres have historically been and to some extent still are studied separately by relatively distinct groups of scientists. The global circulation is one area in which the two should be brought together, and we have done that here. The Tropics and extratropics are given approximately equal weight, with some chapters, particularly chapters 5, 6, and 12, addressing tropical-extratropical interactions. This is not a textbook. Some prior knowledge of the field on the part of the reader is assumed. We hope the book captures the state of the field but communicates it in a more condensed form and with more context, breadth, and perspective than one would find in the primary research literature. We hope the book will be useful to graduate students in atmosphere, ocean, and climate science. When we were graduate students, we enjoyed reading books like this, learned a lot from them, and wished there were more. Of course, more advanced researchers in the field should find the book accessible, and those from related fields may be able to use it to gain a foothold in ours. Many people contributed to this book. First and foremost, of course, are the authors, who have our gratitude. In addition to writing the chapters, the authors also served as reviewers, each reviewing a chapter by another author. We also solicited reviews from a few additional reviewers and thank Michela Biasutti and Edmund Chang for their constructive suggestions on the chapters they reviewed. We are very grateful to William and Sonja Davidow, whose financial support made the conference and this book possible. Adam H. Sobel Tapio Schneider

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The Global Circulation of the Atmosphere

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Chapter 1

Progress and Problems in Large-Scale Atmospheric Dynamics Isaac M. Held

1.1. Introduction A theory for the general circulation of the atmosphere has at its core a theory for the quasi-horizontal eddy fluxes of energy, angular momentum, and water vapor by the macro-turbulence of the troposphere, as well as a theory for the much smaller-scale convective motions that transport heat and water vertically, especially in the Tropics. A few of the many issues related to convective vertical fluxes are discussed in chapters 7, 8, 10, and 11 in this volume. The focus in this chapter, and of chapters 2–6, 9, and 12, is primarily on the large-scale quasi-horizontal component of the problem. In the Tropics, fluxes by large- and small-scale eddies are so tightly coupled that one cannot easily discuss one without simultaneously discussing the other. But outside of the Tropics, one can hope that a focus on large-scale dynamics in isolation is a meaningful starting point, and it is on the extratropical circulation that I concentrate here. All of us would love to find a simple variational principle or “fundamental theorem of climate” that solves this problem in a single stroke, but I suspect that most of us are skeptical that such a principle exists. We assume, instead, that the best way of developing theories for a system of this complexity is to construct a hierarchy of models, of varying levels of comprehensiveness, chosen so as to capture the essential sources of complexity with minimal extraneous detail. When confronted with a theory claiming great generality, we expect to see a demonstration that it explains the behavior seen on a number of different levels of our model hierarchy. An analogy with the use of “model organisms” in biology is informative. Nature has provided us with just the kind of hierarchy, from bacteria to fruit fly to mouse, needed to build up an understanding of our own complex biology. We have no such ready-made hierarchy in climate research, and must instead design and build our own. See Held (2005) for an extended discussion of this analogy and the consequences of the fact that our climate hierarchy is a theoretical construct while the biological hierarchy is provided by nature.

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2 | Isaac M. Held

What hierarchy of models should we study so as to best understand how global climate is controlled by external parameters and boundary conditions? The choice of models is centrally important. Only if, as a community, we have selected appropriate models to study collectively will our understanding accumulate efficiently. I personally do not feel that appropriate models can be selected in a systematic way; our physical intuition must guide us towards the most informative models. In this chapter I will refer to the classic two-layer quasigeostrophic (QG) model, moist QG models, and particular idealized dry and moist primitive-equation models on the sphere. The discussion revolves around the related problems of the poleward eddy heat flux, the effect of latent heat release on midlatitude eddies, and distinctions between the dynamics of the upper and lower troposphere. Considerable space is devoted to the simplest of these models, the two-layer QG model, in an especially simple horizontally homogeneous configuration. I find this model of homogeneous QG turbulence useful from several perspectives, but there is no claim that the theory for the eddy fluxes in this model is of direct quantitative relevance to the atmosphere. When we talk about the need for a model hierarchy, we are implicitly assuming that the more idealized members of this hierarchy are missing some important ingredients, but that, in spite of these limitations, an understanding of these simpler models is a useful stepping stone to an understanding of their more complex relatives.

1.2. The Two-Layer QG Model The two-layer QG system provides us with what may be our simplest turbulent “climate” model. The state of this model is determined by the streamfunctions for the nondivergent component of the horizontal flow in two layers of fluid, meant to represent the flow in the upper (ψ1) and lower (ψ2) troposphere, the (eastward, northward) components of the velocity being (u, v) = (−∂ψ/∂ y, ∂ψ/∂ x). In the meteorological context we can think of two isentropic layers of ideal gas with different entropies, or potential temperatures θ , with θ1 > θ2 so as to represent a gravitationally stable system. Hydrostatic and geostrophic balance combine to create Margules’ relation between the perturbations to the height of the interface between the two layers, η, and the difference between the two streamfunctions. In a Boussinesq fluid (with all potential temperatures assumed to be small perturbations away from a constant θ0) this relation is f (ψ1 − ψ2) = −g ∗η, where g ∗ ≡ g (θ1 − θ2)/θ0 is the reduced gravity and f the Coriolis parameter. The dynamics reduces to the advection by these non-divergent flows of a scalar, the QG potential vorticity q k , within each layer, where q k = ∇ 2ψk + (−1)kλ−2(ψ1 − ψ2) + βy; k = 1, 2

[1.1]

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Large-Scale Atmospheric Dynamics | 3

and λ is the radius of deformation, defined by λ2 = g ∗H/ f 2, with H the resting depth of the two layers (assumed to be equal here). The final term in (1.1), with β a constant, is an approximation to the all-important vorticity gradient due to the increase in the radial component of the vorticity of solid body rotation with increasing latitude y. When relating this two-layer picture to a continuously stratified atmosphere, we think of (g ∗H)1/2 → N H, with N 2 = (g /θ )∂θ/∂z and −η as proportional to the vertically averaged potential temperature. A simple way of creating a statistically steady state is to force the system with mass exchange between the two layers, this model’s version of radiative heating, arranged so as to relax the interface to a “radiative equilibrium” shape with a zonally symmetric meridional slope. This mass exchange can be expressed in terms of potential vorticity sources in the two layers. One also invariably includes two types of dissipation: small-scale diffusion is needed to mop up the vorticity variance that cascades to smallscales; and surface friction, damping the low-level vorticity, is needed to remove energy in a non-scale selective manner. Energy does not cascade to small scales in this model and cannot be removed realistically with horizontal diffusion. Radiative equilibrium is a solution of these equations, with no flow in the lower layer and zonal flow in the upper layer, with the Coriolis force acting on the vertical shear U = u1 − u2 between the two layers balancing the pressure gradients created by the radiative equilibrium interface slope. This flow is unstable in the absence of the dissipative terms, when the isentropic slope is large enough to overcome β and reverse the sign of the north-south potential vorticity gradient in one of the layers. In flows with temperature decreasing (interface slope rising) with increasing y, this reversal occurs in the lower layer. If the relative vorticity gradient of the zonal flow is negligible as compared to β, the criterion is the classic one discussed by Phillips more than half a century ago: ξ ≡ U/(βλ2) > 1. (The supercriticality ξ is the two-layer counterpart to the parameter Sc used in chapter 3.) The existence of this critical slope presents us with a problem, since analogous models of inviscid baroclinic instability in continuously stratified atmospheres are unstable for any nonzero vertical shear (or isentropic slope). (In multilayer models, the critical interfacial slope is simply proportional to the depth of the lowest layer.) We will need to return to this point. Phillips (1956) constructed the first “general circulation model,” or “climate model,” based on two-layer QG dynamics. Nowadays we might instead refer to this work as modeling the statistically steady state of a baroclinically unstable jet on a β-plane. Whatever we call it, this model still captures an impressive subset of the dynamics of the midlatitude storm tracks. Phenomena have been discovered in the solutions of these equations that have then been searched for and found in the atmosphere. The coherent baroclinic wave packets described in Lee and Held (1993) are an example from my own research. (Unfortunately, it is not obvious in reading that paper that we first encountered these wave packets while experimenting with the two-layer QG system.)

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As long as the dissipative terms are linear, a theory for the time-mean geostrophic flow in this model reduces to a theory for the poleward eddy potential vorticity fluxes in the two layers, Pk ≡ vk q k , where an overbar refers to the zonal mean and a prime to deviations from this mean. We can relate these fluxes to the eddy momentum fluxes Mk ≡ vk uk and the thickness (heat) fluxes Tk ≡ vk η (where T1 = T2 ≡ T from Margules’ relation), P1 = −

∂M1 + f T /H; ∂y

P2 = −

∂M2 − f T /H. ∂y

[1.2]

The two potential vorticity fluxes cannot fully determine the three fluxes (M1, M2, T ); therefore, the eddy thickness and momentum fluxes are more than we need to know if we are only interested in the mean zonal flow and the interface displacement (temperature). The most fundamental limitation of QG dynamics is that it assumes a reference static stability; in this two-layer model the potential temperature difference between the two layers is fixed. One is perilously close to throwing the baby out with the bath water in such a theory. What could be more fundamental to a theory of climate than an understanding of the mean stratification of the atmosphere? But perhaps we can develop theories for the QG fluxes, and then use these outside of the QG framework to help as needed in determining the static stability. We illustrate this kind of argument below.

1.3. Eddy Closure in the Two-Layer Model What is the scale of the typical energy-containing eddy in this two-layer QG model? Linear theory points to the radius of deformation, as it is the zonal scale of the most rapidly growing linear waves. A classic assumption (Stone 1972) is that the nonlinearity of the flow isotropizes the eddies in the horizontal and imprints this scale on the meridional as well as zonal eddy structure, and on eddy mixing lengths as well. An interesting implication is that there seems to be potential for scale separation in the horizontal, since this scale would then be independent of the mean flow inhomogeneity in the direction of the flux, in contrast to the situation in most laboratory turbulent flows. If there is scale separation, one is justified in thinking in terms of local rather than global theories for the eddy fluxes. An example of a global theory is an approach referred to as baroclinic adjustment, in analogy with convective adjustment for gravitational instability (e.g., Stone 1978a). Since the instability of the flow can be thought of as due to the reversal in sign of the lower-layer potential vorticity gradient, suppose that the eddy fluxes are just sufficient to bring this gradient back to zero. Given a value of the radiative equilibrium shear and the width of the unstable region LQ , the magnitude of the eddy potential vorticity flux required to destroy the gradient is proportional to L2Q (since the rate of change of the mean gradient is proportional to the second derivative of the eddy flux). LQ is a global piece of information. However, as LQ is increased in numerical

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Large-Scale Atmospheric Dynamics | 5

simulations, the eddy potential vorticity fluxes are found to grow more slowly than L2Q and eventually to asymptote to values independent of LQ (Pavan and Held 1996). While baroclinic adjustment does not work in two-layer QG flows with large LQ , it may very well be an adequate, indeed a very useful, approximation for the case of narrow regions of instability. An example of a local theory is simple diffusion of potential vorticity, with a diffusivity determined by aspects of the local environment. We cannot expect a truly local diffusive theory to be exact. The relationship between eddy flux and environment must be nonlocal over the scale of the eddies at least. Additional nonlocality is introduced if the production and dissipation of the eddies are not colocated. For example, the simplest diffusive picture does not work when applied locally in longitude in the zonally asymmetric midlatitude storm tracks (Marshall and Shutts 1981; Illari and Marshall 1983). Eddies are preferentially generated in the strongly baroclinic zones at the jet entrance regions and decay downstream in the jet exit regions. It is only when one averages zonally over these regions of predominant eddy growth and eddy decay that one has a reason to expect a local, diffusive picture to hold in some approximate sense. Given a diffusivity D and radiative relaxation time τ , we should not expect to reach the LQ-independent asymptotic regime until LQ2 > Dτ , or (L Q/L )2 > (τ/T ), where L and T are eddy length and time scales. The resulting scales are large compared to the radius of the Earth. But we do not need to be in this asymptotic regime to apply a diffusive theory; all that is required is scale separation LQ > L . As in many applications of WKB-like theories, one can even hope that the local theory is adequate when LQ ≈ L . The simplest scaling for the diffusivity is that suggested by Stone (1972): D ∼ V L ∼ U λ, where the eddy velocity scale V has been chosen proportional to the mean vertical shear U over the depth of the atmosphere. The assumption V ∼ U is equivalent to assuming that the eddy kinetic energy is proportional to the mean available potential energy (the increase in potential energy due to the interface slope) within a region of width λ. This diffusivity is itself proportional to the interface slope, or horizontal temperature gradient. If we can use this diffusivity for the sensible heat flux, following Stone, we obtain a heat flux proportional to the square of this gradient. Numerical experiments in the homogeneous limit described below clearly indicate that the eddy fluxes in this two-layer QG model are even more sensitive to the horizontal gradient; they also give us some guidance on how to incorporate β into the theory.

1.4. The Homogeneous Limit Given the potential for a local theory, one is led to artificially create a truly homogeneous environment in which to study eddy fluxes in the simplest possible context. QG theory allows one to do this in an elegant way by assuming that there is a uniform zonal flow

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in both layers, and, therefore, a uniform vertical shear and uniform potential vorticity gradients. One then assumes that the total flow consists of this environment, plus eddies constrained to be doubly periodic. One can think of this geometry as a generalization of the familiar QG β-plane to the case with potential vorticity gradients of opposite sign in the two layers. In this geometry, the eddy fluxes are horizontally homogeneous. Therefore, according to (1.2), the potential vorticity fluxes reduce to the eddy thickness (heat) flux and are equal and opposite in the two layers. The momentum fluxes must also vanish if the climate is unique, since the equations are symmetric with respect to reflection in y, and the momentum fluxes change sign upon reflection. The central simplification is that one can study how the eddy fluxes are controlled by environmental parameters without simultaneously being concerned with the effect of these eddy fluxes on their environment. A problem immediately arises from the inverse energy cascade, a cascade to larger rather than smaller scales. Calculations show unambiguously that the dominant eddy scale in the fully turbulent statistically steady state is generally larger than the radius of deformation due to this inverse cascade. It is useful to rearrange the two vertical degrees of freedom of this model into the barotropic (ψ1 + ψ2) and the baroclinic (ψ1 − ψ2) modes. The picture of the energy flows as a function of wavenumber in this modal basis has been described by Rhines (1977), Salmon (1978, 1980), and Larichev and Held (1995). The barotropic mode is energized by transfer from the baroclinic mode near the radius of deformation. The inverse cascade takes place in the barotropic mode, and energy is dissipated by surface friction on the scales to which this cascade carries the energy. If the cascade is extensive, the barotropic mode dominates the kinetic energy, so that the baroclinic potential vorticity (dominated on large scales by the thickness variations) can then be thought of as advected passively by the barotropic mode (since it does not induce the flow by which it is advected). The available potential energy, or thickness variance, is generated on these large energy-containing scales by extraction of energy from the environmental potential energy through downgradient thickness (heat) fluxes, just as in two-dimensional downgradient turbulent diffusion of a passive scalar, and cascades to smaller scales back towards the radius of deformation, completing the cycle. Albeit directly applicable only for a rather special situation, it is striking how little this homogeneous turbulence picture has left in it that bears any resemblance to the scales and concepts familiar from linear theory. As in Kolmogorov’s classic work on the direct cascade of energy in threedimentional turbulence, the key element of the two-dimensional inverse cascade, as described by Kraichnan (1971), is the rate of transfer of energy through the spectrum, . Together with the wavenumber k, determines the energy level of the flow and the characteristic time scale of the eddies. The key question is the scale at which the inverse energy cascade is halted.

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Large-Scale Atmospheric Dynamics | 7

At this point we take advantage of the insight of Rhines (1975) that the presence of an environmental barotropic vorticity gradient β can effectively stop the cascade, given the property of the Rossby wave dispersion relation that larger waves have larger intrinsic phase speeds (β/k 2). When these phase speeds become comparable to the characteristic velocity of the flow, the eddies morph into linear waves. Stopping the cascade in this way produces a flow that is simultaneously marginally turbulent and marginally wavelike, an elegant qualitative description of midlatitude eddies. From β and one forms length and time, or velocity, scales, or one can proceed directly to a diffusivity with units of length2/time, D ∼ 3/5β −4/5.

[1.3]

Surface friction must eventually remove energy from the model. In the presence of β, the flow forms zonal jets that store the energy until it is dissipated. Sensitivity of the diffusivity to the strength of surface friction might then modify this scaling to the extent that the structure of this jet reservoir feeds back on the eddy statistics. In the absence of β, the strength of surface friction must play a direct role in the scaling (Thompson and Young 2006), since it is then the only process that can stop the inverse cascade. The potential energy extracted from the environment can be written in terms of the eddy potential vorticity flux in either layer,
= UiPi = U P1 = −U P2 = Uβ D1(1 + ξ ) = Uβ D2(ξ − 1), [1.4] i

where U = U1 − U2. We equate this production to the rate of energy transfer through the inverse energy cascade. We define a diffusivity in each layer as the eddy potential vorticity flux divided by the mean potential vorticity gradient. As β → 0, ξ → ∞, and D1 → D2 (see Vallis 1988). Equating D in (1.3) with either D1 or D2, in this limit we have =

D , T2

T≡

NH , fU

[1.5]

where T −1 is often referred to as the Eady growth rate, though it does not enter here through any connection to linear theory. Combining with (1.3) one arrives at 1 β 2T 3

[1.6]

D ∼ ξ 3. βλ3

[1.7]

D∼ or

This is the scaling presented by Held and Larichev (1996). A more accurate fit to numerical experiments is provided by the modified formulation in Lapeyre and Held (2003), for which a satisfactory justification has yet to be provided. The proposal is simply to equate D with the lower-layer diffusivity D2, irrespective of the value of β.

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8 | Isaac M. Held 102

Computed diffusivity

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101

100

101 Predicted diffusivity

102

ξ2 (1 - 1/ ξ )3/2 Figure 1.1. Comparing a theory for eddy heat fluxes in a homogeneous two-layer model with numerical simulations. Dots are the diffusivity, here non-dimensionalized by U λ, computed numerically in a 1024 × 1024 spectral simulation after Lapeyre and Held (2003), compared with the theoretical scaling provided by equation (1.8). The departures at large supercriticality are probably related to the finite size of the domain.

(I return to the motivation for this assumption in section 1.9.) The result is D ∼ ξ 3(1 − 1/ξ )3/2 or βλ3

D ∼ ξ 2(1 − 1/ξ )3/2 Uλ

[1.8]

This has the same ξ → ∞ limit as (1.7). The fit to the numerical results, for fixed strength of surface friction, is shown in Fig. 1.1. This form also has the advantage that the diffusivity vanishes as ξ → 1, consistent with the criterion for instability. This is my best shot at present for a qualitative explanation of the baroclinic eddy fluxes in this idealized homogeneous environment. As β → 0 and ξ → ∞, the eddy length scale increases without bound in this theory, implying that some other scales, determined by the surface friction or the domain geometry, must come into play. Whether or not the details are right, this line of argument points to a flux that is very sensitive to environmental gradients: equation (1.7) yields a diffusivity proportional to the third power, a flux proportional to the fourth power, and an energy cycle proportional to the fifth power of the horizontal temperature gradient. Equation (1.8) only increases this sensitivity to ξ . In practice, this means that it is very hard, in this two-layer model, to change the gradient, to the extent that the system has difficulty supplying energy at the rate required. As the width of the unstable region increases, the flow typically makes a transition from one to two and then to multiple jets, with a storm track associated with each jet. One is tempted to assume that the homogeneous limit cannot be relevant to the one-jet case, but only begins to become appropriate for the case of two or more jets, the former

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being too inhomogeneous. The implication would be that the theory is irrelevant for the Earth, which has only one eddy-driven jet per hemisphere. An argument along similar lines starts with the observation that momentum fluxes vanish in the homogeneous system, averaged over space or, presumably, averaged over time at each point in space. But momentum fluxes appear to play a significant role in the stabilization of flows in the troposphere, as encapsulated in the barotropic governor mechanism of James (1987). The essence of this mechanism is that as barotropic shears are generated by the momentum fluxes, they progressively interfere with the baroclinic production mechanism and thereby limit growth. My impression, in contrast, is that the equilibration mechanisms in one-jet and multiple-jet flows, and in this homogeneous model, are essentially the same, the dominant process being a generalized version of the barotropic governor, in which it is not only zonally averaged barotropic shears but the energy-containing barotropic mode, whether jet-like or not, that interferes with baroclinic production (see Salmon 1980). One does not need time-averaged momentum fluxes to create a barotropic governor; instantaneous shears are adequate. On the other hand, the approach to the homogeneous limit is not likely to be simple. For example, Lee (1997) has shown that eddy statistics undergo non-monotonic evolution as one increases the width of the unstable region so as to make the transition from one jet to two. Relatively little has been achieved with regard to how one might use the homogeneous limit as a starting point for inhomogeneous theory. See in this regard Pavan and Held (1997).

1.5. Static Stability Maintenance A key question in general circulation theory is whether or not the slope of the mean isentropes in the troposphere is strongly constrained. The observed slope is close to the aspect ratio of the troposphere: an isentropic surface that is near the ground in the Tropics rises to the tropopause in polar latitudes. Is this a coincidence, or is this particular slope favored? Using the scaling from the previous section for the diffusivity due to baroclinic eddies, one can, in the spirit of Stone (1972), try to develop a theory for the static stability. In a stratified atmosphere, the expression (1.7) for the diffusivity, for example, −3/2 implies that D ∼ 3H V , where H and V are the horizontal and vertical potential temperature gradients, respectively. To obtain the horizontal eddy heat flux H, one multiplies by another factor of H . To estimate the vertical eddy heat flux V (ignored in QG theory) one can assume that the total flux is aligned along isentropic surfaces, −5/2 averaged over the troposphere, so that V V ∼ H H , or V ∼ 5H V . We next need to assume that the static stability is maintained by a balance between this eddy vertical heat flux and the destabilization by radiation. If we just assume that radiation relaxes V

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to zero on some specified time scale, and that the vertical scale of the eddies is fixed, 10/7 6/7 then V ∼ V, resulting in the estimate V ∼ H and D ∼ H . The point of this manipulation is not to make a case for this specific result, but to illustrate how allowing the stability to adjust to changing eddy fluxes can potentially alter the sensitivity of the fluxes to the horizontal temperature gradient. This scaling suggests that the isentropic slope can be altered by modifying the −3/7 horizontal temperature gradient, albeit with some difficulty: H/ V ∼ H . If we use (1.8) instead of (1.7) the result is a much stronger constraint on the isentropic slope, when the system is in the proximity of the critical slope. But is this legitimate, given the seemingly artificial character of the two-layer model’s critical slope? As an alternative to thinking in two-layer terms, it has been suggested that one needs to couple the prediction of the static stability with a prediction of the tropopause height, and that by doing so one introduces a stronger constraint on the isentropic slope in the continuously stratified case (Held 1982). The essence of this argument can be understood by thinking of a continuously stratified QG model with fixed static stability and vertical shear; in this system the claim is that the distance that the eddy fluxes extend above the surface scales as h ∼ f 2∂U/∂z/(β N 2), which is equivalent to ξ ∼ 1. See Thuburn and Craig (1997) for a critique of this claim, and Schneider (2004) and Schneider and Walker (2006), who provide strong support for a refined version of this argument (while simultaneously calling into question the relevance of continuously stratified QG theory).

1.6. The Entropy Budget In a comparison of theories for the poleward heat flux with various scaling arguments, Barry et al. (2002) combine the Rhines scale-inverse energy cascade relation (1.3) with an estimate of from a global entropy budget, rather than an energy budget. It is useful to understand how these approaches are related, as the entropy perspective may be especially useful in the presence of latent heat release. Consider a dry atmosphere forced by the time-mean heating/cooling Q. The forcing decreases the entropy at a rate determined by averaging Q/T over the atmosphere. (From this point on, the symbol T refers to temperature, not to an eddy time scale.) This is a decrease in entropy because Q creates temperature gradients by warming (cooling) regions that are already relatively warm (cool). In a steady state this entropy destruction is balanced by production due to irreversible processes, the dominant one in a dry atmosphere being the dissipation of kinetic energy (that is, the diffusion of momentum), the rate of kinetic energy dissipation being once again. We ignore radiative damping of transients due to the correlation in time between Q and T , which will create entropy, and we also ignore diffusion of temperature. The latter tends to be small because temperature, in balanced flows, cascades to small scales only at the surface

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and not in the interior of the troposphere. Therefore,  −

Q ≈ , T T

[1.9]

where T is the average temperature at which the energy dissipation occurs and all integrations are over the mass of the atmosphere. We assume small departures of T and T from a reference temperature T0 as needed. Barry et al. (2002) estimate in their model by taking the distribution of Q as given. This may seem like one is giving oneself too much information, in that Q is dominated by the divergence of the eddy heat flux for which one is trying to develop a theory. But suppose one has a theory for the eddy diffusivity, and eddy heat or potential vorticity fluxes, that depends on . Given , one determines the fluxes and temperatures, and therefore Q; one can then iterate to obtain self-consistency. A difficulty with this approach is that one loses the sense of a local theory, being determined by a global integral. But one can regain a local perspective by setting Q = ∇ · F , where F is the flux of dry static energy, and then integrating by parts: −

≈ T



1 F · ∇T = T2



1 F · ∇ ln T = T



 ∂ ln T  1 . FH T ∂ y M

[1.10]

In the final expression, we have assumed that the climate is zonally symmetric, so that F is a vector in the y − z (or y − p) plane with horizontal component F H , and have let M denote a coordinate that is constant on the surface along which F is aligned. One can now apply this locally, setting the local -density equal to the integrand. To see the connection with the QG arguments above, one needs to assume that M ≈ θ . Letting S be the isentropic slope, FH

  ∂ ln T  R ∂ ln p  R ∂ ln p g = FH S. = F = FH S H ∂y θ cp ∂y θ cp ∂z cpT

[1.11]

Substituting for F H ≈ cpv T  and setting S =−

∂y θ ∂z θ

[1.12]

and (g /T )v T  = D f ∂zU , we regain equation (1.5). For later reference, notice that the static stability makes its only appearance in this argument at the point when the mixing slope is set equal to the isentropic slope. Entropy and available potential energy budgets are not equivalent in general, but they are closely enough related that they lead to essentially the same scaling approximations.

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1.7. Moist Eddies We would like our theories for midlatitude eddy fluxes to help us understand the implications of the increase in moisture content in the atmosphere that will accompany global warming. We would also like to make use of the seasonal cycle to test our theories for these eddy fluxes (e.g., Stone and Miller 1980), but these tests are not very convincing as long as one is ignoring the effects of latent heat release, which vary seasonally in tandem with the variations in the large-scale temperature gradients. The length scale of midlatitude eddies is observed to be larger in northern winter than in summer. Is this due to the larger eddy energies in winter, which result in a larger Rhines scale, or is it that eddies are smaller in summer because of a reduction in an effective static stability due to latent heat release? A central theoretical issue is whether there are ways of using concepts like moist entropy (Emanuel and Bister 1996) or moist available potential energy (Lorenz 1978) so as to carry some of the lines of argument developed for dry eddies over to the moist case. Lapeyre and Held (2004) construct a relatively simple moist model by adding a water vapor variable to the two-layer QG model. To obtain consistent energetics, they treat moisture in an analogous way to temperature (or thickness) by requiring the moisture field to be a small perturbation away from a prescribed mean value that is uniform within each layer. Despite this limitation, the form of this model’s energetics is of interest. Here I provide a brief sketch of QG moist energetics more generally, because it has a feature that is counterintuitive (for me) and may have interesting implications for how we think about moist eddies. In a dry QG model the available potential energy (APE) is proportional to the variance of the interface displacement. This form follows from the QG thermodynamic equation of the form ∂b = −N 2w − J (ψ, b). ∂t

[1.13]

We use the Boussinesq approximation for simplicity, with b the buoyancy; the final term represents horizontal advection by the geostrophic flow. The conversion of potential to kinetic energy is [bw], where brackets denote a global mean. One manipulates the buoyancy equation to have the same expression on the right-hand side by multiplying by b/N 2 and averaging: ∂APE = −[wb]; ∂t

 APE ≡

 b2 . 2N 2

[1.14]

In a moist QG model, one has instead, schematically, ∂b = −N 2w − J (ψ, b) + L P , ∂t

[1.15]

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where P is the condensation rate and L the latent heat, and ∂q = −(∂z Q)w − J (ψ, q ) − P , ∂t

[1.16]

where q is now the moisture perturbation (not potential vorticity) and Q = Q(z) the reference moisture. Forming an equation for the buoyancy variance results in the term [P b] on the right-hand side, which we wish to avoid. One can eliminate P by forming a moist enthalpy equation for h ≡ b + L q , but forming an equation for the variance of h generates a term proportional to [hw] rather than [bw]. One can try to remedy this problem by forming an equation for the variance of the moisture, but this reintroduces the precipitation on the right-hand side through [q P ]. The successful manipulation uses the variance of the saturation deficit, d = q s − q , where one assumes that precipitation occurs when saturation occurs, so that [d P ] = 0. Here I describe the simplest case, in which q s is a constant, independent of temperature (the case in which the saturation vapor pressure is a function of temperature or bouyancy (b) is a bit more involved). We can then set this q s = 0 (recall that q is here the departure from the reference Q(z)). We finally obtain an equation of the form ∂QAPE = −[wb]; ∂t

  1 h2 d2 QAPE ≡ +L . 2 (N 2 − L |∂z Q|) |∂z Q|

[1.17]

Thus, our moist available potential energy (QAPE) has one term proportional to the variance of the moist enthalpy, divided by a moist stability, plus an additional term proportional to the variance of the saturation deficit, or dew point depression. This form is presumably related to Lorenz’s general form for moist APE, specialized to the case of small interface displacements and small moisture deficits. The implications for moist energetics of the presence of the term proportional to the saturation deficit variance are obscure but intriguing. There is an energetic cost to an increase in undersaturation. I find it difficult to understand this statement intuitively. See Frierson et al. (2004) for an application of an analogous expression to a shallow-water model. The sources/sinks of QAPE also have additional terms not present in the dry case. Evaporation into unsaturated air and diffusion of water are both important sinks of QAPE and have no direct counterparts in the dry case. Unlike temperature, the water mixing ratio does cascade to small scales in this QG flow, so the diffusive loss of mixing-ratio variance is both significant and energetically important from the perspective of QAPE. There is an intriguing resemblance between these sinks of QAPE and the sources of irreversibility in a moist entropy budget. As discussed by Emanuel and Bister (1996) and Pauluis and Held (2002) for tropical convection, the efficiency of the kinetic energy cycle is reduced by diffusion of moisture, either due to a cascade of variance to small scales, or to evaporation into unsaturated air (microscopically, the latter is simply diffusion down the gradient between the saturated air in contact with

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the liquid and the air a bit further removed). Equation (1.9) is replaced by  Q − R, [1.18] ≈− T T where R is the positive definite generation of entropy due to diffusion of vapor, and Q now includes radiative cooling and surface evaporation plus surface sensible heating (but not latent heat release!). It is likely that the term R reduces the efficiency of midlatitude eddy dynamics substantially (especially in summer) just as it does the efficiency of tropical convection. How would latent heat release modify the kinds of scaling arguments described earlier? We can try to work from either a moist available potential energy or a moist entropy perspective, but the latter may be simpler, especially since the QG version of QAPE is undoubtedly too restrictive. Setting Q equal to the divergence of the eddy moist static energy flux, F , with horizontal component F H equal to the flux of moist enthalpy v h  = cp v T  + L v q , we can write   1 ∂ ln(T )    = vh − R, [1.19] T T ∂ y M where the derivative is taken along the mixing surface, defined by the direction of the eddy moist static energy flux. One can then diffuse moist enthalpy down the mean moist enthalpy gradient, and combine this expression with (1.3) or its equivalent. Thus, moisture and latent heat release affect the theory of Held and Larichev (1996) or Barry et al. (2002) in three ways: by reducing efficiency through the term R, by increasing the mixing slope (i.e., reducing the effective static stability), and by replacing the dry enthalpy by the moist enthalpy as the quantity being diffused. Without expressions for the mixing slope and the efficiency reduction, this is not a closed theory, but it gives us some feeling for what such a theory might look like.

1.8. An Idealized Moist Model on the Sphere Does latent heat release reduce the mean length scale of the energy-containing eddies? The Eady model linear theory of Emanuel et al. (1987) indicates potential for a reduction by a factor of 2 or so. But if one thinks in terms of the Rhines scale, one might guess that a reduction in effective static stability likely increases the scale by increasing the eddy kinetic energy. Frierson et al. (2006, hereafter FHZ) have constructed an idealized moist general circulation model (GCM) on the sphere in part to address questions of this kind. The moist general circulation is generally addressed with comprehensive atmospheric climate models in which clouds, convection, and radiative transfer interact in a host of subtle ways that are only dimly appreciated, and in which there are sensitivities to resolution, time-stepping, and (often undocumented) details in the closure schemes

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4 Flux (PW)

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0

−4

−8

−80°

−40°

0° Latitude

40°

80°

Figure 1.2. Poleward energy fluxes in moist and dry idealized models. Solid: total energy flux in moist model; dashed: dry static energy flux (and, therefore, total flux) in dry model; dash-dot: latent heat flux in moist model. (Provided by D. Frierson.)

that makes it difficult to reproduce model results. In FHZ, the radiation is a function of temperature only, there is no condensate, and the boundary layer and convective closures are kept simple enough to encourage tests of reproducibility and sensitivity to resolution. In the simplest case, the model is run with large-scale condensation only, with no convective closure scheme. The initial results with the FHZ model show surprising insensitivity of the eddy scale to the amount of moisture in the atmosphere, and, therefore, to the amount of latent heat release. There is essentially no difference in the midlatitude eddy spectrum between the dry limit of this model and a control run with realistic moisture content. The dry static stability increases with increasing moisture to prevent large changes in moist stability, so the constancy of the eddy length scale is in disagreement with any theory based on an effective stability that scales with the dry stability. The Rhines scale, on the other hand, does predict the constancy of this scale as the moisture increases if, it turns out, one allows oneself to compute it at the position of the maximum eddy kinetic energy (FHZ). This latitude moves polewards as moisture increases, and the Rhines scale remains unchanged only because of the canceling effects of a reduction in eddy energy and a reduction in β. But this does not explain whether the cancellation is a coincidence or a result of some dynamical constraint. The results in FHZ are also intriguing with respect to the question of the partitioning of the poleward heat flux between latent and sensible parts. As shown in Fig. 1.2, the total poleward flux in this model stays remarkably constant (to within 1%) as the amount of water vapor, and the poleward flux of latent energy, increases from the dry limit to a realistic value. A reduction in sensible flux cancels the increase in latent flux. It is sometimes argued (following Stone 1978b) that the total atmospheric

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flux is more or less as large as it can possibly be, since the profile of outgoing infrared flux is much flatter than that of the absorbed solar flux. But analysis shows that there is substantial room for an increase in the FHZ model. In any case, why should the atmosphere be incapable of reversing the sign of the outgoing longwave gradient? It is not difficult to construct a model that does precisely this (D. Frierson, personal communication). I have recently examined this compensation in the comprehensive climate model when run in aqua-planet mode (over a uniform boundary condition of slab ocean with fixed heat capacity) of the Geophysical Fluid Dynamics Laboratory (GFDL) and find about 80%, rather than near perfect, compensation when the atmospheric CO2 is doubled. My impression is that this level of compensation is typical in comprehensive climate models (e.g., Manabe and Bryan 1985). We suspect that the key to the near-perfect cancellation in FHZ is the fact that the radiation is a function of temperature only. A final question that is addressed by the FHZ model is that of the role of latent heating, and moist convection more specifically, in maintaining the static stability in midlatitudes. The claim is that this idealized model supports the picture of Juckes (2000), who argues that the large-scale eddy fluxes are not capable of stabilizing the atmosphere to the point of preventing moist convection in the warm sectors of extratropical cyclones. A possible implication is that the mean static stability of the extratropical troposphere is maintained by this moist convection so that the favorable sectors of extratropical cyclones are moist neutral. The average moist stability of the atmosphere is then determined by the difference between the average boundary-layer moist enthalpy and the maximum value of this boundary-layer moist enthalpy within the eddies, or equivalently by the rms moist enthalpy in the boundary layer. The latter, in turn, is presumably determined by the large-scale eddy mixing length and the horizontal mean moist enthalpy gradient. Clearly, we have just scratched the surface of many central climatic questions involving the effects of moisture on the large-scale circulation, many of which are important in understanding global warming simulations. Idealized models of the moist general circulation are sorely needed to make contact with our even-more idealized dry models and with the high-end comprehensive models that play the predominant role when we apply climate models to real-world problems.

1.9. Upper vs. Lower Tropospheric Dynamics There is an important qualitative distinction between the upper and lower troposphere that impacts the general circulation in numerous ways: the upper troposphere is more wave-like than the lower troposphere. This distinction must fundamentally be due to β, the environmental vorticity gradient. In the two-layer model, for example, β adds to

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Large-Scale Atmospheric Dynamics | 17

the contribution of vertical shear to the upper-layer potential vorticity gradient, while it tends to cancel this contribution in the lower layer. The result is that potential vorticity gradients are larger in magnitude in the upper than in the lower layer. These gradients create the restoring forces for Rossby waves. A disturbance of a given scale will propagate westward with respect to the upper-level flow more strongly than it will propagate eastward with respect to the lower-level flow (in the lower layer the potential vorticity gradient is negative, causing Rossby waves to propagate eastward rather than westward.) As one consequence, β pushes the steering level for baroclinic instabilities, where the phase speed matches the environmental flow, into the lower troposphere. When waves on shear flows grow, they typically break when the flow perturbations u become comparable to u − c , the phase speed of the wave with respect to the environment. So eddies of the same amplitude will break first in the lower layer, and the upper layer will remain more linear. To the extent that they are determined by this kind of breaking criterion, eddy amplitudes should be larger in the upper than in the lower layer. The flow in each layer can be thought of as induced by the potential vorticity in both layers, but if the eddy amplitudes are larger aloft, the lower-layer flow will be primarily induced by the potential vorticity in the upper layer, while the upper-layer flow will be primarily self-induced, allowing more wave-like evolution. I suspect that this has something to do with the fact that the two-layer closure theory works best when based on lower-layer diffusion of potential vorticity, leading to equation (1.8). The distinction between upper- and lower-troposphere dynamics, with the latter more turbulent and the former more wave-like, is central to any discussion of eddy momentum fluxes. That the eddy momentum fluxes are almost entirely confined to the upper troposphere is a consequence of this distinction. Rossby waves propagating away from their midlatitude source on a positive potential vorticity gradient (as in the upper layer of a two-layer model) converge eastward (positive) angular momentum into midlatitudes; Rossby waves propagating on a negative vorticity gradient (as in the lower layer of the two-layer model) converge negative momentum into the source latitudes. Because almost all of the propagation in fact occurs in the upper troposphere, surface westerlies are generated in midlatitudes to remove the positive momentum flux convergence. If lower-tropospheric propagation were dominant, surface easterlies would be generated in midlatitudes. All of the profound consequences for the atmosphere and the oceans that follow from the existence of midlatitude surface westerlies result from this asymmetry between upper and lower tropospheres. The simplest picture of linear midlatitude eddies in the upper troposphere starts with a barotropic westerly point jet, u(y) = −|y|, the corresponding vorticity distribution being a single contour separating two homogenized regions, with jump

= 2 across the contour. This flow supports the simplest Rossby edge waves with dispersion relation c = U − /(2k). One can usefully speak of a capacity of this jet, the amplitude of the waves that can propagate along this contour without significant breaking. Using the criterion u ∼ u − c for overturning streamlines in the frame of

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reference of the wave, one gets u ∼ /k. The corresponding trajectory displacements are also of the order of the inverse wavenumber k −1. If we now think of the homogenized regions on each side of this contour, and, therefore, the size of the jump , as having been created by the eddies themselves from the environmental gradient β, we are led to assume that ∼ βk −1 = β L , or u ∼ β L 2. The resulting relation between the eddy scale and eddy energy is just that proposed by Rhines, even though there is no association here with an inverse cascade. This picture may help us understand why it is the Rhines scale at the latitude of the jet that seems to be the relevant scale for the eddies in FHZ. The homogeneous turbulence theory outlined above can be thought of as consisting of three relations between three unknowns: the strength of the energy generation/dissipation , an eddy length scale L , and an eddy velocity scale V (or a diffusivity V L ). The three relations are (1) an entropy or available potential energy budget that relates and D, (2) the Rhines scale relation between V and L , and (3) the turbulent cascade scaling ≈ V 3/L . (One can combine (2) and (3) to give equation (1.3).) In light of the results described by Schneider (2004) for a primitiveequation model on the sphere in which the static stability adjusts to prevent a significant inverse cascade, it may be desirable to try to retain (1) and (2), but to replace (3) with a non-turbulent alternative, a choice made palatable by this alternative argument for the Rhines scale. A picture that emerges is of an upper-level waveguide fed by baroclinic eddy production, with an eddy sink given by the sloughing off of excess wave activity and fed by baroclinic eddy production that is, in turn, determined by the diffusion of lowlevel PV (or heat) controlled by the upper-level eddy amplitudes. One can try to expand this picture into a theory for the zonally asymmetric storm tracks (see in this regard Swanson et al. [1997] and chapter 4 in this volume) in which the key new ingredient is the zonally varying capacity of the jet. While we have some useful pictures of upper-level dynamics that help us understand the eddy momentum fluxes, and even some simple linear models that fit the eddy momentum fluxes quantitatively, given the low-level eddy stirring (DelSole 2001), our understanding of the location of the surface westerlies is far from complete. This is evident when we perturb the system and try to understand how and why the surface westerlies (and the associated eddy momentum flux convergence) move. An excellent example is provided by experiments in which the strength of the surface friction is modified; as the friction is weakened, the westerlies move poleward (Robinson 1997). Figure 1.3 is from unpublished work by G. Chen (personal communication, 2005), using the dry dynamical core benchmark of Held and Suarez (1994). The theory for this shift is still undeveloped. Robinson has suggested that a barotropic governor mechanism is the key: as the surface friction is reduced, surface winds and horizontal shears increase, and, it is argued, the resulting stabilization by these shears is larger on the equatorward side. How one would go about making this hypothesis quantitative and then testing it remains a challenge.

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10 U (m s-1)

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−10

0°

30°

60°

90°

Latitude

Figure 1.3. The near-surface zonal-mean wind field in the climate of an idealized dry general circulation model (GCM) for several values of the strength of the surface friction. The surface friction is a linear drag in the lower troposphere, with different relaxation times (0.5, 0.75, 1, 1.25, 1.5 days) in the different cases. The longer relaxation times produce stronger winds and a poleward displacement of the westerlies. (Provided by G. Chen.)

Poleward displacement of the surface westerlies and storm tracks is also seen in global warming simulations. Several alternative explanations have been offered for this shift, some involving the increase in latent heat release. It will be a challenge to our theories, and our ability to develop the appropriate hierarchy of idealized models, to cleanly isolate the dynamics underlying this shift.

Acknowledgments I thank Dargan Frierson, Pablo Zurita-Gotor, and Gang Chen for their insights and help with the figures.

References Barry, L., G. C. Craig, and J. Thuburn, 2002: Poleward heat transport by the atmospheric heat engine. Nature, 415, 774–777. DelSole, T., 2001: A simple model for transient eddy momentum fluxes in the upper troposphere. J. Atmos. Sci., 58, 3019–3035. Emanuel, K. A., and M. Bister, 1996: Moist convective velocity and buoyancy scales. J. Atmos. Sci., 53, 3276–3285.

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20 | Isaac M. Held Emanuel, K. A., M. Fantini, and A. J. Thorpe, 1987: Baroclinic instability in an environment of small stability to slantwise moist convection. Part I: Two-dimensional models. J. Atmos. Sci., 44, 1559–1573. Frierson, D. M. W., A. J. Majda, and O. M. Pauluis, 2004: Large scale dynamics of precipitation fronts in the tropical atmosphere: A novel relaxation limit. Commun. Math. Sci., 2, 591–626. Frierson, D. M. W., I. M. Held, and P. Zurita-Gator, 2006: A gray radiation, aquaplanet moist GCM: Part 1: Static stability and eddy scale. Submitted to J. Atmos. Sci. Held, I. M., 1982: On the height of the tropopause and the static stability of the troposphere. J. Atmos. Sci., 39, 412–417. Held, I. M., 2005: The gap between simulation and understanding in climate modeling. Bull. Amer. Meteor. Soc., 86, 1609–1614. Held, I. M., and V. D. Larichev, 1996: A scaling theory for horizontally homogeneous, baroclinically unstable flow on a beta plane. J. Atmos. Sci., 53, 946–952. Held, I. M., and M. J. Suarez, 1994: A proposal for the intercomparison of the dynamical cores of atmospheric general circulation models. Bull. Amer. Meteor. Soc., 75, 1825–1830. Illari, L., and J. C. Marshall, 1983: On the interpretation of eddy fluxes during a blocking episode. J. Atmos. Sci., 40, 2232–2242. James, I. N., 1987: The suppression of baroclinic instability in horizontally sheared flows. J. Atmos. Sci., 44, 3710–3720. Juckes, M. N., 2000: The static stability of the midlatitude troposphere: The relevance of moisture. J. Atmos. Sci., 57, 3050–3057. Kraichnan, R. H., 1971: Inertial range transfer in two and three dimensional turbulence. J. Fluid Mech., 47, 525–535. Larichev, V. D., and I. M. Held, 1995: Eddy amplitudes and fluxes in a homogeneous model of fully developed baroclinic instability. J. Phys. Oceanogr., 25, 2285–2297. Lapeyre, G., and I. M. Held, 2003: Diffusivity, kinetic energy dissipation, and closure theories for the poleward eddy heat flux. J. Atmos. Sci., 60, 2907–2916. Lapeyre, G., and I. M. Held, 2004: The role of moisture in the dynamics and energetics of turbulent baroclinic eddies. J. Atmos. Sci., 61, 1693–1710. Lee, S., 1997: Maintenance of multiple jets in a baroclinic flow. J. Atmos. Sci., 54, 1726–1738. Lee, S., and I. M. Held, 1993: Baroclinic wave packets in models and observations. J. Atmos. Sci., 50, 1413–1428. Lorenz, E. N., 1978: Available energy and the maintenance of a moist atmosphere. Tellus, 30, 15–31. Manabe, S., and K. Bryan Jr., 1985: CO2-induced change in a coupled ocean-atmosphere model and its paleoclimatic implication. J. Geophys. Res., 90(C6), 11,689–11,707. Marshall, J. C., and G. L. Shutts, 1981: A note on rotational and divergent eddy fluxes. J. Phys. Oceanogr., 11, 1677–1680. Pauluis, O., and I. M. Held, 2002: Entropy budget of an atmosphere in radiative-convective equilibrium. Part II: Latent heat transport and moisture processes. J. Atmos. Sci., 59, 140–149. Pavan, V., and I. M. Held, 1996: The diffusive approximation for eddy fluxes in baroclinically unstable jets. J. Atmos. Sci., 53, 1262–1272. Phillips, N., 1956: The general circulation of the atmosphere: a numerical experiment. Quart. J. Roy. Meteor. Soc., 82, 123–164. Rhines, P. B., 1975: Waves and turbulence on a β-plane. J. Fluid Mech., 69, 417–443. Rhines, P. B., 1977: The dynamics of unsteady currents. In The Sea, Vol. 6, E. A. Goldberg, I. N. McCane, J. J. O’Brien, and J. H. Steele, Eds., Wiley, 189–318.

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Large-Scale Atmospheric Dynamics | 21 Robinson, W. A., 1997: Dissipation dependence of the jet latitude. J. Climate., 10, 176–182. Salmon, R. S., 1978: Two-layer quasi-geostrophic turbulence in a simple special case. Geophys. Astrophys. Fluid Dyn., 10, 25–52. Salmon, R. S., 1980: Baroclinic instability and geostrophic turbulence. Geophys. Astrophys. Fluid Dyn., 15, 167–211. Schneider, T., 2004: The tropopause and the thermal stratification in the extratropics of a dry atmosphere. J. Atmos. Sci., 61, 1317–1340. Schneider, T., and C. C. Walker, 2006: Self-organization of atmospheric macroturbulence into critical states of weak nonlinear eddy-eddy interactions. J. Atmos. Sci., 63, 1569–1586. Stone, P. H., 1972: A simplified radiative-dynamical model for the static stability of rotating atmospheres. J. Atmos. Sci., 29, 405–418. Stone, P. H., 1978a: Baroclinic adjustment. J. Atmos.Sci., 35, 561–571. Stone, P. H., 1978b: Constraints on dynamical transport of energy on a spherical planet. Dyn. Atm. And Ocean., 2, 123–139. Stone, P. H., and D. A. Miller, 1980: Empirical relations between seasonal changes in meridional temperature gradients and meridional fluxes of heat. J. Atmos. Sci., 37, 1708–1721. Swanson, K. L., P. J. Kushner, and I. M. Held, 1997: Dynamics of barotropic storm tracks. J. Atmos. Sci., 54, 791–810. Thompson, A. F., and W. R. Young, 2006: Scaling Baroclinic Eddy Fluxes: Vortices and Energy Balance. J. Phys. Oceanogr., 36, 720–738. Thuburn, J., and G. C. Craig, 1997: GCM tests of theories for the height of the tropopause. J. Atmos. Sci., 54, 869–882. Vallis, G. K., 1988: Numerical studies of eddy transport properties in eddy-resolving and parameterized models. Quart. J. Roy. Meteor. Soc., 114, 183–204.

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Chapter 2

Theories of Baroclinic Adjustment and Eddy Equilibration Pablo Zurita-Gotor and Richard S. Lindzen

2.1. Introduction A basic question in our understanding of the general circulation (and consequently, climate) is what determines the temperature difference between the Tropics and high latitudes. The observed equilibrium must result from the competition between the destabilizing radiative forcing and the stabilizing dynamical tendencies, which are primarily due in the extratropics to the baroclinic eddies. This question has been addressed in the literature at two very different levels of complexity. On the one hand, comprehensive simulations with general circulation models are reasonably successful in reproducing many aspects of the observed circulation (Gates et al. 1999), but may provide limited insight. Additionally, given the “fitting nature” of the physical parameterizations employed by these models, it is debatable that if they reproduce the right climate, they do so for the right reasons—it should at least be worrisome that the same models are far less successful in reproducing certain aspects of past climates. On the other hand, a very different approach attempts to rationalize the extratropical equilibrium using simpler, conceptual models with lower expectations of quantitative agreement. Among them, one should cite the following: (1) diffusive models, (2) baroclinic adjustment models, (3) life cycle studies, and (4) idealized forceddissipative simulations. Although this review is mostly concerned with the second and third approaches, we will also briefly touch on the other two. These are treated in more detail in chapters 1 and 3 in this volume. The aim of these simplified models is to understand and quantify the eddy forcing of the mean flow without actually solving the problem in its full complexity. This requires the provision of what is usually called a closure relation, that is, a simple relation between the eddy forcing and the mean flow state. There have been several attempts in the literature to do so by means of a diffusive approach. Early studies (Stone 1972; Green 1970) used mixing length theory; in this framework, the problem may be reformulated as the specification of the characteristic velocity and length scales. The

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former was in both cases the extratropical jet velocity, based on the equipartition of mean and eddy kinetic energy, while the latter was the planet size in Green’s case and the Rossby deformation radius λ = NH/ f (e.g., Holton 1992) in Stone’s. These early studies were only concerned with the eddy heat fluxes, and provided a single, global-average value for the diffusivity. A different methodology was used by Branscome (1983), who based the determination of the diffusion coefficient on the linear characteristics of the baroclinic normal modes. This allows one to obtain spatially-dependent diffusivities as a function of the local mean flow properties. However, because the underlying instability framework used by Branscome is purely baroclinic, his model cannot take into account the effect of the barotropic shear or predict the eddy momentum fluxes, even if the flow is meridionally sheared. As we shall see, the eddy momentum fluxes may also be important for this problem (James 1987). At the opposite extreme to Branscome’s quasi-linear approach, a third approach uses homogeneous turbulence modeling to derive diffusivity scaling laws. By virtue of homogeneity, the eddy momentum flux also vanishes in these models, and there is a single, constant diffusivity. The key point is that in the presence of an inverse energy cascade, the relevant length scale is the one halting the cascade, either the Rhines scale or the size of the domain (Larichev and Held 1995; Held and Larichev 1996), rather than the Rossby radius as linear theory would predict. Likewise, the velocity scale needs to be predicted. Although the relevance of the homogeneous assumption for real flows is unclear, Pavan and Held (1996) have shown that the diffusivities predicted by the homogeneous model work reasonable well in the inhomogeneous case, and Barry et al. (2002) have obtained good fits to general circulation model (GCM) results using a turbulence closure of this sort. If the homogeneous results are indeed relevant for real three-dimensional flows, one could use the homogeneous turbulence models as a “laboratory device” to “measure” the diffusivity, as conceived by Held (1999). Chapter 1 in this volume provides a more detailed description of homogeneous turbulence models. An alternative parameterization of turbulence is provided by the linear stochastic models (e.g., Farrell and Ioannou 1995). These models parameterize the multiscale transfer that one associates with nonlinearity as a combination of enhanced damping and stochastic excitation. The added damping is sufficient to stabilize the flow (in a normal mode sense), but the stochastically forced turbulence is still amplified due to the non-normality of the linear dynamics (see also Branstator 1995). This allows one to estimate the eddy fluxes associated with a given mean flow and even close the quasilinear problem (Delsole and Farrell 1996). A weakness of this approach is that the fluxes are sensitive to the amplitude of the stochastic forcing. A different approach to the diffusive models is provided by baroclinic adjustment models, the main topic of this review chapter. The rationale behind these models is that the equilibrium between dynamics and radiation is strongly shifted to the side of the dynamics, due to the different magnitude of the time scales involved. Thus, instead of a closure relation, these models directly postulate the equilibrium state, the eddy transport

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being implicit: it is just what is required to produce the postulated adjusted state. One may also interpret this behavior in terms of a steep diffusive law, in which the eddy diffusivity is so strongly dependent on the mean flow that the mean state is allowed to depart only weakly from the adjusted state. The concept of baroclinic adjustment is essentially analogous to that of convective adjustment. As in the convective case, the adjusted state is assumed to be neutral, but in this case to baroclinic instability. However, the concept of a target neutral state is ambiguous in the baroclinic problem because no instability condition is known that is both necessary and sufficient—we shall see that many different versions of baroclinic adjustment exist as a result. We also note that baroclinic adjustment is intimately linked to baroclinic instability theory, and as such relies extensively on quasigeostrophic (QG) concepts. In contrast, it has been argued by Schneider (2004, 2005; see also chapter 3 in this volume) that allowing the eddies to change the basic state stratification makes the extratropical equilibrium fundamentally different. This cannot readily be addressed with the QG framework used throughout this chapter.

2.2. The Concept of Adjustment 2.2.1. Convective Adjustment The primary example of adjustment in the geophysical context is provided by convective adjustment, a concept that dates back to the work by Gold (1913) and Emden (1913) (both of which are well summarized in Brunt [1941] and Goody and Yung [1989]). To motivate this concept, consider the linear stability analysis of the dry Rayleigh-Benard problem in the limit of vanishing viscosity. The necessary and sufficient condition for exponential growth is that the squared Brunt-Väisälä frequency,
 g ∂T g N2 = + , T ∂z cp becomes negative, in which case the growth rate (in the limit of the horizontal wavenumber approaching infinity) is simply σ = |N|. Plugging some numbers, we can see that even for weakly unstable stratifications on the order of 0.01 K km−1 the growth time scale becomes very short, roughly half an hour. This implies that for instabilities generated by large-scale circulations or even internal gravity waves, there is ample time for convection to equilibrate close to a neutral lapse rate. The most common atmospheric application of convective adjustment is in column models of radiative-convective equilibrium (Manabe and Strickler 1964). In these models, the unstable radiative equilibrium profile is replaced in some region by a layer with a prescribed lapse rate that is usually less steep than the dry adiabatic lapse rate, to account for condensation heating and to bring the lapse rate closer to what is observed. To calculate the depth of the adjusted layer, one then needs to

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Baroclinic Adjustment | 25

T2 (°C)

20

10

JULY

0

–10

–20

ANNUAL

JANUARY

–30 15°

30°

45° Latitude

60°

75°

Figure 2.1. Observed midtropospheric temperature (crosses), compared to a prediction based on the two-layer model critical shear (solid). From Stone (1978). (Reproduced with permission c 1978.) from the American Meteorological Society


match this adjusted profile to the overlying layer, the stratosphere, that is assumed to be in radiative equilibrium. Assuming that the outgoing longwave radiation (OLR) is unchanged (convection only redistributes heat vertically), this provides an additional relation between the tropopause temperature and its optical depth (or its height) that can then be used to obtain the depth of the adjusted layer. See chapter 3 in this volume for additional details on how to formulate this so-called “radiative constraint.” An aspect worth emphasizing in this construction is that the adjusted region is always deeper than the original region with unstable lapse rate because resetting the lapse rate to the neutral value wherever this value is exceeded would produce new unstable layers. The implication is that the adjustment is nonlocal, and also extends to regions that were originally stable. As we shall see, the non-locality problem is aggravated in the baroclinic case.

2.2.2. Baroclinic Adjustment Phillips’s (1954) two-layer model for baroclinic instability becomes unstable only when the vertical shear exceeds a critical value. As Thompson (1961) notes, Pocinki (1955) found that the observed mean tropospheric shear was within 25% of this critical shear. By virtue of the thermal wind relation, this critical vertical shear also translates into a critical value of the meridional temperature gradient. Pocinki’s findings were later confirmed by Stone (1978), from which Fig. 2.1 is taken. This figure shows that

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the latitudinal and seasonal structure of the observed midtropospheric temperature (crosses) is accurately captured when using the local temperature gradients predicted by the two-layer model critical shear (solid). The seasonal dependence of the critical shear in this figure reflects seasonal changes in the static stability, while the latitudinal structure is due to rotation. Figure 2.1 implies that the isentropic slope changes much less with season than either the horizontal or vertical temperature gradients because, as explained in next section, the two-layer critical shear is proportional to the stratification. The concept of baroclinic adjustment was introduced by Stone (1978) to explain this finding. Stone proposed that this could be due to the strong feedback exerted by the baroclinic eddies, which would only allow the basic state to become marginally unstable. This would be analogous to the case of convective adjustment discussed above. Later observational and modeling studies also supported the concept of a baroclinic thermostat. For instance, Stone and Miller (1980) found a strong negative correlation in observations between the meridional temperature gradient and the transient eddy heat flux, and GCM modeling studies showed that the total heat flux did not change significantly when some of its components (like the stationary heat transport in Manabe and Terpstra [1974] or the latent heat flux in Manabe and Stouffer [1980]) were changed. This could be explained using the thermostat theory, which requires the baroclinic eddies to transport enough heat to compensate for the reduction in the other fluxes to maintain neutrality. However, the time-scale separation between dynamics and forcing is much less in the baroclinic than in the convective case, if it exists at all. Typical estimates of the dynamical time scale use the modal growth rate, based on the Eady problem (Eady 1949) or more realistic models, and give characteristic times of just a few days, consistent with intense synoptic development. This is faster than typical radiative times in the free troposphere (of order 20 days [Prinn 1977]) but slower than the convective time scales responsible for maintaining the boundary-layer temperature gradients (Swanson and Pierrehumbert 1997). It might then be argued, as it has been done many times in the literature, that baroclinic adjustment is only relevant above the boundary layer. The main problem with this argument is that baroclinic adjustment is inherently a nonlocal process, so that it is questionable whether the local time scale can be used as a measure of the local compliance with baroclinic adjustment. This non-locality is a consequence of the fundamental role that eddy propagation plays for the baroclinic equilibration and, for the mean flow, of the spread of the response by the mean meridional circulation when quasigeostrophic scaling applies. Quasigeostrophic scaling introduces yet another consideration, the interchangeability of heat and momentum sources. As a result, the relevant forcing time scale for baroclinic adjustment may not only be nonlocal, but also non-thermal. For instance, in the presence of a strong barotropic acceleration, friction restores the vertical shear (or horizontal temperature gradient) by damping the surface wind, which may dominate the direct thermal forcing in some circumstances.

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The preceding discussion suggests that the determination of the relevant forcing time scale for baroclinic adjustment is by no means trivial. Similarly, it is also unclear what the most appropriate measure of the dynamical adjustment time scale is. Although eddy growth rates can be large during synoptic development, it may be more appropriate when looking at the maintenance of the mean state to consider the instability of the time-mean flow, which gives much longer dynamical time scales. Another time scale sometimes used, the advective time scale, is also problematic because the bulk of thermal advection is really associated with reversible motions, so that the resulting tendency is largely balanced by the Eulerian time change (in contrast with Lagrangian tendencies, which are balanced diabatically). The question of whether enough separation exists between the dynamical and forcing time scales for baroclinic adjustment to be relevant has been explicitly addressed by Barry et al. (2000) in a recent paper. These authors estimate characteristic spinup and spindown times in their GCM by switching on and off radiation and other physical processes. The evolution that they observe is very different depending on whether surface fluxes are included or not, pointing to the important role played by boundary-layer processes for the whole fluid, but no clear sign of scale separation was found in either case. A generalization of the radiative-convective models described above also exists, with baroclinic adjustment replacing convective adjustment in the midlatitudes (Held 1982). In this framework the adjusted region is still idealized to have a constant (with height) lapse rate, and to meet a stratosphere in radiative equilibrium at the tropopause. The main difference is that the convective constraint of a fixed lapse rate is replaced by a baroclinic adjustment condition, i.e., a stratification everywhere proportional to the meridional temperature gradient. The radiative constraint described above still applies, with the caveat that the local OLR is now allowed to change from radiative equilibrium because there is a meridional heat transport (see chapter 3 in this volume for more details). Thuburn and Craig (1997) tested these ideas by exploring the sensitivity of the tropopause height in a comprehensive GCM when different external parameters were varied. They found that the radiative-baroclinic framework failed to predict the observed changes in tropopause height due to the breakdown of the baroclinic adjustment constraint, while the radiative constraint was much more accurate. The authors tried not only the two-layer constraint introduced above, but also other baroclinic adjustment formulations discussed in next sections. In contrast, Schneider (2004) obtained a much better agreement with Held’s baroclinic-radiative equilibrium construction in an idealized dry GCM. Although Schneider’s considerations are very different from baroclinic adjustment, the dynamical constraint that he uses is still formally similar to Stone’s (1978) baroclinic adjustment condition. Thus, it is unclear why his results differ from those of Thuburn and Craig (1997). One possible explanation is Schneider’s use of a dry model; moisture could also play a role in the more general case considered by Thuburn and Craig, as argued by Juckes (2000). It is also possible that the different results are simply due to the

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small differences between both formulations, particularly to the use of the surface (as opposed to midtropospheric) temperature gradients in Schneider’s case. A more detailed description of Schneider’s framework and Juckes’s arguments can be found in chapter 3 in this volume. Finally, we note that baroclinic instability is simply a special case of parallel shear flow instability. For some forms of parallel shear flow instability, a condition analogous to the Charney-Stern condition exists (Charney and Stern 1962), but for others the conditions differ (for example, the Miles-Howard condition for stratified shear flow). However, all these problems share a universal wave-propagation configuration (Lindzen 1988). This chapter concentrates on the baroclinic problem, but we note in passing that the adjustment concept has also been used with some success for some of these other forms of shear instability (Malkus 1979, 2003).

2.3. The Adjusted State Even accepting that the eddies are so efficient that the time-mean state is only allowed to be marginally unstable, an important difference with convective adjustment is that, in the baroclinic case, the choice of adjusted state is not unique. Lacking a general instability condition that is both necessary and sufficient, different formulations have exploited the specific neutrality conditions of different baroclinic instability models. This has led to many flavors of baroclinic adjustment.

2.3.1. Two-Layer Model Critical Shear Stone (1978) noted that the observed midtropospheric vertical shear agreed well with the two-layer model critical shear, and proposed this state as a paradigm of the adjusted state. The two-layer model critical shear is given by (Phillips 1954) Uc = βλ2,

[2.1]

where λ = N H/ f is the Rossby radius based on the layer depth H. The two-layer model is unstable if the baroclinic component of the zonal wind, Us = U1 − U2, which may be regarded as the two-layer version of a vertical shear, exceeds the critical value Us > Uc, or, equivalently, if the supercriticality ξ=

Us Us = Uc βλ2

[2.2]

exceeds one. (The role of the supercriticality ξ in the equilibration of the two-layer model is discussed in chapter 1 in this volume; ξ is the two-layer model counterpart of the supercriticality Sc discussed in chapter 3.) The results of Stone (1978), who used characteristic upper- and lower-tropospheric levels to estimate Us = U1 − U2, imply

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Baroclinic Adjustment | 29

that the condition ξ = 1 is very nearly satisfied in the extratropical troposphere for all latitudes and seasons. Due to the critical shear’s dependence on the static stability, the criticality condition ξ = 1 is really a constraint on the isentropic slope, rather than the meridional temperature gradient. To see this, combine the definition of λ, λ2 =

N 2H 2 g H 2 = θz, f2 θ0 f 2

with the thermal wind relation g Us θ y. =− H f θ0 We obtain ξ=

f θy Us =− 2 βλ β H θz

→

I =−

θy H = ξ, θz a tan φ

[2.3]

where I is the isentropic slope. Thus, when ξ = 1 the isentropic slope is order O(H/a), which implies that the mean isentropic slope is such that an isentrope leaving the subtropical boundary layer reaches the tropopause at the poles (Held 1982). As noted by Hoskins (1991), this is the case in the Earth’s atmosphere, consistent with Stone’s findings. Despite the good agreement with the theory implied by the observations shown in Fig. 2.1, there are some conceptual problems with this notion of baroclinic adjustment. Two important problems with the theory that have long been recognized are: (1) the critical shear is a peculiarity of the two-layer model that owes its existence to the vertical discretization; and (2) numerical simulations show that the two-layer model itself does not equilibrate near criticality (Salmon 1980). The critical shear of the two-layer model can be understood in terms of the Charney-Stern necessary condition for instability (Charney and Stern 1962). According to this condition, the QG meridional potential vorticity (PV) gradient    ∂ 2U 1 ∂ f 2 ∂U f 2 ∂U  qy = β − 2 − [2.4] ρ 2 − 2  δ(z) ∂y ρ ∂z N ∂z Ns ∂z  z=0

has to change sign for the flow to become unstable. In its original form, the CharneyStern condition involves the interior PV gradient and the temperature gradient at both boundaries. However, Bretherton (1966) showed that it is possible to rephrase this condition in terms of the PV gradient alone, using the generalization in equation (2.4). This requires replacing the surface temperature gradient (or vertical shear) by an isothermal surface and a jump in the shear right above it, which produces the deltafunction contribution on the right-hand side of equation (2.4). In the presence of an upper rigid lid, a similar contribution (not included here) must be added at the upper boundary.

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Figure 2.2. Schematic illustrating how the jump in vertical shear at the surface is smoothed out into a finite curvature for a two-layer model.

This boundary contribution is key to baroclinic instability. In the real atmosphere (and in most classical models of baroclinic instability) the interior PV gradient is always positive due to the β term, with the possible additional contribution of a positive deltafunction PV gradient at the tropopause. Thus, it is only at the surface that the PV gradient changes sign, thereby violating the Charney-Stern condition. For this, it suffices that the flow has a westerly vertical shear at the surface, no matter how small. In contrast, the discretized two-layer PV gradient is given by Us q y,n ≈ β − (−1)n 2 ≈ β(1 − (−1)nξ ), [2.5] λ where n = 1, 2 for the upper and lower layer, respectively, and the relative vorticity contribution to the PV gradient is neglected. Thus, we can see that while the upperlayer PV gradient is always positive, the lower-layer PV gradient only becomes negative when ξ > 1. For smaller values of ξ the PV gradient is one-signed, and the flow is stable. Figure 2.2 shows why this is the case. While in the continuous problem the transition from the interior shear to an isothermal lower surface occurs through a jump at z = 0, in the two-layer model this transition is smoothed across the whole lower layer, due to poor vertical resolution (thick line). Thus, the delta-function of equation (2.4) is replaced by a finite smoothed curvature; only when this curvature exceeds β does the net PV gradient become negative. In other words, the critical shear is such that the integrated delta-function at the surface exactly balances β integrated over half the troposphere. The critical shear is thus essentially a function of resolution. It decreases with the number of layers and disappears in a continuous model. Although the continuous problem is unstable for all shears, the two-layer critical shear may still be meaningful for that problem (Held 1978). By means of a scaling analysis, Held showed that the depth of the most unstable mode in the Charney problem

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(Charney 1947) is given by the lesser of the scale height HS (or tropopause height) and the scale h=

f 2 1 ∂U . N 2 β ∂z

[2.6]

This scale gives the depth over which one needs to integrate the interior PV gradient to exactly balance the integrated delta-function at the surface. In this framework, the two-layer stability condition is rephrased h ≤ HS/2. Held’s scaling analysis implies that when h  HS, the most unstable mode is shallow and only extends over a fraction of the tropospheric depth HS. As a result, when the shear is smaller than the two-layer critical value, the unstable modes are shallower and more weakly growing than the deep modes of the h > HS limit. These modes would be more easily stabilized by friction. Another problem with Stone’s baroclinic adjustment hypothesis is that numerical simulations indicate that the two-layer model itself does not equilibrate in this manner, and is supercritical at equilibration for realistic parameters (e.g., Salmon 1980). Because of this, Cehelsky and Tung (1991) proposed a theory of nonlinear baroclinic adjustment, which has been more recently refined by Welch and Tung (1998a). Cehelsky and Tung argued that the critical shear of the two-layer model is only relevant for the quasilinear equilibration, when the most unstable mode interacts with the zonal flow alone. However, in the fully nonlinear case this wave may still grow at the expense of the zonal flow in equilibrium (i.e., be supercritical) as long as it releases its energy nonlinearly to some other modes. Welch and Tung (1998a) proposed a conceptual model in which the most unstable mode saturates as the forcing is increased and gives its energy to a longer mode with higher carrying capacity. This process may continue further as the forcing continues increasing, so that for any level of forcing it is the most unstable wave among those not yet saturated that dominates the transport. This wave would neutralize the mean flow, which would still be supercritical for more unstable but saturated modes. Welch and Tung (1998b) argued that this could also explain the observed seasonal shift in the heat-transfer spectra. On the other hand, Stone and Branscome (1992) have noted a striking result for the two-layer model: though their model was supercritical, consistent with previous studies, the supercriticality itself was remarkably insensitive to changes in the forcing. In particular, they found that the condition ξ ≈ 2.4 was very accurate over a wide range of parameters. The authors conjectured that this could reflect a weaker form of baroclinic adjustment. The adjusted shear would still be proportional to the static stability as in equation (2.1), but with a proportionality constant that depends on the model used. This conclusion has been recently revised by Zurita-Gotor and Lindzen (2006), who argued that it is the lower-layer PV gradient—rather than the supercriticality—that it is hard to change in the two-layer model. Neglecting the relative vorticity gradient, both are related through the simple expression ξ = 1 − q y/β (cf. equation [2.5]). This point is very clearly made by Fig. 2.3, which shows the equilibrium PV gradient for

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1

β halved (ξ=3.6)

0

f−plane (ξ→∞)

−1

−2

−20

−10

0 y/ λ

10

20

Figure 2.3. Lower-layer PV gradient in two-layer simulations with a typical midlatitude β = 1.6 × 10−11 m−1s −1, with a halved β and on the f -plane.

simulations with three values of β (see Zurita-Gotor and Lindzen [2006] for details on the model and parameter setting). As can be seen, the equilibrium midchannel PV gradient is roughly the same in all cases: q y ≈ −2.1 × 10−11 m−1 s−1. For a typical midlatitude β = 1.6 × 10−11 m−1 s−1, this gives a supercriticality ξ ≈ 2.3, roughly consistent with Stone and Branscome’s results. However, this supercriticality does change to ξ = 3.6 with a halved β, and to ξ = ∞ on the f -plane. In fact, Zurita-Gotor and Lindzen (2006) showed that it is also possible to change the PV gradient, for instance with much stronger changes in the forcing than considered by Stone and Branscome (1992). Zurita-Gotor and Lindzen attributed this behavior to the steepness of the PV diffusivity, which implies that only moderate changes in the PV gradient are needed to produce large changes in the fluxes. Chapter 1 in this volume provides theoretical arguments in support of a steep diffusivity. To conclude, we note that Schneider (2004) has recently found that the vertical shear in his dry model is also very close to the two-layer model critical shear over a wide range of parameters, consistent with Stone (1978). However, Schneider justifies this result using very different arguments. He does not attribute this constraint to neutrality. Although he uses a non-geostrophic framework (see chapter 3 in this volume for details), it is instructive to interpret his arguments in the present QG framework to see how the two theories are related. The key assumption that he makes is that the diffusivity has no vertical structure, as is typically found for highly turbulent flows (e.g., Larichev and Held 1995). Then, the condition that the PV fluxes integrate vertically to zero implies that the negative and positive PV gradients must be, in an integral sense, of the same order (h ∼ HS), which is the essence of the two-layer model constraint.

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However, he argues that adjustments in the static stability are critical to this process (see also Schneider 2005), which might explain why the QG two-layer model fails to equilibrate that way in numerical studies.

2.3.2. Interior versus Boundary PV Mixing Unlike multilayer models, the continuous Charney problem (Charney 1947) is unstable for all westerly shears. Thus, a recurring question in the context of baroclinic adjustment is what prevents the eddies from eliminating the surface temperature gradient in the extratropical troposphere. One possible reason is that the nearly inviscid limit assumed by this theory is after all not appropriate, especially at the surface, where vertical mixing can be very efficient (Swanson and Pierrehumbert 1997). However, other alternatives have also been proposed in the literature. Lindzen and Farrell (1980) suggested a modification of baroclinic adjustment that pertained to the Charney problem rather than the two-layer model, but that gave results very similar to Stone’s. The authors calculated how far above the surface the deltafunction PV gradient at the ground would have to be stretched to eliminate the sign change in q y. They calculated the heat flux required to achieve this, and used this heat flux to predict the surface temperature gradient, implicitly assuming that convection or some other process would restore the temperature gradient and static stability. This procedure led to a reasonable midlatitude temperature distribution, but too small an equator-to-pole temperature difference compared to the present climate, more nearly in line with the ice-free climate of the Eocene. Consistent with this, the agreement was improved when the very large static stabilities over polar ice (Held and Suarez 1976) were taken into account. This suggests the interesting possibility of a positive feedback from the presence of an ice surface that inhibits meridional heat fluxes, at least in terms of their impact on the surface. Note that Lindzen and Farrell’s procedure is similar in many regards to Stone’s original argument because the two-layer critical shear essentially reflects the smoothing of the delta-function PV gradient over some tropospheric depth. The role of the static stability for baroclinic adjustment was further explored by Gutowski (1985). This author noted that the negative delta-function PV gradient at the surface could also be eliminated without eliminating the surface shear. This would be the case if the flow developed an infinite static stability Ns at the surface (cf. equation [2.4]). Although an infinite static stability may not be very realistic, real flows tend to develop large static stabilities at the surface as they equilibrate (see chapter 3). Gutowski argued that the enhancement of the surface static stability by the vertical eddy heat fluxes could be an important component of baroclinic adjustment. Full adjustment (infinite static stability) would be prevented by boundary-layer processes. Note that Gutowski’s mechanism and the elimination of the surface temperature gradient can be rephrased as the same mechanism, the flattening of the isentropic slope h/HS at the surface.

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Another argument attributes the lack of thermal homogenization at the surface to the reduced eddy scale. This was originally proposed by Lindzen (1993), who noted that short waves are stable in the Eady problem (Eady 1949) regardless of the shear. He suggested that the eddies could equilibrate by mixing the interior PV gradient and raising the tropopause. This requires that the scale of the eddies is externally constrained, which Lindzen attributed to the meridional wavenumber imposed by the jet width (Ioannou and Lindzen 1986). Hence, an alternative mechanism to raising the tropopause would be to concentrate the jet and reduce the eddy scale. The rationale for this would be that convergent eddy momentum fluxes are a robust feature of most forms of baroclinic instability (Held and Andrews 1983; Ioannou and Lindzen 1986). Lindzen (1993) argued that observed interior PV gradients are much smaller than surface PV gradients, and noted that only small changes in the vertical curvature of the zonal wind and/or in the static stability are needed to produce PV gradients of order β (see also Lindzen 1994). Lindzen concluded that observational resolution might not allow us to distinguish zero from β (this has been recently questioned by Zurita and Lindzen [2001], as discussed below). The climatic implications of small interior PV gradients were explored by Sun and Lindzen (1994) and Kirk-Davidoff and Lindzen (2000). Sun and Lindzen reconstructed the tropospheric thermal structure assuming convective adjustment in the Tropics and baroclinic adjustment in the extratropics, in the form of interior PV homogenization along the isentropes. With these assumptions, knowledge of the temperature gradient and static stability at the surface suffices to determine the full thermal structure. When the observed surface properties were used, they were able to obtain a climate that compared reasonably with observations. As a further step, Kirk-Davidoff and Lindzen (2000) incorporated the adjustment model into a closed climate model that also predicted the surface temperature using a simple energy balance at the surface. Following Lindzen’s (1993) paper, a number of studies addressed the question of whether and why the interior PV gradient might be better mixed than the surface PV gradient (i.e., the surface temperature gradient). Neglecting the relative vorticity component, the interior PV gradient can be expressed as a function of the local value of h (with h defined by equation [2.6]),  qy = β 1−e

z/HS

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Stone and Nemet (1996) calculated the h profile h ad j required to obtain q y = 0 everywhere, and compared this adjusted profile with the observed values. They found that the agreement was relatively good in the interior, but less so in the boundarylayer, which they attributed to boundary layer dissipation. Swanson and Pierrehumbert (1997) also attributed the lack of surface PV homogenization to boundary-layer dissipation, namely to the vertical mixing of potential temperature over the convective

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boundary layer. Additionally, they argued that another possible reason for the incomplete homogenization could be the smallness of the relevant mixing length for this problem (a Lagrangian mixing length), which is smaller than the width of the baroclinic zone. As a result, the eddies might only be able to homogenize the surface temperature locally, much like in the baroclinic life cycles of Simmons and Hoskins (1978), but not in a climatological sense. However, this argument does not explain why the interior PV gradient would be better mixed, unless the interior mixing length was larger. On the other hand, Zurita and Lindzen (2001) have recently questioned the conjecture put forward by Lindzen (1993) that the PV gradient is better mixed in the interior than at the surface. Zurita and Lindzen argued that the apparent dominance of the negative PV gradients is an artifact of quasigeostrophic theory, which concentrates the boundary PV gradient over an infinitesimal layer (see also Held and Schneider [1999]). However, the fact that the observed vertical shear is of the order of the two-layermodel critical shear implies that the integrated delta-function PV gradient at the surface is roughly equal to β integrated over half the troposphere. From that point of view, β is not really small as argued by Lindzen (1993), or at least it is as small as the surface temperature gradient itself. Zurita and Lindzen (2001) estimated the verticallyintegrated positive and negative PV gradients using Stone and Nemet’s (1996) data, and concluded that both were in fact comparable. They also noted that the choice q y = β gave a better agreement with observations than full PV homogenization (q y = 0) in Sun and Lindzen’s (1994) baroclinic adjustment reconstruction. Another take on this issue is provided by Fig. 2.4, which shows the y-z structure of the meridional PV gradient, normalized by the latitude-dependent β. Remarkably, this profile has little meridional structure and is very robust throughout the seasons (Kirk-Davidoff 1998). Idealized models can reproduce the basic features of this profile (Solomon and Stone 2001a), which is found to be very robust—more than the vertical shear itself—against changes in the forcing (Solomon and Stone 2001b). Fig. 2.4 shows that the PV gradient is only small across a shallow layer centered at 700 hPa, broadly coinciding with the steering level of the modes (Zurita and Lindzen 2001). Based on the fact that the PV flux of a Charney mode peaks at the steering level (Lindzen et al. 1980), and on Bretherton’s (1966) result that PV homogenization at the steering level is a necessary condition for neutrality, Zurita and Lindzen (2001) proposed that short Charney waves might be able to equilibrate by eliminating the PV gradient at the steering level alone. A linear stability analysis confirmed that basic states with zero PV gradients across a localized region surrounding the steering level could be neutral. However, this is not the only possible explanation for the observed PV structure. For instance, Swanson (2001) has noted that one could also get a structure as depicted in Fig. 2.4 if the interior PV gradient were well mixed everywhere except for a jump at the tropopause. The time-mean PV gradient at a certain height would then be a function of the frequency with which the tropopause crosses that height, decreasing away from the time-mean tropopause as in Fig. 2.4. This picture has been confirmed by

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Birner (2006), who showed that at any given time the PV distribution on an isentropic surface is well characterized by a single contour, so that PV is much better mixed away from the instantaneous tropopause than suggested by the time-mean picture. Even so, it is the time-mean picture depicted in Fig. 2.4 that one would use for the linear stability analysis of small-amplitude perturbations (see Morgan 1994), though the relevance of such an analysis would be questionable in view of the observed large amplitudes. In summary, the most recent work suggests that a comparable degree of PV homogenization is found in the interior and at the surface, resulting in observed vertical wind shears that are of the order of the critical shear for the two-layer model. However, this state of affairs is predicted both by Stone’s (1978) baroclinic adjustment theory and by the turbulent diffusion arguments of Schneider (2004). The time-mean PV gradients are small, in the sense that the temperature gradient is significantly reduced from radiative equilibrium, but also unambiguously different from zero. The PV gradient appears to vanish at the steering level, which is a necessary (but not sufficient) condition for neutrality (Bretherton 1966). Thus, it is unclear whether this has any dynamical relevance for the equilibration, or is just a consequence of the kinematic structure of the flow (Zurita-Gotor 2002).

2.4. Equilibration Studies The review in the previous section has shown that there is a great deal of ambiguity with baroclinic adjustment, as several paradigms for the adjusted state have been proposed in the literature. In principle, numerical simulations of eddy life cycles and analyses of the equilibriums of idealized forced-dissipative models should help elucidate whether and in which form baroclinic adjustment is relevant. However, results are ambiguous so far.

2.4.1. Inviscid Life Cycles: Role of Eddy Momentum Flux The rationale of baroclinic adjustment is that the restoration of the mean flow is so slow that the eddies are very efficient in neutralizing the mean state. As discussed in section 2.2.2, the validity of this assumption is questionable. However, one may still usefully pose this question in the context of a numerical model, as a first step toward understanding the maintenance of the equilibrium in the more realistic forced case. This type of analysis is generally known as an eddy lifecyle. Essentially, one perturbs an unstable basic state and allows the perturbation to grow to finite amplitude and to modify the flow. If there is no forcing in the model, a steady state is typically reached (perhaps after additional secondary growth), which implies that the flow must be neutral. The key question is, does this final state resemble any of the adjusted states described above?

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The main difficulty is that baroclinic adjustment is a two-dimensional concept, while the equilibration problem is necessarily three-dimensional as a result of the geostrophic constraints that dominate the midlatitudes. Although the simple conceptual models discussed above consider that momentum is redistributed only vertically, the vertical advection of momentum is negligible under QG scaling. Rather, this vertical redistribution of momentum occurs in quasigeostrophic theory through an eddyinduced mean meridional circulation (MMC) that keeps the wind field in thermal wind balance with the modified temperature field. For instance, when the temperature gradient is reduced, an indirect MMC reduces the vertical shear by imparting a westerly acceleration at lower levels and an easterly acceleration aloft.1 This implies that any vertical redistribution of momentum is intimately associated with a horizontal redistribution of temperature; one cannot change the flow at one latitude without affecting the others. Despite the physical implausibility of a two-dimensional baroclinic equilibration, it is worth noting that a barotropic analog exists (Lindzen et al. 1983) that suggests that the ideas discussed in the previous section would be appropriate in a two-dimensional framework. This analog, the barotropic point jet, is homomorphic with the CharneyBoussinesq problem in the linear regime. It consists of an easterly triangular jet on the beta plane, a configuration that produces positive PV gradient everywhere except at the jet vertex, where the vorticity jump gives a negative delta-function PV gradient (the “surface” of the Charney problem). Schoeberl and Lindzen (1984) and Nielsen and Schoeberl (1984) studied the nonlinear equilibration of that problem. They found that in the absence of dissipation the flow equilibrates by eliminating the vorticity jump and erasing the interior PV gradient over a region with depth given by equation (2.6) (or rather, its barotropic equivalent; see Solomon and Lindzen [2000] for details). More recently, Zurita-Gotor and Lindzen (2004a) have shown that PV homogenization only occurs in the inviscid case when the mode is deep enough. On the other hand, when the scale of the mode is externally constrained (in their case through the channel length), the negative PV gradient cannot be eliminated and the PV gradient is brought to zero at the steering level alone. This would be consistent with the arguments of Zurita and Lindzen (2001). However, things are very different in the baroclinic case due to threedimensionality and the role of the eddy momentum flux. Although in most examples of three-dimensional baroclinic instability the modal eddy momentum flux is convergent over the baroclinic region (Held and Andrews 1983; Ioannou and Lindzen 1986), what seems more relevant is the dominance of the eddy momentum fluxes over the eddy heat fluxes during the nonlinear stage. This was first pointed out by Simmons and Hoskins (1978), who coined the terms baroclinic growth and barotropic decay to refer to the observed behavior. Exploiting the relation between wave propagation and eddy-mean flow interaction (Edmon et al. 1980), the baroclinic growth/barotropic decay paradigm has been reinterpreted by Thorncroft et al. (1993) as consisting of

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three stages: (1) upward wave propagation, (2) initial saturation, and (3) equatorward propagation. Among other effects, this convergent momentum flux concentrates the jet and reduces the meridional scale, an effect that Lindzen (1993) and Swanson and Pierrehumbert (1997) have argued might prevent thermal homogenization at the surface. The impact of the meridional scale is also apparent in the numerical simulations of Simmons and Hoskins (1978), in which the elimination of the surface temperature gradient over the main baroclinic zone is accompanied by enhanced gradients laterally. The concept of baroclinic adjustment has survived the introduction of barotropic effects into the picture, in a generalized version that includes many forms of mixed baroclinic-barotropic adjustment. In other words, many theories still exploit the idea of a mean state that is neutral, even if the adjustment may be more barotropic than baroclinic in character. This is implicit in the theories of Lindzen (1993) and Zurita and Lindzen (2001), which use a purely baroclinic framework but rely on barotropic processes to constrain the meridional eddy scale. More generally, James (1987) has introduced the concept of the barotropic governor to refer to the stabilization of the mean flow through the addition of barotropic shear. This may be due to meridional confinement as in Lindzen (1993), but also to eddy shearing or PV unshielding between upper and lower levels (all these mechanisms can also be explained in terms of changes in the geometry of the critical surface as discussed in Lindzen et al. [1980] and Lindzen [1988]). Despite the important role played by the meridional rearrangement in most inviscid life cycle simulations, we still lack a clear conceptual model of what the barotropic governor mechanism really entails, perhaps because this term has been used in the literature to refer to different concepts. Nakamura (1999) has tried to remedy this problem by de-emphasizing the role played by the horizontal shearing in favor of PV-thinking (Hoskins et al. 1985) and offering an interpretation of Simmons and Hoskins’s (1978) life cycle simulations somewhat different from James (1987). These simulations are characterized by (1) the development of a strong, barotropic jet, (2) the expulsion of the meridional surface temperature gradients to the margins of the original baroclinic zone, and (3) the enhancement of the static stability at lower levels. Although this rearrangement produces a basic state that is consistent with baroclinic adjustment theory over the original baroclinic zone, the full two-dimensional (y-z) state still violates the CharneyStern condition for instability and is only stabilized via the barotropic governor. But while James (1987) emphasized the role of the horizontal shear in preventing the growth of coherent modes, Nakamura (1999) argued that what really matters is the separation between the regions with large PV gradients. This would be similar to Lindzen’s (1993) mechanism, except that in Nakamura’s case the separation between the regions with positive and negative PV gradients occurs meridionally, rather than through changes in the tropopause height. In both cases, the flow must also homogenize PV away from the localized jumps, and indeed Nakamura’s equilibration mechanism only works on the f -plane.

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The hypothesis formulated by Nakamura (1999) agreed well with his twolayer f -plane results, in which the shortwave cutoff appeared to play a key role for neutrality. In the continuous case, Song and Nakamura (2000) used a semi-geostrophic model to investigate the instability of Eady-like basic states with meridionally varying tropopause, a device that allowed them to localize meridionally the region with positive PV gradient. The authors found that the shortwave cutoff could also explain the stability of basic states similar to those obtained in the inviscid life cycle simulations. However, despite this encouraging progress, the three-dimensional problem is complex and the generality of Nakamura’s arguments is still unclear. Additional work is required to understand what other constraints might be important for the mixed barotropicbaroclinic instability problem. Although the y-z eigenvalue problem is computationally intensive, piecewise PV models (Nakamura 1993) provide an efficient tool for addressing this problem.

2.4.2. Role of Forcing and Dissipation Basic states like those produced by the inviscid life cycle calculations are occasionally observed along the oceanic storm tracks (e.g., Hoskins and Valdes [1990]) but are poor descriptors of the extratropical climatology. This suggests that non-conservative processes are important in preventing the mixed barotropic-baroclinic adjustment described above. Because dissipative processes are most important in the planetary boundary layer, a number of studies naturally emphasized the role of surface processes in preventing baroclinic adjustment. For instance, Swanson and Pierrehumbert (1997) argued that the maintenance of a nonzero temperature gradient in the oceanic storm tracks might be related to the strong convective mixing in the unstable boundary layer, which restores the surface temperature gradient on time scales of the order of a day. Similarly, Stone and Nemet (1996) claimed that the isentropic slope is much closer to the adjusted slope (see section 2.3.2) in the free troposphere, which they attributed to boundarylayer dissipation. This is also consistent with the results of the idealized life cycle calculations of Gutowski et al. (1989), who showed that the adjustment of the mean flow by the baroclinic eddies is very sensitive to the inclusion of surface processes. These authors found that when surface heat fluxes were included, the static stability adjustment was limited with respect to the inviscid case. On the other hand, the inclusion of surface friction prevented the barotropic governor mechanism described above. Likewise, the final state in the spindown experiments of Barry et al. (2000) is very different depending on whether only radiation is turned off or the surface fluxes are also eliminated. Even if boundary-layer dissipation is the main factor preventing baroclinic adjustment, it may be misleading to argue that baroclinic adjustment works better in the interior than at the surface. The reason is that the baroclinic equilibration

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is a fundamentally nonlocal process due to the intimate connection between wave propagation and eddy-mean flow interaction (Edmon et al. 1980; Lindzen 1988). Zurita and Lindzen (2001) argue that to the extent that boundary-layer dissipation prevents thermal homogenization at the surface, it must also prevent PV homogenization in the interior because the positive and negative PV fluxes must balance. This is also consistent with the framework of Schneider (2004), who attributed the observed near balance between interior and boundary PV gradients to uniform diffusivity, rather than to neutrality as in Stone (1978). Additionally, the geostrophic constraints linking momentum and temperature imply that factors other than the local thermal damping could be important for preventing thermal homogenization at the surface. In particular, surface friction could also play a role by damping the surface wind and restoring the vertical shear, as in Robinson’s (2000) model for the zonal index. A more complicated question is whether friction may also be important for restoring the baroclinicity on the time-mean. ZuritaGotor and Lindzen (2004b) found this to be the case for the equilibration of short Charney waves, but this problem is not very realistic because the instability is weak and the vertical Eliassen-Palm (EP) convergence small. In contrast, in the actual atmosphere there is little room for frictional control because the convergence of the EP fluxes is dominated by the vertical component (Edmon et al. 1980). Zurita-Gotor and Lindzen (2006) have recently developed a framework that emphasizes the forced-dissipative nature of the baroclinic equilibrium by relating the wave-mean flow interaction to the local forcing of the mean flow. Their framework also distinguishes between the contribution of frictional and diabatic processes to this forcing, allowing one to investigate the impact of surface friction on thermal homogenization. The authors address this question in the context of the two-layer model. They find that, with stronger friction, the eddy momentum fluxes are enhanced but the net EP convergence changes very little due to the dominance of the vertical EP fluxes. To maintain the same eddy PV fluxes, the vertical shear must then be reduced to compensate for the the fact that there is less barotropic stabilization with stronger friction. However, we note that even if the direct contribution of the eddy momentum flux to the net eddy potential vorticity flux is small, barotropic processes may still affect the spatial configuration of the potential vorticity fluxes (at least their location, if not their intensity). For instance, both the strength of surface friction (Robinson 1997) and the intensity of the Hadley cell (Chang 1995) have been found to affect the position of the extratropical jet and storm tracks, as discussed in chapter 5 in this volume. There are many other studies in the literature using forced-dissipative models with various degrees of complexity. Some of these studies address specific issues raised by theories of baroclinic adjustment, like the supercriticality of the two-layer model (e.g., Stone and Branscome 1992; Welch and Tung 1998a; Zurita-Gotor 2007) or the relevance of interior PV homogenization for the continuous problem (e.g., Zurita-Gotor and

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Lindzen 2004a, 2004b). Others consider the equilibration of idealized but more realistic models, for instance Solomon and Stone (2001a, 2001b) in the quasigeostrophic case and Schneider (2004, 2005) in the non-geostrophic case (also discussed in chapter 3). It is beyond the scope of this paper to review these studies.

2.5. Concluding Remarks One of the most striking aspects of the Earth’s climate is the remarkable robustness of the extratropical thermal structure over the seasonal march of the forcing, whether it is in the form of the midtropospheric isentropic slope (Fig. 2.1; Stone 1978) or the interior PV gradient (Fig. 2.4; Kirk-Davidoff and Lindzen 2000). Historically, this has been associated with neutrality, as the observed isentropic slope is close to the critical value of the two-layer model. The most important objection to this argument is that the two-layer model does not equilibrate that way, and is typically supercritical. Yet, to make things more confusing, Stone and Branscome (1992) found a remarkable result: the two-layer model itself tends to exhibit a very robust supercriticality against changes in the forcing (ξ ≈ 2.4). This result has been somewhat overlooked in the literature, perhaps because unlike in the case ξ = 1 there is not a simple argument that explains where this value comes from. Zurita-Gotor and Lindzen (2006) have recently shown that there is indeed nothing special about this supercriticality. One can change it, though this requires fairly large changes in the forcing (larger, for instance, than considered by Stone and Branscome [1992]). Zurita-Gotor and Lindzen attribute this behavior to the steepness of the empirical PV diffusivity, i.e., to the strong dependence of the diffusivity to changes in the mean state. This implies that only moderate changes in the PV gradient (or supercriticality) are required to obtain large changes in the fluxes. Can this also explain the observations of Stone (1978)? In light of the two-layer results described above, it could be argued that the key reading of Stone’s observations should be that the isentropic slope does not change much, while its precise value, ξ = 1, is irrelevant. Although this is a tempting speculation, the results of Schneider (2004, 2005) over a wide range of parameters suggest that criticality is a stronger constraint in the non-geostrophic framework than one might expect based on the simple steepness argument. These results are presented in the next chapter.

Notes 1. Likewise, in the presence of a momentum convergence at upper levels that strengthens the vertical shear, the temperature gradient is also enhanced by means of an indirect MMC that transports heat equatorward. We do not claim that the MMC is just driven by the eddy heat flux; all momentum and thermal forces (dynamical or not) must be considered.

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References Barry, L., G. C. Craig, and J. Thuburn, 2000: A GCM investigation into the nature of baroclinic adjustment. J. Atmos. Sci., 57, 1141–1155. Barry, L., G. C. Craig, and J. Thuburn, 2002: Poleward heat transport by the atmospheric heat engine. Nature, 415, 774–777. Birner, T., 2006: Fine-scale structure of the extratropical tropopause region. J. Geophys. Res., 111, D04104. Branscome, L. E., 1983: A parameterization of transient eddy heat fluxes and radiative heating in an energy balance model. J. Atmos. Sci., 40, 2508–2521. Branstator, G., 1995: Organization of storm track anomalies by recurring low-frequency circulation anomalies. J. Atmos. Sci., 52, 207–226. Bretherton, F. P., 1966: Critical layer instability in baroclinic flows. Quart. J. Roy. Meteor. Soc., 92, 325–334. Brunt, D., 1941: Physical and Dynamical Meteorology. 2nd ed. London: Cambridge Univ. Press. Cehelsky, P., and K. Tung, 1991: Nonlinear baroclinic adjustment. J. Atmos. Sci., 48, 1930–1947. Chang, E. K. M., 1995: The influence of Hadley circulation intensity changes on extratropical climate in an idealized model. J. Atmos. Sci., 52, 2006–2024. Charney, J. G., 1947: The dynamics of long waves in a baroclinic westerly current. J. Meteor., 4, 135–162. Charney, J. G., and M. E. Stern, 1962: On the stability of internal baroclinic jets in a rotating atmosphere. J. Atmos. Sci., 19, 159–172. Delsole, T., and B. Farrell, 1996: The quasi-linear equilibration of a thermally maintained, stochastically excited jet in a quasi-geostrophic model. J. Atmos. Sci., 53, 1781–1797. Eady, E. T., 1949: Long waves and cyclone waves. Tellus, 1, 33–52. Edmon, H. J., B. J. Hoskins, and M. E. McIntyre, 1980: Eliassen-Palm cross-sections for the troposphere. J. Atmos. Sci., 37, 2600–2616. Emden, R., 1913: Über Strahlungsgleichgewicht und atmosphärische Strahlung. Sitz. d. Bayerischen Akad. d. Wiss., Math.-phys. Klasse, 1, 55–142. Farrell, B. F., and P. Ioannou, 1995: Stochastic dynamics of the midlatitude atmospheric jet. J. Atmos. Sci., 52, 1642–1656. Gates, W. L., and coauthors, 1999: An overview of the results of the Atmospheric Model Intercomparison Project (AMIP I). B. Am. Meteorol. Soc., 80, 29–55. Gold, E., 1913: The isothermal layer of the atmosphere and atmospheric radiation. Proc. Roy. Soc. A, 82, 43–70. Goody, R. M., and Y. L. Yung, 1989: Atmospheric Radiation: Theoretical Basis. New York: Oxford University Press. Green, J. S. A., 1970: Transfer properties of large scale eddies and the general circulation of the atmosphere. Quart. J. Roy. Meteor. Soc., 96, 157–185. Gutowski, W. J., 1985: Baroclinic adjustment and midlatitude temperature profiles. J. Atmos. Sci., 42, 1733–1745. Gutowski, W. J., L. E. Branscome, and D. A. Stewart, 1989: Mean flow adjustment during lifecycles of baroclinic waves. J. Atmos. Sci., 46, 1724–1737. Held, I. M., 1978: The vertical scale of an unstable baroclinic wave and its importance for eddy heat flux parameterizations. J. Atmos. Sci., 35, 572–576. Held, I. M., 1982: On the height of the tropopause and the static stability of the troposphere. J. Atmos. Sci., 39, 412–417.

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44 | Zurita-Gotor and Lindzen Held, I. M., 1999: The macroturbulence of the troposphere. Tellus, 51AB, 59–70. Held, I. M., and D. Andrews, 1983: On the direction of the eddy momentum flux in baroclinic instability. J. Atmos. Sci., 40, 2220–2231. Held, I. M., and V. Larichev, 1996: A scaling theory for horizontally homogeneous, baroclinically unstable flow on a beta plane. J. Atmos. Sci., 53, 946–952. Held, I. M., and T. Schneider, 1999: The surface branch of the zonally averaged mass transport circulation in the troposphere. J. Atmos. Sci., 56, 1688–1697. Held, I. M., and M. J. Suarez, 1976: Simple albedo feedback model of ice-caps. Tellus, 26, 613–629. Holton, J., 1992: An Introduction to Dynamic Meteorology. 3rd ed. Academic Press 511 pp. Hoskins, B. J., 1991: Towards a PV-θ view of the general circulation. Tellus, 43AB, 27–35. Hoskins, B. J., M. E. McIntyre, and A. W. Robertson, 1985: On the use and significance of isentropic potential vorticity maps. Quart. J. Roy. Meteor. Soc., 111, 877–946. Hoskins, B. J., and P. J. Valdes, 1990: On the existence of storm tracks. J. Atmos. Sci., 47, 1854–1864. Ioannou, P., and R. S. Lindzen, 1986: Baroclinic instability in the presence of barotropic jets. J. Atmos. Sci., 43, 2999–3014. James, I. N., 1987: Suppression of baroclinic instability in horizontally sheared flows. J. Atmos. Sci., 44, 3710–3720. Juckes, M. N., 2000: The static stability of the midlatitude troposphere: the relevance of moisture. J. Atmos. Sci., 57, 3050–3057. Kirk-Davidoff, D. B., 1998: The implications of potential vorticity homogenization for climate and climate sensitivity. Ph.D. thesis, Massachusetts Institute of Technology. Kirk-Davidoff, D. B., and R. S. Lindzen, 2000: An energy balance model based on potential vorticity homogenization. J. Climate, 13, 431–448. Larichev, V. D., and I. M. Held, 1995: Eddy amplitudes and fluxes in a homogeneous model of fully-developed baroclinic instability. J. Phys. Oceanogr., 25, 2285–2297. Lindzen, R. S., 1988: Instability of plane parallel shear flow (toward a mechanistic picture of how it works). Pure. Appl. Geophys., 126, 103–121. Lindzen, R. S., 1993: Baroclinic neutrality and the tropopause. J. Atmos. Sci., 50, 1148– 1151. Lindzen, R. S., 1994: The Eady problem for a basic state with zero PV gradient but beta not equal zero. J. Atmos. Sci., 51, 3221–3226. Lindzen, R. S., and B. Farrell, 1980: The role of polar regions in global climate, and a new parameterization of global heat-transport. Mon. Wea. Rev., 108, 2064–2079. Lindzen, R. S., B. F. Farrell, and K. K. Tung, 1980: The concept of wave overreflection and its application to baroclinic instability. J. Atmos. Sci., 37, 44–63. Lindzen, R. S., A. J. Rosenthal, and B. Farrell, 1983: Charney’s problem for baroclinic instability applied to barotropic instability. J. Atmos. Sci., 40, 1029–1034. Malkus, W. V. R., 1979: Turbulent velocity profiles from stability criteria. J. Fluid. Mech., 90, 401–414. Malkus, W. V. R., 2003: Borders of disorder: in turbulent channel flow. J. Fluid. Mech., 489, 185–198. Manabe, S., and R. J. Stouffer, 1980: Sensitivity of a global climate model to an increase of CO2 concentration in the atmosphere. J. Geophys. Res., 85, 5529–5554. Manabe, S., and R. Strickler, 1964: On the thermal equilibrium of the atmosphere with a convective adjustment. J. Atmos. Sci., 21, 361–385. Manabe, S., and T. B. Terpstra, 1974: The effects of mountains on the general circulation of the atmosphere as identified by numerical experiments. J. Atmos. Sci., 31, 3–42.

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Baroclinic Adjustment | 45 Morgan, M. C., 1994: Observationally and dynamically determined basic state for the study of synoptic scale waves. Ph.D. thesis, Massachusetts Institute of Technology. Nakamura, N., 1993: An illustrative model of instabilities in meridionally and vertically sheared flows. J. Atmos. Sci., 50, 357–375. Nakamura, N., 1999: Baroclinic-barotropic adjustments in a meridionally wide domain. J. Atmos. Sci., 56, 2246–2260. Nielsen, J. E., and M. R. Schoeberl, 1984: A numerical simulation of barotropic instability. Part II: Wave-wave interaction. J. Atmos. Sci, 41, 2869–2881. Pavan, V., and I. M. Held, 1996: The diffusive approximation for eddy fluxes in baroclinically unstable jets. J. Atmos. Sci., 52, 1262–1272. Phillips, N., 1954: Energy transformations and meridional circulations associated with simple baroclinic waves in a two-level quasigeostrophic model. Tellus, 6, 273–286. Pocinki, L., 1955: Stability of a simple baroclinic flow with horizontal shear. A. F. Cambridge Research Center, Research Paper No. 38. Prinn, R., 1977: On the radiative damping of atmospheric waves. J. Atmos. Sci., 34, 1386–1401. Robinson, W. A., 1997: Dissipation dependence of the jet latitude. J. Climate, 10, 176–182. Robinson, W. A., 2000: A baroclinic mechanism for the eddy feedback on the zonal index. J. Atmos. Sci., 57, 415–422. Salmon, R. S., 1980: Baroclinic instability and geostrophic turbulence. Geophys. Astrophys. Fluid Dyn., 15, 167–211. Schneider, T., 2004: The tropopause and thermal stratification in the extratropics of a dry atmosphere. J. Atmos. Sci., 61, 1317–1340. Schneider, T., 2005: Zonal momentum balance, potential vorticity dynamics and mass fluxes on near-surface isentropes. J. Atmos. Sci., 62, 1884–1900. Schoeberl, M. R., and R. S. Lindzen, 1984: A numerical simulation of barotropic instability: 1. Wave-mean flow interaction. J. Atmos. Sci., 41, 1368–1379. Simmons, A. J., and B. J. Hoskins, 1978: The lifecycles of some nonlinear baroclinic waves. J. Atmos. Sci., 35, 414–432. Solomon, A. B., and R. S. Lindzen, 2000: The impact of resolution on a numerical simulation of barotropic instability. J. Atmos. Sci., 57, 3799–3816. Solomon, A. B., and P. H. Stone, 2001a: Equilibration in an eddy resolving model with simplified physics. J. Atmos. Sci., 58, 561–574. Solomon, A. B., and P. H. Stone, 2001b: The sensitivity of an intermediate model of the midlatitude troposphere’s equilibrium to changes in radiative forcing. J. Atmos. Sci., 58, 2395– 2410. Song, Y., and N. Nakamura, 2000: Eady instability of isolated baroclinic jets with meridionally varying tropopause height. J. Atmos. Sci., 57, 46–65. Stone, P. H., 1972: A simplified radiative-dynamical model for the static stability of rotating atmospheres. J. Atmos. Sci., 29, 405–418. Stone, P. H., 1978: Baroclinic adjustment. J. Atmos. Sci., 35, 561–571. Stone, P. H., and L. Branscome, 1992: Diabatically forced, nearly inviscid eddy regimes. J. Atmos. Sci., 49, 355–367. Stone, P. H., and D. Miller, 1980: Empirical relations between seasonal changes in meridional temperature gradients and meridional fluxes of heat. J. Atmos. Sci., 37, 1708–1721. Stone, P. H., and B. Nemet, 1996: Baroclinic adjustment: a comparison between theory, observations, and models. J. Atmos. Sci., 53, 1663–1674.

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46 | Zurita-Gotor and Lindzen Sun, D. Z., and R. S. Lindzen, 1994: A PV view of the zonal mean distribution of temperature and wind in the extratropical troposphere. J. Atmos. Sci., 51, 757–772. Swanson, K., 2001: Upper-tropospheric potential vorticity fluctuations and the dynamical relevance of the time mean. J. Atmos. Sci., 58, 1815–1826. Swanson, K., and R. T. Pierrehumbert, 1997: Lower-tropospheric heat transport in the Pacific storm track. J. Atmos. Sci., 54, 1533–1543. Thompson, P. D., 1961: Numerical Weather Analysis and Prediction. New York: Macmillan. Thorncroft, C. D., B. J. Hoskins, and M. E. McIntyre, 1993: Two paradigms of baroclinic-wave life-cycle behavior. Q. J. Meteor. R. Soc., 119, 17–55. Thuburn, J., and G. C. Craig, 1997: GCM tests of theories for the height of the tropopause. J. Atmos. Sci., 54, 869–882. Welch, W., and K. Tung, 1998a: Nonlinear baroclinic adjustment and wave selection in a simple case. J. Atmos. Sci., 55, 1285–1302. Welch, W., and K. Tung, 1998b: On the equilibrium spectrum of transient waves in the atmosphere. J. Atmos. Sci., 55, 2833–2851. Zurita, P., and R. S. Lindzen, 2001: The equilibration of short Charney waves: implications for PV homogenization in the extratropical troposphere. J. Atmos. Sci., 58, 3443–3462. Zurita-Gotor, P., 2002: Inhomogeneous potential vorticity homogenization and equilibration in simple models of baroclinic instability with implications for the extratropical circulation. Ph.D. thesis, Massachusetts Institute of Technology. Zurita-Gotor, P., 2007: The relation between baroclinic adjustment and turbulent diffusion in the two-layer model. J. Atmos. Sci., 64, 1284–1300. Zurita-Gotor, P., and R. S. Lindzen, 2004a: Baroclinic equilibration and the maintenance of the momentum balance. Part I: a barotropic analog. J. Atmos. Sci., 61, 1469–1482. Zurita-Gotor, P., and R. S. Lindzen, 2004b: Baroclinic equilibration and the maintenance of the momentum balance. Part II: 3D results. J. Atmos. Sci., 61, 1483–1499. Zurita-Gotor, P., and R. S. Lindzen, 2006: A generalized momentum framework for looking at baroclinic circulations. J. Atmos. Sci., 63, 2036–2055.

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Chapter 3

The Thermal Stratification of the Extratropical Troposphere Tapio Schneider

3.1. Introduction Phillips’s (1956) foundational paper on the general circulation of the atmosphere contains a list “of the most striking features of the atmosphere, calling for explanation.” The first feature on the list is the “general increase of entropy (potential temperature) with height and the existence of the stratosphere,” or, in other words, the thermal stratification of the atmosphere and the existence of the tropopause. Phillips’s paper marks the beginning of general circulation studies as we know them now—experimenting with numerical models of various complexity to develop accounts of the maintenance and variability of observed atmospheric features. But despite decades of experimentation with general circulation models, the dynamics that determine the thermal stratification and the existence of the tropopause in the extratropics are only beginning to be understood. One reason why our understanding of the extratropical thermal stratification is incomplete is that in quasigeostrophic models, such as that used by Phillips, the thermal stratification of the atmosphere is taken to be fixed. Quasigeostrophic models have formed the basis for studies of extratropical dynamics over the past decades and have led to some of the most important insights in dynamical meteorology (e.g., the theory of Rossby waves and baroclinic instability). But because they do not allow dynamics to affect the thermal stratification, quasigeostrophic models are poorly suited for studying the processes that determine the thermal stratification. With today’s computational resources, however, we are in a position to study the maintenance and variability of the extratropical thermal stratification by systematic experimentation with general circulation models that are not based on quasigeostrophic assumptions. This chapter discusses the dynamical mechanisms responsible for the maintenance and variability of the extratropical thermal stratification and tropopause in the zonal mean. Figure 3.1 shows the zonal-mean temperature lapse rate of Earth’s atmosphere for boreal winter and summer. The zonal-mean lapse rate in the free

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Figure 3.1. Zonal-mean temperature lapse rate −∂zT (K km−1) for the DJF and JJA seasons according to reanalysis data for the years 1980–2001 provided by the European Centre for Medium-Range Weather Forecasts (ERA-40 data; see Uppala et al. 2005). Negative contours are dashed. The thick line marks the tropopause, determined as a 2 K km−1 isoline of the lapse rate. The vertical coordinate is pressure normalized by surface pressure, σ = p/ ps .

troposphere is relatively uniform (about 6.5 K km−1) and varies only weakly with season—observations that motivated the assumption of a fixed thermal stratification in quasigeostrophic theory. Regions of smaller lapse rate (statically more stable stratification) are seen near the surface in the subtropics and in high latitudes, particularly in winter. At the tropopause, the lapse rate decreases, in many regions to zero or less, marking the transition from the troposphere to the more stably stratified stratosphere. What distinguishes the troposphere and stratosphere kinematically is that the bulk of the entropy the atmosphere receives by the heating at the surface is redistributed within the troposphere, whereas only a small fraction of it reaches the stratosphere. In fluid-dynamical parlance, the troposphere is the caloric boundary layer of the atmosphere; the tropopause is the top of this boundary layer. The question of what determines the thermal stratification is the question of what determines the dynamical equilibrium between radiative processes and dynamical entropy transport. If one accepts as an observational fact that the redistribution of the entropy received at the surface is largely confined to a well-defined boundary layer, the troposphere, the height of the tropopause can be determined as in classical boundary-layer theories: as the minimum height up to which the entropy redistribution must extend for the flow to satisfy largescale constraints such as energy and momentum balance. Section 3.2 discusses the general form of large-scale constraints on the thermal stratification and tropopause height, arising from radiative and dynamical considerations. Sections 3.3–3.5 discuss dynamical constraints that respectively take slantwise moist convection, moist convection coupled to baroclinic eddies, and baroclinic eddies as central for determining the thermal stratification and tropopause height. Section 3.6 presents simulations with an idealized general circulation model (GCM) that show that an atmosphere can have different dynamical regimes distinguishable according to whether convection or baroclinic eddies dominate the entropy redistribution between surface and tropopause. And section 3.7 concludes this chapter with a discussion of the results presented and of open questions.

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Extratropical Thermal Stratification | 49 40

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Figure 3.2. Temperature in radiative equilibrium (solid line) and in dynamical equilibrium with fixed tropospheric lapse rate  = 6.5 K km−1 (dashed line). The arrow marks the ground temperature in radiative equilibrium, which is greater than the surface air temperature (Tg ≈ 297 K, Ts ≈ 285 K). The ground temperature in dynamical equilibrium is taken to be equal to the surface air temperature (Tg = Ts = 280 K). (Calculations courtesy Paul O’Gorman.)

3.2. Radiative and Dynamical Constraints Held (1982) suggested distinguishing between radiative and dynamical constraints on the thermal stratification and tropopause height. Radiative constraints express the balance of incoming and outgoing radiant energy fluxes in atmospheric columns, plus any dynamical energy flux divergences in the columns. Dynamical constraints express balance conditions based on dynamical considerations, such as that moist convection maintains the thermal stratification close to a moist adiabat (see chapter 7 in this volume) or that baroclinic eddy fluxes satisfy balance conditions derived from the mean entropy and zonal momentum balances. In the simplest model of dynamical equilibrium in an atmospheric column, going back to the concept of radiative-convective equilibrium (cf. Gold 1909; Milne 1922; Manabe and Strickler 1964; Manabe and Wetherald 1967), the dynamical constraint determines a constant tropospheric lapse rate and the radiative constraint determines the tropopause height that is consistent with the lapse rate and with a boundary condition, for example, a given surface temperature. Given a tropospheric lapse rate  and a surface air temperature Ts , taken to be equal to the ground temperature Tg , the radiative constraint determines the tropopause height Ht as the minimum height z at which the temperature profile Ts − z matches a radiative equilibrium temperature profile extending from the height Ht upward (cf. Held 1982; Thuburn and Craig 2000). Figure 3.2 shows such a dynamical equilibrium temperature profile with fixed tropospheric lapse rate and a corresponding radiative equilibrium temperature profile, computed with the column radiation model of the National Center for Atmospheric Research (Kiehl et al. 1996). In dynamical equilibrium, the tropospheric lapse rate is taken to be  = 6.5 K km−1, and the ground and surface air temperatures are taken to

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be Tg = Ts = 280 K. Following Thuburn and Craig (2000), the relative humidity is taken to be constant (50% relative to saturation over liquid water), subject to the constraint that the specific humidity in the stratosphere does not increase with height. Typical midlatitude concentrations are specified for other absorbers. Radiative effects of clouds are ignored, the surface albedo is 0.1, and the solar irradiance incident at the top of the atmosphere is taken to be 260 W m−2, a value about 15% lower than typical annual-mean irradiances in midlatitudes to compensate for the missing albedo of clouds. In radiative equilibrium, the absorber concentrations, solar irradiance at the top of the atmosphere, and albedo are taken to be the same as in dynamical equilibrium, but the ground and surface air temperatures are determined by the conditions of radiative equilibrium, rather than being specified as a boundary condition as in dynamical equilibrium. The dynamical equilibrium profile implies dynamical cooling in the lower troposphere and dynamical warming in the upper troposphere, associated with heat transport, for example, by convection or by baroclinic eddies. Above the tropopause, the dynamical equilibrium is a purely radiative equilibrium. In radiative-convective equilibrium, the outgoing longwave radiant flux is equal to that in radiative equilibrium (where it is equal to the absorbed solar flux); that is, convection redistributes enthalpy only vertically. Unlike in radiative-convective equilibrium, in the dynamical equilibrium in Fig. 3.2 the outgoing longwave flux is not equal to that in radiative equilibrium. The outgoing longwave fluxes for the radiative and dynamical equilibrium profiles in Fig. 3.2 are 228 W m−2 and 239 W m−2, respectively, implying a net flux of enthalpy into the atmospheric column in dynamical equilibrium relative to radiative equilibrium. This implied enthalpy flux may be associated, for example, with convergence of meridional eddy enthalpy fluxes or with fluxes across the lower boundary of the atmosphere. An additional constraint is necessary to relate the temperature profiles in radiative and dynamical equilibrium to each other uniquely. The additional constraint can be an energy-balance condition at the surface, relating the surface temperature to meridional enthalpy fluxes and radiant fluxes. Figure 3.3 shows the tropopause height determined according to the radiative constraint as a function of tropospheric lapse rate and surface temperature. As in Fig. 3.2 and in similar calculations by Thuburn and Craig (2000), the ground and surface air temperatures are taken to be equal, the relative humidity is taken to be constant, subject to the constraint that the specific humidity does not increase with height, and the concentrations of absorbers other than water vapor are taken to be fixed at typical midlatitude values. Figure 3.3 shows that for fixed lapse rate, the tropopause height increases with increasing surface temperature; for fixed surface temperature, the tropopause height increases with decreasing lapse rate (cf. Thuburn and Craig 2000). The dependence of tropopause height on surface temperature and tropospheric lapse rate can be understood qualitatively by considering a semigray atmosphere (transparent to solar radiation and gray for longwave radiation) with an optically thin stratosphere. The stratosphere, according to the radiative constraint, is assumed to

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Figure 3.3. Tropopause height (km) determined according to the radiative constraint as a function of tropospheric lapse rate and surface temperature, with fixed relative humidity. Parameters and concentrations of absorbers other than water vapor as in Fig. 3.2. (Calculations courtesy Paul O’Gorman.)

be in radiative equilibrium. For a semigray atmosphere, the condition for radiative equilibrium in the two-stream approximation is B = (U + D)/2, where B = σbT 4 is the black-body emittance with Stefan-Boltzmann constant σb, and U and D are upwelling and downwelling fluxes of longwave radiant energy (Goody and Yung 1989, chapter 9). If the stratosphere is optically thin, the downwelling longwave flux D in it can be neglected, implying B ≈ U/2 at the tropopause and in the stratosphere. It follows that the upwelling longwave flux U in an optically thin stratosphere in radiative equilibrium is approximately constant, equal to the outgoing longwave flux, and dependent only on properties of the troposphere (Thuburn and Craig 2000). The stratosphere is approximately isothermal, with a temperature Tt ≈ (U/2σb)1/4 that matches the tropospheric temperature profile Ts − z at the tropopause height Ht = (Ts − Tt)/ . In terms of the emission temperature Te and of the emission height He = (Ts − Te)/  at which the tropospheric temperature is equal to the emission temperature, the upwelling longwave flux at the tropopause is U ≈ σbTe4, the tropopause temperature is Tt ≈ αTe with α = 2−1/4 ≈ 0.84 (a relation going back to Schwarzschild [1906]), and the tropopause height is Ts [3.1] + α He .  If the emission height is fixed, the tropopause height increases with increasing surface temperature and with decreasing lapse rate. The rate of increase of tropopause height with surface temperature, ∂ Ht/∂ Ts , decreases with increasing lapse rate; the rate of decrease of tropopause height with lapse rate, −∂ Ht/∂, increases with increasing surface temperature and decreasing lapse rate, qualitatively as seen in Fig. 3.3.1 In the radiative transfer model underlying Fig. 3.3, however, the emission height is not fixed but varies with water vapor concentrations, among other factors. Increasing the surface temperature while keeping the relative humidity and lapse rate fixed increases Ht ≈ (1 − α)

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water vapor concentrations and with them the emission height (see, e.g., Held and Soden 2000), which, according to the estimate (3.1), strengthens the dependence of tropopause height on surface temperature. Similarly, decreasing the lapse rate while keeping the relative humidity and surface temperature fixed increases water vapor concentrations above the surface, which increases the emission height and, according to the estimate (3.1), likewise strengthens the dependence of tropopause height on lapse rate. Such water vapor feedbacks account, for example, for the nonlinearity of the increase of tropopause height with surface temperature seen in Fig. 3.3 (cf. Thuburn and Craig 1997). To be sure, the assumption of a semigray atmosphere with an optically thin stratosphere is not quantitatively accurate for the radiative transfer model underlying Figs. 3.2 and 3.3. For example, absorption of solar radiation by ozone affects the radiative balances near the tropopause. But the tropopause height estimate (3.1) based on this assumption correctly describes the qualitative dependence of tropopause height on surface temperature and lapse rate and represents a rough quantitative estimate. Using representative midlatitude values for surface temperature Ts = 280 K, lapse rate  = 6.5 K km−1, and emission height He = 4 km gives the tropopause height Ht = 10 km, which is roughly consistent with the dynamical equilibrium profile in Fig. 3.2 and with the height of the observed extratropical tropopause in Fig. 3.1. In GCM simulations, radiative constraints of the kind illustrated in Fig. 3.3 account latitude-by-latitude for variations of the extratropical zonal-mean tropopause height with surface temperature and with mean tropospheric lapse rate (Thuburn and Craig 1997, 2000). An exception are polar latitudes with strong temperature inversions near the surface (Fig. 3.1). There, the tropospheric thermal stratification is not well approximated by a constant lapse rate, and the radiative constraint needs to be modified. Taking the lapse rate to be constant, however, was merely done for simplicity. The radiative constraint can easily be modified by replacing the temperature profiles with constant lapse rates by families of temperature profiles with more complicated vertical structures. The way in which the tropopause height is determined given a tropospheric temperature profile and a boundary condition remains unaffected. The essential assumption remains that the stratosphere is approximately in radiative equilibrium. Although Earth’s stratosphere is not in radiative equilibrium (there is a stratospheric circulation), the deviations from radiative equilibrium are small enough to be negligible in a first approximation of the height of the extratropical tropopause. Given the surface temperature and a tropospheric lapse rate, or a different boundary condition and a more complicated tropospheric temperature profile, we can thus determine the tropopause height. In the Tropics, it is well established that moist convection dominates the entropy redistribution between surface and tropopause and maintains the thermal stratification in the interior troposphere close to a moist adiabat (see, e.g., Stone and Carlson [1979], Xu and Emanuel [1989], and chapter 7 in this volume). This provides a dynamical constraint on the tropical thermal stratification and tropopause height. The radiative constraint determines the tropical tropopause height

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as the minimum height up to which the moist adiabat and entropy redistribution by moist convection must extend so that the atmosphere above the tropopause can be approximately in radiative equilibrium.2 The difficulty in accounting for the extratropical thermal stratification and tropopause height has its roots in the lack of dynamical constraints of comparable simplicity for the extratropics. For a closed theory, we need two constraints in addition to the radiative constraint to relate the three unknowns: tropopause height, surface temperature, and tropospheric lapse rate or temperature profile. Here we take the surface temperature as given, returning to the question of what determines it in section (3.7), and discuss dynamical constraints relating it to the tropopause height and to the tropospheric lapse rate or temperature profile.

3.3. Slantwise Convection As in the Tropics, moist convection maintains the thermal stratification close to a moist adiabat in some extratropical regions, for example in continental regions in summer. In the zonal mean and particularly in winter, however, comparison of the lapse rate (Fig. 3.1) with the moist adiabatic lapse rate (Fig. 3.4) shows that the extratropical lower troposphere is more stably stratified than a moist adiabat (Stone and Carlson 1979). At least in winter, mechanisms other than or in addition to moist convection appear to determine the extratropical thermal stratification and tropopause height. Even when the atmosphere is stable with respect to vertical displacements, displacements along slanted surfaces may be unstable with respect to moist symmetric instability (Bennetts and Hoskins 1979; Emanuel 1983a, 1983b). The ensuing slantwise convection may play a role in determining the thermal stratification and tropopause height. Consider inviscid and reversible-adiabatic axisymmetric displacements of a tube of air that extends around a latitude circle and is embedded in a steady balanced flow. The tube is assumed to be thin, so that its displacements do not modify the pressure field of the environment. In inviscid axisymmetric displacements, the tube conserves

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its absolute angular momentum per unit mass about the axis of planetary rotation, m = (a cos φ + u)a cos φ in the approximation of the atmosphere as a thin spherical shell with radius a, angular velocity of planetary rotation , latitude φ, and eastward wind component u. In reversible-adiabatic displacements, the tube conserves its entropy, with entropy understood as saturated moist entropy where condensation/freezing occurs. If, during an upward displacement along a possibly slanted surface on which the angular momentum m of the environment is constant, a tube of air encounters a region with entropy smaller than that of the tube, the tube becomes positively buoyant relative to its environment (assuming that buoyancy fluctuations are proportional to entropy fluctuations), and the displacement is convectively unstable. The tube can continue to move inviscidly upward along the angular momentum surface; slantwise convection ensues. Similarly, if, during a poleward displacement along a surface on which the entropy of the environment is constant (a moist isentrope), a tube of air encounters a region with angular momentum greater than that of the tube, the displacement is inertially unstable. The tube can continue to move adiabatically poleward along the isentrope; slantwise convection likewise ensues. Ordinary convective and inertial instabilities are special cases of moist symmetric instability in barotropic flows, in which angular momentum surfaces are vertical and moist isentropes are horizontal. The concept of slantwise convection unifies inertial and buoyancy forces and inertial and convective instabilities. If angular momentum decreases poleward horizontally and if moist entropy increases upward vertically so that the flow is stable with respect to ordinary convective and inertial instabilities, the flow may nonetheless be unstable with respect to slantwise convection if, somewhere in the atmosphere, the slope of moist isentropes is steeper than or equal to the slope of angular momentum surfaces (Holton 2004, chapter 9; Emanuel 1994, chapter 12). Slantwise convection may occur if there is positive available potential energy for slantwise convection (Emanuel 1983b), which slantwise convection, possibly generating three-dimensional turbulent flows, would reduce by bringing entropy and angular momentum surfaces closer to alignment (Thorpe and Rotunno 1989). In a statistical equilibrium, one would expect that slantwise convection leads to an atmospheric state that is nearly neutral (cf. analogous considerations for ordinary convection in chapter 7 in this volume). In a neutral state, entropy and angular momentum surfaces are aligned, that is, temperature lapse rates along angular momentum surfaces are moist adiabatic. If slantwise convection were to determine the static stability of the extratropical atmosphere, the alignment of entropy and angular momentum surfaces would provide a dynamical constraint on the tropopause height and thermal stratification. The dynamical constraint would relate the temperature lapse rate to the specific (or relative) humidity and to the surface temperature and its derivatives, which, through thermal wind balance, determine the vertical wind shear and thus the slope of angular momentum surfaces.

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Extratropical Thermal Stratification | 55 DJF

JJA 350

340 340

340

340

0

28

0

0

0

29 0.8

350

30

0.2 Sigma

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50°

−50°

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Figure 3.5. Saturation equivalent potential temperature (black, contour interval 10 K) and absolute angular momentum m (gray, contour interval 0.1a 2) according to ERA-40 data for 1980–2001. The thick line marks the tropopause.

Ascending regions of extratropical cyclones are observed to be nearly neutral with respect to slantwise convection, suggesting that slantwise convection may occur there, with the displaced tube of air extending along a front, rather than around a latitude circle, and with the angular momentum to be considered correspondingly changed (Emanuel 1983b, 1988). But analyses are lacking that would establish the frequency and large-scale effects of slantwise convection in baroclinic eddies. Figure 3.5 shows zonal-mean moist adiabats (saturation equivalent potential temperature surfaces) and angular momentum surfaces calculated from reanalysis data. The mean stratification of the interior tropical troposphere is close to a moist adiabat. In summer in the Northern Hemisphere midlatitudes (∼ 45◦ N), the mean stratification appears to be nearly neutral with respect to slantwise or ordinary moist convection. Higher latitudes and the Southern Hemisphere midlatitudes, in the zonal mean, appear to be stable with respect to slantwise or ordinary moist convection. However, slantwise convection, if it occurs, is a mesoscale phenomenon that is typically neither resolved nor parameterized in GCMs, so reanalysis data have to be used cautiously in establishing the potential importance of slantwise convection for the extratropical thermal stratification and tropopause height. More detailed analyses with different data may yield different results. And even if it is not a dominant process for the zonal-mean stratification, slantwise convection may still play a role regionally, for example, over continents in summer.

3.4. Moist Convection Coupled to Baroclinic Eddies Moist processes such as moist convection and latent-heat release in large-scale condensation doubtlessly influence the extratropical static stability and tropopause height, likely as processes that are coupled to baroclinic eddies. The dynamics of baroclinic eddies in the presence of moisture, however, are poorly understood. The linear stability, nonlinear life cycles, and turbulent statistically steady states of moist baroclinic eddies

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have been investigated in several studies, using models idealized to various degrees and with various ways of coupling moist processes to eddy dynamics (e.g., Mak 1982; Bannon 1986; Emanuel et al. 1987; Gutowski et al. 1992; Fantini 1993; Lapeyre and Held 2004). But there is no general theory accounting for the scales of baroclinically unstable waves in the presence of moisture, or for the relative importance of dry and moist processes in statistically steady states of baroclinic eddies. Absent a general theory, Juckes (2000) proposed heuristic arguments for how moist convection coupled to baroclinic eddies may determine the extratropical thermal stratification. Moist convection in baroclinic eddies occurs preferentially in regions of low static stability—in the warm sectors of cyclones. To the extent that moist convection reaches the tropopause and prevents the stratification from becoming significantly less stable than a moist adiabat, the minimum of the bulk moist stability ve = θet − θes , the equivalent potential temperature difference between tropopause (θet) and surface (θes ), is approximately zero. Juckes assumes that the distribution of bulk moist stabilities in baroclinic eddies is well characterized by this minimum (zero) and the standard ¯ ve = θ¯et − θ¯es deviation, and that the temporal- and zonal-mean bulk moist stability  can be estimated as the minimum plus a multiple of the standard deviation, ¯ ve ∼ min(ve) + d std(ve), 

[3.2]

where d is an empirical factor. (For this assumption to result in a predictive theory, the empirical factor d needs to be universal, holding for a range of standard deviations. It is unclear whether this is an adequate assumption for distributions of bulk moist stabilities. For example, it is generally not an adequate assumption for log-normal distributions, according to which one might expect bulk moist stabilities to be distributed. It would be an adequate assumption for exponential distributions.) Juckes assumes fluctuations of bulk moist stability to be generated by approximately adiabatic meridional advection of equivalent potential temperature at the tropopause and at the surface by baroclinic eddies, such that the standard deviation scales as p

std(ve) ∼ L e|∂ y θ e |,

[3.3] p

where L e is an eddy length scale (or the width of the storm track) and ∂ y θ e is the mean meridional equivalent potential temperature gradient at constant pressure. (Juckes actually proposed that the standard deviation scales with the potential temperature gradient in place of the equivalent potential temperature gradient; however, in the presence of moisture, it is equivalent potential temperature rather than potential temperature that is materially conserved in adiabatic displacements, so using the equivalent potential temperature gradient appears to be preferable.) Juckes proposes that the equivalent potential temperature gradient is to be evaluated at a midtropospheric pressure level. If the standard deviation of bulk moist stability is dominated by near-surface fluctuations of equivalent potential temperature, however, it may be more appropriate to evaluate the

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Bulk Moist Stability (K)

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40

30

30 40 50 Equivalent Potential Temperature Contrast (K)

¯ ev and Figure 3.6. Extratropical mean bulk moist stabilities  σ meridional equivalent potential temperature contrasts d L e|∂ y θ e | according to ERA-40 data for 1980–2001. The length scale d L e was chosen to be constant (0.9a with Earth’s radius a). The bulk stabilities and equivalent potential temperature gradients are seasonal and zonal averages over latitudes between 35◦ and 65◦ in the Northern and Southern Hemisphere, respectively, with the equivalent potential temperature gradient evaluated at the midtropospheric level σ = 0.5 (cf. Juckes 2000). Bulk moist stabilities are equivalent potential temperature differences between the tropopause and the near-surface level σ = 0.9. Each plotting symbol corresponds to a year and a season (squares: DJF; diamonds: MAM; circles: JJA; triangles: SON). Filled symbols represent Northern Hemisphere averages; open symbols represent Southern Hemisphere averages.

equivalent potential temperature gradient near the surface. Combining the estimates (3.2) and (3.3) and assuming that the minimum bulk moist stability is approximately zero, Juckes obtains the estimate ¯ ve ∼ d L e|∂ y θ ep|, 

[3.4]

which relates the mean bulk moist stability to the meridional equivalent potential p temperature contrast d L e|∂ y θ e |.3 Figure 3.6, similar to Fig. 6 of Juckes (2000) but with equivalent potential temperature gradients at a sigma level in place of potential temperature gradients at a pressure level, shows the relation between extratropical mean bulk moist stabilities and meridional equivalent potential temperature contrasts for different seasons and different years according to reanalysis data. The length scale d L e was chosen to be constant (eddy length scales do not vary strongly interannually or seasonally in Earth’s atmosphere). Consistent with Juckes’s arguments, there is a positive correlation between seasonal

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variations of bulk moist stabilities and meridional equivalent potential temperature contrasts. However, interannual variations of bulk moist stabilities and meridional equivalent potential temperature contrasts correlate less clearly. (These results do not change substantially if equivalent potential temperature contrasts are evaluated at different levels or if averages are taken over different extratropical latitude zones.) If it can be more generally substantiated, and provided that an estimate of the length scale d L e is available, the estimate (3.4) of the bulk moist stability provides a dynamical constraint on the extratropical tropopause height and thermal stratification, relating the temperature lapse rate and tropopause height, via the bulk moist stability, to the distribution of specific (or relative) humidity and to the meridional gradient of equivalent potential temperature. Although seasonal variations of the bulk moist stability are approximately linearly related to meridional equivalent potential temperature contrasts, it is an open question whether Juckes’s proposed mechanism is indeed acting. The next section shows that dynamical constraints of a structure similar to that of the estimate (3.4) also arise from considerations of balance conditions on dry baroclinic eddies.

3.5. Baroclinic Eddies Another dynamical constraint on the extratropical tropopause height and thermal stratification can be derived from balance conditions on baroclinic eddy fluxes without considering moist processes. To derive it, we need to estimate the vertical extent of the baroclinic entropy flux, which can modify the thermal stratification and tropopause height. The vertical extent of the baroclinic entropy flux will give a lower bound on the tropopause height. With the notion of the troposphere as the atmospheric layer within which entropy received at the surface is redistributed, the tropopause cannot lie below the height up to which significant baroclinic entropy fluxes extend, although it may lie above it, for example, if convective entropy fluxes extend to higher altitudes than baroclinic entropy fluxes. The vertical extent of the baroclinic entropy flux will depend on the meridional surface potential temperature gradient and on a measure of the tropospheric static stability, and will thus provide a dynamical constraint on the tropopause height and thermal stratification. Since considerations of potential vorticity dynamics form a good basis of descriptions of the interaction between a mean flow and rapid, nearly adiabatic and inviscid fluctuations such as occur in baroclinic eddies, and since the potential vorticity flux has components only along isentropes, but not across isentropes (Haynes and McIntyre 1987, 1990), it is convenient to consider the interaction between the eddies and the mean flow and thermal stratification in isentropic coordinates. In isentropic coordinates, entropy fluxes correspond to mass fluxes along isentropes, so the question of the vertical extent of the baroclinic entropy flux is the question of the vertical extent of the baroclinic

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mass flux along isentropes. Hence we need to discuss the structure of the mass flux along isentropes. We will ignore moist processes to investigate the extent to which dry extratropical dynamics alone can provide constraints on the thermal stratification and tropopause height and can provide limits to which any future theory accounting for moist processes must be reducible. The developments, based on a series of papers (Held and Schneider 1999; Schneider 2004, 2005; Schneider and Walker 2006), will be discussed in some detail to expose assumptions and approximations and to show how they differ from similar quasigeostrophic developments (cf. chapter 2 in this volume).

3.5.1. Balance Condition on Mean Mass Fluxes The mean entropy balance in isentropic coordinates is equivalent to the mean mass balance, ∗

∂ y(ρ¯ θ v ∗) + ∂θ(ρ¯ θ Q ) = 0,

[3.5]

where the meridional derivative is to be understood as a derivative at constant potential temperature θ , Q = Dθ/Dt is the diabatic heating rate, and overbars denote ∗ temporal and zonal means: (·) along isentropes, and (·) = (ρθ ·)/ρ¯ θ along isentropes weighted by the isentropic density ρθ = −g −1∂θ p H(θ − θs ), where H(·) is the Heaviside step function and the subscript s marks surface quantities (cf. Andrews et al. 1987, chapter 3). The Heaviside step function represents the convention that the isentropic density vanishes on isentropes “inside” the surface, that is, on isentropes with potential temperature θ < θs (x, y, t) less than the instantaneous surface potential temperature. This convention ensures that the mean mass balance (3.5) holds throughout the flow domain in isentropic coordinates, including the surface layer of isentropes that, at any given latitude, sometimes intersect the surface (cf. Lorenz 1955). The meridional velocity v = vg + va can be decomposed into a geostrophic component vg and an ageostrophic component va, where the geostrophic component in isentropic coordinates is vg = f −1∂x M H(θ − θs ), with Montgomery streamfunction (dry static energy) M = c pT + g z and with the zonal derivative understood as a derivative at constant potential temperature. The Heaviside step function represents the convention that the geostrophic meridional velocity vanishes on isentropes inside the surface, a convention that is not necessary at this stage but will be used later. This decomposition leads to the alternative form of the mean mass balance ∗

∂ y(ρ¯ θ v ∗g ) = −∂θ(ρ¯ θ Q ) − ∂ y(ρ¯ θ v a∗).

[3.6]

Divergence of geostrophic mass fluxes along isentropes is balanced by vertical convergence of diabatic mass fluxes and by convergence of ageostrophic mass fluxes along isentropes.

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60 | Tapio Schneider

Pot. Temperature (K)

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13

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Figure 3.7. Isentropic mass fluxes in simulation with idealized GCM. (a) Mass flux streamfunction (109 kg s−1); (b) meridional mass flux ∂θ = 2πa cos(φ)ρ¯ θ v ∗ with its (c) geostrophic and (d) ageostrophic components (109 kg s−1 K−1). The dotted lines represent the 10%, 50%, and 90% isolines of the cumulative distribution of surface potential temperatures.

Figure 3.7, based on a simulation with the idealized GCM described in section 3.6, illustrates the mean mass balance in isentropic coordinates. The mass flux along and across isentropes can be represented by a streamfunction
θ

(φ, θ ) = 2πa cos(φ) ρ¯ θ v ∗ dθ , θb

where the potential temperature θb(φ) is a nominal lower boundary, less than or equal to the lowest surface potential temperature that occurs at the latitude φ under consideration. The streamfunction forms an overturning cell in each hemisphere (Fig. 3.7a). Cross-isentropic rising corresponds to diabatic heating, and cross-isentropic sinking to diabatic cooling. Included in Fig. 3.7 are the 10%, 50%, and 90% isolines of the cumulative distribution of surface potential temperatures; that is, at any given latitude, the surface potential temperature is 10% of the time less than that indicated by the lowermost dotted line and 90% of the time less than that indicated by the uppermost dotted line. The 50% isoline, the median surface potential temperature, is approximately equal to the mean surface potential temperature. The 10% and 90% isolines can be taken as demarcating the surface layer of isentropes that, at any given latitude, sometimes intersect the surface. The meridional mass flux is equatorward near the surface, roughly in the surface layer, and poleward in the above-lying interior atmosphere (Fig. 3.7a, 3.7b). A large fraction of the equatorward mass flux at any given latitude occurs at potential temperatures below the median surface potential temperature, that is, in coldair outbreaks (Held and Schneider 1999).

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The geostrophic mass flux dominates the meridional mass flux in midlatitudes (Fig. 3.7c); the ageostrophic mass flux dominates in the Tropics (Fig. 3.7d). The geostrophic mass flux is poleward in the interior troposphere and equatorward in the surface layer. It is this geostrophic mass flux that is directly associated with baroclinic eddies; we need to estimate its vertical extent. The geostrophic mass flux along isentropes implies a net poleward and (because of the slope of isentropes) upward transport of entropy. The ageostrophic mass flux in the interior troposphere is poleward in the Tropics and equatorward in midlatitudes. It represents the upper branches of the Hadley cells and of the thermally indirect Ferrel cells. The ageostrophic mass flux in the surface layer is equatorward in the Tropics and poleward in midlatitudes. It represents the lower branches of the Hadley cells and of the Ferrel cells, or the Ekman transports due to the frictional stresses on surface easterlies in the Tropics and on surface westerlies in midlatitudes. The structures of the isentropic mass flux components in the idealized GCM resemble those of Earth’s atmosphere (Johnson 1989). To estimate the vertical extent of the baroclinic entropy flux, we need to estimate the vertical extent of the geostrophic mass flux along isentropes. Assume that the geostrophic mass flux extends up to a potential temperature θg above which it can be taken to vanish. Since the Coriolis force on the geostrophic mass flux in a layer balances the form drag or pressure drag on the layer, f ρ¯ θ v ∗g = ρθ∂x M, the Coriolis force on the geostrophic mass flux integrated vertically up to potential temperature θg can be written as
θg s ρ¯ θ v ∗g dθ = ps ∂xz s − p∂xz|θg , f [3.7] θb

s

where (·) denotes an average along the surface. The first term on the right-hand side is the surface pressure drag at mountains; the second term is the pressure drag of the above-lying atmosphere on the upper boundary of the layer extending from the nominal lower boundary θb to θg (Andrews 1983; Held and Schneider 1999; Koh and Plumb 2004). If the top of the atmosphere is taken as the upper boundary of the integration (θg → ∞), the second term on the right-hand side vanishes. The Coriolis force on the vertically integrated geostrophic mass flux in an atmospheric column balances the surface pressure drag at mountains. By splitting the integral to the top of the atmosphere into the sum of one segment from θb to θg , given by equation (3.7), and a second ∞ segment from θg to the top of the atmosphere, f θg ρ¯ θ v ∗g dθ = p∂xz|θg , one sees that the assumption that the geostrophic mass flux is negligible above θg implies that the pressure drag − p∂xz|θg at the upper boundary of the layer extending from θb to θg is negligible. Therefore, it is consistent with our assumption of negligible geostrophic mass fluxes above θg to neglect the second term on the right-hand side of equation (3.7). Let us also assume that surface topography and the associated mountain pressure drag are negligible for estimating the vertical extent of the baroclinic entropy flux, such that the total pressure drag on the layer between θb and θg can be taken to vanish.

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Then at each latitude the geostrophic mass flux integrates to zero over the layer (cf. Juckes et al. 1994),


θg

θb

ρ¯ θ v ∗g dθ ≈ 0.

[3.8]

To the extent that surface topography is negligible, the geostrophic mass flux alone forms a closed circulation, and we assume that this geostrophic circulation extends to potential temperature θg . From the perspective of the entropy or mass balance (3.6), this implies that the convergence of the ageostrophic mass flux vertically integrated up to potential temperature θg balances the diabatic heating at potential temperature θg . In the Tropics, the diabatic heating at the potential temperature θg up to which the geostrophic mass flux extends (if it is significant at all) is generally nonzero. Consequently, the convergence of the vertically integrated ageostrophic mass flux below that level is nonzero, and the ageostrophic mass flux extends to higher levels (cf. Fig. 3.7d). In midlatitudes, the diabatic heating at the potential temperature θg up to which the geostrophic mass flux extends may be close to zero and there may be no significant ageostrophic mass flux above that level if the atmosphere above that level is in an approximate radiative or radiative-convective equilibrium (for example, if the geostrophic mass flux extends to the tropopause and if the stratosphere, as in section 3.2, can be taken to be in an approximate radiative equilibrium).

3.5.2. Balance Condition on Eddy Fluxes The balance condition (3.8) on the geostrophic mass flux along isentropes implies a balance condition on potential vorticity and surface potential temperature fluxes, fluxes of adiabatically approximately conserved quantities for which we can make closure assumptions to obtain an estimate of the vertical extent of the baroclinic entropy flux. To obtain the balance condition on eddy fluxes, we reformulate the geostrophic zonal momentum equation as a balance equation between the geostrophic mass flux and eddy fluxes. If one expresses the planetary vorticity f = ρθ Pg as the product of isentropic density and of potential vorticity Pg = f /ρθ in the planetary-geostrophic limit, the definition of the geostrophic meridional velocity vg = f −1∂x M H(θ − θs ) leads to the relation ∗

f v g = ρ¯ θ vg Pg = ∂x M H, with the shorthand H = H(θ − θs ) for the Heaviside step function. Decomposing the ˆ = (·) − (·)∗ denoting potential vorticity flux into mean and eddy components, with (·) fluctuations about the density-weighted mean, and dividing by the mean potential

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vorticity yields

ρ¯ θ v ∗g =

−

ρ¯ θ vˆ g Pˆ g Pg

∗

∗

+

∂x M H Pg

∗

.

[3.9]

∗

The mean potential vorticity Pg on isentropes inside the surface involves the indefinite expression ρθ Pg = fρθ/ρθ with ρθ = 0, for which we adopt the convention ρθ/ρθ = 1, such that ρθ Pg = f on isentropes inside the surface as well as above the surface. (The developments in what follows depend crucially on this somewhat arbitrary convention; see Schneider [2005] for a rationale for this convention and Koh and Plumb [2004] and Schneider [2005] for alternative conventions.) With this convention, the mean ∗ potential vorticity is Pg = f /ρ¯ θ on isentropes in the surface layer as well as in the interior atmosphere. It then follows from the definition of the geostrophic meridional velocity that the second term on the right-hand side of the geostrophic mass flux decomposition (3.9) is, throughout the flow domain in isentropic coordinates, the mean component ρ¯ θv g of the geostrophic mass flux, and the first term, consequently, is the eddy component ρθ vg of the geostrophic mass flux, ∂x M H Pg

∗

= ρ¯ θv g

and

−

ρ¯ θ vˆ g Pˆ g Pg

∗

∗

= ρθ vg ,

with primes denoting fluctuations (·) = (·) − (·) about the isentropic mean. The relation between geostrophic eddy fluxes of potential vorticity and mass resembles similar relations in quasigeostrophic theory but, unlike in quasigeostrophic theory, in the present isentropic-coordinate framework it is only necessary to consider the planetarygeostrophic limit; it is not necessary to assume that fluctuations of the isentropic density are small. Also unlike in quasigeostrophic theory, the mean component ρ¯ θv g of the geostrophic mass flux does not vanish in the surface layer because the zonal average of the pressure gradient force per unit mass −∂x M H does not vanish but gives rise to a mean zonal pressure drag per unit mass, similar to the pressure drag at mountains appearing in the mean momentum balance in pressure or height coordinates (cf. Peixoto and Oort 1992, chapter 11). This mean surface pressure drag can contain topographic contributions, but it does not require topography; it is the mean zonal pressure drag per unit mass that the flow along isentropes experiences at intersections of isentropes with the surface, whether at topographic obstacles or at a flat surface (see Koh and Plumb [2004] and Schneider [2005] for details). One obtains a balance condition on baroclinic eddy fluxes by integrating the geostrophic mass flux decomposition (3.9) from the nominal lower boundary θb to the potential temperature θg up to which the geostrophic mass flux along isentropes extends. The integral of the surface pressure drag term, which is only nonzero in the surface layer,

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a

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Figure 3.8. Components of geostrophic mass flux (109 kg s−1 K−1) along isentropes in the ∗ ∗ same GCM simulation as in Fig. 3.7. (a) Geostrophic eddy mass flux ρ  v  = −ρ¯ θ vˆ g Pˆ g /Pg ; ∗

θ g

(b) geostrophic mean mass flux ρ¯ θ v g = ∂x M H/Pg . As in Fig. 3.7, the dotted lines represent the 10%, 50%, and 90% isolines of the cumulative distribution of surface potential temperatures.

becomes approximately
θg ∂x M H 0  s ∗ dθ ≈ −ρ¯ θ vg s θs , Pg θb

[3.10]

where ρ¯ 0θ = ρ¯ θ(θ¯s ) is the mean isentropic density at the temporal- and zonal-mean surface potential temperature θ¯s (φ), and vg s is the geostrophic meridional eddy velocity at the surface (Schneider 2005). (Overbars now denote general temporal and zonal means, which are understood as means along isentropes if the argument depends on a vertical coordinate; primes denote fluctuations about surface or isentropic means.) Combining the constraint (3.8) on the geostrophic mass flux with the relations (3.9) and (3.10) yields a balance condition on eddy fluxes,


θg

θb

ρ¯ θ v ∗g dθ


≈−

θg θb

ρ¯ θ vˆ g Pˆ g Pg

∗

∗ s

dθ − ρ¯ 0θ vg s θs ≈ 0.

[3.11]

This balance condition makes manifest that the geostrophic mass flux along isentropes is directly associated with eddy fluxes of potential vorticity and of surface potential temperature. It states that, upon vertical integration, the geostrophic mass flux associated with the eddy flux of potential vorticity balances the geostrophic mass flux associated with the eddy flux of surface potential temperature. The balance condition is a statement of zonal momentum balance for baroclinic eddy fluxes and is the point of departure for estimating the vertical extent of the baroclinic entropy flux. Figure 3.8 shows the eddy and mean components of the geostrophic mass flux along isentropes, or the components of the geostrophic mass flux along isentropes that are associated with the eddy fluxes of potential vorticity and, upon vertical integration, of surface potential temperature. In the interior atmosphere, only the geostrophic eddy mass flux, or the mass flux associated with the eddy flux of potential vorticity, contributes to the geostrophic mass flux along isentropes. In the surface layer,

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both the eddy and mean components contribute to the geostrophic mass flux along isentropes, with contributions of the same sign and of the same order of magnitude (which can be shown to be generally the case; see Schneider [2005]). That both eddy and mean components (or eddy fluxes of potential vorticity and of surface potential temperature) contribute to the geostrophic mass flux in the surface layer contrasts with the representation of the near-surface mass flux in quasigeostrophic models, in which either the surface-layer potential vorticity flux (in continuously stratified models) or the surface potential temperature flux (in layer models) is neglected. See Schneider (2005) for reasons for this difference. The presence of surface-layer mass fluxes associated both with the potential vorticity flux and with the surface potential temperature flux is essential for the estimate of the vertical extent of the baroclinic entropy flux to be discussed below. Qualitatively, the direction of the geostrophic mass flux can be understood by assuming that baroclinic eddies mix potential vorticity and surface potential temperature ∗ downgradient. The potential vorticity gradient ∂ y Pg = β/ρ¯ θ − f ∂ yρ¯ θ/ρ¯ 2θ changes sign from the interior troposphere to the surface layer. It is generally positive in the interior troposphere because the Coriolis parameter increases poleward and the isentropic density usually decreases poleward along an isentrope. For sufficiently large surface potential temperature gradients, it is generally negative in the surface layer because the frequency with which an isentrope lies above the surface increases as one moves poleward along an isentrope, and so the mean isentropic density increases poleward along surface-layer isentropes. (The role the mean isentropic density plays here is similar to that of the layer thickness in quasigeostrophic two-layer models, in which the potential vorticity gradient, for a baroclinically unstable flow, likewise changes sign from the upper to the lower layer; see chapters 1 and 4 in this volume.) Consequently, eddy fluxes of potential vorticity are generally southward in the interior troposphere and northward in the surface layer and so are associated with poleward mass fluxes in the interior troposphere and with equatorward mass fluxes in the surface layer. Similarly, the eddy flux of surface potential temperature is generally poleward and hence is associated with a vertically integrated equatorward mass flux in the surface layer (Schneider 2005). Figure 3.9 illustrates the direction of the mass and eddy fluxes. Having related the geostrophic mass flux to eddy fluxes of quantities that are approximately materially conserved in adiabatic fluctuations, we are now in a position to derive an estimate of the vertical extent of the geostrophic mass flux with the help of semi-empirical eddy flux closures that quantify the above qualitative arguments for the direction of eddy fluxes. 3.5.3. Vertical Extent of Baroclinic Entropy Flux and Supercriticality4 ∗

Assume that the eddy fluxes can be modeled as downgradient diffusive fluxes vˆ g Pˆ g ∼ ∗ s −D∂ y Pg and vg s θs ∼ −D∂ y θ¯s . Since it is not the planetary-geostrophic potential

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350

ˆg * vˆg P 300

v´gs θ´s

270 0°

s

50° Latitude

Figure 3.9. Sketch of isentropic mass flux and eddy fluxes (based on the idealized GCM simulation in Figs. 3.7 and 3.8). Streamlines are contours of the isentropic mass flux streamfunction shown in Fig. 3.7a. Wavy lines indicate eddy fluxes. The dotted lines represent the 10% and 90% isolines of the cumulative distribution of surface potential temperatures; that is, they demarcate the surface layer of isentropes within which, at any given latitude, the instantaneous surface potential temperature lies 80% of the time.

vorticity Pg = f /ρθ that is materially conserved in adiabatic fluctuations but the potential vorticity P = ( f + ζθ)/ρθ, which includes the relative vorticity ζθ normal to isentropes, the closure assumption for the planetary-geostrophic potential vorticity flux is tantamount to assuming that relative vorticity fluxes can be modeled separately from the planetary vorticity fluxes; that is, meridional fluxes of wave activity are either negligible or can be modeled separately from vertical fluxes of wave activity that are local in latitude (see chapter 1 in this volume for a discussion of conditions for the adequacy of diffusive eddy flux closures). If we further assume that the eddy diffusivity D may vary with latitude but is taken to have no essential vertical structure below the upper bound θg of the integration, the diffusivity drops out of the balance condition (3.11), and one obtains the estimate (Schneider and Walker 2006) p¯ s − p¯ g ∼

f ∂y θ¯s . β 2 ∂p θ s

[3.12]

The difference between the mean surface pressure p¯ s and the mean pressure p¯ g = ¯ g ) up to which the baroclinic entropy flux extends depends on the meridional p(θ s potential temperature gradient ∂ y θ¯s and on the static stability −∂p θ at the surface (or immediately above a near-surface mixed layer). It does not depend explicitly, for example, on the eddy diffusivity and planetary rotation rate, although these parameters enter implicitly in determining the surface potential temperature gradient and the nearsurface static stability. The pressure difference p¯ s − p¯ g on the left-hand side arises from a vertical integration of the mean isentropic density ρ¯ θ = −g −1 ∂θ p H in the potentialvorticity flux term in balance condition (3.11). The factor of two in the denominator on the right-hand side arises from the mean isentropic density ρ¯ 0θ at the mean surface

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potential temperature, which is about half a near-surface average of the isentropic density, since the isentrope with the mean surface potential temperature at any given latitude is about half the time inside the surface. Neglected in the estimate (3.12) is a term involving the slope of isentropes at the upper boundary θg —a term that is small in the extratropics of Earth’s atmosphere if the upper boundary θg is identified with the tropopause (Schneider 2004; Schneider and Walker 2006). In addition to the assumption of separability of potential vorticity flux components, the adequacy of the estimate (3.12) depends on the validity of the assumption that the eddy diffusivity has no essential vertical structure, implying vertically uniform mixing, in the pressure range over which the baroclinic entropy flux extends. Eddy diffusivities may be poorly defined or large in or near regions where the potential vorticity gradient vanishes (e.g., near the top of the surface layer, where the potential vorticity gradient changes sign). But as long as the potential vorticity fluxes in those regions vanish or are small, the structure of the eddy diffusivity in those regions is not essential for the adequacy of the estimate (3.12). Consider the limiting cases of macroturbulence with weak nonlinear eddy–eddy interactions and macroturbulence with strong nonlinear eddy–eddy interactions. In strongly nonlinear macroturbulence, the inverse energy cascade to large horizontal and vertical scales would lead to barotropization of the energy-containing eddies (Smith and Vallis 2002). Therefore, since the energy-containing eddies are expected to dominate the advection of potential vorticity and surface potential temperature fluctuations (Held and Larichev 1996), the assumption of vertically uniform mixing is justifiable. In weakly nonlinear macroturbulence, mean flow properties such as the thermal stratification determine the vertical structure of baroclinic eddies. In Earth’s atmosphere, however, the streamfunctions of the most unstable linear or weakly nonlinear waves vary only weakly in the pressure range over which baroclinic eddies redistribute entropy (Simmons and Hoskins 1976, 1977; Valdes and Hoskins 1988). To the extent that the vertical structure of the eddy diffusivity is similar to that of the streamfunction—the streamfunction scales with the product of an eddy length scale and an eddy velocity scale and so scales like an eddy diffusivity (Holloway 1986; Kushner and Held 1998)—the assumption of vertically uniform mixing thus remains plausible for rough estimates (cf. Held 1978). Thus, the estimate (3.12) can be expected to hold quite generally, at least in a scaling sense.5 A dynamical constraint on the extratropical thermal stratification and tropopause height is implied by the estimate (3.12) of the vertical extent of the baroclinic entropy flux. If the baroclinic entropy flux is strong and extends to the tropopause, possibly determining the height of the tropopause and the thermal stratification, the estimate (3.12) holds with the mean tropopause pressure p¯ t in place of p¯ g . If baroclinic eddies are weak and the baroclinic entropy flux does not extend to the tropopause but the tropopause height is determined by other processes (such as radiation and convection), the pressure difference p¯ s − p¯ g is less than the pressure difference between surface and

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tropopause. That significant baroclinic entropy fluxes extend above the tropopause is inconsistent with the notion of the troposphere as the atmospheric layer within which entropy received at the surface is redistributed. Therefore, defining the bulk stability ¯ v = −2 ∂p θ s ( p¯ s − p¯ t) 

[3.13]

and the supercriticality Sc = −

f ∂y θ¯s , ¯v β 

[3.14]

a nondimensional measure of the slope of near-surface isentropes or a ratio of pressure ranges Sc ∼ ( p¯ s − p¯ g )/( p¯ s − p¯ t), we conclude Sc
1 (Schneider and Walker 2006). The supercriticality is less than one if the baroclinic entropy flux is shallower than the tropopause, as is typically the case in the Tropics; the supercriticality is approximately equal to one if the baroclinic entropy flux extends to the tropopause, as may be the case in the extratropics. The constraint Sc
1 provides a dynamical constraint on the thermal stratification and tropopause height, relating the temperature lapse rate near the surface and the tropopause height, via the bulk stability, to the meridional gradient of surface potential temperature and to the Coriolis parameter and its gradient. One may use a radiative constraint of the kind discussed in section 3.2 with a constant tropospheric lapse rate to obtain a first approximation of the tropopause height; for a more accurate account of the tropopause height, one should take the vertical structure of the thermal stratification into account, which the scaling arguments presented here do not provide. p For Earth-like atmospheres with approximately constant static stability ∂z θ and with an extratropical tropopause height somewhat greater than the scale height (with a quotient of extratropical tropopause height and scale height of about 1.5, as in Earth’s atmosphere), the bulk stability in the supercriticality (3.14) is approximately equal ¯ v ∼ θ¯t − θ¯s . to the potential temperature difference between tropopause and surface,  (The decrease of density with height just compensates the factor of two in the bulk stability [3.13]; see Schneider and Walker [2006].) Hence, if baroclinic entropy fluxes determine the tropopause height (Sc ∼ 1), the supercriticality constraint, aside from the ignored moist processes, differs from Juckes’s dynamical constraint (3.4) primarily by the length scales that enter: | f /β| for the supercriticality constraint and d L e for Juckes’s dynamical constraint. In midlatitudes, the length scale for the supercriticality constraint is approximately | f /β| ≈ a, which, for Earth’s atmosphere, is of the same order of magnitude as the length scale for Juckes’s dynamical constraint (d L e = 0.9a was used in Fig. 3.6), so the two constraints are difficult to distinguish if the (equivalent) potential temperature gradient in Juckes’s constraint scales similarly as the surface potential temperature gradient in the supercriticality constraint. If baroclinic entropy fluxes determine the tropopause height, the supercriticality constraint implies θ¯t − θ¯s ∼ a|∂ y θ¯s | in midlatitudes, which means that the potential temperature difference between

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Extratropical Thermal Stratification | 69 100 Sc < 1 50 Earth

∆v (K)

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Sc > 1

γ=0.7 γ=0.8 γ=0.9 2Ω e 4Ω e 2a e 4a e

10

10

50

100

(K)

¯ v and scaled surface Figure 3.10. Extratropical bulk stability  potential temperature gradient −( f /β)∂ y θ¯s in dynamical equilibria of idealized GCM simulations. Included are results from simulations with terrestrial rotation rate e and radius ae and with different convective lapse rates γ d (γ = 0.7, . . . , 0.9) and from simulations with twice and four times the rotation rate and radius of Earth. For each set of parameters, the figure shows a series of simulations obtained by varying the pole-equator surface temperature difference in radiative equilibrium. Displayed quantities are averages over extratropical baroclinic zones. The corresponding quantities for Earth’s atmosphere (northern hemisphere annual mean according to ERA-40 data for 1980–2001) are shown for comparison. The dashed line represents supercriticality Sc = 1, with Sc < 1 above it and Sc > 1 below it. (Adapted from Schneider [2006].)

equator and pole is similar to that between tropopause and surface, as is indeed the case in Earth’s atmosphere. The supercriticality constraint depends on the assumption that the planetary vorticity gradient β is dynamically significant. It becomes singular in the limit β → 0 (e.g., in the limit of infinite planet radius). It is unclear how baroclinic eddies would affect the thermal stratification and tropopause height if the energy-containing baroclinic eddies were so small compared with the planet radius that the planetary vorticity gradient β could effectively be set to zero.

3.6. Simulations with an Idealized GCM Figure 3.10 shows the results of a test of dynamical constraints with an idealized GCM. The GCM is idealized in that, among other simplifications, radiative heating and cooling are represented by Newtonian relaxation of temperatures toward statically unstable

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radiative-equilibrium states and in that it has no explicit hydrologic cycle. However, if a layer is statically unstable relative to a specified convective temperature lapse rate, a convection scheme mimics latent heat release by relaxing temperatures in that layer toward an enthalpy-conserving profile with the convective lapse rate. The convective lapse rate is γ d , where d = g /cp = 9.8 K km−1 is the dry adiabatic lapse rate and γ ≤ 1 is a rescaling parameter. The scheme mimics the stabilizing effect of latent heat release in moist convection, with the implied latent heat release increasing with decreasing γ (see Schneider and Walker [2006] for a model description). The mean lapse rate of 6.5 K km−1 in Earth’s tropical free troposphere corresponds to a rescaling parameter γ = 0.66. The simulated circulations span a very wide range of possible planetary circulations, including circulations with multiple jets and belts of surface westerlies in each hemisphere. Consistent with the supercriticality constraint, all simulations in Fig. 3.10 condense onto the line Sc ∼ 1 for sufficiently large surface potential temperature gradients. For small surface potential temperature gradients, there is a regime in which Sc < 1, in which baroclinic eddies are weak and the extratropical thermal stratification and tropopause height are determined by convection, similar to the tropical thermal stratification and tropopause height. In this regime, the extratropical lapse rate is approximately equal to the convective lapse rate (see Fig. 3.11, which shows dynamical-equilibrium lapse rates for representative simulations with convective lapse rate 0.9d = 8.8 K km−1). As the surface potential temperature gradient increases, the simulations approach the line Sc ∼ 1 (Fig. 3.10). Consistent with the supercriticality constraint, the larger the convective lapse rate γ d, and hence the smaller the radiativeconvective bulk stability, the smaller is the scaled surface potential temperature gradient −( f /β)∂ y θ¯s at which the line Sc ∼ 1 is reached. When the line Sc ∼ 1 is reached, baroclinic entropy fluxes stabilize the extratropical thermal stratification and modify ¯ v increases in proportion with the tropopause height, such that the bulk stability  the scaled surface potential temperature gradient −( f /β)∂ y θ¯s . The stabilization of the thermal stratification is concentrated near the surface, but for sufficiently large surface temperature gradients or large convective lapse rates, the stratification throughout the troposphere is stabilized by eddies (Fig. 3.11). However, convection only ceases to provide a significant portion of the dynamical heating in the upper troposphere for larger convective lapse rates (γ ≥ 0.8) and for the largest surface potential temperature gradients we simulated. It is potentially significant that, consistent with surface or nearsurface quantities playing a prominent role in the theory of baroclinic entropy fluxes, the stratification stabilization is concentrated near the surface, similar to what is seen in Earth’s atmosphere (cf. Fig. 3.1). To obtain the close agreement between theory and simulations in Fig. 3.10, it is important that the potential temperature gradient near the surface is considered rather than a midtropospheric potential temperature gradient, such as would appear in quasigeostrophic baroclinic-adjustment constraints (cf. endnote 5 and chapter 2 in this volume). This difference between quasigeostrophic baroclinic-adjustment constraints

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Extratropical Thermal Stratification | 71 ∆ h =30 K

∆ h =75 K

Sigma

0.2

0.8

9

∆ h=150 K

300

8

270

8

9

7

7

∆ h=360 K 460

6

7

8

340

7

8

0.2 Sigma

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6

9 −50°

0° Latitude

50°

6 −50°

0°

50°

Latitude

Figure 3.11. Zonal-mean temperature lapse rate, potential temperature, and tropopause in GCM simulations with terrestrial rotation rate and radius and convective lapse rate 0.9d = 8.8 K km−1. The pole-equator surface temperature difference in radiative equilibrium increases from h = 30 K in the upper left panel to h = 360 K in the lower right panel. The contour intervals are 10 K for potential temperature (gray) and 1 K km−1 for lapse rate (black, negative contours dashed). The heavy line marks the tropopause. The increase of tropopause height with h is primarily caused by an increase of the mean surface temperature with h, in accordance with the “radiative” constraint for the GCM. The increase of the mean surface temperature with h was necessary to make possible simulations with large h with physically realizable polar temperatures (Schneider and Walker 2006).

and the supercriticality constraint may, at least in part, be responsible for the lack of agreement Thuburn and Craig (1997) found between quasigeostrophic baroclinicadjustment constraints and simulations with a complex GCM (additional complications due to comparing a theory that ignores moist processes with simulations with a GCM with a hydrologic cycle may also have contributed). If Juckes’s mechanism for determining the bulk stability were acting, one would ¯ v with convective lapse rate, since expect to see systematic variations in bulk stability  the convective lapse rate would change the minimum stability entering a dry variant of Juckes’s mean bulk stability estimate (3.2). However, no such systematic variations are evident in Fig. 3.10. Instead, the bulk stabilities condense onto the line Sc ∼ 1 irrespective of convective lapse rate. Similarly, extratropical angular momentum surfaces are generally steeper than neutral surfaces for convection with the specified convective lapse rate. There is no evidence for adjustment to a state that is neutral with respect to slantwise convection, as would not have been expected given that slantwise convection was neither resolved nor parameterized in the idealized GCM.

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3.7. Discussion and Open Questions The thermal stratification and tropopause height of the extratropical atmosphere can be maintained by at least two mechanisms, convection and baroclinic eddies. If the baroclinicity is small, convective entropy fluxes maintain the thermal stratification and tropopause height. To the extent that convection is in quasi-equilibrium with its large-scale environment, as it is in the simulations with the idealized GCM, the tropospheric lapse rate is equal to the convective lapse rate. In this convectiondominated regime, radiative constraints of the kind discussed in section 3.2 determine the tropopause height. If the baroclinicity is sufficiently large, baroclinic entropy fluxes stabilize the thermal stratification and modify the tropopause height, such that the supercriticality Sc does not significantly exceed one. In this eddy-dominated regime, radiative constraints likewise determine the tropopause height, with the lapse rate entering radiative constraints of the kind discussed in section 3.2 obtainable from the bulk stability and the tropopause height (that is, the radiative and dynamical constraints have to be solved simultaneously for the unknown lapse rate and tropopause height). The scaling theory for the vertical extent of the baroclinic entropy flux does not provide the vertical structure of the thermal stratification, which may have to be taken into account in radiative constraints if the lapse rate cannot be approximated by a constant. Whether slantwise convection plays a role in determining the thermal stratification and tropopause height was not addressed with the idealized GCM, since slantwise convection was neither resolved nor parameterized in the GCM. A variant of Juckes’s mechanism, coupling baroclinic eddies and the parameterized convection of the idealized GCM, did not appear to be acting in the GCM. The dynamical constraints discussed relate measures of the tropospheric lapse rate and the tropopause height via, for example, a bulk stability to the surface potential temperature and, in the case of dynamical constraints involving baroclinic eddies or slantwise convection, to the meridional gradient of surface potential temperature. In addition to the radiative constraint, a third constraint is necessary to obtain a closed theory for the tropopause height, thermal stratification, and surface temperature. This third constraint is given by an energy balance condition at the surface, which determines the surface temperature given the differential heating of the surface and a theory of how the eddy flux of surface (potential) temperature depends on other mean-field quantities. For example, if the eddy flux of surface potential temperature is related to mean-field quantities via a diffusive eddy flux closure, one needs to know how the eddy diffusivity depends on mean-field quantities. The supercriticality constraint does not depend on eddy diffusivities, but the meridional surface potential temperature gradient will depend on the eddy diffusivity. A theory of how the eddy flux of surface potential temperature depends on mean-field quantities is lacking, even leaving out moist processes, although the supercriticality constraint for baroclinic eddies points to how baroclinic-eddy

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length and energy scales depend on mean-field quantities (Schneider and Walker 2006; Schneider 2006). Given the prominence of large-scale condensation and moist convection in the extratropics of Earth’s atmosphere, developing a theory of how moist processes affect baroclinic eddies is obviously important. The simulations with the idealized GCM with different convective lapse rates, mimicking different degrees of latent heating, suggest that the supercriticality constraint may be generalizable to moist atmospheres. The seasonal (Stone and Nemet 1996) and decadal (Schneider 2004) variability of Earth’s atmosphere satisfies the constraint Sc
1 or similar constraints (Stone 1978), despite considerable variability of the thermal structure. And the fundamental mechanism underlying the supercriticality constraint—macroturbulence reduces the surface potential temperature gradient and stabilizes the thermal stratification—also acts in a moist atmosphere. What is unclear, however, is what eddy flux closures are adequate in a moist atmosphere. The balance condition (3.11) is derived from the zonal momentum balance and thus also holds in a moist atmosphere. What may not hold in a moist atmosphere is the closure assumption of modeling the eddy fluxes of surface potential temperature and of potential vorticity along isentropes as diffusive fluxes. Other balance conditions, involving quantities that are materially conserved in moist adiabatic fluctuations, may be a more fruitful starting point for a theory taking moist processes into account.

Acknowledgments I thank Chris Walker for carrying out the simulations shown in sections 3.5 and 3.6 and Paul O’Gorman for carrying out the radiative transfer calculations shown in section 3.2. Chris Walker and Paul O’Gorman also provided the ERA-40 data in a format that helped to produce the figures. Many of the presented results from my own work are based on collaborations, over several years, with Isaac Held and Chris Walker. I am grateful to Simona Bordoni, Kerry Emanuel, Robert Korty, Paul Kushner, Paul O’Gorman, and Adam Sobel for careful readings of and helpful comments on drafts of this chapter. Part of the material presented here is based upon work supported by the National Science Foundation under Grant No. 0450059.

Notes 1. Alternatively, in terms of the emission temperature, the tropopause height is Ht ≈ (Ts − αTe)/ , an expression that, if the emission temperature and with it the tropopause temperature Tt ≈ αTe are proportional to the surface temperature, has the same qualitative dependence on surface temperature and lapse rate as the estimate (3.1) with fixed emission height. However, like the emission height, the emission temperature depends on water vapor concentrations and on other factors, and, although it increases with surface temperature, it is not proportional to the surface temperature.

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74 | Tapio Schneider 2. In the Tropics the tropopause has been variously associated with the coldest point in an atmospheric column, with a critical value of temperature lapse rate, and with the height of the main convective outflow—different conventions that yield different tropopause heights. The radiative constraints discussed here can only be expected to account for the height up to which the bulk of the dynamical entropy redistribution extends, that is, in the Tropics approximately the height up to which significant entropy transport by moist convection extends (cf. Sarachik 1985). The radiative constraints do indeed seem to account for this height (Thuburn and Craig 1997; Highwood and Hoskins 1998). Depending on the radiative-equilibrium profile above this height, the coldest point in an atmospheric column may be considerably higher (Forster et al. 1997). 3. Similar considerations for slantwise convection coupled to baroclinic eddies lead to a minimum bulk moist stability that scales approximately with the product of equivalent and virtual potential temperature gradients, hence is quadratic in potential temperature gradients, with virtual potential temperature gradients arising from thermal wind balance and the slope of angular momentum surfaces (K. A. Emanuel, personal communication). 4. This subsection follows parts of Schneider and Walker (2006) closely, in some passages verbatim. 5. Estimates of the vertical extent of baroclinic eddy fluxes that structurally resemble (3.12) also arise within quasigeostrophic theory but differ from the estimate (3.12) fundamentally in the theoretical reasoning underlying them. For example, estimates resembling (3.12) arise if one considers the pressure range over which baroclinic eddies would need to modify the atmospheric thermal structure to stabilize it with respect to baroclinic instability (Lindzen and Farrell 1980). The estimate (3.12) differs from Lindzen and Farrell’s and similar quasigeostrophic baroclinic-adjustment estimates (e.g., Stone 1978; Lindzen 1993) in that it neither presupposes nor implies that the atmosphere is neutral with respect to baroclinic instability. In contrast with quasigeostrophic baroclinic-adjustment estimates (cf. chapter 2 in this volume), neither potential vorticity nor surface potential temperature gradients are assumed to be small in the theory presented here; rather, they are sufficiently large that the mass fluxes associated with the eddy fluxes of potential vorticity and surface potential temperature, shown in Fig. 3.8, balance each other upon vertical integration. Moreover, the estimate (3.12) differs from quasigeostrophic baroclinic-adjustment estimates in that temperature gradients and static stabilities at the surface appear in place of the interior-atmosphere averages in the quasigeostrophic estimates. See Schneider (2004, 2005) and Schneider and Walker (2006) for a discussion of similarities and differences between the estimate (3.12) and quasigeostrophic estimates.

References Andrews, D. G., 1983: A finite-amplitude Eliassen-Palm theorem in isentropic coordinates. J. Atmos. Sci., 40, 1877–1883. Andrews, D. G., J. R. Holton, and C. B. Leovy, 1987: Middle Atmosphere Dynamics. International Geophysics Series, Vol. 40. Academic Press, 489 pp. Bannon, P. R., 1986: Linear development of quasi-geostrophic baroclinic disturbances with condensational heating. J. Atmos. Sci., 43, 2261–2274. Bennetts, D. A., and B. J. Hoskins, 1979: Conditional symmetric instability—a possible explanation for frontal rainbands. Quart. J. Roy. Meteor. Soc., 105, 945–962. Emanuel, K. A., 1983a: The Lagrangian parcel dynamics of moist symmetric instability. J. Atmos. Sci., 40, 2368–2376.

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Extratropical Thermal Stratification | 75 Emanuel, K. A., 1983b: On assessing local conditional symmetric instability from atmospheric soundings. Mon. Wea. Rev., 111, 2016–2033. Emanuel, K. A., 1988: Observational evidence of slantwise convective adjustment. Mon. Wea. Rev., 116, 1805–1816. Emanuel, K. A., 1994: Atmospheric Convection. Oxford University Press, 580 pp. Emanuel, K. A., M. Fantini, and A. J. Thorpe, 1987: Baroclinic instability in an environment of small stability to slantwise moist convection. Part I: Two-dimensional models. J. Atmos. Sci., 44, 1559–1573. Fantini, M., 1993: A numerical study of two-dimensional moist baroclinic instability. J. Atmos. Sci., 50, 1199–1210. Forster, P., R. S. Freckleton, and K. P. Shine, 1997: On aspects of the concept of radiative forcing. Climate Dyn., 13, 547–560. Gold, E., 1909: The isothermal layer of the atmosphere and atmospheric radiation. Proc. Roy. Soc. Lond. A, 82, 43–70. Goody, R. M., and Y. L. Yung, 1989: Atmospheric Radiation: Theoretical Basis. 2nd ed. Oxford University Press, 519 pp. Gutowski, W. J., Jr., L. E. Branscome, and D. A. Stewart, 1992: Life cycles of moist baroclinic eddies. J. Atmos. Sci., 49, 306–319. Haynes, P. H., and M. E. McIntyre, 1987: On the evolution of vorticity and potential vorticity in the presence of diabatic heating and frictional or other forces. J. Atmos. Sci., 44, 828–841. Haynes, P. H., and M. E. McIntyre, 1990: On the conservation and impermeability theorems for potential vorticity. J. Atmos. Sci., 47, 2021–2031. Held, I. M., 1978: The vertical scale of an unstable baroclinic wave and its importance for eddy heat flux parameterizations. J. Atmos. Sci., 35, 572–576. Held, I. M., 1982: On the height of the tropopause and the static stability of the troposphere. J. Atmos. Sci., 39, 412–417. Held, I. M., and V. D. Larichev, 1996: A scaling theory for horizontally homogeneous, baroclinically unstable flow on a beta-plane. J. Atmos. Sci., 53, 946–952. Held, I. M., and T. Schneider, 1999: The surface branch of the zonally averaged mass transport circulation in the troposphere. J. Atmos. Sci., 56, 1688–1697. Held, I. M., and B. J. Soden, 2000: Water vapor feedback and global warming. Annu. Rev. Energy Environ., 25, 441–475. Highwood, E. J., and B. J. Hoskins, 1998: The tropical tropopause. Quart. J. Roy. Meteor. Soc., 124, 1579–1604. Holloway, G., 1986: Estimation of oceanic eddy transports from satellite altimetry. Nature, 323, 243–244. Holton, J. R., 2004: An Introduction to Dynamic Meteorology. International Geophysics Series, Vol. 88, 4th ed. Elsevier, 529 pp. Johnson, D. R., 1989: The forcing and maintenance of global monsoonal circulations: An isentropic analysis. In B. Saltzman, Ed., Advances in Geophysics, Vol. 31, Academic Press, 43–304. Juckes, M. N., 2000: The static stability of the midlatitude troposphere: The relevance of moisture. J. Atmos. Sci., 57, 3050–3057. Juckes, M. N., I. N. James, and M. Blackburn, 1994: The influence of Antarctica on the momentum budget of the southern extratropics. Quart. J. Roy. Meteor. Soc., 120, 1017–1044.

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76 | Tapio Schneider Kiehl, J. T., J. J. Hack, G. B. Bonan, B. A. Boville, B. P. Briegleb, D. L. Williamson, and P. J. Rasch, 1996: Description of the NCAR Community Climate Model (CCM3). Tech. rep., National Center for Atmospheric Research, Boulder, Colorado. NCAR Technical Note NCAR/TN-420+STR, www.cgd.ucar.edu/cms/ccm3/TN-420. Koh, T.-Y., and R. A. Plumb, 2004: Isentropic zonal average formulation and the near-surface circulation. Quart. J. Roy. Meteor. Soc., 130, 1631–1654. Kushner, P. J., and I. M. Held, 1998: A test, using atmospheric data, of a method for estimating oceanic eddy diffusivity. Geophys. Res. Lett., 25, 4213–4216. Lapeyre, G., and I. M. Held, 2004: The role of moisture in the dynamics and energetics of turbulent baroclinic eddies. J. Atmos. Sci., 61, 1693–1710. Lindzen, R. S., 1993: Baroclinic neutrality and the tropopause. J. Atmos. Sci., 50, 1148– 1151. Lindzen, R. S., and B. Farrell, 1980: The role of the polar regions in global climate, and a new parameterization of global heat transport. Mon. Wea. Rev., 108, 2064–2079. Lorenz, E. N., 1955: Available potential energy and the maintenance of the general circulation. Tellus, 7, 157–167. Mak, M., 1982: On moist quasigeostrophic baroclinic instability. J. Atmos. Sci., 39, 2028– 2037. Manabe, S., and R. F. Strickler, 1964: Thermal equilibrium of the atmosphere with a convective adjustment. J. Atmos. Sci., 21, 361–385. Manabe, S., and R. T. Wetherald, 1967: Thermal equilibrium of the atmosphere with a given distribution of relative humidity. J. Atmos. Sci., 24, 241–259. Milne, E. A., 1922: Radiative equilibrium: the insolation of an atmosphere. Phil. Mag., 44, 872–896. Peixoto, J. P., and A. H. Oort, 1992: Physics of Climate. American Institute of Physics, 520 pp. Phillips, N. A., 1956: The general circulation of the atmosphere: a numerical experiment. Quart. J. Roy. Meteor. Soc., 82, 123–164. Sarachik, E. S., 1985: A simple theory for the vertical structure of the tropical atmosphere. Pure Appl. Geophys., 123, 261–271. Schneider, T., 2004: The tropopause and the thermal stratification in the extratropics of a dry atmosphere. J. Atmos. Sci., 61, 1317–1340. Schneider, T., 2005: Zonal momentum balance, potential vorticity dynamics, and mass fluxes on near-surface isentropes. J. Atmos. Sci., 62, 1884–1900. Schneider, T., 2006: The general circulation of the atmosphere. Ann. Rev. Earth Planet. Sci., 34, 655–688. Schneider, T., and C. C. Walker, 2006: Self-organization of atmospheric macroturbulence into critical states of weak nonlinear eddy–eddy interactions. J. Atmos. Sci., 63, 1569–1586. Schwarzschild, K., 1906: Über das Gleichgewicht der Sonnenatmosphäre. Nachrichten von der Königlichen Gesellschaft der Wissenschaften zu Göttingen. Math.-phys. Klasse, 195, 41–53. Simmons, A. J., and B. J. Hoskins, 1976: Baroclinic instability on the sphere: Normal modes of the primitive and quasigeostrophic equations. J. Atmos. Sci., 33, 1454–1477. Simmons, A. J., and B. J. Hoskins, 1977: Baroclinic instability on the sphere: Solutions with a more realistic tropopause. J. Atmos. Sci., 34, 581–588. Smith, K. S., and G. K. Vallis, 2002: The scales and equilibration of midocean eddies: Forceddissipative flow. J. Phys. Oceanogr., 32, 1699–1720. Stone, P. H., 1978: Baroclinic adjustment. J. Atmos. Sci., 35, 561–571.

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Extratropical Thermal Stratification | 77 Stone, P. H., and J. H. Carlson, 1979: Atmospheric lapse rate regimes and their parameterization. J. Atmos. Sci., 36, 415–423. Stone, P. H., and B. Nemet, 1996: Baroclinic adjustment: A comparison between theory, observations, and models. J. Atmos. Sci., 53, 1663–1674. Thorpe, A. J., and R. Rotunno, 1989: Nonlinear aspects of symmetric instability. J. Atmos. Sci., 46, 1285–1299. Thuburn, J., and G. C. Craig, 1997: GCM tests of theories for the height of the tropopause. J. Atmos. Sci., 54, 869–882. Thuburn, J., and G. C. Craig, 2000: Stratospheric influence on tropopause height: The radiative constraint. J. Atmos. Sci., 57, 17–28. Uppala, S. M., and coauthors, 2005: The ERA-40 reanalysis. Quart. J. Roy. Meteor. Soc., 131, 2961–3012. Valdes, P. J., and B. J. Hoskins, 1988: Baroclinic instability of the zonally averaged flow with boundary layer damping. J. Atmos. Sci., 45, 1584–1593. Xu, K.-M., and K. A. Emanuel, 1989: Is the tropical atmosphere conditionally unstable? Mon. Wea. Rev., 117, 1471–1479.

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Chapter 4

Storm Track Dynamics Kyle L. Swanson

4.1. Introduction It has long been appreciated that mobile, O(1000 km) spatial scale high and low pressure systems comprise much of the day-to-day weather variability in the midlatitudes. Coupled with the central role of these systems in the general circulation via their transports of sensible heat, momentum, and moisture, it is natural that the geographic organization of these transients is an important topic if one is to understand the climate of the Earth’s atmosphere. Areas of preferred transient or storm (cyclone) activity are referred to as storm tracks. Historically, there have been two basic approaches to diagnosing storm tracks (see Hoskins and Hodges [2002] for a review). The more traditional approach identifies individual weather systems, tracks their position as a function of time, and produces statistics for their distributions, e.g., track densities, storm life spans, etc. An alternative approach determines statistics at a set of grid points in analyzed atmospheric fields, usually in a frequency band associated with what are considered to be synoptic time scales (e.g., Blackmon 1976; Blackmon et al. 1977). The latter bandpass filtering approach has the advantage that it can be carried out at all levels of the atmosphere, providing a three-dimensional picture of storm tracks. Further, it is more consistent with the view that storm tracks are statistical features of the climate, in the same manner, for example, that the global mean temperature distribution is a statistical feature of the climate. This approach provides a definition of storm tracks as geographically localized maxima in bandpass transient variance. An example of a storm track structure that emerges from such an analysis is shown in Fig. 4.1 (top), where the storm tracks are marked in the bandpass standard deviation of the 250-hPa geopotential height field by enhanced variability off the east coasts of Asia and North America. The structure of the bandpass standard deviation field in Fig. 4.1 (top) suggests the primary question that has arisen time and time again in the study of storm track dynamics, namely what features of the background time-mean flow determine the

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Figure 4.1. Top: 250-hPa geopotential height bandpass (3–10 day) standard deviation for January/February (m). Bottom: 250-hPa mean zonal winds for January/February (m s−1).

magnitude and degree of zonal localization of storm tracks? Since the synoptic transients that compose the storm tracks feed off of the available potential energy stored in the meridional pole-to-equator temperature gradient, it is natural to associate storm tracks with localized regions of large meridional temperature gradients, or baroclinic zones. A suitable measure of that baroclinicity is given by the Eady growth-rate maximum σBI = 0.31 f |∂v/∂z|N −1,

[4.1]

where f is the Coriolis parameter, N the Brunt-Väisälä frequency, and v the time-mean wind field. Lindzen and Farrell (1980) show that this formula provides an accurate estimate of growth rate in a range of baroclinic instability problems, suggesting an intimate link between the vertical shear and disturbance growth. Figure 4.1 (bottom) shows that during the Northern Hemisphere winter the baroclinic zones, marked by the climatological jet streams in the upper troposphere via thermal wind balance, are located off the east coast of the Asian and North American continents. The storm tracks are located just downstream (eastward) from these baroclinic zones, hinting at spatio-temporal amplification of synoptic eddies as they travel eastward through these zones. However, this observation raises more questions than it answers: What determines how far downstream of the region of maximum baroclinicity the maximum transient variance occurs? What is the physical mechanism responsible for terminating the storm tracks? Is it consistent to view the storm tracks as being determined solely by the properties of the background flow? The importance of answering these questions is highlighted by the fact that storm tracks exhibit stunning modes of variability on intra-annual, interannual, and decadal time scales, as discussed by Chang et al. (2002). For example, Fig. 4.2 shows the 250-hPa

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Figure 4.2. Top: 250-hPa geopotential height bandpass (3–10 day) standard deviation for March/April (m). Bottom: 250-hPa mean zonal winds for March/April (m s−1).

geopotential height bandpass standard deviation and mean zonal wind for the months of March/April. Comparison between this figure and Fig. 4.1 reveals that in spite of substantially weaker mean zonal winds (and hence weaker baroclinicity) over the Pacific during March/April, transient variance is actually larger over the Pacific than it is during January/February. This so-called “midwinter minimum” in Pacific storm track activity was first noted by Nakamura (1992); a concise theoretical explanation has remained elusive. Given this and other types of observed storm track variability, it is presumed that changes in storm tracks will play a major role in shaping future climate change (Hall et al. 1994). Significant progress has been made over the past three decades in explaining how observed storm tracks function and diagnosing modes of storm track variability. A range of idealized models have been used to explore different modes of observed storm track variability with success. However, intercomparison between various theoretical and idealized storm track models and observed structures remains difficult. This leaves aside the difficult questions posed by the paleoclimate record, where evidence is consistent with multiple modes of storm track variability, variability not captured by the current generation of climate models (see Broecker [1997] and chapter 12 in this volume). Ideally, a theory of storm track dynamics would provide straightforward answers to the set of storm track questions raised above. However, the issues raised when considering observed storm track dynamics are by nature difficult, and it is not clear that direct comparison of theoretical ideas with observations is the best manner in which to proceed. Observed storm track dynamics have a number of complicating factors that may well be important, in particular the role of moisture, that are missing from current theoretical models. Given this state of affairs, in this chapter a different approach is followed to the dynamics of storm tracks, namely the comparison of theoretical ideas

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and simple models against models of comparable complexity. As discussed by Held and Hoskins (1985), such an approach provides unambiguous answers to the underlying dynamical issues addressed, allowing one to build experience in treating dynamical issues that ultimately may be extended to the observational arena.

4.2. Overview Since storm tracks consist of a statistically steady composite of synoptic eddies at various points in their life cycles, the synoptic eddy life cycle is a natural starting point to understand storm track dynamics. As outlined in studies such as Simmons and Hoskins (1980), the synoptic eddy life cycle consists of a baroclinic growth stage, followed by ultimate saturation and decay primarily by barotropic processes. Baroclinic growth is marked by cyclogenesis at the surface, with the eddy growth at upper levels following, with saturation occurring first at the surface followed by the upper levels. From the perspective of the Eliassen-Palm flux (Andrews et al. 1987; see also chapter 5 in this volume), this is consistent with downgradient heat fluxes during the early part of the eddy life cycle, followed by radiation of eddy activity onto the flanks of the jet via meridional momentum fluxes, where dissipation occurs by Rossby wave breaking-like processes (Killworth and McIntyre 1985). Of course, the synoptic eddy life cycle above describes disturbance growth and decay as a function of time. The most basic idea of storm track structure equates this disturbance life cycle in time with a disturbance life cycle in space within the storm track. Disturbances are introduced at the upstream end of the storm track, and develop in space and time as they extract energy from the enhanced baroclinicity of the background flow. The downstream end of the storm track is then marked by the decay stage of the synoptic eddy life cycle. An estimate of the storm track length then is simply the time span of the synoptic eddy life cycle divided by the phase speed of the individual eddies. While this picture is appealing in that it retains the synoptic eddy life cycle as the distinguishing element, it has a number of faults. First, the estimate of storm track length described above is about a factor of two too small compared to observed Northern Hemisphere winter storm tracks. Nonlinear life cycle experiments give disturbance growth to decay stages of about 5 days (Simmons and Hoskins 1980), which when coupled with a 7–8 m s−1 phase speed (Chang 1993) suggests a storm track length of roughly 3000 km, whereas the observed storm tracks of Figs. 4.1 and 4.2 are more like 8–10,000 km in length. In addition, statistical decomposition reveals that there is not really a bottom-trapped, baroclinic growth phase for the transients (Lim and Wallace 1991; Chang 1993). This is most readily shown by examining regressions of storm track transients, an example of which is shown in Fig. 4.3. This regression of meridional wind along the latitude circle 40◦ N against the meridional wind at the point (175◦ E, 300 hPa) provides a picture of the characteristic scale of transients, as well as of the

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Figure 4.3. Regression of meridional velocity along 40◦ N against the point at (175◦ E, 300 hPa). The contour interval is 2 m s−1, with negative contours dashed.

wavy structure of disturbances. Notably, there is little difference between disturbance structures upstream or downstream of the regression point. Disturbances are somewhat tilted more strongly against the shear upstream of the regression point, but there is not a clear counterpart to the surface cyclogenesis found in traditional life cycle experiments of the type described by Simmons and Hoskins (1980). A more fundamental difference between transients in storm tracks and traditional life cycles is the sense of group propagation within storm tracks, which suggests that the phase speed (the speed of individual cyclones) is not the relevant speed for storm track structure. Figure 4.4 shows the correlation of the meridional wind field against the meridional wind at a point in the center of the Pacific storm track, along with the analogous fields that lag and lead that point by one day. There is an obvious sense of group propagation—disturbances upstream of the base disturbance are stronger in the day−1 correlation, while those downstream are stronger in the day +1 correlation. As shown by Chang (1993), this group speed is more characteristic of the upper-tropospheric background flow, roughly 40 m s−1, in contrast to the speed of individual cyclones, which resembles the flow at the steering level of about 700 hPa, roughly 7–8 m s−1. The picture that emerges from this analysis is one of modulated wave packets of synoptic disturbances, exemplified by the nonlinear baroclinic wave packets that have been noted in both the Northern and Southern Hemispheres and in simple models (Lee and Held 1993; Chang 1993, 1999, 2000, 2001). The dominance of the group propagation dynamic in the regression analysis indicates that the manner in which these packets of eddies interact with streamwise variations in the background flow ultimately determines storm track structure. The key issue then boils down to how these packets amplify and decay as they propagate through a streamwise varying background flow.

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Storm Track Dynamics | 83 60 °N

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Figure 4.4. Correlation of meridional velocity on the 300 hPa surface against the point (40◦ N, 175◦ E). In the top panel, the correlated fields lag that point by 1 day, in the middle they are simultaneous, and in the bottom panel that point leads the fields by 1 day. The contour interval is 0.1, with negative contours dashed and the zero contour omitted.

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The potent nature of the group dynamic within these packets is illustrated by Chang and Orlanski (1993), who show that there is no intrinsic downstream limit to storm tracks—once the transients are in place, the group dynamic allows the track to extend an indefinite distance downstream. However, sufficient barotropic deformation in the background flow can act to localize storm tracks (Lee 1995). This deformation is visible in the tendency of disturbances to become stretched in the meridional direction and compressed in the zonal direction in the eastern Pacific as shown in Fig. 4.4. This stretching marks the barotropic transfer of disturbance energy into the background flow (Cai and Mak 1990). However, this process need not be irreversible, and may simply mark an elastic interaction of the propagating wave packet with the streamwise varying background flow (Swanson et al. 1997). Other processes, such as storm track tilt (Frisius et al. 1998), surface friction, and the stability of the near-surface atmospheric layers no doubt also play a part in determining the details of storm track structure. This analysis suggests that a minimal storm track theory must explain the storm track length in relation to an imposed background flow, along with the associated disturbance fluxes of heat and momentum, while retaining the sense of group dynamics. It is difficult to formulate such a theory for a model containing complete physical processes (moisture, etc.), let alone in the observed setting. In general, theoretical approaches are most valuable when they are applied in a setting in which the assumptions underlying the theory are satisfied. In this way, one develops experience in judging the strengths and weaknesses of a given theoretical approach in a clean setting, which provides the comfort that if the theory is adequate in such a setting, it may scale to provide explanations of behaviors found in more complete models or the observations. This is how the problem of storm track dynamics is approached here, first examining an intermediate model that possesses realistic storm track dynamics, followed by a comparison of the various theories of storm track behavior against model behavior.

4.3. Storm Track Variability: A Case Example 4.3.1. Dynamical Setting Consider the nondimensional equations describing the quasigeostrophic flow of a twolayer fluid on the beta plane. Let λ = (g δρ D/ρ0 f 02)1/2 be the deformation radius based upon the total fluid depth D, where ρ0 is the average fluid density and δρ the density difference between the two layers, and let U be an as yet unspecified velocity scale. If we nondimensionalize lengths by λ, velocities by U , and times by λ/U , the governing dynamical equation can be written ∂tq n + J (ψn, q n) = Sn ,

[4.2]

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where n = 1 corresponds to the upper layer and n = 2 corresponds to the lower layer. Here, ψn is the streamfunction and q n is the potential vorticity, which is given by q n = ∇ 2ψn − (−1)n(ψ2 − ψ1)/2 + βy.

[4.3]

In (4.3), β = β ∗λ2/U is a nondimensional parameter, where β ∗ is the dimensional planetary vorticity gradient. The source/sink term Sn generally includes radiative relaxation of the temperature θ = (ψ1 − ψ2)/2 to some radiative equilibrium profile θeq with some time scale τrad , and Ekman damping of the vorticity in the lower layer on a timescale τek . The velocity scale U is chosen consistent with the maximal imposed zonal-mean radiative equilibrium shear , leaving ξ = β −1 as a measure of the supercriticality of the zonal-mean flow (cf. chapter 1 in this volume; the supercriticality measure in chapter 1 differs from that here by a factor 2). To study storm track dynamics, it is necessary to introduce zonal asymmetry in some form. If we restrict our focus to the Northern Hemisphere winter, two physical mechanisms are of particular importance in forcing the observed asymmetry. First, there are substantial zonal variations in the baroclinicity, with strong baroclinic zones off the east coasts of Asia and North America, where warm oceanic currents abut cold wintertime continental interiors. These temperature contrasts at the surface are communicated into the interior of the atmosphere by longwave radiative energy transfer, along with latent and sensible heat fluxes in the turbulent boundary layer and convective processes, resulting in the effective damping of the depth-integrated atmospheric profile to a zonally varying temperature distribution. Also, there are the prominent stationary wave patterns forced by topography and zonally varying heating; see the review by Held et al. (2002). To simplify the issue, for the cases examined herein we consider a zonal asymmetry in the planetary-scale flow forced by radiative relaxation to a zonally asymmetric temperature profile of the form 1 2 θeq = σ tanh(y/σ )(1 − γ e −(y/5σ ) cos 2π x/L ), 2

[4.4]

where L is the domain length, σ is the jet width, and is the shear of the radiative equilibrium jet. Note that caution must be applied in interpreting the results of any model response to simplified forcing such as this—as detailed by Andrews (1984), forcing that interacts with disturbances complicates the dynamical interpretation of zonally asymmetric flows. Suppose that the zonally averaged flow in each layer is Un. In the absence of zonal asymmetry (γ = 0) and eddies, the flow will be zonal, U2 will vanish, and U1 = Ueq = −2

∂θeq ≡ sech2(y/σ ). ∂y

[4.5]

If we choose as the velocity scale, a horizontally uniform vertical shear U1 − U2 ≡ 1 is baroclinically unstable by Phillips’s criterion, in the absence of damping, if β < 1/2.

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Given this situation, there are a number of parameters that in general will influence storm track structure and amplitude; the supercriticality ξ = β −1, the zonal asymmetry γ , the jet width σ , and the thermal and mechanical damping parameters τrad and τek. For reference, one nondimensional time unit is roughly one-half day for the values λ = 800 km and U = 20 m s−1. In general, we will speak in terms of dimensional quantities for ease of interpretation.

4.3.2. Base Case Figure 4.5 shows the upper-layer time-mean streamfunction and bandpass streamfunction standard deviation for the moderately supercritical situation β = 0.1, where the jet width σ = 1.5λ and the zonal asymmetry parameter γ = 3.5. The storm track system evolves in a wide channel with zonal length L = 30λ. The radiative and Ekman damping times are τrad = 30 days and τek = 5 days, respectively, and the model is initialized from a state of rest perturbed by small amplitude vorticity disturbances in both layers. Since storm tracks are statistical creatures by nature, in principle one might expect the precise nature of this initial condition to be immaterial. This assumption shall be discussed further in section 4.6, but for the examples in the next two sections, the statistical steady state is independent of the initial conditions. Certain aspects of observed storm tracks are well captured by this model. Localized storm tracks are found in both layers, with the upper-layer track displaced roughly λ downstream from the upper-layer jet maxima, and the standard deviation in the upper layer roughly twice that in the lower layer. The model’s fluxes also resemble

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Figure 4.6. Correlation of meridional wind in the upper layer against the point 22λ at the channel center. In the top panel, the correlated fields lag that point by 1 day, in the middle they are simultaneous, and in the bottom panel that point leads the fields by 1 day. The contour interval is 0.1, with negative contours dashed and the zero contour omitted.

the observed, with the heat flux largest over the upstream end of the storm track and the momentum fluxes largest on the downstream end of the storm track on the flanks of the jet. The ability of this model to mimic observed storm track structures carries over to its spatio-temporal behavior. Figure 4.6 shows the one-point lagged correlations for a point located at the channel center at the zonal distance 22λ. As in the observed case, there is a distinct signature of group propagation different from phase propagation. This is marked by the larger magnitude correlations upstream of the base disturbance for the leading case (day − 1), and downstream of the base disturbance for the lagging case (day + 1). As in the observations, this group speed is more characteristic of the upper-layer flow, while the phase speed is similar to the lower-layer flow. In the upper layer, the transients result in a downgradient flux of potential vorticity, particularly on the flanks of the jet. Such mixing is consistent with an acceleration of the flow along the jet axis, which itself is counteracted by dissipation of momentum in the lower-layer Ekman layer, resulting in mean westerlies in both layers. Geographically, this mixing is concentrated downstream of the baroclinic zone. Locally, the potential vorticity fluxes have substantial components both down the basic state potential vorticity gradient as well as along basic state potential vorticity contours, complicating the interpretation of the interaction of transients with the time-mean flow

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Figure 4.7. Time-mean zonal wind along the channel center for the perturbed storm track cases.

in comparison to the zonal-mean case. The issue of transient-mean flow interaction in storm tracks becomes involved rather quickly; Kushner and Held (1999) contains a discussion of these issues at an advanced level. 4.3.3. Storm Track Perturbations A classic problem in storm track dynamics is the statistically steady response of a storm track to an imposed forcing to the basic state flow, such as heating due to an El Niño sea surface temperature anomaly. For the situation here, perhaps the simplest basic state perturbation involves adjusting the zonal asymmetry parameter γ . Other parameters, of course, are also of interest; Zurita-Gotor and Chang (2005) provide an in-depth examination of the response of a storm track to changes in the barotropic wind in a model similar to that here. Figure 4.7 reveals that the response of the time-mean zonal wind component along the channel center to changes in γ is quite substantial. For the value γ = 1, the jet in the upper layer is nearly in phase with the radiative forcing, but increasing the asymmetry γ beyond this level results in a jet roughly 3λ downstream from the radiative forcing maximum, with the maximum jet speed increasing more or less consistently with γ . Corresponding weakening of the zonal flow away from the jet as γ is increased results in only small changes in the zonal-mean zonal flow as a function of γ . This contrast between the substantial local increases in jet strength (and hence baroclinicity) and the relatively benign global change in zonal-mean zonal wind is important to keep in mind when the discussion turns to the dynamics linearized around time-mean flows in section 4.4. Figure 4.8 shows that the bandpass standard deviation associated with the variation in γ scales in a manner similar to that of the wind itself, increasing more where the time-mean zonal wind is large, with a weaker response where the time-mean zonal wind is small. This response is not surprising; since the lower-layer flow is small, the upper-layer time-mean zonal wind along the channel center provides a measure of the baroclinicity of the system, with stronger time-mean zonal winds associated with

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Figure 4.8. Bandpass standard deviation of upper-layer streamfunction along the channel center for the perturbed storm track cases as a function of γ .

higher baroclinicity. It is natural that disturbances should amplify more in response to this enhanced baroclinicity as they traverse the storm track. Additionally, this behavior highlights the importance of the zonal waveguide; where that waveguide is well formed, i.e., locations where the zonal flow is large, and upper-layer potential vorticity gradients are large, is where significant changes in transient variance occur. This variation in bandpass standard deviation in response to changes in γ provides a reasonable set of test cases for storm track theories, which we explore in detail in the next section.

4.4. Linear Theories Synoptic eddies and storm tracks are almost always discussed in relation to a background flow, whether that flow is steady, time mean, slowly evolving, or representative of some “instantaneous” weather regime. Such a separation of the flow into mean and eddy parts often presents conceptual difficulties, and also raises the issue of whether, and to what extent, the mean flow actually determines the eddy statistics or vice versa. However, in the context of storm track dynamics, it seems reasonable to inquire to what extent one can deduce storm track structure given an appropriate time-mean flow. This is the question addressed by mean field theories, of which linear theories are an important subset. Insofar as one can successfully develop a linear theory linking storm tracks to the basic state flow, such a theory would be quite powerful. It is by now understood that changes in storm track structure force a substantial component of interannual stationary planetary wave activity (Lau and Holopainen 1984; Branstator 1992, 1995). A theory for storm track structural shifts could then be coupled with a stationary wave model, for example, providing a theory for the extratropical response to forced interannual variability. Linear theories for storm tracks are linked to the initial-value problem of disturbance evolution and differ primarily over what time scales are important to the

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overall storm track structures. Normal mode theory, with its emphasis on exponentially amplifying disturbances, assumes that disturbance structures in the long time limit are most important. In contrast, various stochastic theories either examine disturbance structures at a presupposed time, or add sufficient artificial dissipation to stabilize the linear system and, as such, effectively consider the time integral of the disturbance response. Let us consider how well these theories do at explaining the base and perturbed storm track structures from the nonlinear model simulations described above. 4.4.1. Normal Modes Following the success in explaining certain aspects of synoptic eddy structure by examining linearly growing disturbances on zonally symmetric flows, as discussed by Pierrehumbert and Swanson (1995), it is reasonable to inquire whether examining the three-dimensional stability problem with a zonally varying flow provides insight into storm track structure. Upon substituting ψn = ψn (x, y) + ψ (x, y, t) q n = qn(x, y) + q (x, y, t)

[4.6]

and neglecting terms quadratic in the perturbation quantities, the linearized equations of motion are found to be ∂t q n + J (ψn, q n ) + J (ψn , qn) = Sn .

[4.7]

In principle, one can solve this problem analytically or semi-analytically in the limit of slow streamwise variations in the imposed flow, following Pierrehumbert (1984). However, for more realistic models and flows, such a procedure becomes tedious. In such situations, one is left with solving the linear problem (4.7) directly for the zonally varying background flow. For a specific first example, consider the linear response to the base-case timemean flow shown in Fig. 4.5, neglecting (for the time being) forcing and dissipation by setting Sn = 0. Integrating this condition forward for a sufficiently long time isolates a wavy normal mode, one phase of which is shown in Fig. 4.9. Consistent with Fredricksen (1983), this normal mode has structural characteristics similar to the nonlinear model’s base-case highpass transient variability, and the scale of the waves themselves is similar to transient structures obtained from constructing point correlations in the evolving fully nonlinear flow, as shown in Fig. 4.6. In particular, it appears to capture the zonal structure of the variance, with peak variance between x = 20λ and x = 25λ for the domain here. The qualitative agreement between the structure of this unstable normal mode and the model’s storm track is intriguing. However, there are a number of reasons to be cautious about using such a mode as a “theory” for storm tracks. First, in general there will be a spectrum of unstable normal modes, particularly for more complicated

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Figure 4.9. Upper-layer streamfunction for one phase of the most unstable mode for dynamics linearized about the base-case time-mean flow. Contours are arbitrary, with negative contours dashed and the zero contour omitted.

flows with multiple storm tracks. Unless the growth rate of the most unstable mode is well separated from that of other modes, there is no reason to suspect that the most unstable normal mode should be preferred in such a situation. Secondly, it is now generally believed that in models of this class, as well as in the atmosphere, unstable normal modes are global, rather than local, in character. As discussed at length by Pierrehumbert (1984), this distinction is vital. Local modes have a number of desirable structural properties that in principle make them ideal candidates to explain storm track structures. Their amplitude peaks downstream of the point of maximum baroclinicity of the basic state flow, and decays to zero both downstream and upstream of the peak. Most importantly, local modes do not require a re-entrant domain for their existence. This compact nature ensures that they are not influenced by factors remote from a given localized baroclinic zone. In addition, their growth rate is determined by the growth rate at the point of maximum baroclinicity for a given local baroclinic zone, rather than some domain-averaged measure of baroclinicity. In contrast, global modes require a re-entrant domain for their existence, and because of this, take very long to become established in any given region. Their growth rate is sensitive not to the peak baroclinicity in the domain, but rather to the baroclinicity averaged throughout the domain. The global modes form a near continuum, and owing to the large time required for a single mode to emerge from random initial conditions, they are physically relevant only insofar as they can be used in combination to represent the evolution of transient wave packets. For the case here, as well as for the atmosphere, dynamics linearized about the time-mean flow do not support local modes (Lin and Pierrehumbert 1993). Such local modes appear to require rather robust easterlies at the surface to exist, and these are found neither in the model here nor in the atmosphere. In the model here, the lack of local modes may be directly confirmed by inserting a “sponge” well downstream of the storm track that inhibits disturbance recirculation. Doing so eliminates exponentially amplifying disturbances in this particular system. This leaves us in a quandary—since the most unstable mode is global, there is no reason to expect that this mode in isolation should be more important than other modes with similar, albeit smaller, growth rates. Indeed, the entire question of growth rates is

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troublesome, as their link to storm track structure is not immediately obvious—i.e., do larger magnitude linear disturbance growth rates necessarily lead to larger-magnitude storm tracks? The difficulties with the use of unstable normal modes to describe storm tracks may be illustrated by considering the dynamics of the most unstable mode for the sequence of mean flows generated by varying γ . Consistent with the discussion above regarding the dependence of global modes upon the zonally averaged baroclinicity, which as noted in section 4.3 does not vary substantially as a function of γ , there is little variation in the disturbance growth rates with γ . Those growth rates range from an e-folding time of 20 days for γ = 1 to an e-folding time of 15 days for γ = 5. The sense of these growth rates is consistent with the changes in bandpass standard deviation as γ is varied, i.e., higher growth rates do correspond to larger bandpass standard deviations. However, this sensitivity of the growth rates is much smaller than it would be if these were local modes (Zurita-Gotor and Chang 2005). Comparison of the growth-rate variation as a function of γ suggests that normal mode growth must operate over a time scale of 50 days to explain the changes in bandpass standard deviation that occur as γ is varied (cf. Fig. 4.8). This time scale is long compared to the de-correlation time scale for this system, casting doubt on the strict application of normal mode theory as an explanation of storm track behavior for the case here. Further, this result is for inviscid disturbance growth rates; incorporating radiative damping and surface friction neutralizes these normal modes, consistent with the transients driving the time-mean flow to the point of stabilization in a manner similar to a baroclinic adjustment (see chapter 2 in this volume). While normal-mode analysis does not provide a theory for storm tracks, it provides insight into certain aspects of storm track structure. As outlined above, storm tracks result from the amplification of transient wave packets. This leads to the question of whether such amplification can be understood in terms of linear spatiotemporal growth of wave packets. Assuming a flow is sufficiently zonal, and varies slowly in the streamwise direction, it will support a spectrum of unstable waves. An initially localized disturbance will evolve into a wave packet with time, and it can be shown that the peak of this wave packet will travel at the velocity
∂ω

cg = , [4.8] ∂k
max σ where ω denotes the real component of the amplifying disturbance frequency and σ the imaginary component. For a packet developing linearly from an initially localized disturbance, this means the peak of the packet travels at the group speed of the most unstable mode (Swanson and Pierrehumbert 1993). Assuming a baroclinic zone of length L , the quantity σ L/c g , i.e., the growth rate multiplied by the time it takes to traverse the baroclinic zone, is the factor by which one expects a given disturbance will amplify as it traverses the baroclinic zone. Larger disturbance growth may occur not only due to rapid growth (σ large), but also for slow propagation (cg small). Note that

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these concepts are based upon disturbance growth in purely zonal flows. This picture appears to capture the amplification of storm track transients as they propagate through the storm track region, as noted by Harnik and Chang (2004). By and large, this is a product of the fact that growth rates for purely zonal flows roughly scale with the shear (Lindzen and Farrell 1980). In conclusion, normal-mode thinking has its uses in explaining certain details of storm track structure. However, it offers limited information regarding the relative magnitude of storm tracks in response to perturbations in the background mean flow. As such, it is desirable to examine the extent to which other linear approaches might provide a more complete theory for storm tracks.

4.4.2. Stochastic Models A different approach to the problem of simulating the statistics of storm track transients follows from the work of Farrell and collaborators (Farrell 1984; Farrell and Ioannou 1994). From this perspective, storm track transients are best viewed as stochastically forced disturbances evolving on a baroclinically stable flow. Eddies can still grow on such a flow, for a finite time, through local energy extraction from the background shears. The effects of quadratic nonlinearities are approximated as stochastic excitation plus an augmented dissipation, similar to that done in early studies of homogeneous turbulence, and the governing equations reduce to a multi-dimensional linear Markov process. This approach has shown certain successes when models linearized about either the observed time-mean flow (Whitaker and Sardeshumkh 1998) or mean flows generated from a general circulation model (GCM) (Zhang and Held 1999) are stochastically forced. For our purposes here, we consider stochastically forced, artificially damped linear dynamics using the base-case time-mean flow of Fig. 4.5, as well as the time-mean flows resulting from the variation of γ as highlighted in Fig. 4.7. The equation solved is (4.7), where the forcing term Sn includes not only Ekman damping and radiative relaxation, but also stochastic forcing and an artificial Newtonian relaxation of the disturbance potential vorticity fields. The latter two processes are intended to mimic nonlinear wave-wave dynamics, which scatter energy between different wavenumber bands. As noted above, Ekman damping and radiative relaxation of disturbances by itself results in time-mean profiles that are nearly neutral to linear disturbance growth. Hence, there is substantial flexibility in the choice of the level of artificial Newtonian relaxation. Two situations are examined, one with a relaxation time scale of 7 days and the other with a relaxation time scale of 30 days. The model is forced by independently and randomly exciting the vorticity and temperature components of the quasigeostrophic potential vorticity, following Zhang and Held (1999). By adjusting the relative strength of these two forcings, it is relatively straightforward to tune the stochastic model to reproduce not only the storm track structure but also the transient heat and momentum (or potential vorticity) fluxes.

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94 | Kyle L. Swanson 1.0 Bandpass Deviation

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γ=5 γ=4 γ=3 γ=2 γ=1

0.8 0.6 0.4 0.2 0

0

5

10

15 Zonal Distance (λ)

20

25

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Figure 4.10. Stochastically modeled bandpass standard deviation of upper-layer stream function along the channel center for the perturbed storm track cases under the strong damping scenario.

The key question regarding such stochastically forced models concerns what is truly determining the transient statistics. For this, it is useful to define “strong” and “weak” interpretations of stochastic storm track models. The “strong” interpretation follows Whitaker and Sardeshmukh (1998), and is characterized by artificial disturbance relaxation over time scales similar to the de-correlation time scale for storm track systems, roughly one week. Storm track structure and amplitude is then determined primarily by the rapid non-modal growth of disturbances (Farrell 1984). Under this scenario, there is substantial dependence on the manner in which the system is forced, i.e., whether the applied forcing is spatially white in the vorticity, energy, or streamfunction measures. This follows from the work of Farrell and Ioannou (1994), where forcings that have substantial vertical structure preferentially lead to large responses. In contrast, a “weak” interpretation of stochastic storm tracks is marked by relatively weak Newtonian relaxation, exploiting the fact that the action of transients in a model yields a time-mean flow that is quasi-neutral. In this setting, it is natural to expect the dynamics to be much more normal-mode in character, and for the most part insensitive to the structure of the stochastic forcing. For this simple model, both strong and weak Newtonian relaxation results in adequate reproduction of the storm track structure for the base case shown in Fig. 4.5, as measured, for example, by pattern correlation coefficients. However, as shown in Figs. 4.10 and 4.11, situations with stronger damping (7 days versus 30 days) are quite insensitive to variations in the time-mean flow that occur when γ is varied. As γ increases from 1 to 5, the maximum storm track standard deviation increases only 10% in response to the changes in the flow for the strong relaxation case, and roughly by a factor of 2 for the weak relaxation case. Curiously, Zhang and Held (1999) used relatively large damping in their attempt to explain the midwinter minimum. Their study showed hints of a minimum, but the overall response was much too weak, at least qualitatively resembling the strong damping case shown in Fig. 4.10.

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0.8

γ=5 γ=4 γ=3 γ=2 γ=1

0.6 0.4 0.2 0 0

5

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15 Zonal Distance (λ)

20

25

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Figure 4.11. Stochastically modeled bandpass standard deviation of upper-layer stream function along the channel center for the perturbed storm track cases under the weak damping scenario.

This result highlights some of the unresolved questions regarding the application of stochastically forced storm track models. More experience is necessary to understand the relative roles of forcing structure and disturbance damping before the application of theories of this type can be considered understood. However, the ability of these models to produce statistically steady models of storm track behavior ensures such models will play an important role in furthering our understanding of storm tracks in the future.

4.5. Heuristic Models While much of what is understood about storm tracks is rooted in linear models, more qualitative models have much to offer as well, with the additional benefit of sharpening the focus on those aspects of storm track dynamics that are truly vital. Over the past decade, it has become increasingly apparent that the fundamental structure in storm track dynamics is not baroclinic growth or conversion per se, but rather the nature and dynamics of the upper-tropospheric waveguide. This is apparent in the paper of Branstator (1995), where changes in transient momentum fluxes are shown to be consistent with the anomalous steering of short wavelength, non-dispersive Rossby waves by anomalous background time-mean flows. Anomalies in transient barotropic vorticity fluxes in the upper troposphere result from the response of transients to changes in that waveguide itself, rather than as some “source” not linked to the largescale flow that defines that waveguide. The starting point for any heuristic theory of storm structure is to understand the source of storm track transients. The most promising approach is to note that transient variations in the lower troposphere act primarily to diffuse heat downgradient, with a coefficient of diffusivity equal to the transient streamfunction standard deviation. This is dimensionally consistent, as the streamfunction, like diffusivity, has units of a length multiplied by a velocity, so it provides a simple way to estimate the eddy length and time

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scales needed for computing the diffusivity. This point has been made by Kushner and Held (1998) using observed lower-tropospheric winds and heat fluxes. This heat flux provides a source of transient wave activity (more formally pseudomomentum [see Andrews et al. 1987]) to the upper troposphere. For the upper layer of the two-layer quasigeostrophic system, this wave activity conservation relation has the form ∂t A 1 +

∂ (U1A1) + ∇ · F = v1 θ , ∂x

[4.9]

where A1 ≡ 12 q 12/∂ y Q 1 is the pseudomomentum, θ  = (ψ1 − ψ2)/2, F ≡ [(v12 − u2 1 )/2, −u1v1 ] is the two-dimensional Eliassen-Palm flux vector, and Q 1 is the upper-layer zonal-mean potential vorticity distribution. When discussing storm tracks, interest is usually focused on the existence of a zonally oriented upper-tropospheric waveguide. That waveguide will generally be imperfect, with loss of wave activity due to wave breaking in critical layers on the jet flanks, as described, for example, by Killworth and McIntyre (1985), and leakage due to meridional Rossby wave propagation away from the waveguide. These effects are found in the momentum flux −u1v1 in the Eliassen-Palm flux vector F. Hence, it is useful to qualitatively rewrite the expression (4.9) as ∂t A1 + ∂x(c g x A1) = v1 θ  − breaking − leakage,

[4.10]

where c g x = U1 + (v12 − u2 1 )/2 is the group speed in the zonal direction, and the terms “breaking” and “leakage” incorporate the loss of wave activity in the waveguide. The development of a heuristic theory of storm track dynamics thus is reduced to two parts: a link between the measure of disturbance amplitude A1 and the “source” of wave activity −v1 θ , and a theory for the breaking and leakage by the waveguide. Regarding the former, Kushner and Held (1998) have shown that to a very good approximation the downgradient heat flux is diffusive, where the diffusivity is simply the transient streamfunction standard deviation. This transient streamfunction standard deviation can be related to the wave activity, providing a closure theory for the baroclinic   source. In terms of an upper-layer disturbance streamfunction standard deviation ψ1 =  2 1/2 ψ1 , where the bar indicates time-mean, following Kushner and Held (1998) the     heat flux will be v1 θ  = − ψ1 ∂y θ = ψ1 U1/2, provided that the lower-layer winds are small (i.e., the shear is approximately U1/2 = −∂y θ by thermal wind). With a theory for the source of wave activity in hand, the next challenge is to understand what that wave activity will do when it is in the upper troposphere. The upper troposphere in general will act as a waveguide and reservoir of wave activity, with wave activity within that waveguide being deformed both reversibly and irreversibly. Irreversible wave breaking and/or Rossby wave radiation away from the waveguide will act as sinks of that wave activity. However, for streamwise varying flows, it is the pseudoenergy rather than the pseudomomentum that is the relevant conserved quantity. As shown by Held (1985), these quantities are related by P = c A, where c is the phase

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–15

–10

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Figure 4.12. Loss of disturbance amplitude due to irreversible wave breaking in the simple storm track model of Swanson et al. (1997). The weak flow region is in the center of the domain; disturbances break there first, and their amplitude is reduced after exiting that region. The coordinates are nondimensionalized.

speed of disturbances. In a time-mean setting, wave activity conservation for P1 then will have the form   ∂x(c g xP1) = c ψ1 U1/2 − c (breaking + leakage).

[4.11]

Writing P1 in terms of the disturbance streamfunction ψ1 closes this equation in the absence of breaking and leakage. In the absence of leakage or radiation away from the waveguide, wave activity will increase in the waveguide due to the downgradient flux of heat. However, in a re-entrant domain the magnitude of storm track disturbances will be determined by the nature of the leakage and radiation of wave activity away from the waveguide. While theoretical guidance on these effects is limited, the study of Swanson et al. (1997) provides a particularly simple picture. As shown in Fig. 4.12, as disturbances propagate from a region of strong flow into a region of weak flow, their amplitudes are “clipped” due to wave breaking. It is straightforward to show that clipping effectively sets the disturbance amplitude where the flow along the waveguide is weakest. This is similar to the result found in the perturbation experiments of Fig. 4.8; the disturbance standard deviation varies in response to changes in the flow much more in the heart of the storm track than it does in the weak-flow region upstream of the storm track where complicated wave breaking processes occur. Hence, it appears that under certain circumstances regions of weak zonal flow may act as the “gatekeeper” for storm tracks, setting the disturbance

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amplitude at the upstream end of the track. Baroclinic growth via downgradient heat fluxes is vital to the structure of storm tracks, but arguably less so for their overall amplitude. This type of expression suggests why linear techniques are effective at capturing the basic structure of storm tracks, as that structure is primarily determined by the growth of disturbances, which is effectively a linear process. However, the breaking and leakage of disturbances in storm track exit regions is not linear, and nonlinearity may lead to a more substantial decrease in disturbance magnitude than the linear modulation predicted by stochastic theories or normal modes. Perhaps the most important challenge as we move forward in understanding storm track dynamics is to better understand dynamics in storm track exit regions.

4.6. Nonlinear Behavior Given the ability of linear and quasi-linear theories to explain certain aspects of storm track structure, it may at first glance appear unnecessary to delve too deeply into the role of nonlinearity in storm track dynamics. However, nonlinearity in storm track dynamics has multiple guises. The first guise, that of baroclinically forced turbulence, appears to be adequately captured by the assumptions that turbulent wave-wave interactions can be modeled as an artificially enhanced Newtonian relaxation of disturbances coupled with stochastic forcing of disturbances as noted in section 4.4. However, there is another, lesswell discussed aspect of storm track dynamics, namely, whether the global dynamics of storm tracks are robust to changes in the radiative forcing, topography, or damping. In general, it is assumed that there is sufficient “noise” in a storm track system generated by the recirculation of disturbances to maintain robust storm track variability. However, the extent to which this is actually true remains poorly understood, with potentially important ramifications for our understanding of climate variability. To examine some global properties of storm track systems, consider the base-case forcing of section 4.3, but now varying the supercriticality ξ = β −1. In all cases, the model is started from an initial condition of radiative equilibrium with small wavy disturbances. For values of β ≤ 0.19, a “normal” climate regime with respect to the observed climate is found similar to that discussed in section 4.2, consisting of a robust storm track displaced downstream from an intense upper-layer jet. The storm track and resultant mean temperature gradient for β = 0.19 are shown in Fig. 4.13. However, as the supercriticality is further reduced, a bifurcation occurs in the system’s dynamics. As shown in Fig. 4.14, decreasing the supercriticality to β = 0.21 reveals a markedly different climate regime, with a broader, more intense baroclinic zone compared to the “normal” climate found for higher supercriticalities. Transient variability is substantially weaker, with a dipole of transient variability occurring downstream of the jet with standard deviation roughly one-quarter that of the normal climate regime of Fig. 4.13.

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Meridional Distance (λ)

4 2

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Figure 4.13. Top: Time-mean temperature distribution for β = 0.19 case. Bottom: Total transient standard deviation of upper-layer streamfunction for β = 0.19 case. Units are λU .

–0.8

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Figure 4.14. Top: Time-mean temperature distribution for β = 0.21 case. Bottom: Total transient standard deviation of upper-layer streamfunction for β = 0.21 case. Units are λU .

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Domain Average Bandpass RMS (λU)

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0.30

0.20

0.10

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1000

5000 Time (λ/U)

10000

Figure 4.15. Time series of domain-averaged bandpass standard deviation of streamfunction variations in simulations started with initial conditions taken from the final state of the β = 0.19 and β = 0.21 cases discussed in the text. Both simulations have β = 0.2.

The variability in this situation is low frequency in character, with the baroclinic zone spawning low-frequency, larger-scale transients on the flanks of the jet. Synoptic scale variability is also present, but mobile transients are substantially weaker in magnitude than for higher supercriticalities. The traditional method to isolate a bifurcation of this type is to continue solutions from above and below the bifurcation value, making small incremental changes in β, and examine where the jump in dynamical properties occur. Following this procedure, it is found that over the parameter range β ∈ (0.19, 0.21), multiple climates occur in this storm track system for different initial conditions. For example, consider two model simulations with the model supercriticality fixed at β = 0.2, with initial conditions respectively taken from the respective final state of the simulation with β = 0.19 of Fig. 4.13 and the simulation with β = 0.21 of Fig. 4.14. Other than the initial condition, the simulations are identical. Figure 4.15 shows the time series of the domain-averaged bandpass standard deviation (time scales less than 10 days) for the two simulations resulting from these initial conditions. After a period of some transient adjustment, the two simulations maintain very different levels of synoptic transient variability for what appear to be arbitrarily long times. It appears that for this particular forcing this idealized storm track system possesses a hysteresis, insofar as the model’s system state depends upon where the initial condition was taken, whether above or below the bifurcation point. Further, since these solutions are separately chaotic and irregular, it appears as if the system also possesses two coexisting strange attractors, each with quite different climates. While multiple attractors in simple atmospheric models have been noted in a number of different contexts, the presence of baroclinic instability is generally assumed to eliminate such behavior, leading instead to regime-like behavior in the underlying dynamics.

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Complicated behavior associated with these two attractors is found for this storm track system—sensitive dependence of the ultimate climate upon initial conditions and “bursts” of synoptic transient activity that appear and disappear suddenly are part of this system’s phenomenology. One must of course take results arising from a model without full dynamical complexity with a grain of salt. However, the presence of this type of behavior does raise a question not often asked in our current examination of climate dynamics: is climate unique? If this type of behavior is found in more complete models, it would open the possibility of strong climate changes in response to rather innocuous triggers, as well as threshold-type behavior, all mediated by storm track dynamics. Significantly, such behavior does appear in the paleoclimate record, in the apparent sensitivity of the hemispheric and even global atmosphere to forcing localized in the North Atlantic basin arising from alterations to the thermohaline circulation in the paleoclimate record (see Dansgaard et al. 1989 and chapter 12 in this volume).

4.7. Conclusions Where do we stand with understanding storm tracks? The best current theoretical models of storm track dynamics suggest that storm tracks may be adequately modeled as the response of linear dynamics to incoherent stochastic forcing, with baroclinic development of certain preferred disturbances giving the storm tracks their unique character. However, in reality the “forcing” in such stochastic linear storm track models is the result of internal nonlinear dynamics, and there is no reason that such forcing need to be simple or consistent given changes to either external parameters or in response to internal variations in the large-scale flow. The effects of this can be as subtle as a stochastic model’s inability to capture changes in storm track amplitude due to an altered basic-state flow, or as profound as the multiple strange attractor behavior shown in section 4.6. The fundamentally nonlinear transient response to deformation at the downstream end of baroclinic zones emerges as a problem of particular importance, as the ability of transients to transit this zone appears to provide a limit on storm track amplitudes for the entire system. Unfortunately, theoretical guidance of the physics governing how disturbances evolve through such diffluent flow environments is extremely limited. Much more remains to be discovered about storm track dynamics in this region before our understanding of even current-day storm track fluctuations can be considered complete. Above all, intermediate models still have a role to play in furthering our understanding of storm track dynamics, and hence the climate system. There are always many choices that must be made in the study of storm track dynamics. Is it better to fully explore the parameter space of an intermediate model, or apply that computer power

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toward simulations with a more complete model? Is it better to test theories completely on storm track models of the same complexity as the theory itself, or attempt to use them to explain variations in the observed storm tracks? While these are questions that have no easy answers, the state of understanding storm tracks may well be better served by inquiries along the lines of the first part of each question. There is still much work that remains in studying the dynamics of storm tracks before their role in the climate can be considered understood.

References Andrews D.G. (1984). On the stability of forced non-zonal flows. Q. J. Roy. Met. Soc., 110, 657–662. Andrews D.G., Holton J.R., and Leovy C.B. (1987). Middle Atmosphere Dynamics. Academic Press, 489 pages. Blackmon M.L. (1976). A climatological spectral study of the 500 mb geopotential height of the Northern Hemisphere. J. Atmos. Sci., 55, 1607–1623. Blackmon M.L., Wallace J.M., Lau N.-C., and Mullen S.L. (1977). An observational study of the Northern Hemisphere wintertime circulation. J. Atmos. Sci., 34, 1040–1053. Branstator G. (1992). The maintenance of low-frequency atmospheric anomalies. J. Atmos. Sci., 49, 1924–1945. Branstator G. (1995). Organization of storm track anomalies by recurring low-frequency circulation anomalies. J. Atmos. Sci., 52, 207–226. Broecker W.S. (1997). Thermohaline circulation, the Achilles Heel of our climate system: Will man-made CO2 upset the balance? Science, 278, 1582–1588. Cai M. and Mak M. (1990). Symbiotic relation between planetary waves and synoptic scale waves. J. Atmos. Sci., 47, 2953–2968. Chang E.K.M. (1993). Downstream development of baroclinic waves as inferred from regression analysis. J. Atmos. Sci., 50, 2038–2053. Chang E.K.M. (1999). Characteristics of wave packets in the upper troposphere. Part II: Hemispheric and seasonal differences. J. Atmos. Sci., 56, 1729–1747. Chang E.K.M. (2000). Wave packets and life cycles of troughs in the upper troposphere: Examples from the Southern Hemisphere summer season of 1984/85. Mon. Wea. Rev., 128, 25–50. Chang E.K.M. (2001).The structure of baroclinic wave packets. J. Atmos. Sci., 58, 1694–1713. Chang E.K.M. and Orlanski I. (1993). On the dynamics of a storm track. J. Atmos. Sci., 50, 999–1015. Chang E.K.M., Lee S., and Swanson K. (2002). Storm track dynamics. J. Climate, 15, 2163–2183. Dansgaard W. et al. (1989). The abrupt termination of the Younger Dryas climate event. Nature, 339, 532–534. Farrell B.F. (1984). Modal and non-modal baroclinic waves. J. Atmos. Sci., 41, 668–673. Farrell B.F. and Ioannou P. (1994). A theory for the statistical equilibrium energy and heat flux produced by transient baroclinic waves. J. Atmos. Sci., 51, 2685–2698. Fredricksen J.S. (1983). Disturbances and eddy fluxes in Northern Hemisphere flows: Instability of three-dimensional January and July flows. J. Atmos. Sci., 40, 836–855. Frisius T., Lunkeit F., Fraedrich K., and James I. (1998). Storm-track organization and variability in a simplified atmospheric global circulation model. Q. J. Roy. Met. Soc., 124, 1019–1044.

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Storm Track Dynamics | 103 Hall N.M.J., Hoskins B.J., Valdes P.J., and Senior C.A. (1994). Storm tracks in a high-resolution GCM with doubled carbon dioxide. Q. J. Roy. Met. Soc., 120, 1209–1230. Harnik N. and Chang E.K.M. (2004). The effects of variations in jet width on the growth of baroclinic waves: Implications for midwinter Pacific storm track variability. J. Atmos. Sci., 61, 23–40. Held I.M. (1985). Pseudomomentum and the orthogonality of modes in shear flows. J. Atmos. Sci., 42, 2280–2288. Held I.M. and Hoskins B.J. (1985). Large-scale eddies and the general circulation of the troposphere. Adv. Geophysics, 28A, 3–31. Held I.M., Ting M., and Wang H. (2002). Northern winter stationary waves: Theory and modeling. J. Climate, 15, 2125–2144. Hoskins B.J. and Hodges K.I. (2002). New perspectives on the Northern Hemisphere winter storm tracks. J. Atmos. Sci., 59, 1041–1061. Killworth P.D. and McIntyre M.E. (1985). Do Rossby-wave critical layers absorb, reflect or overreflect? J. Fluid Mech., 161, 449–492. Kushner P.J. and Held I.M. (1998). A test, using atmospheric data, of a method for estimating oceanic eddy diffusivity. Geophys. Res. Lett., 25(22), 4213–4216. Kushner P.J. and Held I.M. (1999). Potential vorticity fluxes and wave-mean flow interaction. J. Atmos. Sci., 56, 948–958. Lau N.-C. and Holopainen E.O. (1984). Transient eddy forcing of the time-mean flow as identified by geopotential tendencies. J. Atmos. Sci., 41, 313–328. Lee S. (1995). Localized storm tracks in the absence of local instability. J. Atmos. Sci., 52, 977–989. Lee S. and Held I.M. (1993). Baroclinic wave packets in models and observations. J. Atmos. Sci., 50, 1413–1428. Lim G.H. and Wallace M. (1991). Structure and evolution of baroclinic waves as inferred from regression analysis. J. Atmos. Sci., 48, 1718–1732. Lin S.-J. and Pierrehumbert R.T. (1993). Is the midlatitude flow absolutely unstable? J. Atmos. Sci., 50, 505–517. Lindzen R.S. and Farrell B.F. (1980). A simple approximate result for the maximum growth rate of baroclinic instabilities. J. Atmos. Sci., 37, 1648–1654. Nakamura H. (1992). Midwinter suppression of baroclinic wave activity in the Pacific. J. Atmos. Sci., 49, 1629–1642. Pierrehumbert R.T. (1984). Baroclinic instability. J. Atmos. Sci., 41, 2141–2162. Pierrehumbert R.T. and Swanson K.L. (1995). Baroclinic instability. Ann. Rev. Fluid Mech., 27, 419–467. Simmons A.J. and Hoskins B.J. (1980). Barotropic influences on the growth and decay of nonlinear baroclinic waves. J. Atmos. Sci., 36, 1239–1254. Swanson K.L., Kushner P.J., and Held I.M. (1997). Lower-tropospheric heat transport in the Pacific storm track. J. Atmos. Sci., 54, 1533–1543. Swanson K.L. and Pierrehumbert R.T (1993). Nonlinear baroclinic wave packet evolution on a baroclinically unstable jet. J. Atmos. Sci., 51, 384–396. Whitaker J.S. and Sardeshmukh P.D. (1998). A linear theory of extratropical synoptic eddy statistics. J. Atmos. Sci., 55, 237–258. Zhang Y. and Held I.M. (1999). A linear stochastic model of a GCM’s midlatitude storm tracks. J. Atmos. Sci., 56, 3416–3435. Zurita-Gotor P. and Chang E.K.M. (2005). The impact of zonal propagation and seeding on the eddy-mean flow equilibrium of a zonally varying two-layer model. J. Atmos. Sci., 62, 2261–2273.

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Chapter 5

Eddy-Mediated Interactions Between Low Latitudes and the Extratropics Walter A. Robinson

5.1. Introduction Large-scale changes in the Tropics, such as occur during El Niño episodes, affect the global atmosphere. While this has been known since early in the twentieth century, it was not until 1981 that dynamical understanding of these teleconnections was achieved. In that year, Horel and Wallace published one of the most famous pictures in climate science. It showed a train of alternating stationary highs and lows emanating from the tropical Pacific Ocean in response to anomalous warmth in that basin. This observational picture resembled a theoretical one published in the same year by Hoskins and Karoly. They showed that the steady linear response to equatorial forcing is a train of Rossby waves following nearly a great-circle path (also Webster 1981). Together, these papers demonstrated that the most important feature of interannual variability in the global climate—and, as it has subsequently turned out, perhaps the only such feature with useful predictability—can be understood and modeled using the linear theory of Rossby waves. There have been important elaborations over the past two decades due to Simmons et al. (1983), who showed that important effects, including local amplification of the disturbance, result when the zonal asymmetry in the basic flow on which the waves propagate is included, and due to Sardeshmukh and Hoskins (1988), who showed how tropically forced signals generate extratropical Rossby wave trains. Of particular relevance for this chapter is the work of Hoerling and Ting (1994), who showed that transient eddies are important for the extratropical response to tropical forcing. More than half the strength of the anomalous upper-air low south of Alaska in an El Niño winter could, in some cases, be attributed to the transport of vorticity by transient eddies. Hoerling and Ting’s analysis, like most earlier work on tropical teleconnections, excluded the zonal-mean response in midlatitudes. The zonal-mean component is, however, a strong and significant part of the response of the midlatitude atmosphere

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Figure 5.1. Changes following the 1997–98 El Niño, shown as differences between December– May 1998–99 and December–May 1997–98: (a) sea surface temperature, (b) 200 hPa geopotential height, (c) zonally averaged atmospheric temperature. The contour interval is 1◦ C in (a), 30 m in (b), and 0.25◦ C in (c). Negative contours are dashed.

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to tropical warming or cooling. This has been demonstrated by several observational studies. Most dramatic are the changes that accompanied the tropical cooling in 1998 and 1999 following the 1997–98 El Niño (Hoerling et al. 2001). Here we reproduce some of their results using data from the National Center for Environmental Prediction/National Center for Atmospheric Research (NCEP/NCAR) reanalysis project (Kalnay et al. 1996).1 Figure 5.1a shows the difference in sea surface temperature from winter/spring (Dec.–May) 1997–98 to winter/spring 1998–99. As is expected in a transition from El Niño to La Niña, there was strong cooling—as much as 4◦C— along the equator from the International Dateline to the west coast of South America. Slight warming occurred in the far western tropical Pacific, with weak cooling along the equator in the Indian and Atlantic Oceans and stronger cooling in the western Indian Ocean. The atmospheric change accompanying this transition is shown in Fig. 5.1b. Geopotential heights at 200 hPa fell throughout the Tropics, and heights increased in nearly zonal bands in both the Northern and Southern Hemispheres, though not at the same latitudes. A cross section of zonally averaged temperature changes (Fig. 5.1c) reveals cooling throughout the tropical tropopause, with deep narrow bands of warming in both hemispheres. Such zonal features in tropospheric temperatures are generally associated with El Niño and La Niña, as was shown earlier (1994) by Yulaeva and Wallace using 13 years of Microwave Sounding Unit (MSU) data. They found that tropospheric temperatures, measured by radiances in MSU channel 2, were in roughly zonal bands in both hemispheres that were anti-correlated with temperatures in a tropical belt. Similar features are evident in NCEP/NCAR reanalysis data. The panels of Fig. 5.2 are similar to those of Fig. 5.1, but show correlations for December–May averages with the Niño 3.4 Index, the spatially averaged sea surface temperature (SST) anomaly (deviation from climatology) between 5◦ N and 5◦ S from 120◦ to 170◦ W. Niño 3.4 is an index of El Niño; strongly positive values indicate El Niño conditions, while strongly negative values indicate La Niña. Data from the entire reanalysis period 1949–2003 are used. When the surface of the equatorial Pacific Ocean east of the International Dateline is warm, there are warm waters throughout the tropical oceans (Fig. 5.2a) and the 200 hPa surface is elevated at all longitudes throughout the Tropics (Fig. 5.2b) with broken wavy bands of lowered heights in the midlatitudes of both hemispheres. In the zonal average, there are high temperatures throughout the depth of the tropical troposphere, with much reduced warming or even cooling in deep layers to the north and the south (Fig. 5.2c). While these correlations include linear trends over this period, the same signals appear when the analysis is done for shorter sub-periods. Thus, it appears these are robust features of the global response to tropical warming or cooling. Seager et al. (2003) explored the dynamics of this tropical-midlatitude teleconnection. They demonstrated that the intensification in El Niño conditions of the poleward decrease in temperatures across the subtropics—the deep warming in the off-equatorial Tropics and the similarly deep cooling in the subtropics—is not a direct response to

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Figure 5.2. Correlations of December–May seasonal averages with the Nino 3.4 SST Index: (a) sea surface temperature, (b) 200 hPa geopotential height, (c) zonally averaged atmospheric temperature. The contour interval is 0.2 in (a) and (b), and 0.1 in (c). Negative contours are dashed.

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equatorial warming, but rather is an indirect response, mediated through changes in the fluxes of heat and momentum by transient eddies. We reproduce in Fig. 5.3 some of their key results. The first two panels (5.3a and 5.3b) show the zonally averaged temperature and zonal wind anomalies that result from taking the point-by-point regression of these quantities (December, January, February [DJF] averages) against a time series that is nearly an El Niño index,2 over the years 1979 to 2000. Data from the NCEP/NCAR reanalysis are used. The third panel (Fig. 5.3c) shows the contribution of momentum transports by transient eddies to the tendency of the zonally averaged zonal wind, where “transient” is defined as a deviation from the monthly mean. This contribution, like the temperatures and winds, is calculated by regressing this quantity against the index time series. The momentum flux convergence by transient eddies in middle and high latitudes reinforces the zonal wind anomalies associated with anomalous warmth in the Tropics. The stationary-eddy term is much less important, and the Coriolis term (neither are shown) opposes the transient-eddy forcing in the upper troposphere and supports the zonal wind anomalies near the surface, presumably balancing surface drag. Figure 5.3d shows the contribution of the transient-eddy heat flux to the zonally averaged thermal anomalies. The convergence of the transient-eddy heat flux opposes the thermal anomalies in Fig. 5.3a—i.e., the regressed transient-eddy heat flux convergence is positive at 40◦ N and 50◦ S, coincident with the deep regions of lower temperatures. Diabatic cooling cannot account for these thermal anomalies; rather, they are associated primarily with adiabatic cooling resulting from rising motion in the anomalous mean meridional circulation. The picture that emerges is of zonal-mean circulation anomalies that are not direct responses to tropical warming, but instead are driven by anomalies in the transienteddy fluxes of heat and momentum. Tropospheric cooling occurs where rising motion develops poleward and below regions of anomalous momentum-flux convergence. These convergent momentum fluxes also produce westerly wind anomalies: the eddyinduced secondary circulations yield anomalous temperatures in thermal wind balance with the anomalous zonal winds, as is expected from quasigeostrophic theory. The cooling by eddy-pumped lifting is opposed by the eddy transport of heat, but the momentum fluxes “win.” Why this should be is addressed later in this chapter. Additional significant features of this zonal teleconnection from the Tropics are: • Because the climatological maxima of the zonally averaged zonal wind in DJF

are at 30◦ N and 45◦ S, the anomalies shown in Fig. 5.3b represent an equatorward shift of the mean jets. There is an accompanying equatorward shift in the latitude of strongest eddy activity, as measured by the variance of the transient-eddy meridional wind. • The zonally averaged response to tropical warmth (Figs. 5.1c, 5.2c, and 5.3) displays considerable symmetry between the Northern and Southern Hemispheres. This is remarkable because these results are for the Northern

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Figure 5.3. Regressions against an interannual El Niño-like index for December–February: (a) zonally averaged temperature, (b) zonally averaged zonal wind, (c) tendency of the zonally averaged zonal wind due to transient-eddy momentum fluxes, (d) tendency of the zonally averaged temperature due to transient-eddy heat fluxes. The contour interval is 0.1 K in (a), 0.3 m s−1 in (b), 0.1 m s−1 day−1 in (c), and 0.025 K day−1 in (d). From Seager et al. (2003). c 2003.) (Reproduced with permission from the American Meteorological Society


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Hemisphere cold season, in which the Hadley circulation is not at all symmetric about the equator. There is northward flow at the equator in the upper troposphere and a much stronger circulation in the Northern than in the Southern Hemisphere, with corresponding asymmetry in the strengths of the zonally averaged subtropical jets, The hemispheric symmetry of the response is even more pronounced and extends to higher latitudes when the influence of the Southern Annular Mode (SAM) is removed (see section 5.6 below). • The rising motion responsible for the bands of cooling in Fig. 5.3a also creates, in the zonal mean, bands of increased moisture flux convergence and precipitation (Seager et al. 2005a), and these zonally symmetric features contribute to precipitation anomalies, such as increased rainfall along the Gulf Coast of North America, in El Niño years. The moistening of the subtropical atmosphere when the Tropics are warm appears in observations of atmospheric water vapor over the southeast United States, and there is some evidence, primarily from models, that the resulting effect on cloud cover may influence surface temperatures (Robinson et. al. 2002). • Atmospheric general circulation models (GCMs) reproduce these zonally symmetric teleconnections from the Tropics in temperatures, winds, and precipitation when the models are driven by observed sea surface temperatures (Seager et al. 2003, 2005a). This includes the observed association of decadal fluctuations in tropical Pacific temperatures with decadal droughts and pluvials in the Great Plains and Southwest of the United States (Schubert et al. 2004; Seager et al. 2005b). The data indicate that there is an important eddy-mediated zonally symmetric component to tropical-extratropical teleconnections and that one can consider the overall extratropical response to, say, El Niño, roughly as a superposition of this zonal response and the more familiar Rossby-wave response of Hoskins and Karoly and of Horel and Wallace. This is the approach taken in the remainder of this chapter. In dealing with a zonally symmetric feature we take advantage of the simplicity and advanced development of the theory of interactions between zonal mean flows and eddies. This chapter focuses on the roles of eddies. The dynamical arguments begin by postulating the existence of a background state of the circulation, in which temperatures are determined by a radiative-convective equilibrium, there are no surface winds, and therefore no surface drag, and the winds aloft are in thermal wind balance with temperatures determined by radiative-convective equilibrium. This background state is presumed to be unstable to baroclinic eddies, which transport heat and momentum, and in so doing, drive temperatures away from radiative-convective equilibrium. In this eddy-centric view of the circulation, all net heating or cooling of the atmosphere is a consequence of the eddies. However idealized this model may be, its suitability for

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understanding the extratropical circulation is demonstrated by the success of models and theories that include only these dynamics in capturing features of the circulation and its variability as observed in nature and in more comprehensive models. The postulated primacy of eddies fails, however, in the Tropics, where even in the absence of eddies, the Hadley circulation redistributes heat and angular momentum. The Hadley cell responds to heating in a narrow equatorial belt by distributing that warmth throughout the Tropics. A nearly inviscid model of the Hadley cell (Held and Hou 1980) predicts that warming the Tropics, relative to higher latitudes, should lead to a more vigorous Hadley cell, expanded poleward, with a subsequent poleward shift of the subtropical jets. This is not the observed response to tropical warming (Figs. 5.1–5.3), which argues for the importance of the eddies. The redistribution of heat by the Hadley cell, however, is a crucial first “step” in any extratropical response to tropical heating. Another limitation of our approach is that decomposing the behavior of the circulation into a tropical “forcing” and an eddy-mediated extratropical “response” is artificial. Some substantial portion of the poleward motion in the upper branch of the Hadley circulation is eddy driven. If that eddy driving increases, and the Hadley cell strengthens, the low-level convergence of moisture into the rising branch of the Hadley cell will likewise increase and presumably will produce increased latent heating, further strengthening the circulation. Looking at the eddy-mediated extratropical response to tropical forcing, with the latter viewed as a boundary condition on the extratropics, should, therefore, be seen as only a beginning in understanding a more complex set of tropical-extratropical interactions. The next section presents a quasigeostrophic eddy-mean flow interaction theory that provides a context for interpreting the observational results described above, as well as those from numerical models of varying complexity, to be described later. These results include some that are directly comparable to the observed extratropical responses to tropical warming or cooling, and some that do not correspond to anything as yet observed in Earth’s atmosphere—changes in the basic structure of the extratropical jets and circulation.

5.2. Theory Our goal is a simple theory that can be used to understand the eddy-mediated responses of the midlatitude atmosphere to tropical warming. Quasigeostrophic dynamics, specifically the conservation of quasigeostrophic pseudo potential vorticity following the geostrophic motion, is used to derive results for the steady and zonally averaged responses to the generation and dissipation of baroclinic eddies. The implications and limitations of this theory are discussed, before it is deployed, in subsequent sections, to explain the results of numerical models.

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In pressure coordinates (e.g., Holton 2004), quasigeostrophic conservation of potential vorticity can be written, ∂q + J (ψ, q ) = S, ∂t

[5.1]

where ψ is the geostrophic streamfunction, and q is the pseudo-potential vorticity (henceforth PV) given by,
2  f 0 ∂ψ ∂ 2 q = βy + ∇ ψ + . [5.2] ∂p σ ∂p The usual static-stability parameter, σ , is given by σ = −RT0 p −1dln θ0/dp, where T0 and θ0 are the background temperature and potential temperature; both depend only on pressure. The sources and sinks of PV are represented by S in (5.1). At the surface, or more accurately, at the top of the boundary layer, the thermodynamic equation can be written identically to (5.1), ∂qb + J (ψb , qb) = Sb , ∂t

[5.3]

with the subscript b indicating that quantities are evaluated at the lower boundary, and with the boundary PV defined as 
2 f 2p f ∂ψ qb = − 0 + 0 ψ . [5.4] σ ∂p RT p= pb Here and throughout the pressure pb, of this lower boundary, is assumed to be constant. The focus of this chapter is on the quasi-steady and zonally symmetric responses to tropical forcing. Thus we take the zonal average (5.1) and neglect the time derivative, leaving a steady balance between the zonally averaged source of PV and its zonally averaged transport by eddies, ∂ν q  = S, ∂y

[5.5]

where an overbar indicates a zonal average, and primes denote deviations there from. As before, this equation applies at the lower boundary with the appropriate definitions for PV and its source. Equation (5.5) excludes advection of PV by the mean meridional circulation, which, because it is ageostrophic, is assumed to be an order Rossby number weaker than the geostrophic eddy winds. This approximation breaks down in the Tropics, where advection by ageostrophic winds flushes PV from regions of divergent horizontal flow, concentrating horizontal gradients of PV at the boundaries of such regions (Held and Hou 1980; Sardeshmukh and Hoskins 1988). Sources and sinks of PV result from radiative heating and cooling in the free atmosphere and from radiative or boundary-layer heating and mechanical drag at the lower boundary. If heating is parameterized as Newtonian thermal damping, with a pressure-dependent rate D, then the associated source of PV in the interior

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is given by, S =−

∂ ∂p




f 02D ∂ψ σ ∂p

 .

[5.6]

At the lower boundary, the PV source also includes mechanical drag, and is given by
2  f 0 D ∂ψ Sb = − Ep∇ 2ψ , [5.7] σ ∂p p= pb where E is the rate of Ekman drag. Including these sources of PV in (5.5) and taking the y derivative yields  
2  ∂ 2 ν q  f 0 D ∂u ∂ [5.8] = ∂y 2 ∂p σ ∂p in the free atmosphere and   ∂ 2 ν q  ∂ 2u f 2D ∂u + Ep 2 , =− 0 2 ∂y σ ∂p ∂y

p = pb

[5.9]

at the lower boundary. We consider solutions to these equations forced by interior and surface eddy fluxes of PV. For simplicity, it is assumed that the interior flux of PV is concentrated at a single level, pf . This makes the present setup much like a two-level model, and, in fact, identical results can be obtained for the two-level quasigeostrophic model. We define the boundary and interior fluxes of PV as   ν q b = pb B(y), p= pb [5.10]   ν q = pb F (y)δ( p − pf ). Potential vorticity balance in the interior becomes
2  f 0 D ∂u d 2F ∂ = pb 2 δ( p − pf ) , ∂p σ ∂p dy

[5.11]

with the lower boundary condition ∂ 2u d 2B f 02D ∂u − Ep 2 = − pb 2 , σ ∂p ∂y dy

p = pb .

[5.12]

With the condition that the vertical shear vanish at p = 0, the interior equation gives f 02D ∂u d 2F = pb 2 , σ ∂p dy

p > pf ,

[5.13]

so at the lower boundary E

∂ 2u d 2(B + F ) = , ∂y 2 d y2

p = pb .

[5.14]

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Integrating twice, and assuming that the constants of integration may be taken to vanish, yields u p= pb = Ub =

B+F . E

[5.15]

Thus, there are westerly winds at the surface only where the pressure-integrated eddy flux of potential vorticity is poleward. Integrating over the depth of the atmosphere, there must be a net northward transport of PV to produce the convergence of the eddy momentum flux that maintains the low-level westerlies. This is a familiar result, going back at least to the work of Green (1970). It was shown by Bretherton (1966; also Charney and Stern 1962) that the domain integral of B + F must vanish. Thus, the domain-averaged lower-boundary zonal wind must vanish in the absence of externally imposed torques or momentum transport across the boundaries. Integrating (5.13) with respect to pressure yields u p= p f = Uf =

L 2 d 2F B+F − ,  dy 2 E D

[5.16]

where the ratio of the internal deformation radius to a weighted average of the Newtonian damping rate is defined as L2 pb ≡ 2  f0 D

pb

σ dp. D

[5.17]

pf

According to (5.15) and (5.16), which were derived for the two-layer model by Pavan and Held (1996), the zonal flow is set by the interior and boundary eddy fluxes of potential vorticity. Because the problem is linear, we may add an arbitrary baroclinic radiative equilibrium zonal flow UR , with the constraint that it vanish at the lower boundary so it is not acted upon by surface drag. Defining a mean temperature between the level of forcing and the lower boundary, pb = T

T d(ln p)

pf

ln( pb/p f )

,

and integrating the thermal wind equation meridionally yields
 f 0L 2 pb dF  T (y) = T0 + . ln  p f dy RD

[5.18]

[5.19]

Again, this is only the eddy-driven portion of the temperature—the deviations from radiative equilibrium. In the steady quasigeostrophic limit described here, the temperatures (5.19) and, by thermal wind balance, the difference between the zonal winds aloft and at the surface are determined solely by F , the meridional distribution of the PV flux aloft.

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Figure 5.4. Schematic of eddy-zonal flow interactions in the quasigeostrophic model. The shaded closed curve depicts the transformed Eulerian mean (TEM) circulation, the horizontal arrows indicate the direction of eddy fluxes of potential vorticity, and the dark arrows show the propagation of baroclinic eddy activity, as represented by the Eliassen-Palm (EP) flux.

The physical picture is depicted schematically in Fig. 5.4. Eddies are generated baroclinically at the lower boundary, where the eddy transport of PV is northward; this is equivalent to a divergent flux of Eliassen-Palm eddy activity (Edmon et al. 1980), or EP flux. Eddy activity propagates vertically and meridionally to some location aloft where it is dissipated, in so doing producing a southward flux of PV. The westward force exerted by convergent EP flux aloft is balanced by the Coriolis force due to the transformed Eulerian mean (TEM) meridional wind (Andrews and McIntyre 1976). By mass continuity there is necessarily rising motion equatorward and below, and sinking motion poleward and below, the region of convergent EP flux. This vertical motion causes adiabatic cooling on the equatorial side and warming on the poleward side of regions of convergent EP flux, balanced by radiative cooling or warming, as represented by the Newtonian thermal damping. This is closely analogous to the conventional Eulerian responses to regions of positive and negative vorticity advection aloft, as described by the quasigeostrophic omega equation. A region of equatorward vorticity flux aloft implies the existence of a meridional couple in vorticity advection, with positive vorticity advection to the south causing rising motion and cooling at lower levels, and negative vorticity advection to the north causing subsidence and warming. It is useful to describe the atmosphere using only two levels, the surface and a single level aloft, only if there is some qualitative difference in the dynamics at these two levels. The behavior of eddies aloft may be considered wave-like, in the sense that their propagation is consistent with the prediction of linear dynamics (see chapter 1 in this volume). Such wave-like behavior is demonstrated by the propagation towards and absorption nearby critical latitudes (Randel and Held 1991). Critical-latitude absorption, as well as more subtle changes in propagation associated with uppertropospheric waveguides, explains some of the eddy behavior associated with tropical

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warming (section 5.4). At the same time, PV fluxes in the lower atmosphere appear to be less wave-like, more turbulent, and more diffusive. A result of this dichotomy is seen (section 5.4) in a two-level model, where distributions of B are found to be smooth and centered on the latitudes of strongest thermal gradients, while distributions of F are more highly structured, reflecting the influences of the zonal flow on wave propagation and dissipation. The treatment of the lower boundary in quasigeostrophic theory, in which it is assumed that the surface is at a single pressure and that all eddy PV fluxes in this layer are described by eddy heat fluxes, is problematic. This is not an accurate treatment of the atmosphere because the potential temperature of the solid lower boundary varies widely. The surface layer properly includes all isentropic surfaces that intersect the ground, and the the flux of potential vorticity in this layer includes the eddy flux of potential temperature at the surface and the flux of potential vorticity within the surface layer above the solid boundary, the latter neglected in quasigeostrophic theory (Schneider 2005). When the zonally averaged quasigeostrophic PV balance is integrated over the depth of the atmosphere and again from the surface to some level, taking the difference and applying hydrostatic and geostrophic balance yields
 p   dln θ0 ∂v q − dp T = T0( f 0D) dp ∂y −1

[5.20]

0

for the deviation of the zonally averaged temperature from its radiative equilibrium values. This equation applies only above the surface layer, which typically excludes the lowest 250 hPa of the atmosphere (Schneider 2005). At a level outside of the surface layer, however, temperatures are determined by where, above that level, eddy activity is dissipated. This, in turn, depends on the baroclinic sources of upwelling eddy activity in the lower atmosphere and the propagation of that eddy activity aloft. Different conditions for eddy propagation aloft, such as a shift in the critical latitude, therefore can affect the temperature. Because this result does not apply all the way down to the lower boundary, it cannot be used to predict changes in the surface baroclinicity. It should, however, apply to deep thermal anomalies, such as were found by Seager et al. (2003) in the response to El Niño. Some examples help fix ideas about the eddy modification of the zonal flow. We take B and F as Gaussians in y. Flanking a region where wave activity is generated at the surface, B > 0, are two equal regions of eddy activity absorption aloft—i.e., where F < 0. Thus,
2



 y A wB (y − yF )2 (y + yF )2 B = A exp − 2 , F =− exp − +exp − . [5.21] 2 wF wB w2F w2F

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The boundary and upper-level winds and the temperature can be determined from (5.15), (5.16), and (5.19). Examples are shown in Fig. 5.5, all with the same width of the eddy generation region w B equal to the internal deformation radius L . Three different distributions of F are considered; in the first case (solid curves in Figs. 5.5a and 5.5b, and Fig. 5.5c), F (y) = B(y), so eddy activity is dissipated immediately above where it is generated. There is no vertically integrated momentum flux convergence and no surface wind. Winds aloft are easterly at y = 0, and the eddy-induced zonally averaged temperatures increase poleward. If the width of the region of southward PV flux aloft (−F ) is doubled, w F = 2w B = 2L , (dashed curves in Figs. 5.5a and 5.5b, and Fig 5.5d) the eddy momentum flux is convergent, and there are surface westerlies at y = 0. The eddy-induced winds aloft are still easterly, however, and the eddy-induced temperatures increase poleward. Despite the momentum flux convergence, the full influence of the eddy driving is to make the winds aloft more easterly because the easterly form drag implicit in the vertically propagating eddies exceeds the horizontal momentum flux convergence. The presence of convergent eddy momentum fluxes is not, therefore, sufficient for eddies to strengthen the jet. If, however, the distribution of −F aloft is split (dotted curve in Figs. 5.5a and 5.5b, and Fig. 5.5e), then the eddies drive westerlies aloft at y = 0 and generate a poleward decrease in temperatures; the eddies strengthen the jet in this case. These last two cases indicate the strong sensitivity of the resulting zonal flow and temperatures to the details of the distribution of −F , a consequence of their dependence (equations [5.16] and [5.19]) on its meridional derivatives. There are midlatitude surface westerlies in nature, so something like one of the latter two cases must apply. Observed climatologies of the EP flux and especially of the TEM circulation (Edmon et al. 1980; Karoly 1982; Tanaka et al. 2004) suggest that the eddies enhance the poleward decrease in temperature across portions of the middle latitudes. This appears, in TEM plots such as Fig. 5.6, as TEM descent in the subtropics of both hemispheres and ascent near 50◦ S and 40◦ N. The eddies, as represented in the TEM circulation, thus appear to warm the subtropics and cool the high middle latitudes, consistent with the idealized case shown in Fig. 5.5d, though this requires two qualifications. First, the TEM circulation is, overall, clockwise in the Northern Hemisphere and counterclockwise in the Southern Hemisphere, so that the effect of the circulation is to reduce the temperature contrast between the equator and the poles, as must be the case if baroclinic eddies derive their energy from the zonalmean available potential energy. The circulation concentrates the thermal gradient only across a portion of the midlatitudes. Second, in Fig. 5.6 it is incorrect to determine the influence of the circulation on the temperature solely from the vertical motion. Isentropes slope upward towards the poles, an effect that is neglected in the quasigeostrophic theory, which includes only the vertical advection of potential temperature by the secondary circulation. In order to produce midlatitude cooling, the circulation must cross isentropes from below. In Earth’s atmosphere, where both isentropes and

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Figure 5.5. Responses of the zonally averaged circulation to different distributions of eddy driving in the quasigeostrophic model. All are the same at the lower boundary, but there are three different distributions aloft: (a) eddy driving, (b) temperatures, (c)–(e) zonal winds at the lower boundary and aloft. The line types in (b)–(e) correspond to the three different line types for F in (a). Units of temperature and zonal wind are arbitrary, but are the same for all cases. The meridional coordinate, y, is in units of deformation radius.

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Figure 5.6. The transformed Eulerian mean (TEM) streamfunction, averaged over the years 1981–2000 from European Centre for Medium-Range weather Forecasting reanalysis (ERA-40) data. The contour interval is 2 × 1010 kg s−1. The vertical coordinate σ = p/ ps is pressure p normalized by surface pressure ps.

typically the midlatitude secondary circulation slope upwards toward the poles, it is not certain whether or not, in the climatological mean, the eddies actually strengthen the thermal gradient across some regions of the middle latitudes or merely concentrate the gradient in these regions, while reducing it overall. That eddies can, in principle, locally enhance the temperature gradient, however, is important for how the circulation responds to tropical warming or cooling, as will be seen in the next sections.

5.3. Responses to Tropical Warming in Idealized Models The responses to tropical warming and cooling displayed in the observations and in ensembles of GCM experiments forced by observed sea surface temperatures (SST) are readily reproduced by a suite of much simpler models. This leads to some understanding of the relevant dynamical mechanisms, because it tells us the necessary dynamical mechanisms are among those that can be represented by models that are simpler than a full GCM or the real atmosphere. The most idealized relevant model is a dry two-level primitive equation model on the sphere, such as that used over many years to explore mechanisms of extratropical variability (Hendon and Hartmann 1985; Robinson 1991a). Radiative heating and cooling are parameterized by Newtonian relaxation back to a reference profile, which is a function of latitude and pressure. In the present version, the horizontal resolution is coarse (spectral truncation of rhomboidal 15, corresponding approximately to a 7.5◦ × 4.5◦ latitude-longitude grid), and there is no topography. The model includes horizontal hyper-diffusion, but no vertical diffusion or mixing. The deviation of the radiative equilibrium temperature from its global mean

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value at the middle level of the model, 500 hPa, is given by
 φ 2 Trad =  cos(2φ) + δ exp − , φw

[5.22]

where for the runs described here,  = 30◦C, φw = 10◦, and δ is varied from case to case. Results shown are averages over the last 1500 days of 1700-day model runs. Zonally averaged 500 hPa temperatures for the control run (δ = 0) are shown in Fig. 5.7a, along with the temperature difference between tropically warmed and tropically cooled runs (δ = 10, −10◦C) and the control. For higher radiative equilibrium temperatures in the Tropics, the Tropics warm and the extratropics cool, and vice versa. This is unsurprising. Because the globally averaged temperature is fixed in this model, tropical warming must be compensated by cooling elsewhere. Of greater interest is the meridional structure of these temperature changes. The warming is spread from the narrow region of increased Trad throughout the Tropics, and midlatitude cooling is greatest at 45◦ N and S. Thus, when the deep Tropics are warmed, there is a marked increase in the poleward decrease of temperature across the subtropics. Consistent with these changes in temperature and with thermal wind balance, as the Tropics warm, the upper-level zonal jet (Fig. 5.7b) strengthens and shifts equatorward, and there is a small equatorward shift of the lower-level westerlies. Given the simplicity of this model, these results are remarkably similar to the observed responses to tropical warming shown in Fig. 5.3. That many of these changes in the zonally averaged circulation are driven by the eddies is shown by the distributions of upper- and lower-level PV fluxes (Fig. 5.7c). In all three model runs, the upper level PV fluxes (F ) have regions in the middle latitudes where d 2F /dy 2 < 0. In these regions, according to (5.19), the eddies act to increase the rate of poleward decrease of temperature, or, by thermal wind balance, the vertical shear in the zonal wind. This region strengthens and shifts equatorward as the Tropics are warmed. Although baroclinic eddy generation at the lower level (B in Fig. 5.7c) strengthens nearly everywhere as the Tropics warm, the midlatitude region of eddy-induced cooling (where dF /dy < 0) widens and strengthens. The overall response to tropical warming is a strengthening and equatorward shift of the climatological eddy PV fluxes and their resulting influences on the zonally averaged circulation. Warming does not occur where the eddy heat fluxes are increasingly convergent, consistent with the fact that in the climatology of this model, convergent eddy heat fluxes do not necessarily lead to local warming when the full effects of the eddies are considered. These responses to tropical changes are robust, not only in the two-level model, in observations, and in full GCMs, but also in earlier idealized multi-level model experiments conducted by Hou (1993, 1998), Hou and Molod (1995), and Chang (1995). This line of research was motivated (Hou 1993) by an interest in exploring the tropical-extratropical connections that are mediated by zonally averaged dynamical

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Figure 5.7. Results from the two-level primitive equation model, comparing a control run with runs with added tropical heating or cooling: (a) climatological zonally averaged 500 hPa temperature in the control run, and temperature deviations from the control run for experiments with added tropical heating and cooling; (b) zonally averaged zonal winds at 250 and 750 hPa for all three experiments; (c) zonally averaged eddy fluxes of potential vorticity at 250 hPa (F ) and 750 hPa (B).

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122 | Walter A. Robinson westerly acceleration region of by a stronger increased Hadly cell baroclinicity midlatitude cooling

highlatitude warming

region of reduced PV gradient EQUATOR

WINTER POLE

Figure 5.8. Schematic of how changes in the Hadley cell can affect temperatures at higher latitudes. From Hou (1998). (Reproduced with permission from the American Meteorological Society c 1998.)


transports of heat. Studies of the zonally symmetric dynamics of the Hadley circulation by Lindzen and Hou (1988) and Hou and Lindzen (1992) had shown that when tropical heating is concentrated and shifted off the equator into one (the “summer”) hemisphere, the subtropical jet in the opposite (“winter”) hemisphere strengthens. This was expected to influence eddy transports out of the subtropics in the winter hemisphere, an expectation that was confirmed in a number of idealized GCM experiments conducted by Hou (1993). In all cases, a broadly similar result was obtained, whether the tropical heating was shifted into the Northern or Southern Hemisphere and whether the tropical heating was zonally uniform or was confined to specific longitudes. Within the Tropics, the summer hemisphere warmed and the winter hemisphere cooled. The temperature anomalies immediately poleward of the Tropics had a sign opposite to those within the Tropics. This suggested it is the tropical temperature anomalies rather than the meridional and vertical flow of the Hadley circulation that communicates the tropical changes to the extratropics, though the Hadley circulation plays an essential role in distributing warming within the Tropics. In his 1993 and subsequent papers, Hou emphasized the middle-latitude cooling and high-latitude warming in the winter hemisphere that occurs when the tropical heating is moved away from the equator in the summer hemisphere. His proposed mechanism for these changes, summarized in a schematic at the conclusion of Hou (1998), is reproduced here as Fig. 5.8. Hou argued that moving the heating away from the equator in the summer hemisphere strengthens the cross-equatorial Hadley cell and thus the subtropical jet in the winter hemisphere. The increased vertical shear of the subtropical jet and the associated increase in subtropical baroclinicity lead to stronger poleward eddy heat transports, warming the high latitudes, and cooling the subtropics.

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Figure 5.9. Response, in the two-level model, to shifting enhanced tropical heating off of the equator: (a) temperature differences (zonal averages at 500 hPa) between experiments with the tropical heating centered at 10◦ or 15◦ S with an experiment in which tropical heating of the same strength is centered on the equator; (b) zonally averaged eddy fluxes of potential vorticity at 250 hPa (F ) and 750 hPa (B) for runs with the heating centered at 15◦ S and on the equator.

The two-level model produces results similar to those of Hou (1993), when tropical heating is shifted away from the equator. Figure 5.9a shows differences in temperatures between runs with the additional tropical heating centered at 10◦ or 15◦ S with a run where it is centered on the equator. For these experiments, δ = 10◦C. Such a shift, whether 10◦ or 15◦ off the equator, causes similar changes: temperatures increase throughout the Southern Hemisphere extratropics, most strongly between 30◦ and 40◦ S. There is cooling in the Tropics and in the northern subtropics, weakening into high northern latitudes. The eddy driving that leads to this response is shown (only for the 15◦ S case) in Fig. 5.9b. In the Southern Hemisphere the eddy generation (B) and the upper-level dissipation of eddy activity (F ) strengthen and shift poleward

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slightly—the sense of this meridional shift is opposite to that which results from increased tropical heating centered on the equator. These changes in F indicate that the midlatitude warming in the Southern Hemisphere is eddy-driven, a consequence of a shift and strengthening of the descending branch of the TEM circulation poleward of the enhanced peak in −F . The cooling in northern low latitudes results from the shift of the tropical heating away into the opposite hemisphere, while the reduction in baroclinicity across northern middle latitudes occurs because the eddy-driven strengthening of baroclinicity is slightly reduced. High-latitude temperature changes in the Northern Hemisphere are weak and not robust. Note, for example, the difference in sign between the two cases in Fig. 5.9a. Similarly, Hou (1993) found that different members of his ensemble produced different thermal responses in the high latitudes of the winter hemisphere. While the two-level model results are similar to those from Hou’s GCM, it does not appear to follow the mechanism shown in Fig. 5.8. In the two-level model, moving the extra heating into the Southern Hemisphere cools the northern Tropics. The resulting reduced temperature gradient across the northern subtropics leads to reduced production of baroclinic eddies—smaller B—in the Northern Hemisphere, and the full effect of this reduced eddy production is a further reduction in the temperature gradient, increased d 2F /dy 2 at 40◦ N in Fig. 5.9b. Chang (1995) performed similar experiments in a dry idealized anelastic GCM at rhomboidal 21 truncation and with 10 levels. In his perturbation experiment, tropical heating, as represented by the radiative equilibrium temperature, was concentrated in a narrow band off the equator. His results more resemble those described earlier for models forced by El Niño or by tropical warming than they do Hou’s. There was tropical warming and subtropical cooling, the latter being stronger in the winter hemisphere. Chang performed extensive diagnoses of the dynamical processes leading to these temperature changes. He found that the dynamical heating due to transient eddy heat fluxes opposed that due to the (Eulerian) mean meridional circulation (MMC) throughout the extratropics, but that the effect of the mean meridional circulation was consistently greater, so that the “MMC wins.” Taking the zonal average of the model equations and linearizing about the unperturbed climatology of the model allowed Chang to diagnose the full, albeit linearized, effects of eddy heat and momentum fluxes separately, including the direct effects of the heat and momentum fluxes as well as flux-induced changes in the MMC. He found that when the tropical heating is concentrated, the full effects on the extratropical temperature of changes in the heat and momentum fluxes opposed each other, but that the “momentum flux wins.” His results were consistent with a strengthening and equatorward shift of the eddy-driven jet and its associated storm track, especially in the winter hemisphere. Chang also found weak but significant warming in subpolar latitudes of that hemisphere. Were these extratropical responses induced by changes in the zonal-mean flow within the Tropics, or did they depend on the dynamics of the Hadley circulation? To address this, Chang applied a baroclinic acceleration of the westerlies at 20◦ S to

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mimic the acceleration of the winds by the change in the Hadley circulation that results when the tropical heating is concentrated and shifted into the Northern Hemisphere. The responses of the extratropical zonally averaged temperatures and winds were nearly identical to those produced by changing the tropical heating. Chang concluded that the Hadley-circulation-forced changes in the winds and temperatures, rather than changes in the meridional circulation of the Hadley cell, drove the extratropical response. While the zonal momentum forcing was applied at 20◦ S, the maximum increase in the westerlies appeared between 30◦ and 35◦ S, again signaling the dominant role of the eddies in shaping the extratropical response to tropical forcing.

5.4. Dynamical Mechanisms In observations, in simulations by realistic models, and in idealized models tropical warming leads to a strengthening and equatorward shift of the jet and storm track. To understand how this happens, it is helpful to separate the direct effects on the circulation of tropical warming from those mediated by eddies, denoted the “eddy feedback.” In a simple model, this is accomplished by constructing a zonally averaged version of the model, in which the effects of eddies are replaced by their climatological average fluxes taken from a control run of the full model. When unperturbed radiative forcing is applied to this zonally symmetric model, it reproduces by construction the climatology of the control run. When tropical warming is applied as before, the results show the response in the absence of any eddy feedback. Such results, for δ = 10, are shown in Fig. 5.10. The direct response to tropical warming is an acceleration of the westerlies aloft between 10◦ and 35◦ N (Fig. 5.10a, dotted curve). When this same heating is applied to the full model (long dashes), there is less intensification of tropical and subtropical westerlies, while the midlatitude jet strengthens. The response to tropical warming in the zonally averaged meridional wind at 250 hPa, with and without eddy feedback, is shown in Fig. 5.10b. Without eddy feedback, tropical warming strengthens the Hadley circulation, redistributing the added heating throughout the Tropics. In the full model, tropical warming induces equatorward flow between 20◦ and 40◦ N and reinforces the strengthening of the Hadley circulation. Thus, the effects of the eddies (Fig. 5.10c) are to (1) cool the Tropics, where the rising branch of the Hadley cell strengthens, (2) cool the midlatitudes, where the rising branch of the Ferrel cell strengthens, and (3) warm the subtropics, where equatorward flow aloft from midlatitudes and the enhanced poleward branch of the Hadley circulation combine to produce stronger subsidence. What is the dynamics of these eddy effects? From the changes in the eddy fluxes of PV between the tropically warmed and control runs (Fig. 5.10d), it appears two different mechanisms are relevant. The equatorward shift and strengthening of the lower-level production of eddy activity (B) is the expected response to the increase in subtropical baroclinicity produced by stronger tropical heating. Second, the absorption of eddy

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Figure 5.10. Diagnoses of the effects of enhanced tropical heating in the two-level model: (a) zonally averaged zonal wind at 250 hPa in the control run, in the control run perturbed by enhanced tropical heating but without eddy feedback, and in the full model with enhanced tropical heating; (b) responses of the 250 hPa zonally averaged meridional wind with and without eddy feedback; (c) effects of eddy feedback on the zonally averaged temperature (500 hPa) and zonal wind (250 hPa); (d) responses of the zonally averaged fluxes of potential vorticity at both model levels.

activity aloft (−F ) strengthens and shifts towards the equator. Greater absorption of eddy activity is a necessary consequence of greater production, but the shift in F can come about, again from two different mechanisms. Given that the propagation of baroclinic eddies is primarily upward, it is expected there will be a concomitant shift in their absorption when the region of their generation is displaced. Second, the acceleration of low-latitude westerlies produced directly by tropical warming should allow the baroclinic eddies to propagate further into the Tropics before they approach their critical latitudes and are absorbed. Diagnostic calculations alone, however, cannot readily tease apart the relative importance of changes in eddy generation and in eddy propagation. One approach to understanding the eddy response is to treat baroclinic eddies as linear Rossby waves in the latitude-height plane. Harnik carried out calculations of this type, based on the approach of Harnik and Lindzen (2001), and these calculations were described by Seager et al. (2003). A test wave with a zonal wavelength, phase speed,

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and growth rate appropriate to baroclinic eddies was introduced into a quasigeostrophic model linearized about a zonal flow, and results were compared for the climatological zonal flow and for a zonal flow with perturbations intended to mimic the direct, as opposed to eddy-mediated, effects of tropical warming. The latter was a nearly equivalent barotropic strengthening of westerlies poleward of 10◦ latitude and equatorward of the climatological subtropical jet in both hemispheres. This qualitatively resembled the zonal flow changes, in the absence of eddy feedback, shown for the two-level model in Fig. 5.10a. A key diagnostic quantity was the meridional wavenumber, defined from the complex streamfunction of the steady wave solution:

2  ∂ψ l 2 = −Re /ψ , [5.23] ∂y 2 where ψ is the geostrophic streamfunction of the wave. Harnik argued that the direct changes in the zonal flow induced by El Niño increased meridional wavenumbers in low latitudes, while decreasing them in the subtropics. This leads to greater separation between the tropical and extratropical “waveguides” for baroclinic waves. Vertically propagating baroclinic waves were refracted poleward and equatorward away from this refractive index gap (near 35◦ N), causing a reduced convergence of the EP flux aloft. The cooling in middle latitudes then arose from the changes in the eddy-driven meridional circulation. In Harnik’s linear model, the amplitude of the wave forcing at the lower boundary was fixed, but this did not fix the generation of wave activity—in other words, her model could capture changes in both B and F . It is, therefore, once again difficult to say whether the realistic changes in wave driving, and thus in vertical motion and in temperature, that she obtained were primarily a result of changes in the generation of eddy activity at the lower boundary or of changes in eddy propagation. Regarding the latter, the she drew particular attention to the increase in the meridional wavenumber, in low latitudes, and its decrease in the subtropics. It appears that the anomalous EP vectors avoided this region of reduced l 2, though the causality of such a change is unclear because the wavenumber was calculated diagnostically from the wave solution and was not an intrinsic property of the zonal flow itself.

5.5. Structural Changes in the Circulation The observed and modeled responses of the extratropical circulation to tropical warming and cooling associated with the natural variability of the Tropics are of modest amplitude, though with important practical implications for regional climates. The jets and stormtracks change their latitudes and intensities only slightly, while the basic structure of the circulation remains unaltered. At least in models, however, more drastic changes occur as the strength of tropical warming is increased. Figure 5.11 shows an example, obtained using the same two-level model as before. As δ in (5.22) is increased,

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Figure 5.11. Variations in jet latitude (the latitude of strongest 250 hPa zonally averaged zonal wind) with the strength of the enhanced tropical heating (or cooling) in the two-level model. The added heating is centered on the equator.

there is a steady and gradual increase in the strength of the westerly jet aloft, and a similarly gradual, within the limits of model resolution, equatorward migration of the jet. At a critical value, however, the jet jumps to a much greater strength, while at the same time shifting almost discontinuously towards the equator. Figure 5.12 shows the zonal winds and the eddy driving of the zonal flow on both sides of this critical value. The winds aloft, for δ above the transition, are strongly westerly and nearly constant throughout the Tropics. The baroclinic generation of eddy activity B is shifted toward the Tropics, following the baroclinicity, as is the region of negative d 2F /dy 2 aloft. The strong eddy deceleration of the winds aloft, near 30◦ latitude in the “normal” circulation, is replaced by a weak acceleration in the anomalous one. The meridional winds of the Hadley cell (not shown) are reduced in strength by nearly an order of magnitude. This phenomenon is denoted “equatorial superrotation,” and it was described in models similar to the present one by Suarez and Duffy (1992) and by Saravanan (1993), though it was first observed in models many years earlier, as described in the excellent review by Held (1999). Superrotation can occur when strong zonally asymmetric thermal forcing is applied in the Tropics. This perhaps is not surprising, as the resulting poleward radiation of Rossby waves is accompanied by an equatorward flux of eastward momentum. It is of greater interest, in the present context, that equatorial superrotation can be induced simply by intensifying zonally symmetric tropical warming, as in the results shown above. In this case the baroclinic eddies are generated at much lower latitudes than in a normal state (Fig. 5.12b). They provide the needed equatorward flux of westerly momentum as they propagate to higher latitudes, where they dissipate. Close examination of the near-equatorial momentum budget, however, reveals that the

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Figure 5.12. Results from two-level model runs with enhanced equatorial heating—one (δ = 15◦C) with a “normal” circulation and the other (δ = 20◦C) exhibiting equatorial superrotation: (a) zonally averaged zonal winds at 250 hPa (thick curves) and 750 hPa (thin curves), (b) zonally averaged eddy fluxes of potential vorticity at 250 hPa (F ) and 750 hPa (B).

sub-gridscale diffusion of zonal momentum is important in maintaining the upper-level westerlies. The dependence of this result on an unphysically parameterized process raises the question of whether equatorial superrotation is a model artifact. The horizontal diffusion of momentum is important in these simple models because of their coarse resolution and because they lack physical mechanisms that can transport momentum vertically in the real atmosphere, such as momentum transport by convection. The first of these issues is addressed in experiments by Williams (2003) using a dry general circulation model, with thermal forcing similar to that used in the two-level model, but with much higher resolution: spectral truncation at rhomboidal 30 or higher, and 30 levels. When the radiative equilibrium temperature profile is concentrated near the equator, a superrotating state emerges, closely resembling that in the two-level

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Figure 5.13. Potential vorticity “snapshots” in the upper level (250 hPa) of the two-level model: (a) a run with a “normal” circulation (δ = 15◦C); (b) a run with equatorial superrotation (δ = 20◦C). The contour interval is 5 × 10−5 s−1.

model. The principal difference is that in Williams’s model, baroclinic eddies are much shallower in the superrotating state than in the normal state; this cannot happen in a two-level model. The transition to the superrotating state occurs over a very small range of forcing parameters (Fig. 5.11). Saravanan (1993) found that the superrotating regime exhibits hysteresis—the superrotating regime was maintained, once initiated, when the value of the key forcing parameter was reduced below the value at which it first appears, though this result appeared to depend on the model resolution. The present model, which has approximately the same resolution as Saravanan’s coarser model, exhibits similar hysteresis. The existence of such sharply defined regimes implies the existence of positive feedbacks that maintain the flow in one regime or the other. In fact, the structures of eddies are very different in the superrotating and the normal regimes, as shown in Fig. 5.13. Each panel is a snapshot of the upper-level PV in midlatitudes. The normal regime (Fig. 5.13a) is represented by a state with strong tropical heating, δ = 15◦C, while for the superrotating case (Fig 5.13b) tropical heating is increased, δ = 20◦C. In the normal regime, eddies primarily break anticyclonically. High values of PV advect southward from high latitudes and then wrap around anticyclones to the southwest. The result is a southwest-to-northeast tilt of PV structures south of 50◦ N. In the superrotating regime eddies tend to break and wrap up more cyclonically and mostly tilt from southeast to northwest.

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These differences in the eddy structures are reminiscent of the baroclinic eddy life cycles described by Thorncroft et al. (1993). Their LC2 life cycle, which occurs in an environment of barotropic cyclonic shear in the preexisting zonal flow, features cyclonic wrapping up of PV, and leads late in the life cycle to strong EP flux convergences in latitudes poleward of 40◦ N. By contrast, in their LC1 life cycle, which occurs in the absence of such shear, PV contours wrap up anticyclonically in the subtropics, and EP fluxes are strongly convergent in the subtropics, especially at the tropopause. Because convergent EP fluxes decelerate the zonal wind, these distinctions between the LC1 and LC2 life cycles are potentially self-reinforcing if it is imagined that in a sustained flow each life cycle pre-conditions the zonal flow for subsequent ones. That a shear-induced transition from one type of baroclinic eddy life cycle to another could explain the sharp transition between the normal and superrotating regimes is supported by experiments performed by Hartmann and Zuercher (1998). As the barotropic shear in the initial zonal flow was increased, they found a sharp transition in the nature of the life cycle, from LC1-like behavior to that resembling LC2. It is difficult, however, to make a definite connection between these life cycle calculations and the transition between regimes of a sustained circulation. Life cycles lack a restoring force on the zonal flow and they start from highly unstable states. These effects exaggerate the responses of the zonal flow to differences in eddy structures. The differences in the outcomes of life cycles occur primarily during their barotropic decay. In sustained flows, however, the distributions of the near-surface source of baroclinic eddies (B) are important. Finally, Hartmann (2000) obtained the surprising result that it is the meridional shear in the zonal wind near the surface, and not that at upper levels where the momentum fluxes are strongest, that is most important for the subsequent evolution of the baroclinic life cycle. There is no evidence that Earth’s atmosphere has ever entered a state of equatorial superrotation, and such a state has not been achieved in any GCM with a full complement of physical processes. One possible reason for this was pointed out by Kraucunas and Hartmann (2005), who found that realistic hemispheric asymmetry in the Hadley circulation prevented equatorial superrotation that otherwise arose in their model when zonally asymmetric equatorial heating was applied. It is plausible, therefore, that the superrotating state is an artifact of the over-simplification of the models in which it appears. This does not, however, preclude the possibility of more subtle regimes. Such regimes exist at least in one model, as shown by Son and Lee (2005). They used a dynamical-core model, much like that of Williams, with 10 levels and a horizontal resolution corresponding to a rhomboidal 30 truncation, but with only 15 zonal wavenumbers retained.3 The model is driven by radiative relaxation of temperatures to a zonally symmetric equilibrium. Figure 5.14a (their Fig. 2a) shows their control-run equilibrium temperature profile, together with two perturbations: one that increased the cooling rate at high latitudes, and the other, of more interest here, that increased the strength of tropical heating within 10◦ of the equator. As the

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tropical heating, distributed as shown in Fig. 5.14a, was increased in equal increments, the upper-level westerlies strengthened, but remained in what the authors denoted an “intermediate jet” (IJ) structure, with an eddy-driven jet near 45◦ N, and only a hint of a subtropical jet. As a critical value of tropical heating was exceeded, however, the upper-level winds made a regime-like transition (Fig. 5.14b) to a state with a single subtropical jet (SJ). The differences between the SJ and IJ states regimes were diagnosed (Figs. 5.14c, 5.14d, 5.14e), and the pattern of changes in the zonal wind and in wave driving were similar to those found for the responses to tropical warming induced by El Niño. The magnitude of the changes in Son and Lee’s calculations were an order of magnitude larger, however. While the authors argued that the subtropical jet in their SJ state was driven primarily by the Hadley circulation—i.e., by the Coriolis

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torque acting on the zonally symmetric poleward flow aloft—the transition to the SJ state was strongly reinforced by eddies (Figs. 5.14d and 5.14e): a strong enhancement and equatorward shift of the baroclinic generation of eddy activity, as indicated by increased poleward eddy heat fluxes between 30◦ and 35◦ N (Fig. 5.14e). Recalling that the meridional component of the EP flux is proportional to the eddy momentum flux, with a change of sign, and that the upward EP flux is proportional to the northward eddy heat flux, Figs. 5.14d and 5.14e indicate that from the intermediate to the subtropical jet cases the difference in the EP fluxes splits at upper levels. A region of reduced EP flux convergence at the latitude of the subtropical jet is bracketed to its north and south by increased convergence. This, of course, is the pattern that will reinforce the baroclinic subtropical jet. There is no indication of regime-like behavior in the observed response to El Niño, or, for that matter, in model responses to global warming (e.g., Huang et al., 2001). This behavior, unlike the response to modest tropical warming or the transition to superrotation, defies simulation in the two-level model. When tropical heating is increased in the two-level model, the jet moves smoothly between states resembling Son and Lee’s IJ and SJ states, and the only abrupt transition is to the superrotating state. Of course, absence from a two-level model does not preclude the appearance of a phenomenon in nature! The robustness of distinct IJ and SJ regimes in Son and Lee’s model is supported by the fact that they reappeared in a series of experiments using a moist version of the model in which the influence of moisture was made weaker or stronger by a parameter that sets the background specific humidity. As this parameter was increased in uniform small increments, their model made an abrupt transition from an IJ to an SJ state. This abrupt transition appeared to be governed by the dynamics of extratropical eddies; there was no accompanying abrupt transition in the rates or distributions of tropical latent heat release. This result raises the intriguing, and alarming, possibility that such abrupt changes could lurk in our future as the climate warms. Son and Lee sought dynamical understanding of these abrupt regime transitions in the behavior of baroclinic life cycles, specifically those in the work of Lee and Kim (2003). They found, much as did Hartmann and Zuercher, abrupt transitions in the outcomes of baroclinic life cycles. Lee and Kim’s initial conditions, however, were obtained from the output of a zonally symmetric model run with different strengths of additional tropical heating. The end states of these life cycles yielded a single strong subtropical jet only when the tropical heating was sufficiently strong. Interestingly, they could predict the outcomes of the life cycle calculations from the momentum fluxes of the most unstable baroclinic mode on the initial zonal flow. When the same approach was applied to the steady-state GCM results of Son and Lee, it was found that linear baroclinic instability theory still predicted a transition, but it could not quantitatively predict the strength of tropical heating at which the transition would occur.

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5.6. Tropical Forcing of the Annular Modes A feature of the extratropical response to low-latitude warming, present in observations and in realistic GCMs but not in simple models, is a strong response in the Southern Annular Mode (SAM). SAM is a dominant mode of intra-seasonal and interannual variability in the Southern Hemisphere (Thompson and Wallace 2000), featuring deep, oppositely signed anomalies in the zonal wind centered at approximately 35◦ and at 60◦ S. SAM is defined as positive when the westerlies are stronger at 60◦ S and weaker at 35◦ S, so that positive SAM represents a poleward shift in the eddydriven jet. Observational analyses show a robust tendency for SAM to be negative during El Niño events (Karoly 1989; L’Heureux and Thompson 2006). L’Heureux and Thompson’s analysis, using data only from the era of satellite data (1978–2004), indicated that SAM was most negative early in the austral summers of El Niño years. Following their work, we show (Fig. 5.15a) the regression of zonally averaged zonal winds, averaged from November through January, against the Nino 3.4 Index of equatorial Pacific sea surface temperature (SST) over the years 1978–2004. As before, NCEP/NCAR reanalysis data are used. The zonal wind anomalies in Fig. 5.15a are those that would accompany a one-standard-deviation positive perturbation in the Nino 3.4 Index. The easterlies at 60◦ S, together with the westerlies around 40◦ S, project nearly perfectly on the zonal wind anomalies associated with SAM. L’Heureux and Thompson suggested that tropical sea surface temperatures explain roughly a quarter of the month-to-month variability in SAM in austral summer. Although only 27 years of data are used for this analysis, there is good reason to believe the effect is real. First, very similar results are obtained when the entire NCEP reanalysis record is used, extending back to 1948. Second, Karoly (1989) found similar results from a composite of three El Niño episodes (1972, 1976/77, and 1982) using Australian operational analyses. Finally, L’Heureux and Thompson showed that a very similar response to El Niño, albeit with a delayed seasonal response, is found in the output of a GCM. L’Heureux and Thompson then removed that portion of their El Niño response that projects on SAM. What remained was a pattern with striking interhemispheric symmetry: a narrow band of equatorial easterlies, subtropical westerlies, and easterlies in middle latitudes. This resembles the response to tropical warming found in the two-level model (Fig. 5.7b), although contracted towards the equator. Their analysis can, therefore, be considered an observational confirmation of the idealized model depiction of the hemispherically symmetric response to tropical warming. Idealized GCMs typically have robust annular modes, and the two-level model is no exception (Robinson 1991b). The dynamics through which eddies sustain the annular modes against dissipation, are, moreover, essentially the same as those by which eddy feedback produces the midlatitude response to tropical warming (Robinson 2000).

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Yet the two-level model produces no annular-mode response to tropical warming. In fact, the annular mode in the model is in nearly perfect meridional quadrature with its response to tropical warming. There is in observations a Northern Hemisphere annular mode with nearly identical structure (NAM) to the SAM, though its significance, given the greater zonal asymmetry in the Northern Hemisphere circulation, is more controversial. Observations indicate that the NAM is in quadrature with the zonally symmetric

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response to El Niño, so the Northern Hemisphere more nearly resembles the idealized model. These results raise two related questions: What is the source of the SAM response to El Niño? Why is there no similar response in NAM? Answers likely involve the zonally asymmetric response to El Niño and the zonally asymmetric climates of both hemispheres; these effects are not included in the simple model. Figure 5.15b shows the regression of austral summer geopotential heights at 250 hPa against the Nino 3.4 Index, using the same years as before. The patterns here are similar to those shown in Fig. 5.2b, but the zonal means have been removed, so what remains is the “stationary eddy” response to El Niño. In both hemispheres, trains of stationary Rossby waves appear to emerge from the Tropics near 120◦ E, longitudes of anomalous atmospheric cooling during El Niño (cf. DeWeaver and Nigam 2004). These wave trains follow different paths in the two hemispheres, presumably a consequence of the very different climatological flows on which they propagate. The strongest features, in geopotential height, are the low at 40◦ N, 150◦ W and the high at 55◦ S, 120◦ W. Hoerling and Ting (1994) demonstrated that the former feature is strongly reinforced by transient-eddy vorticity fluxes, and it is plausible that this is true of the latter. Both lie in the stormtrack, the latitude of strongest transient-eddy activity, and both are east—downwind—of local maxima in stormtrack intensity. How can these asymmetric features lead to a response in SAM? There are zonally averaged momentum fluxes associated with the wave train, but these momentum fluxes cannot cause a zonally averaged response that is proportional to the tropical SST anomaly because the same momentum flux anomalies would occur for the La Niña pattern, when the signs of the geopotential anomalies are reversed (Weickmann and Sardeshmukh 1994). Rather, the SAM response must arise from interactions between the El Niño-associated wave train in Fig. 5.15b and zonal asymmetries in the climatological circulation. The interference of the anomalous and climatological stationary eddies can produce stationary-eddy momentum fluxes. Alternatively, a signal in the zonally averaged transient-eddy momentum fluxes could come about if, because of zonal asymmetries in the storm track, transient-eddy momentum fluxes are particularly sensitive to the large-scale flow in the region around 120◦ W. Because SAM is sustained by transient-eddy momentum fluxes, there is always a close association between these fluxes and the appearance of the annular mode. This was confirmed by L’Heureux and Thompson in their SAM response to El Niño. It may, therefore, be difficult to determine from observations alone how the SAM response is initiated. Another perplexing aspect of the observed association of SAM with interannual variability in the Tropics is that in a number of GCMs (Kushner et al. 2001; Huang et al. 2001; Brandefelt and Källén 2004; Fyfe and Saenko 2006) anthropogenic global warming causes SAM to increase, opposite in sign to its response to El Niño. Given that both greenhouse warming and El Niño cause a warming of the Tropics, this suggests, as discussed above, that the SAM response to El Niño may depend on the zonal asymmetries in the extratropical atmospheric response, which are either absent

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in or different from the response to global warming. Alternatively, while the Tropics warm under greenhouse warming, it could be that some other aspect of global warming drives the increase in SAM. One possibility is that global greenhouse warming induces stratospheric cooling and that the subsequent downward influence on SAM from the stratosphere (Polvani and Kushner 2002) in late winter persists into the austral summer. Kushner et al. (2001) computed the internally generated SAM in their GCM, and then used this to decompose their modeled responses to greenhouse warming into a component that projects on the SAM and a residual. The residual shows some similarity to the modeled and observed responses to tropical warming: a strengthening and equatorward shift of the subtropical jet that is, at least in part, driven by eddy momentum fluxes. Once again, then, it appears that the SAM response to tropical warming is a separate dynamical phenomena from the responses at lower latitudes.

5.7. Concluding Remarks Observations show that coexisting with the familiar stationary Rossby wave train response to El Niño there is a zonally symmetric response, comprising a strengthening and equatorward shift of the zonally averaged jet and storm track and cooling poleward of the subtropical jet. This zonally symmetric response to tropical warming is robust. It appears in GCMs of varying realism and in the simplest model that could be expected to capture this phenomenon, a two-level model on the sphere. Because this response to tropical warming projects strongly on the zonal mean, it is reasonable to consider this as a problem in eddy-zonal flow interaction, for which there is a familiar and well developed theory. When such theory is applied to this phenomenon, it becomes apparent that the response is what would be expected if tropical warming induces a strengthening and equatorward displacement of the extratropical eddy-driven zonal mean circulation. The “MMC wins” in shaping this response, in the sense that the intensified generation of baroclinic eddies in the lower troposphere leads to cooling poleward of the latitudes of enhanced eddy heat fluxes. This is the result expected from boosting baroclinic eddy generation if, in the climatological circulation, the MMC again “wins,” producing a stronger thermal gradient across a band of middle latitudes than in the absence of eddies. For the MMC to “win” in the climatological mean or in the anomalous response to tropical warming requires that the baroclinic eddies propagate far enough meridionally from their source latitudes that there is a local minimum in the absorption of eddy activity (EP flux convergence) aloft at those source latitudes. Thus, one can think of the response to tropical warming as a strengthening and equatorward shift of the dynamical arrangement of the climatological eddy-driven circulation, depicted schematically in Fig. 5.16a. The strengthening of the circulation is the unsurprising response when more heat is put into the Tropics. The equatorward shift comes about because the strongest baroclinicity is shifted equatorward, but it is

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important to recognize that, as shown in Fig. 5.10a, the direct effect of near-equatorial heating is to increase baroclinicity within the Tropics. That baroclinicity is increased, instead, across the subtropics is a result of the eddy-driven circulation that causes subtropical warming and midlatitude cooling. Global greenhouse warming leads to warming in the tropical atmosphere, so a response may be expected that is similar, at least in its zonally symmetric aspects, to that induced by El Niño. Consistent with this expectation, transient global warming experiments show strengthening of the subtropical jets (Kushner et al. 2001; Huang et al. 2001). While Kushner et al. focused on the annular mode response to global warming, close examination of their results reveals that, independently of the SAM response, the subtropical jet in their model intensified (though at a low latitude, around 20◦ S), it was reinforced by eddy momentum fluxes, and, consistent with this, the zonally averaged surface flow became more westerly at the same latitude at which the jet intensified. The thermal response to these circulation changes was, however, overwhelmed by the direct affect of increasing greenhouse warming. Thus, there is some model evidence suggesting that global warming will bring about the same sort of circulation changes that occur in response to interannual tropical warming. If, as has been suggested (Cane et al. 1997), global warming produces a bias towards La Niña conditions, then the subtropical response may be more complicated. A more precipitous and alarming response to global warming would be a change in the dynamical regime of the circulation. It is possible such changes have occurred in the past and were responsible for the abrupt changes in the global climate inferred from paleoclimatic records (see chapter 12 in this volume). The most extreme version of such a change, the transition to equatorial superrotation, comes about in idealized circulation models as tropical warming is intensified or the region of tropical warming

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is narrowed. Rather than a shift in the eddy-driven circulation, superrotation involves a reorganization of its structure. Where baroclinic eddies are generated, their propagation, and where they are dissipated are all significantly altered. This is depicted, schematically, in Fig. 5.16b. To date, no realistic GCM with a full complement of physical parameterizations has generated a sudden transition to a superrotating state, though easterlies did appear gradually in the tropical upper troposphere in transient greenhouse simulations analyzed by Huang et al. (2001). This does not preclude the possibility of more subtle regime shifts in the circulation as climate warms, such as the transition to a dominant subtropical jet found by Son and Lee (2005), though again, no shifts this dramatic have yet been simulated in realistic models.

Acknowledgments The author acknowledges the support of his research by the Climate Dynamics Program of the National Science Foundation and the CLIVAR Program of the National Oceanographic and Atmospheric Administration. Drs. Tapio Schneider and Richard Seager provided helpful comments on an earlier draft of this chapter.

Notes 1. Figures 5.1 and 5.2 were created using online data and tools from the Web site of the NOAA-CIRES Climate Diagnostics Center, Boulder Colorado, USA, http://www.cdc.noaa.gov 2. It is the principal component of the leading empirical orthogonal function of the zonally averaged zonal winds at 300 hPa, computed in the calendar month (June–May) latitude domain. See Seager et al. (2003) for details. 3. That this is a reasonable modeling approach for eddy-zonal flow interaction problems was shown by Held and Phillipps (1993).

References Andrews, D. G., and M. E. McIntyre, 1976: Planetary waves in horizontal and vertical shear: The generalized Eliassen-Palm relation and the mean zonal acceleration. J. Atmos. Sci., 33, 2031–2048. Brandefelt, J., and E. Källén, 2004: The response of the Southern Hemisphere atmospheric circulation to an enhanced greenhouse gas forcing. J. Climate, 17, 4425–4442. Bretherton, F. P., 1966: Critical layer instability in baroclinic flows. Quart. J. Roy. Meteor. Soc., 92, 325–345. Cane, M. A., A. C. Clement, A. Kaplan, Y. Kushnir, D. Pozdnyakov, R. Seager, S. E. Zebiak, and R. Murtugudde, 1997: Twentieth century sea surface temperature trends. Science, 275, 957–960. Chang, E. K. M., 1995: The influence of Hadley circulation intensity changes on extratropical climate in an idealized model. J. Atmos. Sci, 52, 2006–2024.

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140 | Walter A. Robinson Charney, J. G., and M. E. Stern, 1962: On the stability of internal baroclinic jets in a rotating atmosphere. J. Atmos. Sci., 19, 159–172. DeWeaver, E., and S. Nigam, 2004: On the forcing of ENSO teleconnections by anomalous heating and cooling. J. Climate, 17, 3225–3235. Edmon, H. J., Jr., B. J. Hoskins, and M. E. McIntyre, 1980: Eliassen-Palm cross sections for the troposphere. J. Atmos. Sci., 37, 2600–2616. Fyfe, J. C., and O. A. Saenko, 2006: Simulated changes in the extratropical Southern Hemisphere winds and currents. Geophys. Res. Lett. 33, L06701, doi:10.1029/2005GL025332. Green, J. S. A., 1970: Transfer properties of the large-scale eddies and the general circulation in the atmosphere. Quart. J. Roy. Meteor. Soc., 96, 157–185. Harnik, N., and R. S. Lindzen, 2001: The effect of reflecting surfaces on the vertical structure and variability of stratospheric planetary waves. J. Atmos. Sci., 58, 2872–2894. Hartmann, D. L., 2000: The key role of lower-level meridional shear in baroclinic wave life cycles. J. Atmos. Sci., 57, 389–401. Hartmann, D. L., and P. Zuercher, 1998: Response of baroclinic life cycles to barotropic shear. J. Atmos. Sci., 55, 297–313. Held, I. M., 1999: Equatorial superrotation in Earth-like atmospheric models. Bernhard Haurwitz Memorial Lecture for 1999. Available at: http://www.gfdl.noaa.gov/∼ih/. Held, I. M., and A. Y. Hou, 1980: Nonlinear axially symmetric circulations in a nearly inviscid atmosphere. J. Atmos. Sci., 37, 515–533. Held, I. M., and P. J. Phillipps, 1993: Sensitivity of the eddy momentum flux to meridional resolution in atmospheric GCMs. J. Climate, 6, 499–507. Hendon, H. H., and D. L. Hartmann, 1985: Variability in a nonlinear model of the atmosphere with zonally symmetric forcing. J. Atmos. Sci., 42, 2783–2797. Hoerling, M. P., and M. Ting, 1994: Organization of extratropical transients during El Niño. J. Climate, 7, 745–766. Hoerling, M. P., J. S. Whitaker, A. Kumar, and W. Wang, 2001: The midlatitude warming during 1998–2000. Geophys. Res. Lett., 28, 755–758. Holton, J. R., 2004: An Introduction to Dynamic Meteorology: Fourth Edition. Elsevier Academic Press, 535 pp. Horel, J. D., and J. M. Wallace, 1981: Planetary-scale atmospheric phenomena associated with the Southern Oscillation. Mon. Wea. Rev., 109, 813–829. Hoskins, B. J., and D. J. Karoly, 1981: The steady linear response of a spherical atmosphere to thermal and orographic forcing. J. Atmos. Sci., 38, 1179–1196. Hou, A. Y., 1993: The influence of tropical heating displacements on the extratropical climate. J. Atmos. Sci., 50, 3553–3570. Hou, A. Y., 1998: Hadley circulation as a modulator of the extratropical climate. J. Atmos. Sci., 55, 2437–2457. Hou, A. Y., and R. S. Lindzen, 1992: The influence of concentrated heating on the Hadley circulation. J. Atmos. Sci., 49, 1233–1241. Hou, A. Y., and A. Molod, 1995: Modulation of dynamic heating in the winter extratropics associated with the cross-equatorial Hadley circulation. J. Atmos. Sci., 52, 2609–2626. Huang, H.-P., K. M. Weickmann, and C. J. Hsu. 2001: Trend in atmospheric angular momentum in a transient climate change simulation with greenhouse gas and aerosol forcing. J. Climate, 14, 1525–1534.

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Tropical-Extratropical Interactions | 141 Kalnay, E., M. Kanamitsu, R. Kistler, W. Collins, D. Deaven, L. Gandin, M. Iredell, S. Saha, G. White, J. Woollen, Y. Zhu, A. Leetmaa, B. Reynolds, M. Chelliah, W. Ebisuzaki, W. Higgins, J. Janowiak, K. C. Mo, C. Ropelewski, J. Wang, R. Jenne, and J. Dennis, 1996: The NCEP/NCAR 40-year reanalysis project. Bul. Amer. Meteor. Soc., 77, 437–471. Karoly, D. J., 1982: Eliassen-Palm cross sections for the Northern and Southern Hemispheres. J. Atmos. Sci., 39, 178–182. Karoly, D. J., 1989: Southern Hemisphere circulation features associated with El Niño-Southern Oscillation events. J. Climate, 2, 1239–1252. Kraucunas, I., and D. L. Hartmann, 2005: Equatorial superrotation and the factors controlling the zonal-mean zonal winds in the tropical upper troposphere. J. Atmos. Sci., 62, 371–389. Kushner, P. J., I. M. Held, and T. L. Delworth, 2001: Southern Hemisphere atmospheric circulation response to global warming. J. Climate, 4, 2238–2249. Lee, S., and H.-K. Kim, 2003: The dynamical relationship between subtropical and eddy-driven jets. J. Atmos. Sci., 60, 1490–1503. L’Heureux, M. L., and D. W. J. Thompson, 2006: Observed relationships between the El-Niño/Southern Oscillation and the extratropical zonal-mean circulation. J. Climate, 19, 276–287. Lindzen, R. S., and A. Y. Hou, 1988: Hadley circulations for zonally averaged heating centered off the equator. J. Atmos. Sci., 45, 2416–2427. Pavan, V., and I. M. Held, 1996: The diffusive approximation for eddy fluxes in baroclinically unstable jets. J. Atmos. Sci., 53, 1262–1272. Polvani, L. M., and P. J. Kushner, 2002: Tropospheric response to stratospheric perturbations in a relatively simple general circulation model. Geophys. Res. Lett., 29, 2001GL014284. Randel, W. J., and I. M. Held, 1991: Phase speed spectral of transient eddy fluxes and critical layer absorption. J. Atmos. Sci., 48, 688–697. Robinson, W. A., 1991a: The dynamics of low-frequency variability in a simple model of the global atmosphere. J. Atmos. Sci., 48, 429–441. Robinson, W. A., 1991b: The dynamics of the zonal index in a simple model of the atmosphere. Tellus, 43A, 295–305. Robinson, W. A., 2000: A baroclinic mechanism for the eddy feedback on the zonal index. J. Atmos. Sci., 57, 415–422. Robinson, W. A., R. Reudy, and J. E. Hansen, 2002: GCM simulations of recent cooling in the east-central United States. J. Geophys. Res., 107, 4748, doi:10.1029/2001JD001577. Saravanan, R., 1993: Equatorial superrotation and maintenance of the general circulation in two-level models. J. Atmos. Sci., 50, 1211–1227. Sardeshmukh, P. D., and B. J. Hoskins, 1988: The generation of global rotational flow by steady idealized tropical divergence. J. Atmos. Sci., 45, 1228–1251. Schneider, T., 2005: Zonal momentum balance, potential vorticity dynamics, and mass fluxes on near-surface isentropes. J. Atmos. Sci., 62, 1884–1900. Schubert, S. D., M. J. Suarez, P. J. Pegion, R. D. Koster, and J. T. Bacmeister, 2004: Causes of long-term drought in the U.S. Great Plains. J. Climate, 17, 485–503. Seager, R., Y. Kushnir, N. Harnik, W. A. Robinson, and J. Miller, 2003: Mechanisms of hemispherically symmetric climate variability. J. Climate, 16, 2960–2978. Seager, R., N. Harnik, W. A. Robinson, Y. Kushnir, M. Ting, H.-P. Huang, and J. Velez, 2005a: Mechanisms of ENSO forcing of hemispherically symmetric precipitation variability. Quart. J. Royal. Meteor. Soc., 131, 1501–1527.

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142 | Walter A. Robinson Seager, R., Y. Kushnir, C. Herweijer, N. Naik, and J. Velez, 2005b: Modeling of tropical forcing of persistent droughts and pluvials over western North America: 1856–2000. J. Climate, 18, 4065– 4088. Simmons, A. J., J. M. Wallace, and G. W. Branstator, 1983: Barotropic wave propagation and instability, and atmospheric teleconnection patterns. J. Atmos. Sci., 40, 1363–1392. Son, S.-W., and S. Lee, 2005: The response of westerly jets to thermal driving in a primitive equation model. J. Atmos. Sci., 62, 3741–3757. Suarez, M. J., and D. G. Duffy, 1992: Terrestrial superrotation: A bifurcation of the general circulation. J. Atmos. Sci., 49, 1541–1554. Tanaka, D., T. Iwasaki, S. Uno, M. Ujiie, and K. Miyazaki, 2004: Eliassen-Palm flux diagnosis based on isentropic representation. J. Atmos. Sci., 61, 2370–2383. Thompson, D. W. J., and J. M. Wallace. 2000: Annular modes in the extratropical circulation. Part I: Month-to-month variability. J. Climate, 3, 1000–1016. Thorncroft, C. D., B. J. Hoskins, and M. E. McIntyre, 1993: Two paradigms of baroclinic wave life-cycle behavior. Quart. J. Roy. Meteor. Soc., 119, 17–55. Webster, P. J., 1981: Mechanisms determining the atmospheric response to sea surface temperature anomalies. J. Atmos. Sci., 38, 554–571. Weickmann, K. M., and P. D. Sardeshmukh, 1994: The atmospheric angular momentum cycle associated with a Madden-Julian oscillation. J. Atmos. Sci., 51, 3194–3208. Williams, G. P., 2003: Barotropic instability and equatorial superrotation. J. Atmos. Sci., 60, 2136–2152. Yulaeva, E., and J. M. Wallace, 1994: The signature of ENSO in global temperature and precipitation fields derived from the Microwave Sounding Unit. J. Climate, 7, 1719–1736.

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Chapter 6

On the Relative Humidity of the Atmosphere Raymond T. Pierrehumbert, Hélène Brogniez, and Rémy Roca

6.1. Introduction Water is a nearly miraculous molecule that enters into the operation of the climate system in a remarkable variety of ways. A wide-ranging overview can be found in Pierrehumbert (2002). In the present chapter, we will be concerned only with the radiative effects of atmospheric water vapor on climate, and with the kinematics of how the water vapor distribution is maintained. These effects are important because feedbacks due to changes in atmospheric water vapor amplify the climate system’s response to virtually all climate forcings, including anthropogenic and natural changes in CO2, changes in solar luminosity, and changes in orbital parameters. In contrast to cloud feedbacks, which differ greatly amongst general circulation models, clear-sky water vapor feedback is quite consistent from one model to another. Essentially all general circulation models (GCMs) yield water vapor feedback consistent with that which would result from holding relative humidity approximately fixed as climate changes (Held and Soden 2000; Colman and McAvaney 1997). This is an emergent property of the simulated climate system; fixed relative humidity is not in any way built into the model physics, and the models offer ample means by which relative humidity could change. Why should models with such diverse representations of moist processes yield such similar results when it comes to water vapor feedback? The answer to this question has a considerable bearing on the extent to which one can trust models to faithfully reproduce the water vapor feedback occurring in nature, and in particular in climates that haven’t yet been directly observed. There is much indirect evidence that the water vapor feedback in models is correct, and indeed no compelling reason has emerged to doubt it. Nonetheless, it has proved difficult to articulate cleanly and convincingly from basic principles exactly why one should have confidence in this aspect of the models. If the atmosphere were saturated at all levels, understanding water vapor feedback would offer few challenges. The difficulty arises from the prevalence of highly unsaturated air in the atmosphere. To make progress, one needs better conceptual models of the factors

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governing the distribution of subsaturated air in the atmosphere. Once mechanisms are understood, prospects for determining whether key processes operate in the same way in models as in the real world become brighter. Another reason for seeking a better mechanistic understanding of the relative humidity distribution is that such an understanding is a prerequisite for credible incorporation of water vapor feedback in idealized climate models. Despite advances in computer power, energy balance models and their somewhat embellished cousins the “intermediate complexity models” continue to have an important role in exploration of new ideas. When simulations of tens or hundreds of thousands of years are called for, they are indispensable. In thinking about representation of the hydrological cycle in idealized models, one must distinguish between the requirements of modeling water for such purposes as computing precipitation and latent heat transport, and the requirements of modeling water for the purpose of determining its radiative impacts. The former is a challenging enough task, but involves primarily low-altitude processes where most of the atmosphere’s water is found. The latter is yet more formidable, as it requires modeling the relatively small proportion of water found in the upper reaches of the troposphere. Even in idealized models that attempt some representation of the hydrological cycle (Petoukhov et al. 2000; Weaver et al. 2001), the dynamic influences on water vapor feedback are generally neglected. The aim of this chapter is to present some simple but quantifiable ideas concerning the way in which the atmosphere’s population of unsaturated air emerges from the interplay of transport and condensation. The key insight is to think of the water content of a parcel of air as resulting from a stochastic process operating along air parcel trajectories. The various incarnations of this idea that will be discussed in this chapter account for the prevalence of dry air, offer some insight as to how the relative humidity distribution may change in response to a changing climate, and yield some diagnostic techniques that can be used as a basis for comparing moisture-determining processes in models and the real world. Through detailed analysis of an idealized version of the stochastic problem, we expose some of the mathematical challenges involved in formulating moisture parameterizations for idealized climate models, and suggest a possible approach to stochastic modeling of moisture. The class of theories we consider here deals only with those aspects of moisture that can be inferred on the basis of largescale wind and temperature fields such as can be explicitly resolved by general circulation models and global analysis/assimilation datasets. This is only a piece of the puzzle, but we think it is a large piece. The whole complex of important issues concerning the way deep convection injects moisture into the atmosphere, and the sensitivity of such processes to microphysics and parameterization (e.g., Tompkins and Emanuel [2000]; see also chapters 7 and 11 in this volume) is left untouched for the moment, but must at some point be brought back into the picture. We begin by reviewing some basic material concerning the radiative importance of water vapor in sections 6.2 and 6.3. The essential ideas governing the generation of

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subsaturated air are introduced in section 6.4. A simple class of models embodying these ideas is analyzed in section 6.5, where we also analyze a more conventional diffusive model for the sake of comparison and contrast. In section 6.6 the concepts are applied to diagnosis of moisture dynamics in the present climate, and in GCM simulations of warming in response to elevation of CO2.

6.2. Basic Properties of Water Vapor There are two basic aspects of the physics of water vapor that give it a distinguished role in determining the sensitivity of the Earth’s climate. The first aspect is common to all substances that undergo a phase change: the maximum partial pressure of water vapor that can be present in a volume of atmosphere in thermodynamic equilibrium is a strongly increasing function of temperature. This maximum is known as the saturation vapor pressure, e s , and is governed by the Clausius-Clapeyron relation. For water vapor at modern terrestrial temperatures, the Clausius-Clapeyron relation is well approximated by the exponential form e s (T ) = e s (To)e

− RLv




1 T

− T1o



,

[6.1]

where L is the latent heat of the appropriate phase transition (vapor to liquid at warm temperatures, vapor to solid at sufficiently cold temperatures), Rv is the gas constant for water vapor, and To is a reference temperature. At the freezing point, e s is 614 Pa or 6.14 mb; L /Rv = 5419 K for condensation into liquid and 6148 K for condensation into ice. The formula predicts a very strong dependence of saturation vapor pressure on temperature. At 300 K, e s rises to 3664 Pa, whereas at 250 K, e s drops to a mere 77 Pa. Once the partial pressure of water vapor reaches the saturation vapor pressure, any further addition of water vapor will lead to condensation sufficient to bring the vapor pressure back down to saturation. The condensed water may remain in suspension as cloud droplets, or it may aggregate and be removed from the atmosphere in the form of precipitation. Because specific humidity is a materially conserved quantity until condensation occurs, it is convenient to define the saturation specific humidity, q s , which is the ratio of the mass of water vapor that could be held in saturation to the total mass of   saturated air. In terms of saturation vapor pressure, q s = 0.622e s (T )/ pa + 0.622e s (T ) , where pa is the partial pressure of dry air. Condensation occurs whenever the specific humidity q exceeds q s .1 The Clausius-Clapeyron relation provides a powerful constraint on the behavior of water vapor, but it is not at all straightforward to tease out the implications of this constraint for climate, for the reason that it only gives an upper bound on the water vapor content for any given temperature, and tells us nothing about how closely that bound might be approached.

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The second key aspect of water vapor is that it is a potent greenhouse gas. Like most greenhouse gases, the effect of water vapor on outgoing longwave radiation (OLR) is approximately logarithmic in specific humidity, once the concentration is sufficiently large to saturate the principal absorption bands. The logarithmic effect of water vapor is somewhat more difficult to cleanly quantify than is the case for wellmixed greenhouse gases like CO2, but if one adopts a base-case vertical distribution and changes water vapor by multiplying this specific humidity profile by an altitudeindependent factor, one finds that each doubling of water vapor reduces OLR by about 6 W/m2 (Pierrehumbert 1999). This is about 50% greater than the sensitivity of OLR to CO2. A comprehensive analysis of the latitude and altitude dependence of the sensitivity of OLR to water vapor can be found in Held and Soden (2000). The idea that small quantities of water vapor can have a lot of leverage in climate change has a fairly long history, and is now widely recognized. Water vapor feedback was included in the very first quantitative calculations of CO2-induced warming by Arrhenius, and the importance of water vapor aloft was implicit in such calculations, as it was in the first comprehensive radiative-convective simulation of the problem by Manabe and Wetherald (1967). However, consciousness of the importance of free tropospheric humidity, and of the difficulty of understanding it, did not really awaken until the early 1990s. Significant early work on the subject appeared in Soden and Bretherton (1993) and Shine and Sinha (1991), partly in response to some salutary, if ultimately easily dismissed, skepticism about global warming expressed in Lindzen (1990). Related points have been discussed in Spencer and Braswell (1997), Pierrehumbert (1999), and Held and Soden (2000), among others. One important consequence of the logarithmic dependence is that the relatively small amount of water vapor aloft nonetheless has a great influence on the radiation budget. Another consequence is that the degree of dryness of dry air also has a great effect on the radiation budget: reducing free tropospheric humidity from 5% to 2.5% in a dry zone has nearly the same radiative impact as reducing humidity from 40% to 20%, even though the former reduction involves a much smaller quantity of water. A related consequence of the logarithmic dependence is that the radiative effect of water vapor cannot be accurately determined on the basis of the mean specific humidity alone. Fluctuations in specific humidity, e.g., fluctuations at scales smaller than resolved in a climate model, affect the OLR to some extent. If we denote the specific humidity by q and time or space averages by angle brackets, then
log q  = log
q . Specifically, writing q =
q  + q ,     

 q 1 q 2 log
q  1 + ≈ log
q  − < log
q  [6.2]
q 
q  2 Since OLR is proportional to − log q , fluctuations in water vapor increase the OLR and have a cooling effect. Having some very dry air and some very moist air allows more infrared cooling than would the same amount of water spread uniformly over the

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atmosphere. To provide some idea of the magnitude of this effect, consider a region of the atmosphere within which the temperature is horizontally homogeneous, and where the relative humidity is 50% everywhere. Next, increase the humidity in one half of the region to 87.5% while keeping the total water in the system constant. This requires a reduction of the rest of the region to 12.5% relative humidity. One not-quite doubles the humidity in half of the region, while reducing the humidity in the rest by a factor of 1/4, yielding a net increase of OLR relative to the uniform state of 3.6 W/m2, based on the sensitivity factor given above. If we make the dry air still drier, reducing it to 6.25% while increasing the moist air to 93.75%, the OLR increase becomes 11 W/m2. Water vapor affects climate sensitivity through its effect on the slope of OLR vs. surface temperature. Following Held and Soden (2000), the change
T in temperature caused by a change
F in radiative forcing can be written
T =  ·
F , where the sensitivity coefficient is given by =

1 . (1 − β) ∂∂T OLR|specific humidity fixed

[6.3]

If free tropospheric water vapor increases with temperature, the additional greenhouse effect reduces OLR compared to what it would be with water vapor fixed, whence the water vapor feedback factor β is positive, and climate sensitivity is increased. General circulation model simulations of the change in the present climate caused by increasing CO2 universally show polar amplification, in the sense that the temperature increase at high latitudes is greater than that at low latitudes. Observations of climate change over the past century seem to conform to this expectation. It is important to note that water vapor feedback does not contribute to polar amplification. In fact, for the present pole-to-equator temperature range, water vapor feedback makes the sensitivity  smaller at cold temperatures than at warm temperatures (see Pierrehumbert [2002]), and hence would lead to tropical rather than polar amplification. Certainly, ice albedo feedback plays a role in polar amplification, but dynamical heat transport and clouds may also contribute. These feedbacks must be sufficiently strong to overcome the tendency of water vapor feedback to put the greatest warming in the Tropics.

6.3. The Importance of Water Vapor to the Radiation Budget In this section we present a few reminders of the central role that water vapor radiative effects play in determining the Earth’s climate. We begin by examining the way in which CO2, water vapor, and the presence of subsaturated air individually affect the radiation balance of the Earth, as measured by OLR. To do this, we used the NCAR CCM3 radiation model (Kiehl et al. 1998) to compute what the Earth’s OLR would be under various assumptions. For each case, clear-sky OLR was computed on a latitude-longitude grid, with the 3-D temperature pattern held fixed at the January monthly mean climatology derived from NCEP reanalysis data (Kalnay et al. 1996) for

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500 450 400 OLR (W m–2)

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350 300 250 200 150 –90

–60

–30

0 Latitude

30

60

90

Figure 6.1. The effect of atmospheric composition on OLR, with atmospheric and surface temperature held fixed at January climatological values, 1960–1980.

1960–1980. The zonal mean results, shown in Fig. 6.1, are derived from averages of latitude-dependent OLR, rather than being computed on the basis of zonal-mean temperature fields. This is important, because radiation is nonlinear in temperature and humidity. We have not, however, taken the effect of daily fluctuations into account. This introduces some error into the analysis, particularly in the midlatitudes. As compared to the calculation with no atmospheric greenhouse effect whatsoever, CO2 by itself brings the tropical OLR down by 60 W/m2. The OLR reduction decreases with latitude in the extratropics, falling to 25 W/m2 at 60◦ S in the summer hemisphere, and even smaller values in the winter extratropics. The latitudinal variation in the CO2 greenhouse effect derives from the vertical structure of the atmosphere: in the Tropics there is more contrast between surface and tropopause temperature than there is in the extratropics, and the summer extratropics has more contrast than the winter extratropics. When OLR is recalculated with the observed humidity content of the atmosphere (based on NCEP) in addition to the CO2 from the previous case, OLR drops by an additional 100 W/m2 in the Tropics, and a lesser amount in the extratropics. In fact, at each latitude the greenhouse effect of water vapor is approximately twice that of CO2. As noted previously, the atmosphere is highly unsaturated. If we recompute the OLR assuming a saturated troposphere, the OLR is further reduced. The effect is considerable in the Tropics, amounting to over 20 W/m2 of additional greenhouse effect. In the summer midlatitudes, the effect of saturating the atmosphere is much weaker, amounting to only 7 W/m2 at 45◦ S. This is because the monthly mean relative humidity above the boundary layer is already on the order of 50% in the summer extratropics,

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so one is only doubling the assumed moisture content there. The winter extratropics shows a somewhat weaker effect, on the order of 5 W/m2. These results underestimate the effect of dry air on midlatitude OLR because the more extreme dry-moist contrast seen in daily data is associated with mobile synoptic systems that average out in the monthly means. Thus, a much drier atmosphere has the potential to be considerably colder than the present climate, whereas a saturated atmosphere has the potential to be considerably warmer. To translate this statement into more quantitative terms, we have carried out a series of GCM experiments in which the water vapor content of the atmosphere has been artificially altered in the radiation calculation alone. This approach allows one to study the radiative impacts of water vapor in isolation, without needing to confront the formidable task of disentangling them from the myriad other complex influences of the hydrological cycle on climate. It is essentially the same technique employed by Hall and Manabe (1999) in their study of water vapor feedback. Our experiments were carried out with the Fast Ocean-Atmosphere Model (FOAM) GCM coupled to a mixed-layer ocean and thermodynamic sea-ice model, employing realistic present-day geography.2 Each simulation started from a control-run modern climate with unaltered water vapor. The evolution of the climate following the imposition of each altered water vapor assumption is shown in Fig. 6.2. When the radiative effect of water vapor is eliminated, the Earth falls into a globally glaciated snowball state after 8 years. To probe the opposite extreme, we forced the radiation to be calculated under the assumption that the entire troposphere is saturated. Note that this does not mean that the specific humidity was fixed at the value the control climate would have had in saturation. Rather, each time the radiation module is called, the saturation specific humidity is computed using the model’s temperature field at that instant of time; this specific-humidity field is then used in the radiation calculation. This technique yields additional warming because the radiative forcing caused by saturating the atmosphere at any given temperature is itself subject to amplification by water vapor feedback, as it should be. In the saturated experiment the tropical temperature rises to an impressive 320 K after 8 years (left panel). It appears to have leveled off at this time, but we do not know if the climate would continue to warm if the integration were carried out for a longer time, since the simulation halted owing to numerical instabilities arising from extreme temperature contrast between the Antarctic glacier and the surrounding warm ocean waters. The tropical warming in the saturated GCM simulation is similar to what one of us predicted earlier on the basis of an idealized two-box model of the Tropics (Pierrehumbert 1999). The saturated case shows the potential for water vapor feedbacks to make the climate response to CO2 much more extreme than found in the extant climate models; there is plenty of dry air around, and if it somehow became saturated, the consequences for climate would be catastrophic. The saturated case provides only an upper bound on how bad things could get, but it is nonetheless an upper bound that should be kept in mind when thinking about climate prediction uncertainties.

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300 280 260 Dry

240 0

2

4

6 8 10 12 14 16 Time (years)

0.1

300 0.08 299 298

0.06

Temperature

297 296

0

5

10

15

0.04 20

Time (years)

Figure 6.2. Results of GCM simulations in which the radiative impact of water vapor has been artificially altered. Left panel: time evolution of equatorial temperature for completely saturated and completely dry atmospheres. Right panel: time evolution of equatorial temperature and global ice cover for a case in which the water vapor used in computing radiation is reduced by a factor of two compared to the standard parameterization.

We also examined the impact of a less-extreme assumption about water vapor, results for which are shown in the right panel of Fig. 6.2. In this case, the water vapor was computed using the GCM’s standard explicit and parameterized physics package, but the resulting specific humidity was multiplied by one-half before being handed to the radiation scheme. In this run, the equatorial temperature drops 4 K, to 296 K. Moreover, the fractional sea-ice cover doubles, to nearly 10%. The summer snow line on land (not shown) also moves considerably equatorward. All in all, the resulting climate rather resembles the Earth’s climate during the Last Glacial Maximum. Removing the atmosphere’s water provokes a Snowball Earth. Saturating the atmosphere provokes an extreme hothouse climate warmer than any encountered in the past several hundred million years. Reducing water to half that produced by standard parameterizations is sufficient to provoke an ice age. Water vapor is a dynamic and fluctuating quantity with very little intrinsic persistence beyond the monthly time scale. It is striking that, despite this fact, our climate doesn’t undergo massive fluctuations in response to spontaneously generated fluctuations in water vapor content. The lack of such fluctuations is a testament to the precision with which water vapor statistics are slaved to the large-scale state of the climate. Water vapor is the atmosphere’s single most important greenhouse gas, and it is correct to say that an accurate prediction of climate change hinges on the ability to accurately predict how the water vapor content of the atmosphere will change. It would be a gross error, however, to conclude that the effect of CO2 is minor in comparison to that of water vapor. The greenhouse effect of CO2 accounts for fully a third of the total, and this is a very considerable number. Eliminating the 50 W/m2 of tropical CO2 greenhouse effect would drop the tropical temperature by about 25 K, once amplified by water vapor feedback. When further amplified by ice-albedo feedback, this would certainly cause the Earth to fall into a snowball state. A warmer, water-rich world

Fractional Sea Ice Cover

Saturated

320

220

Sea Ice Cover

301 Equatorial Temperature (K)

340 Equatorial Temperature (K)

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subjected to much greater solar radiation than the Earth could indeed be in a state where the addition or removal of a few hundred ppm of CO2 would make little difference to the climate, but this state of affairs does not by any stretch of the imagination apply to the Earth. Another important difference between CO2 and water vapor is that the former has a much longer response time because on Earth it is removed only by slow biogeochemical processes as opposed to the rapid condensation and precipitation processes affecting water vapor. Water vapor responds quickly to temperature changes induced by CO2, whereas CO2 would take thousands of years to respond to a change in the hydrological cycle.

6.4. How Saturated Is the Atmosphere? Above the boundary layer, much of the atmosphere is highly unsaturated, with relative humidities below 10% occurring frequently in both the Tropics and the extratropics. The satellite snapshot of midtropospheric humidity on 16 January 1992, shown in Fig. 6.3, serves as a reminder of the prevalence of dry air. The prevalence of dry air can be quantified in terms of the probability distribution of relative humidity. This diagnostic has been extensively studied in the Tropics (Spencer and Braswell 1997; Zhang et al. 2003; Brogniez 2004). The midlatitude distribution has not received as much attention, but there is ample evidence that highly undersaturated air is common there as well (e.g., Soden and Bretherton 1993). Early work suggested a lognormal distribution of relative humidity (Soden and Bretherton 1993) but further study has revealed a greater variety of shapes of the relative humidity histogram, including even bimodality (Zhang et al. 2003; Brogniez 2004). For now, the detailed shape of the probability distribution will not concern us; it is enough to know that most of the free troposphere is significantly undersaturated. This widespread occurrence of unsaturated air is a manifestation of the fact that from a thermodynamic standpoint, the atmosphere is a nonequilibrium system. In a state of thermodynamic equilibrium with the oceanic moisture reservoir, the entire atmosphere would be saturated. The prevalence of subsaturated air, and the sorting of air into moist and dry regions, yield a state of relatively low thermodynamic entropy. Evaporation of liquid water or ice into the subsaturated regions, or diffusion of moisture from regions of high vapor pressure into subsaturated regions of lower vapor pressure, would increase the entropy of the atmosphere (Pauluis and Held 2002). It is essential to understand the processes that determine the degree of subsaturation of the atmosphere, and to come to an understanding of how the net result of these processes might change as climate changes. In thinking about atmospheric water vapor content, one must draw a distinction between the net column water content and the water content in the mid to upper troposphere. The latter accounts for a small fraction of the total water content of the atmosphere because the lower warmer parts of the atmosphere have much higher saturation vapor pressure and are moreover sufficiently

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90 45°N 80 30°N

70

15°N

60

0°

50

15°S

40

30

30°S

20 45°S 10

45°W

30°W

15°W

0°

15°E

30°E

45°E

Figure 6.3. Free tropospheric relative humidity on 16 January 1992, retrieved from the Meteosat water vapor channel, based on retrieval in Brogniez (2004). Cloudy regions where the infrared retrieval could not be done were filled in with 100% relative humidity for the purposes of this figure.

close to the surface water reservoir that they tend to stay relatively saturated. Yet water vapor, or any greenhouse gas, placed near the surface has little effect on OLR because the low-level air temperature is not much lower than the surface temperature. To get a significant greenhouse effect, one must increase the infrared opacity of a portion of the atmosphere that is significantly colder than the surface. Because the radiative effect of water vapor is logarithmic in its concentration, small quantities of water vapor can accomplish this task aloft. This leads to the concept of free tropospheric humidity (FTH) or upper tropospheric humidity (UTH), which may be loosely defined as the water content of the portion of the atmosphere where water vapor has a considerable effect on the radiation budget. Diversion of a tiny proportion of the atmosphere’s net water vapor content would be sufficient to saturate the mid to upper troposphere and radically warm the climate. For example, consider a 50 mb thick layer of saturated air near the surface of the tropical ocean, having a temperature of 295 K. Less than 3% of the water content of this layer would suffice to completely saturate a layer of equal mass having a temperature of 250 K, such as would be encountered in the tropical mid troposphere, or at lower altitudes in the extratropics. Clearly, the magnitude of the boundary layer water vapor reservoir is not the limiting factor in determining the free tropospheric humidity.

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We pause now to dismiss two fallacies, which unfortunately have not yet completely disappeared from discussions of the subject at hand. The first is the evaporation fallacy, typified by statements like: “In a warmer world there will be more evaporation, which will increase the supply of moisture to the atmosphere and increase its humidity.” This is wrong on at least two counts. It is not inevitable that warmth increases evaporation, since evaporation is determined by wind speed and boundarylayer relative humidity, as well as the boundary-layer saturation vapor pressure (which does unquestionably go up with temperature). It is entirely possible for a warmer climate to have less evaporation. An even more serious objection to the evaporation fallacy is that it hopelessly confuses fluxes and reservoirs. Evaporation gives the rate at which moisture fluxes through the atmosphere, which is distinct from, and even has different units from, the amount of water left behind in the atmosphere (or more specifically the free troposphere). In this regard, the evaporation fallacy is as absurd as saying that, in comparing the water content of two vessels, the one with the higher rate of filling will contain more water, regardless of whether one of them is a sieve while the other is a watertight bucket. The second fallacy is the saturation fallacy, which states flatly that a warmer atmosphere will tend to become moister because saturation vapor pressure increases with temperature. That is like saying that one always expects to find more water in a bigger bucket than a smaller bucket, no matter how leaky either might be. ClausiusClapeyron does indeed govern the relation between temperature and water content, but in a far subtler way that will begin to become clear shortly. So what really determines the water vapor content of the free troposphere? It is easiest to think about this problem in a Lagrangian sense, tracking the water content of an air parcel as it wanders about the atmosphere. The fluctuating water content of the parcel results from a balance between the rate at which water is added to the parcel against the rate at which water is removed. Water vapor is removed either by condensation or by diffusion into a neighboring drier air parcel. Let us suppose for the moment that diffusivity is so low that the latter mechanism is unimportant. In that case, water vapor is removed when the air parcel wanders into a region where the local saturation specific humidity is lower than the current specific humidity of the parcel, at which time the specific humidity is reset to the lower local saturation value and the balance is rained out. The net result is that the specific humidity of an initially saturated parcel after time τ is equal to the minimum q s encountered along the trajectory during that time. By definition, this is a nonincreasing function of τ , though there will be long periods of time over which the minimum remains constant between those times   at which new minima are encountered. Define q min(to, τ ) ≡ min q s (t); to < t < to + τ . Then the rate at which q min decreases, which is different for each trajectory, determines the drying rate on that trajectory. The drying is not a continual process, but occurs in fits and starts as new minima are encountered. If no new water is added to the parcel, new condensation events eventually become very infrequent, or cease altogether if the global minimum is encountered. The drying rate determined by this process must be balanced

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against whatever processes episodically add new water to the air parcel, determining the statistically fluctuating water content of the parcel. The importance of nonlocal control of tropospheric humidity was clearly revealed in the case study described in Kelly et al. (1991). Drawing on this insight, Yang and Pierrehumbert (1994) examined the general problem of midlatitude dry air production from a Lagrangian standpoint, and pointed out the importance of the q min statistic. The most prominent dry air pool in the atmosphere is found in the subtropics, and over the next several years a number of investigators developed an interest in largescale control of subtropical humidity, employing Lagrangian techniques (Emanuel and Pierrehumbert 1996; Salathé and Hartmann 1997; Soden 1998; Pierrehumbert 1998; Pierrehumbert and Roca 1998; Sherwood and Dessler 2000; Galewsky et al. 2005). It was an idea whose time had come, and these studies provided a reassuring counterpoint to the then-emerging suspicion that everything was controlled by parameterized microphysics, and everything was inscrutable. In recent years, the term time of last saturation model has gained currency as a name for the general approach. We prefer the broader term advection-condensation model. If one knows the specific humidity of the air parcel at its destination, then one can determine the time of last saturation by following the back trajectory until that specific humidity becomes saturated. However, if the aim is to predict the specific humidity at the destination, determining the time of last saturation requires some assumption about the processes which, at some time in the past, added moisture to the air parcel. Figure 6.4 shows temperature and pressure following a typical midlatitude trajectory. The temperature and pressure are nearly perfectly correlated, so minima in q s are very nearly coincident with minima in T or p. The trajectory executes a wave-like motion under the influence of midlatitude synoptic eddies, swinging between the subtropical surface where the temperature reaches 290 K and the high-latitude tropopause where the temperature drops to 230 K. If there were no moisture sources in the free troposphere, then we would know that when the trajectory reaches the 800 mb level on day 5, its specific humidity would be no greater than 1.4 · 10−4, the value corresponding to the minimum at day 0 (computed with 232 K and 500 mb). If the parcel becomes saturated when it encounters the ground at day 7, then it remains at saturation through day 13 since the temperature and saturation specific humidity decrease monotonically through that part of the trajectory. However, when the parcel lands on the 800 mb surface on day 18, its specific humidity would be 6.6 · 10−4, corresponding to the minimum at day 16. Based on the local saturation specific humidity on day 18, the relative humidity would be 25%. This trajectory illustrates an important principle: in order to know the specific humidity at a given point on the trajectory, one only needs to know the history going back to the time of last saturation. Information about earlier times is erased whenever saturation occurs. The trajectory also illustrates the fact that midlatitude trajectories typically dip into the subtropical boundary layer every 15 days or so, where they have an opportunity to pick

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Relative Humidity of the Atmosphere | 155 NCEP Trajectory, January 2003; Midlatitude Mid-Pacific Launch 300 Pressure (mb)

1000

280 800 260

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Temperature (K)

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10

20

400 30

Time (days) Figure 6.4. Temperature and pressure along a 3-D air parcel trajectory launched at 500 mb in the midlatitude Pacific. The trajectory was computed on the basis of 4 times daily NCEP horizontal and vertical velocity fields. The gray boxes indicate encounters with the boundary layer, where moistening is presumed to occur.

up new moisture. This is probably the main moisture source for the midlatitude free troposphere. Most midlatitude trajectories have a character qualitatively like that shown in Fig. 6.4, but tropical trajectories are quite different. Isentropes are flat there, and there is less baroclinic eddy activity. Trajectories are dominated by organized rising motion in convective regions, ejection due to steady outflow or tropical transient eddies, and slow descent in the large-scale organized subsidence regions (see Pierrehumbert 1998; Pierrehumbert and Roca 1998). The tropical model of Minschwaner and Dessler (2004) provides perhaps the simplest realization of this class of models, in that the trajectories consist simply of ejection of saturated air from convective regions, followed by monotone subsidence at a rate determined by infrared radiative cooling. This idealized trajectory model neglects the lateral mixing incorporated in Pierrehumbert and Roca (1998), but nonetheless yields a number of insights concerning the nature of water vapor feedback. Generally speaking, one can expect the statistics characterizing trajectories to differ considerably between the Tropics and midlatitudes. If we consider an ensemble of trajectories launched from a given location, the behavior of the minimum q s statistic can be characterized by the probability density Q min(q min|q o, τ, t), which gives the probability that the minimum q s encountered between time t and t + τ is near q min, given that q s = q o at the place where the particle is located at time t. Obviously Q min = 0 for q min > q o. If the process is statistically

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stationary, Q min will be independent of t. If one is trying to understand the water vapor at a specified point, it is most convenient to deal with back trajectories, corresponding to negative τ . We are more interested in where the air arriving at the target point came from than in where it is going. If the trajectory process is statistically reversible, all statistics of back trajectories have the same behavior as the corresponding statistics of forward trajectories, and in particular Q min(q min|q o, τ, t) = Q min(q min|q o, −τ, t). Somewhat counter-intuitively, Brownian motion is statistically reversible in this sense. It is widely assumed, and probably true, that the atmospheric trajectory problem is statistically reversible, though we will not explicitly make use of the assumption in our calculations below. As an example of the use of back trajectory statistics, let’s suppose that we wish to know the probability distribution of specific humidity q in some patch of the 500 mb surface at a given time t, and are willing to assume that the entire atmosphere was saturated at time t − τ , and moreover that there were no moisture sources in the intervening time. To solve this problem, we use Q min for the ensemble of trajectories that arrive at the patch at the specified time. For simplicity, we will assume that q o is constant within the patch, though this is an assumption that can be easily relaxed. Because we assumed that all parcels are saturated at time t − τ (though each has a different q s , appropriate to its location at that time), the specific humidity each parcel winds up with by time t is simply the minimum q s encountered along the trajectory. Hence, the probability density function (PDF) of q in the patch is Q min(q |q o, −τ, t), which is concentrated on progressively drier values as τ is made larger. Note that this distribution is entirely distinct from the distribution of initial saturations at time t − τ . These could all be in the tropical boundary layer and have very high values, and the final humidity at time t would still become small. A passive tracer, with no sources or sinks, would retain its initial value, so that its PDF at later times is determined solely by its initial PDF, with no knowledge of the nature of the intervening path required. In contrast, the statistics of moisture are sensitive to the entire history of the path taken. The sensitivity to probabilities that depend on entire paths is one of the chief mathematical novelties of the water vapor problem, and the source of most of the theoretical challenges. The probability density Q min characterizes the drying process, and one needs a corresponding probabilistic description of the moisture source in order to complete a theory of the atmospheric humidity. A moisture source such as evaporation of precipitation falling through dry air could add just a bit of moisture to an air parcel without saturating it. However, let’s idealize the moisture source as a series of saturation events, which occur randomly in time, with the chance that a saturation event will occur after waiting for a time τ described by a probability distribution Psat(τ ). Then, the PDF of specific humidity is given by the convolution

∞

Q(q |q o, t) = 0

Psat(τ )Q min(q |q o, −τ, t)dτ,

[6.4]

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where q o is the saturation specific humidity at the point under consideration. If the trajectory process is statistically stationary, Q will be independent of t. As an example of the application of equation (6.4) in the simplest possible context, we consider a uniform subsidence model similar to that in Minschwaner and Dessler (2004). In this case, since the trajectories always go downward to regions of larger q s , the minimum q s in the back trajectory always occurs at the time of saturation. To be definite, let’s assume that the vertical coordinate is pressure p, that trajectories subside at a constant rate ω, and that q s = q s ( p), i.e., that the saturation field is horizontally homogeneous, as is approximately the case in the Tropics. Then Q min is a δ-function concentrated on the value of q s at the pressure the parcel was at when it was resaturated. If po is the pressure at the target point (i.e., q o = q ( po)) then the pressure at the saturation point is po − ωτ and hence Q min(q |q o, −τ, t) =   δ q − q s ( po − ωτ ) . Substituting this into equation (6.4) and using the relation    −1 δ g (τ ) dτ = g (τ ) δ(g )dg , we find Q(q |q o, t) =

ω

1 

dq s  d p  p −ωτ ∗ o

Ps at(τ ∗),

[6.5]

where τ ∗(q ) is the solution to q s ( po − ωτ ∗) = q . In this case, the specific humidity PDF is determined by the saturation statistics and the vertical structure of q s .3 In the general case, the moisture dynamics is characterized by the two PDFs Ps at and Q min. In order to fully understand water vapor feedbacks, we need to understand how these two PDFs change as climate changes. This is a tall order. Some further examples of the trajectory statistics in action will be given in section 6.5 for idealized trajectory models and section 6.6 for realistic trajectories. There are three difficulties with the trajectory approach, two of them technical and one of a more fundamental nature. The first difficulty is that there may be mixing of moisture amongst air parcels arising from small-scale turbulent motions. Because large-scale resolved strain causes exponential amplification of gradients (Yang and Pierrehumbert 1994; Pierrehumbert 1998), even a weak effective diffusivity would eventually cause significant mixing. The mixing is likely to be dominated by vertical rather than horizontal mixing processes, for the reasons discussed in Haynes and Anglade (1997). Incorporation of mixing greatly complicates the calculation because the moisture evolution on one trajectory becomes dependent on the moisture evolution of all other trajectories that pass in its vicinity during the past. Explicit calculation then calls for either Eulerian methods (at the expense of the need to confront the difficult problem of unwanted numerical diffusion) or simultaneously integrating the problem on a sufficiently large swarm of trajectories. Mixing between moist and dry air parcels is important because it both reduces the frequency of very dry air and because the dilution increases the subsaturation of the moist air, and delays condensation. It is of utmost importance to determine how much small-scale mixing enters into the tropospheric

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moisture problem in the real atmosphere, and comparison of observations with results of the non-diffusive trajectory calculation can provide a means of doing so. There has been considerable attention to the diagnosis of small-scale mixing in the stratosphere (see Haynes and Anglade [1997]; Legras et al. [2003] and references therein), but the analogous question for tropospheric water vapor is unresolved. The second difficulty with the trajectory method is similar to the mixing problem: precipitation generated by large-scale condensation along a trajectory may evaporate as it falls through dry air, adding moisture to trajectories it encounters. This, too, couples trajectories. It can be regarded as just another form of vertical mixing. The third difficulty comes at the cloud scale, where we confront a much harder problem, particularly in tropical convective regions. Models and analyses produce a large-scale rising motion in such regions, diagnostically associated with the moisture convergence required to feed the convective precipitation. This upward motion, which lifts all trajectories, must be regarded as wholly fictitious. In reality, most of the air in the convective region is subsiding, and the upward mass flux is concentrated in cumulus towers covering only a small fraction of the area of the convective region. The resolved large-scale vertical velocity correctly represents the net upward mass flux, but is not typical of the velocity of individual fluid parcels in the convective region. The net result is that the detrainment obtained from large-scale trajectory simulations is always saturated, whereas the cloud-scale motions offer ample opportunities for the detrained air to be substantially undersaturated. The mixing of moisture between moist and dry air at cloud and sub-cloud scales engages microphysical issues of the sort discussed by Tompkins and Emanuel (2000), and ultimately determines the degree of subsaturation. The success of models with saturated detrainment at reproducing water vapor observed in nonconvective regions (e.g., Minschwaner and Dessler 2004; Pierrehumbert and Roca 1998) suggests that the subsaturation must not be too extreme in the typical case, but the whole matter requires further study.

6.5. A Few Illustrative Models It is instructive to consider some simple one-dimensional models of the connection between mixing, transport, and drying. These models are offered in order to highlight some of the basic issues in cases that can be solved completely. We do not pretend that any of these models yield good representations of the atmosphere’s actual water vapor distribution. Both models that we consider in this section are formulated in terms of an abstract spatial coordinate y. The only thing we need to know about y is the saturation specific humidity q s as a function of y. We may think of y as the northsouth distance following an isentropic surface, which yields a decreasing q s because the surface becomes generally higher (hence colder) as the pole is approached. The coordinate could represent north-south distance at a fixed midtropospheric pressure

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level, as in typical one-layer energy balance models. With a suitable increase in mixing rates, y could equally well represent altitude, with latitude and longitude held fixed; in this case the mixing parameterization is to be thought of as a surrogate for convection rather than the larger scale, more ponderous, large-scale advection. If temperature decreases linearly with y, then the Clausius-Clapeyron relation implies that q s decreases approximately exponentially with y. A decrease of pressure with y somewhat offsets the exponential decay, but not so much as to prevent us from using q s (y) = exp(−y) as a useful conceptual model of the atmosphere’s saturation specific humidity. 6.5.1. The Diffusion-Condensation Model We wish to study the interplay of transport and condensation. Diffusion is the simplest model of transport, and is moreover employed as a surrogate for eddy transport of water vapor in many idealized climate models (e.g., Weaver et al. 2001; Petoukhov et al. 2000). Hence, we first examine a model in which moisture is represented by the mean specific humidity q (y, t), which is stipulated to satisfy a diffusion equation ∂t q − D∂ yy q = −S(q , q s ).

[6.6]

In this equation, the sink S(q ) instantaneously resets q to q s whenever diffusion causes it to exceed q s (y, t). The instantaneous sink may be taken as a limiting form of the function    1 q − q (y, t) for q > q s s S(q , q s ) = τ [6.7] 0 for q ≤ q s as τ → 0. The mathematical novelty that a sink of this form adds to the problem is that it makes the problem nonlinear. Condensation can be thought of as a particularly simple form of unary chemical reaction, and in fact many of the issues we encounter in the condensation problem are generic to a broad class of nonlinear chemical reactions. In general, q s could be a function of both y and t, but henceforth we shall assume q s = q s (y). In the instantaneous removal limit (τ → 0), q (y) = q s (y) is a steady solution to equation (6.6) in any region where d 2q s /dy 2 > 0. If this condition is satisfied at time t, then S = 0 at that time but the diffusion term makes ∂t q > 0, so that the moisture sink will be activated at the next instant of time, and reset the moisture to q s , keeping the moisture in the region fixed at q s (y).4 The moisture loss is balanced by moisture flux from regions of larger q . Next, we consider a simple initial-value problem. Suppose that y extends from 0 to ∞, and impose the no-flux boundary condition ∂ y q = 0 at y = 0 so that there are no sources of new moisture. In this case, the condensation caused by diffusion of moisture from regions of large q s into regions of small q s will deplete the moisture content. We shall assume that d 2q s /dy 2 > 0 throughout the domain, and that q s attains its maximum value at y = 0. Given the curvature assumption, this implies that q s is

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monotonically decreasing. If the atmosphere is initially saturated, then there is some Y (t) such that q (y) = q s (y) for y > Y (t); this is the region in which condensation is taking place. For y < Y (t) the air is undersaturated, and satisfies the diffusion equation with no sources or sinks. As moisture is drawn out of the system by condensation, Y (t) increases. Under the assumptions on q s , this quantity approaches a minimum value C from above as y → ∞. As t → ∞, we have Y (t) → ∞ and q → C . The question of the long-time asymptotic behavior of Y is thus well posed, and in fact has a fairly simple and general answer. Since the advection-condensation equation is invariant under the transformation q → q − C, q s → q s − C , we can assume that C = 0, without any loss of generality. Examination of some numerical solutions suggests that q is approximately parabolic in the non-condensing region after a sufficiently long time has passed. Let us then look for a solution with q = a − by 2 for y < Y (t). What we shall present here is not a complete and rigorous asymptotic analysis, for we shall not derive the conditions under which the assumed form of q in the non-condensing region is valid. The matching conditions at Y and the requirement that q satisfy the diffusion equation in the subsaturated region imply a − bY 2 = q s (Y )

[6.8]

−2bY = q s(Y ) da = −2b D. dt Given the assumed form of q , we can only exactly satisfy the diffusion equation at one point, and in writing the third of these relations we have chosen to do so at y = 0. Taking the time derivative of the first equation and substituting from the other two results in the following differential equation for Y : 1 dY 1  dY dY −2b D + q s(Y ) + q s (Y )Y = q s(Y ) 2 dt 2 dt dt or equivalently   q s(Y ) dY Y · 1−Y  = 2D. q s (Y ) dt

[6.9]

[6.10]

The quantity q s/q s has dimensions of inverse length. For q s = exp(−y 2/σ 2) it is −2Y/σ 2 at large Y , for q s = exp(−y/L ) it has the constant value −1/L , and for q s = A/y α it is −(α + 1)/Y . Suppose that q s/q s = AY n at large Y . Then, if n ≤ −1 the second term multiplying the derivative on the left-hand side of equation (6.10) is at most order unity, and the equation can be integrated to yield Y (t) ∼ (Dt)1/2. On the other hand, if n > −1 the second term dominates and we find Y ∼ (Dt)1/(n+3). It is interesting that rapid decay of q s can cause the thickness of the subsaturated region to grow more slowly than the diffusive length scale (Dt)1/2.

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Relative Humidity of the Atmosphere | 161 1

2

0.7 0.6 Total Water

1

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3 t

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0.9

Y

1

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q

3

Y

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0.1 0

0 0.5 1 1.5 2 2.5 3 3.5 4 y

Figure 6.5. Numerical results for the freely decaying diffusion-condensation model with a no-flux barrier at y = 0. Left panel: time evolution of the point Y (t) bounding the subsaturated region, and of total moisture in the system. The short-dashed line gives the fit to the asymptotic result Y ∼ t 1/3. Right panel: the profile of specific humidity at the times indicated on the curves.

Once Y (t) is known, the humidity decay curve is obtained using   1 1 q (0) = a = q s (Y ) − Y q s(Y ) = 1 − Y q s(Y )/q s (Y ) q s (Y ). 2 2

[6.11]

At large Y , q s/q s will generally have the same scaling as q s/q s; by assumption, it is always negative. If q s decays exponentially, then q s/q s is a negative constant and the second term in the factor multiplying q s dominates; in this case q (0) ∼ Y q s (Y ), and hence decays like t 1/3 exp(−c t 1/3) for some constant c . If q s decays algebraically, then both terms in the factor are order unity, so q (0) ∼ q s (Y ) and hence decays like t −α/2, assuming q s decays like y −α at large y. In either case, it can easily be shown by integrating q (y) that the total water vapor remaining in the atmosphere decays like Y (t)q (0, t) provided q s has a finite integral. Thus, moisture is drawn out of the system by diffusion from subsaturated regions neighboring the boundary (the ground, in many cases) into regions of saturated air with lower specific humidity at larger y, where it can condense. The rate of decay is determined by the rate at which q s decays with increasing y. Even for exponentially decaying q s , the moisture decay is slower than exponential, owing to the time required for moisture to diffuse into the ever-retreating condensing region. Figure 6.5 shows numerical solutions to the problem with q s = exp(−y), together with the theoretical result for Y (t). The agreement between simulation and theory for this quantity is excellent. As the next stage in our exploration of the diffusion-condensation equation (6.6), we exhibit the steady-state response to a moisture source. We introduce the moisture source by holding q = rh · q s (0) at y = 0 (with rh < 1). If we assume d 2q s /d y 2 > 0 as before, the equilibrium moisture distribution has the simple form  q s (y) for y > ys   [6.12] q (y) = q (y )−rh·q (0) y s s s rh · q (0) + for y < y , s s ys

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where ys is chosen to make the flux continuous, i.e.,   q s (ys ) − rh · q s (0) dq s (ys ) = . dy ys

[6.13]

All condensation is at y > ys , and ys → 0 as rh → 1, in which case the whole atmosphere becomes saturated. Further, although the undersaturation for y < ys relies on mixing, q (y) is independent of the magnitude of D. However, the rate of moisture flux into the condensation region, and hence the precipitation rate, is proportional to D. This solution underscores the point we made earlier with regard to the evaporation fallacy: the factors governing the atmospheric relative humidity distribution are quite distinct from those governing the rate at which water fluxes through the system. Note also that the moisture sink produces air that has a lower specific humidity than the source, even in regions where no condensation is taking place. In effect, this is due to dilution of moist air with drier air from larger y. Still, the diffusive model cannot produce an undersaturated layer unless the source air at y = 0 is undersaturated. Another interesting configuration is the cold trap, in which q s (y) = q c < 1 in a small region near y = 0, while q s (y) = 1 elsewhere. A local minimum of this sort could be taken as a crude representation of the minimum occurring at the tropopause, with y measuring altitude. However, the cold trap is also relevant to the case in which y represents horizontal distance on a fairly level surface, such as the tropical tropopause; in this case, the minimum corresponds to the cold conditions occurring over the high tropopause region above the west Pacific warm pool, and we seek to understand the global drying effect of this region. In the freely decaying case subject to initial condition q (y) = 1 in an unbounded domain, the cold trap creates a zone around itself in which q approaches q c. The width of this dry zone increases in proportion to (Dt)1/2. Alternately, we can seek an equilibrium solution by imposing q = 1 at y = −1 and a no-flux condition ∂ y q = 0 at y = 1. The steady solution in this case is simply q = 1 − (1 − q c)(y + 1) for −1 ≤ y ≤ 0 and q = q c for y ≥ 0. In accord with intuition, moisture is reset to the cold trap specific humidity when air passes through it. We can also impose a moisture source at the right-hand boundary by replacing the no-flux condition by q = q 1 at y = 1. In this case the solution is  1 − (1 − q )(y + 1) c q= q c + (q 1 − q c)y

for −1 ≤ y ≤ 0 for 0 ≤ y ≤ 1

[6.14]

provided that q c ≤ 12 (1 + q 1). If this condition is met, the solution takes the form of a bent stick, with the cold trap depressing the value at the center of the domain. However, if q c > 12 (1 + q 1), diffusive dilution of the moist air reduces the humidity at the midpoint of the domain to such an extent that no condensation occurs; in this case, the solution is simply a straight line linking the limiting values at the two boundaries, and is a solution of the conventional diffusion equation. In section 6.5.3 we will treat the

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same configuration using a stochastic model of water vapor, and find some intriguing differences in the behavior. There can be no sustained condensation in a region where d 2q s /d y 2 < 0. In such a region, if q = q s initially, then diffusion will immediately reduce q to below its saturation value everywhere, halting condensation. For example, suppose q s = exp(−y 2) in an infinite domain, and that q = q s initially. After the initial instant, condensation halts for |y| < 1, though it continues for |y| > 1 where the curvature is positive. The summary behavior of the one-dimensional diffusion-condensation problem is quite simple. It creates saturated regions embedded in regions where q s is positive and q s is small. These drain moisture out of the surrounding air, creating subsaturated regions in the vicinity. Apart from the possible inadequacies of diffusion as a representation of the mixing, the main shortcoming of the diffusion-condensation model is that it represents the moisture field in terms of a single concentration q at each y. In reality, a small box drawn about y will contain an intermingling of moist and dry air. The diffusion model is incapable of predicting a probability distribution function for moisture; worse, the neglect of fluctuations fundamentally misrepresents the drying process itself, owing to the nonlinear nature of condensation. The next class of models we shall study rectifies this shortcoming of the diffusion-condensation model.

6.5.2. Stochastic Drying: Initial-Value Problem As a counterpoint to the diffusion-condensation problem we pose a simplified random walk version of the stochastic drying process introduced in section 6.4. Suppose that an ensemble of particles execute independent random walks in an unbounded onedimensional domain with coordinate y. Particle j is located at point y j (t), and is tagged with a specific humidity q j , which can change with time if condensation occurs. The saturation specific humidity field q s (y) is assumed to be monotonically decreasing with y. The particles are initially saturated (i.e., q j (0) = q s (y j (0))), but whenever they find themselves at a place where q j > q s (y j ), then q j is instantaneously reset to the local q s . Under these conditions, what is the probability distribution of specific humidity for the particles located at point y at time t? This problem is readily solved in terms of the maximum-excursion statistics for Brownian motion (i.e., a random walk with velocity δ-correlated in time). We make use of the fact that the statistics for forward trajectories are identical to those for backward trajectories, and pose the question as follows: for a particle that is located at yb at time t = 0, what is the probability that it was located at ya at time t = −τ ? What is the probability that the maximum y visited in the time interval [−τ, 0] was ymax? Since q s is monotonically decreasing, the latter probability gives us the probability of q min. If we define the random walk as satisfying d y/dt = v(t) with
v(t)v(t ) = 2Dδ(t − t ), then the probability that the parcel landing at yb came from ya is given by the familiar Gaussian form, just as if we were running the trajectory forward rather than backward

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(Karatzas and Shreve 1991; Lesigne 2005): p(yb|ya, τ ) = p(yb − ya, τ ) = √

1 4π Dτ

e −(yb−ya) /(4Dτ ). 2

[6.15]

This probability density satisfies a diffusion equation with diffusivity D. It is a remarkable fact that the probability of the maximum excursion along the path, which is a probability distribution on the space of paths, can be determined in terms of the probability of the endpoint given in equation (6.15). This is a consequence of the reflection principle, discussed in Karatzas and Shreve (1991) and Lesigne (2005), and holds for a broad class of random walk processes. For the unbounded Brownian motion, the maximum excursion probability is found to be simply  2 p(y − y , τ ) for y ≥ y max b max b Pmax(ymax|yb, τ ) = [6.16] 0 otherwise, where p is the probability defined in equation (6.15). We know that the maximum position visited in the past can be no less than the position at which the particle finds itself at t = 0, which is why the probability vanishes for ymax < yb . For ymax > yb , the reflection principle implies that the probability that ymax is the maximum position visited is twice the probability that the particle originated at ymax . This seems too good to be true, but the mathematics is incontrovertible. Somewhat surprisingly, the most probable ymax is the terminal position yb . In other words, the most probable situation is that the value of y where the particle is found at t = 0 is the greatest it has visited over the past time τ . Going backwards in time, most trajectories wander to smaller values of y and never come back to the starting position. This is a consequence of the domain being unbounded below, which allows plenty of room in the domain y < yb for the trajectories to get lost in. Note, however, that the probability of the most probable excursion decays like (Dτ )−1/2 as τ → ∞, so that trajectories with ymax = yb become rather improbable. For back trajectories of length τ , values of ymax as large as yb + (Dτ )1/2 are nearly as probable as the most probable one. Back trajectories with ymax = yb are saturated when they reach the target position yb , since the particle has never visited a place with lower q s than it has at its destination. Because the probability of such trajectories decays very slowly with τ , the mean moisture content of an ensemble of trajectories landing at yb also decays very slowly. To make these ideas more explicit, let’s consider the case q s = exp(−y). In this case, a trajectory with maximum excursion ymax will produce a specific humidity q = exp(−ymax) when it lands   at yb. Solving for ymax yields ymax = − ln(q ). In addition, yb = − ln q s (yb) , whence   ymax − yb = − ln q /q s (yb) . This can be substituted for the argument of p in equation (6.16). From this, we learn that the PDF of − ln(q /q s ) at a given point yb is in fact the PDF of ymax − yb for back trajectories landing at that point. The PDF of humidity may be described as a “truncated lognormal” in the sense that − ln(q ) is normally distributed for q ≤ q s , but the probability is identically zero for q > q s .

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Relative Humidity of the Atmosphere | 165

Probability

1.6

τ = .1

1.2 0.8

τ = .5 τ=1

0.4

τ=2

0 5

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2 τ=1

1

τ = .5 0 0

0.2

0.3

0.6

0.8

1

q/qs   Figure 6.6. Probability density functions for − ln q /q s (yb) (top panel) and for q /q s (bottom panel) for particles executing an unbounded random walk in a saturation field q s (y) = exp(−y). Results are shown at t = 0, under the assumption that the entire domain was saturated at the time t = −τ indicated on each curve. Results are shown for τ = 0.1, 0.5, 1, and   2. For the exponential saturation profile used in this calculation, the PDF of − ln q /q s (yb) is identical to the PDF of the maximum excursion of back trajectories, i.e., ymax − yb.

If we want the PDF of q itself, we must transform the density using −dln(q /q s ) = −q dq . Then, letting Q(q , yb , τ ) be the PDF of q at point yb , at a time τ after an everywhere-saturated state, we have  2q −1 p  − ln q /q (y ), τ  for q ≤ q (y ) s b s b Q(q , yb, τ ) = [6.17] 0 otherwise. −1

The behavior of Pmax and Q are shown in Fig. 6.6. Note that while the probability of ymax − yb (or equivalently of − ln(q /q s )) has its peak at zero, corresponding to saturated air, the peak probability of q shifts toward dry air as time progresses. This is a consequence of the q −1 factor involved in transforming the probability density from a density in ln(q ) to a density in q . The population shifts toward unsaturated air as time progresses, though there is a long moist tail owing to the persistent high probability of trajectories that do not visit cold places. The stochastic model generates dry air in the unbounded domain, whereas the diffusion-condensation model does not. The difference in the two models lies in the fact that the latter represents moisture by a single mean value at any y, and therefore cannot reflect the fact that amongst the ensemble of

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166 | Pierrehumbert, Brogniez, Roca 1.2

τ = .22

Probability

1 0.8 0.6

τ = 1.76

τ = .44 τ = .88

0.4 τ = 3.52

0.2 0

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τ = 3.52

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τ = 1.76

τ = .88

3

τ = .44

2

τ = .22

1 0 0

0.2

0.4

0.6 q/q

0.8

1

s

Figure 6.7. As for Fig. 6.6, but for the case with a reflecting barrier at y = 0. In this case, the results depend on yb. These PDFs were computed at yb = 0.5. Results are shown for τ = 0.22, 0.44, 0.88, 1.76, and 3.52. As before, the top panel can be regarded as the PDF of ymax − yb.

particles making up such an average, some have visited very cold places in the past, and therefore have become very dry. The mean moisture for the ensemble of particles landing at yb is obtained by carrying out the integral

qs

qs    
q  = q Q(q , yb, τ )dq = 2 p − ln q /q s (yb) , τ dq . [6.18] 0

0

At large τ , p becomes nearly constant outside of an infinitesimal interval near q = 0, which contributes nothing to the integral in the limit of τ → ∞. Since p ∼ (4π Dτ )−1/2 in this limit, we conclude that
q  ∼ 2q s (yb)/(4π Dτ )1/2 at large times. The humidity in the unbounded random walk model exhibits a slow algebraic decay owing to the high probability of back trajectories wandering off to regions with large q s and never returning. The fact that the moisture decays at all nonetheless stands in stark contrast to the diffusion-condensation model in an unbounded domain, for which q = q s (y) is an exact steady state, and the air remains saturated for all times. The unbounded case is instructive, but it is very unlike the real atmosphere, because in the real atmosphere the temperature (and hence saturation specific humidity) is strictly bounded above by the maximum values prevailing at the tropical surface. Real air parcels do not have the liberty of wandering off into arbitrarily warm regions. The random walk model can be made more realistic by imposing a reflecting barrier at y = 0,

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Relative Humidity of the Atmosphere | 167 0.7 Random Walk with Barrier Diffusion Condensation Fit, q = A exp(–B (Dt)1/2)

0.6 0.5 0.4 q

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0.3 0.2 0.1 0 0

1

3

2

4

5

Dt

Figure 6.8. Decay of ensemble mean specific humidity at y = 0.5 for the bounded random walk with a barrier at y = 0. The thin black curve gives the fit to a stretched exponential decay of the form √ A exp(−B Dt). The upper thick, solid curve gives the moisture decay for the diffusion-condensation model computed in the same domain, and with the same diffusivity. The saturation specific humidity profile is q s (y) = exp(−y).

which causes the saturation specific humidity to be bounded above. The maximum excursion statistics for a bounded random walk of this type can also be obtained by application of the reflecting principle, but the answer comes in the form of an infinite sum of shifted Gaussians, which is rather tedious to work with. The results we present here are based instead on Monte Carlo simulation, with ensembles of 10,000 particles or more. The PDF does not sense the presence of the barrier until such time that (Dτ )1/2 becomes comparable to yb and particles have had enough time to frequently reach the barrier. For longer times, the chief effect of the barrier is to shift the most probable maximum excursion to values greater than yb ,which moreover increase without bound as time progresses. It can be shown that the maximum excursion scales with (Dτ )1/2 at large τ . The easiest way to see this is to note that the trajectories with a reflecting barrier at y = 0 are identical to those in an unbounded domain, transformed by reflecting the negative-y part of the trajectories about y = 0. The only trajectories where ymax remains small are those that stay near the barrier, and these become improbable relative to farther-wandering trajectories as time goes on. The expected behavior can be clearly seen in the upper panel of Fig. 6.7, where we show the time evolution of the PDF of ymax − yb computed at yb = 0.5. Recall that this PDF is identical to the PDF of −(ln q /q s ) if q s (y) = exp(−y). Thus, with a barrier the most probable value of − ln(q) shifts towards drier values as τ increases. In the lower panel of the figure, we show the PDF of q itself. As in the no-barrier case, the peak shifts

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168 | Pierrehumbert, Brogniez, Roca 1 Time = 1.44 0.8

qs(y) 0.6 DiffusionCondensation

q

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0.4 Ensemble Mean q Random Walk Model 0.2

0

0

1

2 y

3

4

Figure 6.9. Profile of ensemble mean q at time Dτ = 1.44 for the random walk model with a barrier at y = 0. The corresponding profile for the diffusion-condensation model is shown for comparison.

toward dry values as τ increases. However, the pronounced moist tail disappears after a moderate time has passed and nearly-saturated air becomes extremely rare. The difference with the no-barrier case shows up even more strongly in the decay of ensemble mean humidity, shown in Fig. 6.8. Instead of the slow algebraic decay found produced by the unbounded random walk, the bounded case yields a rapid √ decay of the form A exp(−B Dt), in accordance with the fact that the most probable maximum excursion increases with time. In Fig. 6.8 we also compare the stochastic result with the moisture decay yielded by solving the diffusion-condensation equation subject to a no-flux boundary condition at y = 0. The diffusivity was chosen so as to correctly reproduce the rate of spread of a cloud of particles in the random walk case. We see that the stochastic process dries air much faster than the corresponding diffusion-condensation process. The roots of this difference lie in the nonlinearity of condensation. To shed further light on the comparison between diffusion-condensation and the random walk model, we show the profile of the ensemble mean moisture at a fixed time in Fig. 6.9, together with the corresponding moisture profile from the diffusioncondensation calculation. At every y, the stochastic process yields drier air than the diffusion-condensation process. This happens because some of the particles in each ensemble are significantly moister than the mean, and can therefore lose water by condensation upon being slightly displaced. We can also see an important distinction already familiar from the unbounded case: the stochastic process generates subsaturated air in situ even in a region with d 2q s /d y 2 > 0, whereas such regions remain saturated in the diffusion-condensation model until such time as dry air invades from smaller values

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Relative Humidity of the Atmosphere | 169

of y. The stochastic model thus lacks the sharp moving front separating condensing from noncondensing regions. To those familiar with the derivation of the diffusion equation in terms of random walk processes, it may come as some surprise that the random walk model of humidity has different behavior from the diffusion-condensation model. If q is a passive tracer, the ensemble average value obtained at each point by running a series of Brownian back trajectories to some initial time does satisfy a diffusion equation. Indeed, this technique has been used to obtain Monte Carlo solutions to the diffusion equation evaluated along aircraft tracks, without the need to solve the diffusion equation globally (Legras et al. 2003). Condensation destroys the means by which the diffusion equation for the ensemble mean field is derived, precisely because the operation of coarse-graining (taking the average of concentrations from many trajectories) does not commute with the operation of condensation. A saturated parcel and dry-air parcel, when averaged together, will not condense until subjected to substantial cooling. The same two air parcels, tracked separately, will yield condensation immediately on the slightest cooling because one is saturated. The difficulty that emerges from the condensation process is in fact endemic to all systems where the tracer evolves according to some nonlinear process, including most chemical reactions. The difficulty of describing the evolution of the coarse-grained concentration field by means of a partial differential equation calls into question the very notion of an “eddy diffusivity” when nonlinear processes and unresolved small-scale fluctuations are involved. We suggest that the random walk model with a barrier provides a minimal conceptual model for the generation of dry air in the atmosphere. It generates a rate and a profile; it generates unsaturated air in the interior of regions with exponentially decaying saturation specific humidity, without waiting for dry air to invade from the boundary; it predicts the probability distribution of subsaturation, which is a necessary input to radiation and other nonlinear processes; it has much in common with the way the real atmosphere generates dry air. Thus, it is a much better conceptual model than diffusion-condensation. As we saw in section 6.5.1, when the diffusion-condensation model is supplied with moisture from a boundary layer that (like the observed one) is nearly saturated, it saturates the entire atmosphere except for a thin strip near the ground. In the next section, we will see that the random walk model is free from this shortcoming.

6.5.3. Forced Equilibria in the Stochastic Problem We now examine the statistical equilibria obtained by balancing the drying processes of the preceding section against some simple models of the moisture source. As a first example, we introduce a moisture source into the bounded random walk model by supposing that parcels are reset to saturation when they encounter the boundary at

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170 | Pierrehumbert, Brogniez, Roca Realization 1 Realization 2

ymax 2

ymax Y

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1

0 –8

–6

–4 Time

–2

0

Figure 6.10. Two realizations of a random walk between the time of most recent encounter with the moisture source at the boundary and the time of arrival at the target position.

y = 0. At large times, the humidity at the target point is no longer determined by the maximum y (minimum q s ) encountered over an arbitrarily long time τ . The statistic of interest is now the maximum y attained since the most recent encounter with the boundary. This statistic is illustrated in Fig. 6.10. Back trajectories that take a long time to hit the boundary have a higher probability of significantly overshooting the target position, and therefore yield drier values. The equilibrium specific humidity PDF is given by equation (6.4), where the moistening probability Ps at(τ ) is the probability that a trajectory starting at y first encounters the boundary after time τ . This encounter-time probability is also a classic statistic studied in the theory of Brownian motion, where it is sometimes referred to as a “stopping time problem.” It can be derived from a maximum negative excursion statistic analogous to equation (6.16), giving the probability that the smallest value visited was ymin, which necessarily is less than yb. There is a characteristic time τ D = yb2/D in the problem. For t  τ D, few particles have had time to encounter the boundary because the width of spread of the cloud starting at yb is smaller than the distance to the boundary. For τ  τ D, most particles have already encountered the boundary, and the probability of a new first encounter decays like τ −3/2. When plugged into the convolution (6.4), these results imply that the equilibrium moisture PDF looks something like Q min(q |q (yb), τ D) , that is, like a freely decaying drying process that has only been allowed to run for a time τ D. However, because of the fat τ −3/2 tail of the encountertime PDF, there is also a considerable additional population of dry air, arising from trajectories that have taken much longer then τ D to encounter the boundary. Results of an equilibrium Monte Carlo simulation are shown in Fig. 6.11. In this simulation, 20,000 particles started at various yb are random-walked while keeping track of their moisture, until almost all have reached the boundary. The moisture PDFs were computed assuming q s = exp(−y), as before. Recall that in this case, the PDF of − ln(q /q s ) is also the PDF of the maximum excursion relative to yb. As seen in

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Relative Humidity of the Atmosphere | 171 1.4 y = .25

Probability

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y = .5 y=1

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y = 1.5 y=2

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y = .5 y=1 y = 1.5

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1 0

y=3 0

0.2

0.4

0.6

0.8

1

q/qs Figure 6.11. Equilibrium probability distributions for − ln(q /q s ) (top panel) and q /q s (bottom panel) at the various positions indicated on the curves. The PDFs are shown at y = 0.25, 0.5, 1, 1.5, 2, and 3. Particles execute a random walk in the domain [0, ∞], and the moisture tagged to each particle is reset to saturation when they encounter the boundary at y = 0. Calculations were carried out with q s (y) = exp(−y). Plotting of the full height of the dry spike in the lower panel has been suppressed to make the remainder of the behavior more visible.

the top panel, as we increase the distance from the source, the proportion of particles whose overshoot is near zero reduces, and the most probable overshoot moves slightly toward larger values. The most prominent feature at larger distances, though, is that the distribution develops a fat tail, indicating a fairly high probability of large overshoots (large values of − ln(q /q s )). If q /q s were lognormal, the tails in the distribution shown would be Gaussian. Replotting the data with a logarithmic ordinate (not shown) reveals that the tails in the equilibrium distribution are exponential rather than Gaussian. In the PDFs of q itself (lower panel), the fat tails in the overshoot probability create a pronounced dry spike near q = 0. The overall evolution of the q PDF as y is increased can be described as a shift of probability from nearly saturated values to very dry values. For positions moderately close to the source, the PDF of q is distinctly bimodal, with a moist peak and a dry peak; the moist peak disappears at larger distances. At all distances, there is a broad shoulder of intermediate humidities, over which the probability is nearly uniform. The cold-trap configuration provides a very clear-cut example of the contrast between mean field theories like diffusion that model everything in terms of the field

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172 | Pierrehumbert, Brogniez, Roca 1

0.8

0.6 q

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0.4

0.2 Stochastic Solution 0

-1

-0.5

0 y

0.5

1

Figure 6.12. The equilibrium profile of mean humidity for a coldtrap configuration subjected to boundary conditions q = 1 at the left-hand boundary and q = 0 at the right. The cold trap is at the center of the domain, and resets the moisture to q = 12 . The dashed line gives the result for the diffusion-condensation equation, whereas the solid line gives the result of a Monte Carlo simulation of the random walk model.

of an ensemble average quantity and models like the random walk model that retain information about fluctuations. The specific configuration we consider is subjected to boundary conditions q = 1 at y = −1 and q = 0 at y = 1. The saturation specific humidity is unity everywhere, except for a narrow region near y = 0 where q s = 12 . For the diffusion-condensation model, the equilibrium profile of q is the straight-line pure conduction profile joining the boundary values, indicated by the dashed line in Fig. 6.12. The cold trap causes no condensation, and has no effect on the moisture profile in the diffusion-condensation model. We also solved this problem using the random walk model, by random-walking 5000 particles, resetting their moisture values to 1 or 0 upon encounters with the left or right boundary, and resetting the moisture to 12 upon encounter with the cold trap. The ensemble mean moisture profile for this model is given by the solid line in Fig. 6.12, and has a kink indicating dehydration by the cold trap. It is as if the cold trap exerts an “action at a distance” caused by the fluctuations about the mean moisture, allowing it to have an effect on moisture that cannot be determined on the basis of the knowledge of the mean alone. The origins of this behavior are no great mystery. In the stochastic model, there are only three possible values for q : 1, 12 , and 0. A particle with q = 1 will always lose moisture upon crossing the cold trap, so there are no such particles to the right of the cold trap. Moreover, particles with q = 1 to the left of the cold trap but not too far from it have a high probability of crossing over and back again, reducing the population of such particles to the left of the cold trap. This behavior is illustrated in Fig. 6.13, which shows how the population of particles depends on y. A population consisting of half q = 1 particles and half q = 0 particles

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Relative Humidity of the Atmosphere | 173 N(1) N(0.5)

Stochastic Simulation: Humidity Probabilities

N(0) 5000

4000

3000

2000

1000

0

–1

–0.8 –0.6 –0.4 –0.2

0 y

0.2

0.4

0.6

0.8

1

Figure 6.13. Spatial dependence of the number of particles of each type in the random walk simulation of the cold-trap configuration shown in Fig. 6.12.

would not condense on the basis of the average q for the ensemble, but does nonetheless lose moisture since each individual q = 1 particle will condense when crossing the cold trap. What we have just presented is but one instance of a large class of possible stochastic water vapor models kept in equilibrium by a moisture supply. Many refinements or variations are possible. For example, one can include a second reflecting barrier, at yb > 0; this would represent some of the effects of the confinement of trajectories within the troposphere. Another variant is the random-walk/steady-subsidence model, which approximates the tropical situation. In this case, we think of y as latitude; particles execute a random walk in y through a horizontally homogeneous q s field, but q s is allowed to vary with pressure p and the trajectories undergo a steady subsidence in p as they random-walk in y. Parcels are reset to saturation when they encounter the boundary at y = 0, which is thought of as the region of the Tropics that is maintained near saturation by deep convection. This model is essentially equivalent to the subsidence model described by equation (6.5), with Ps at(τ |yo) taken to be the probability distribution of time required for particles starting at y = yo to encounter the boundary. Since the mean waiting time gets longer as distance from the boundary increases, the air at any given p becomes drier with distance from the boundary because it has subsided more in the time it takes to get there. Still a further refinement of this model would be to replace the assumption of resaturation at y = 0 with an assumption that the parcels are resaturated when they encounter a more general, spatially complex

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set  in (x, y) space. One would correspondingly replace the one-dimensional random walk with either a two-dimensional random walk in (x, y) or a random walk in y and steady sheared advection in x. This model now begins to approach the model in Pierrehumbert (1998), where trajectories were modeled using observed winds on isentropic surfaces, but with a steady subsidence across isentropic surfaces, and were resaturated upon encounters with the actual tropical convective region.

6.6. How Will Water Vapor Change in Reaction to a Changing Climate? It would be fruitful to tinker with the stochastic water vapor model in search of improved, non-Brownian statistical descriptions of the trajectories that might better reproduce the salient properties of real trajectories. The key characteristic of mixing to compute is the maximum excursion probability distribution. Time-correlated random walks can still be treated using the reflection principle so long as they are Markov processes. However, there are few general methods for computing such path statistics for non-Markov random walk processes, such as Levy flights, and it is likely that one would need to resort to Monte Carlo simulations to get the needed PDF, if such processes turn out to be needed to capture the salient characteristics of atmospheric trajectories. We believe this is the correct path to take in the quest for water vapor parameterizations suitable for incorporation in idealized climate models. In this section, however, we leap ahead to the direct use of trajectories computed from observed or simulated wind fields, without any further attempt at a reduced statistical description. Our goal in this section is to study how temperature affects the relative humidity of the free troposphere, taking into account the Lagrangian nature of the problem. In particular, we shall attempt to provide some precise justification for the expectation that free tropospheric humidity will increase as the climate becomes warmer. The procedure we employ here is identical to that used in the equilibrium stochastic model, except that the back trajectories are computed using simulated or observed/analyzed three-dimensional global wind fields. In particular, the effect of internal mixing and moisture sources are neglected; trajectories are tracked back to the boundary layer, and assumed to be saturated at that point. The moisture at the terminal point of the trajectory is then the minimum saturation specific humidity experienced since the encounter with the boundary layer. This approach was employed in Pierrehumbert and Roca (1998), who found excellent agreement with the observed tropical dry-zone humidity. The same advection-condensation model has since been applied to much larger datasets in Roca et al. (2005) and Brogniez (2004). We will concentrate our efforts on the midlatitude moisture distribution, since this region has received less attention than the subtropical problem. The region of study is the midlatitude Atlantic region bounded by 100◦ E to 0◦ E longitude and 30◦ N to 60◦ N latitude.

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Relative Humidity of the Atmosphere | 175

Figure 6.14. 500 mb relative humidity at 12Z on December 3, 1994, reconstructed using the back trajectory method driven by NCEP winds and temperature.

5 Control -1K 4

Probability

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Control Control +1K

3 ERA40 2

1

0

0

0.2 0.4 0.6 0.8 Relative Humidity (Fraction)

1

Figure 6.15. Probability distribution of relative humidity for December 1994 over the region shown in Fig. 6.14, computed 4 times daily using NCEP winds and temperatures. Results for experiments with temperature uniformly increased or decreased by 1 K are also shown, but the curves are barely visible because they lie almost exactly on the control case. For comparison, the relative humidity PDF over the same time and region for the ERA-40 analysis is also shown.

First, we computed the back trajectory reconstructed relative humidity fields using NCEP winds and temperatures for December 1994. The reconstruction was performed four times daily at a spatial resolution of 0.25 degrees in latitude and longitude. To provide a general idea of what the fields look like, a map of relative humidity at 12Z on 3 December 1994 is shown in Fig. 6.14. It shows the filamentary structure familiar from earlier work, with small-scale alternations of moist and dry air. The PDF computed over the region of study for the entire month of data is shown in Fig. 6.15. It has a dry spike and a moist spike, with a broad region of nearly uniform

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but weakly decreasing probability in between. In this sense the computed PDF looks qualitatively like the PDF for the equilibrium random walk model forced by a saturated boundary layer, at the point y = 0.5 (Fig. 6.10). For the sake of comparison with an observationally based moisture estimate, we also show the relative humidity PDF calculated from 500 mb ERA-40 analyses covering the same time period and region.5 As compared to the trajectory reconstruction, the ERA-40 PDF has a much less pronounced dry spike, but a greater population of air with intermediate saturation. A pattern like this could be obtained by allowing some mixing between moist and dry filaments in the trajectory model, possibly representing a physical effect left out of the trajectory model. It could also result from moistening of dry air by evaporation of precipitation falling through it. On the other hand, the lack of a dry spike in the ERA-40 result may be an unphysical artifact of excessive numerical diffusion in the model used to do the analysis/assimilation cycle. Is reality more like ERA-40 or more like the trajectorybased reconstruction? We do not know of any humidity dataset that can unambiguously answer this question at present. Now we pose the question of what happens if the trajectories are kept fixed but the temperature at all points in the atmosphere is increased by a uniform amount
T . In a real climate change, such as caused by the doubling of CO2, one can expect the statistics of trajectories to change somewhat, and it is well known also that the warming is not uniformly distributed in latitude and altitude. In posing the simplified form of the problem, we are supposing for the moment that such effects are of only secondary importance in determining the changes in relative humidity. The problem stated this way forms a kind of null hypothesis about the behavior of water vapor, which can serve as a launching point for more sophisticated extensions. The effect of uniform warming on the relative humidity PDF can be determined by straightforward reasoning. Consider a point with temperature and pressure ( p, T ). The humidity here is determined by the point ( pm , Tm) where the most recent minimum saturation specific humidity following a resaturation event occurred (i.e., the position of the “time of last saturation”). Using the fact that the specific humidity is conserved in the     absence of condensation, we find that the relative humidity is e s (Tm)/ pm / e s (T )/ p =   ( p/ pm) e s (Tm)/e s (T ) . If we increase temperatures uniformly and keep trajectories  fixed, pm stays the same and the new relative humidity is ( p/ pm) e s (Tm +
T )/  e s (T +
T ) . Now, if
T is small compared to T and Tm , as is usually the case, the Clausius-Clapeyron relation implies that the new relative humidity is approximately  (
T ) ≈

p pm

  2  
T T L −1 . ≈ rh(0) 1 + RvT Tm2 T e s (T ) + e s (T ) RvLT 2
T

 e (T ) + e (T ) L
T s m s m RvT 2 m

[6.19] This expression assumes the perfect gas law, but otherwise proceeds directly from Clausius-Clapeyron without the need to assume the approximate exponential form

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Relative Humidity of the Atmosphere | 177 5 Control T(4x) W(control) 4xCO2 W(4x) T(control)

4

Probability

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2

1

0

0

0.2

0.4

0.6

0.8

1

Relative Humidity (Fraction) Figure 6.16. Relative humidity PDFs reconstructed by the back trajectory technique driven by various combinations of winds and temperature fields from a control and a 4 × CO2 GCM simulation. See text for definition of the cases.

given in equation 6.1. This result implies that the relative humidity increases with warming, but the coefficients are such that the changes are quite small for moderate values of
T . For example, with T = 260 K, Tm = 240 K and
T = 1 K, the increase in relative humidity at each point is only 0.014rh(0). Hence, it is expected that moderate uniform warming or cooling should leave the PDF of relative humidity essentially unchanged. To confirm this reasoning, we recomputed the NCEP back trajectory PDFs with temperature uniformly increased by 1 K and also with temperature uniformly decreased by 1 K. These curves are plotted in Fig. 6.15, and overlay the control back trajectory PDF almost exactly. We now apply the same technique to diagnosis of humidity changes in a GCM simulation of climate change. This enables us to probe the effects of changes in the temperature structure, and changes in the statistics of the trajectories. We consider two equilibrium simulations with the FOAM GCM coupled to a mixed-layer ocean model. Two simulations were carried out, both with realistic geography: the first with 300 ppmv CO2 (the “control” case) and the second with 1200 ppmv CO2 (the “4 × CO2” case). Figure 6.16 shows the relative humidity PDFs constructed using the back trajectory method applied to the December GCM wind and temperature fields for both the control case and the 4 × CO2 case. Despite the changes in temperature pattern and wind fields, the PDFs have almost the same shape for both simulations. The warm simulation has a slightly greater population of air with relative humidity in the vicinity of 30%, at the expense of a reduction in more saturated air. We performed two additional back

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3

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0.4 0.6 Relative Humidity (Fraction)

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Figure 6.17. Relative humidity PDFs for the internally computed GCM moisture fields (control and 4 × CO2) compared with results from the ERA-40 analysis, the NCEP trajectory reconstruction, and the trajectory reconstruction driven by GCM-simulated winds and temperature for the control case.

trajectory reconstructions in an attempt to isolate the importance of changes in temperature structure and trajectory statistics. The reconstruction labeled “T(4×) W(control)” was carried out with the temperature field from the 4 × CO2 run but wind fields from the control run. The reconstruction labeled “W(4×) T(control) ” was carried out with the wind field from the 4 × CO2 run but temperature fields taken from the control run. Both reconstructions generate more dry air and less saturated air than the previous two reconstructions. Changes in the temperature structure in a warmer climate, taken in isolation, enhance the production of dry air. However, the trajectories in the warmer climate adjust so as to avoid the colder regions, leaving the PDF invariant when both effects are applied in conjunction. Why this cancellation of effects should take place is at present wholly mysterious. The cancellation is particularly difficult to understand in light of the fact that trajectory changes, taken in isolation, also enhance dry air production. It will be very interesting to see whether other GCMs exhibit similar behavior. How does the relative humidity PDF reconstructed from back trajectories compare with that based on the humidity computed internally to the model? These may be expected to be different, since the former reconstructs what the humidity would be in the absence of mixing and internal moisture sources, whereas the low resolution GCM allows a considerable degree of mixing. As seen in Fig. 6.17, the dry spike seen in both NCEP and model Lagrangian reconstructions is completely absent from the

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internally computed PDF. This is a result of probably excessive mixing in the GCM. Significantly, the wind statistics in the GCM do not seem to be a problem, for the Lagrangian reconstruction based on model winds is reasonably similar to that based on NCEP winds. The PDF from the ERA-40 analysis is also shown in the figure to underscore that the GCM is deficient in dry air. This deficiency is not surprising, since the leakage of a small proportion of moisture from saturated air is enough to eliminate a considerable population of dry air. We do not wish to imply that this deficiency is characteristic of all GCMs, though it may indeed be endemic to low resolution GCMs. It is interesting that, despite the considerable effect of mixing on the internally computed humidity, the GCM humidity PDF is nonetheless invariant between the control and 4 × CO2 case. In contrast with the GCM and trajectory-based diagnostic results presented above, the idealized humidity model of Minschwaner and Dessler (2004) predicts a modest decrease of free tropospheric relative humidity as the climate warms; this slightly reduces the positive water vapor feedback, but does not eliminate it. Since Minschwaner and Dessler (2004) is a tropical model, whereas the analyses above were applied to a midlatitude case, a more meaningful comparison must await the application of the trajectory-based diagnostics to tropical regions. This is straightforward, but will be deferred to future work. If the disagreement persists, it might be due to the neglect of lateral mixing in Minschwaner and Dessler (2004), or it might be due to the low vertical resolution of the simulated and analyzed tropical climate as compared to the resolution employed in Minschwaner and Dessler (2004).

6.7. Conclusions The behavior of water vapor in the climate system is complex and multifold, and there will probably never be any one simple theoretical framework that accounts for everything one would like to understand about water vapor. The problem in its full complexity, as manifest in either atmospheric observations or comprehensive GCMs, defies human comprehension. Understanding, as opposed to mere simulation, requires simple models whose behaviors can be grasped in their entirety, even if they are wrong or incomplete in some particulars. This would be true even if GCM simulations were perfect, and the necessity is even more pressing in the face of simulations that are both imperfect and in certain regards suspect. We have presented some simple ideas pertinent to the proportion of subsaturated air in the atmosphere, and the degree of its subsaturation. The emphasis on highly subsaturated (“dry”) air arises from its importance in determining the radiative feedback of water vapor. In particular, based on an analysis of the way large-scale atmospheric trajectories influence subsaturation, we have provided a concise and defensible statement of why one should expect atmospheric water vapor to increase as climate gets warmer: The

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specific humidity at a given point in the atmosphere is determined by the saturation specific humidity at the point of minimum temperature encountered along the trajectory extending backward in time from this point until it encounters a moisture source sufficiently strong to saturate it. If the statistics of the trajectories do not change too much as the atmosphere warms, this minimum temperature increases, leading to an increase in the water vapor content of the target point. This is a somewhat cartoonish statement that is modified in its details by closer study, but one that survives a fair amount of scrutiny. It is intended to replace wholly indefensible statements which simply invoke the Clausius-Clapeyron relation. Clausius-Clapeyron is indeed at the root of the behavior of water vapor, but the proper use of the relation hinges on identifying the temperature to which the relation should be applied; it’s not the surface temperature, and the effect of the relation on evaporation is of little relevance to water vapor feedback. If one takes the cartoon picture to its idealized extreme, it predicts that if trajectories are held fixed while the atmosphere warms uniformly in space and time, the relative humidity probability density function (PDF) will remain nearly invariant for small or moderate warming. This behavior was confirmed in a midlatitude case, using trajectories and unperturbed temperatures from NCEP analyses. The Lagrangian viewpoint lends itself to the formulation of diagnostics that can be applied to atmospheric data and GCM simulations, and used to compare the operation of processes in GCM simulations to that in atmospheric data by applying the same diagnostics to both. In essence, one uses three-dimensional wind and temperature fields to study the pattern of an idealized moisture substance that does not diffuse from one air parcel to another, and that is maintained by an idealized source (typically in the boundary layer). Applied to midlatitude NCEP data, the diagnostic yields a relative humidity PDF whose dominant features are a dry spike at 5% relative humidity, a lesser spike at saturation, and a broad tail of intermediate degrees of saturation in between. The corresponding PDF computed from ERA-40 data has a much broader and less pronounced dry peak, centered on 10% relative humidity, and a greater probability of air with intermediate humidity. A key unresolved question at this point is whether reality looks more like the ERA-40 analysis, or the nondiffusive trajectory reconstruction. The ERA-40 analysis, which incorporates assimilated satellite data, could be indicative of the importance of mixing processes in the real atmosphere; likely candidates include vertical turbulent diffusion at small scales (Haynes and Anglade 1997) and vertical redistribution of moisture owing to evaporation of precipitation falling through very dry air. On the other hand, the broad dry maximum in the ERA-40 PDF may be symptomatic of excessive numerical diffusion in the assimilation process. The Lagrangian moisture reconstruction diagnostic is an instance of the general idea of studying the behavior of an idealized water-like substance that isolates some key feature of the problem one would like to understand and which is easier to understand than the real thing. The isentropic study of Yang and Pierrehumbert (1994) was one of the earlier examples of this approach. That study examined, in effect, a water-like

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tracer that condensed, but had zero latent heat so that condensation would not cause a trajectory to leave a fixed dry isentropic surface. The recent work by Galewsky et al. (2005) also makes use of an idealized water substance that doesn’t release latent heat, but this time in an Eulerian framework that allows for the effects of mixing amongst air parcels. In the same spirit, Frierson et al. (2006) have used a synthetic water vapor with adjustable latent heat but no radiative impact, in order to focus on the influence of moist processes on energy transport. The Lagrangian moisture diagnostics were also applied to a GCM simulation of warming induced by a quadrupling of CO2. The midlatitude relative humidity PDF reconstructed on the basis of trajectories driven by model winds was found to be invariant between the control and 4 × CO2 cases, even though the model warming is not spatially uniform and the trajectory statistics are allowed to change. However, the invariance of the PDF was found to result from a mysterious cancellation between the effects of nonuniform warming and changes in the excursion statistics of the trajectories, each of which individually causes a moderate increase in the population of subsaturated air. The humidity PDF reconstructed on the basis of the model wind and temperature closely resembles that computed on the basis of observed winds and temperatures. However, the PDF of relative humidity computed internally to the GCM is very different from the Lagrangian diagnostic, and is completely missing the dry peak. We believe this is probably due to excessive mixing in the low-resolution model. Nonetheless, the PDF of the GCM internally computed humidity is still invariant under warming, indicating that the basic invariance inferred from the nondiffusive Lagrangian calculation survives the addition of a considerable degree of mixing. In order to illustrate the manner in which certain trajectory statistics govern the probability distribution of humidity, we formulated and analyzed a family of idealized models of water vapor in which trajectories are modeled as random walk processes. The calculation predicts the shape of the PDF and the way the PDF depends on distance from the humidity source. With some further refinements, it is even possible that models of this class could be made suitable for use in idealized climate models, enabling such models to treat a broader range of questions concerning water vapor feedback. Without a moisture source, the stochastic model predicts that, in midlatitude conditions, the √ mean humidity at any given point decays like exp(− Dt) for some constant D if the random walk is bounded above in temperature space. The humidity PDF quickly develops a peak at dry values, with an approximately exponential moist tail. In the presence of a moisture source at the boundary, the equilibrium PDF develops a dry spike at sufficiently large distances from the source, with a “fat tail” extending to more saturated values. At moderate distances from the source, the PDF is bimodal, with a secondary peak at saturation. This shape is similar to that found in midlatitudes using realistic trajectories. We have compared the predictions of the stochastic model with those of a more conventional approach based on diffusion of moisture. Diffusion creates monolithic

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regions of saturated air and dries the atmosphere in a manner very different from the idealized stochastic model, which more closely reproduces the mechanisms operating in the real atmosphere. Diffusion is not a suitable approach to representation of water vapor in simplified climate models, at least not if one’s aim is to treat water vapor feedback. The comparison between the stochastic model and the diffusion model also highlights the mathematical novelty of the former. Viewed as a stochastic problem, the distribution of subsaturation depends on probability distributions on the space of paths, rather than just on the space of endpoints of paths encountered in conventional linear diffusion problems. Lagrangian history probability problems of this type cannot be adequately modeled with a mean field theory like the partial differential equation for diffusion. One needs a Monte Carlo approach, or some other approach that retains information about fluctuations as well as ensemble averages. This calls into question the whole utility of eddy diffusivity as a means of parameterizing mixing of tracers subject to a nonlinear removal process. The trajectory approach, as usually formulated, does not treat regions of deep tropical convection explicitly. These are detected through the large-scale wind field as regions of persistent upward motion, and are almost always saturated with moisture. This has the virtue of not requiring information about convection that is not implicit in the wind field, but it does amount to an assumption of saturated detrainment from convective regions. This leaves out all the mechanisms of the sort discussed by Tompkins and Emanuel (2000), which could keep convective regions significantly subsaturated. The trajectory approach could be improved by making explicit use of information about convection, and allowing for subsaturated detrainment. The logarithmic dependence of outgoing longwave radiation on specific humidity means, however, that the subsaturation would have to be very substantial before it had much effect on water vapor feedback. We have sidestepped the issue of cloud radiative effects by speaking throughout of “clear sky” radiation when translating water vapor into outgoing longwave radiation. However, the minute a separation between clear-sky and cloudy-sky radiation is invoked, one is implicitly assuming that such a distinction is meaningful. The concept of “clear sky” outgoing long-wave radiation makes sense when applied to large coherent regions free of mid- or high-level clouds, and one should in fact understand the term to allow the inclusion of parts of the scene including boundary-layer clouds (which do not affect the outgoing longwave radiation). In contrast, tropical convective regions and the midlatitude storm tracks are typically characterized by a more-or-less continuous and spatially complex distribution of cloudy and clear air, and all types in between. Since condensed water is so opaque to infrared radiation, the escape of infrared radiation to space is more controlled by the size of the clear-sky holes between clouds than by fluctuations in their humidity. This engages the whole subject of fractional cloud cover, which may well be the most problematic aspect of cloud representation in climate models. Learning to characterize and predict the radiative effect of intermingled,

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coupled assemblages of cloudy and clear air is one of the greatest challenges to the understanding of climate.

Acknowledgments The research upon which this chapter is based was funded by the National Science Foundation under grants ATM-0121028 and ATM-0123999. We are grateful to the editors of this book for numerous valuable suggestions.

Notes 1. q /q s is not precisely equal to the relative humidity, which is defined as the ratio of the mixing ratio of water vapor to the saturation mixing ratio (equivalently, the ratio of the partial pressure of water vapor to the saturation partial pressure). However, for the purposes of this chapter, where water vapor is assumed to be a minor constituent of the atmosphere, mixing ratio and specific humidity can be regarded as practically interchangeable. 2. When coupled to a mixed-layer ocean, the FOAM GCM is essentially a portable, Beowulf-oriented re-implementation of CCM3 (Kiehl et al. 1998). All simulations reported in this chapter were carried out at R15 (4.5◦ × 7.5◦) resolution. Further details on FOAM can be found at www.mcs.anl.gov/foam. 3. The recent work of Sherwood and Meyer (personal communication) on relative humidity PDFs employs a special case of this class of solutions. 4. In the regularized case with small but finite τ , the removal process is less singular and the steady solution has q very slightly greater than q s and nonzero removal rate S. 5. We used ERA-40 analyses in preference to NCEP analyses because we found that the ERA-40 analyses seem to give better agreement with patterns seen in satellite retrievals. We used the analyses in preference to satellite data itself because the latter represent averages over a fairly deep atmospheric layer, whereas the analyses are available at an individual model level.

References Brogniez, H., 2004: Humidité de la Troposphere Libre Africaine: Elaboration d’une Archive Meteosat, Analyse Climatique et Evaluation de Modeles. PhD thesis, Univ. Paris VI, 250 pp. Colman R. A., and McAvaney B. J., 1997: A study of general circulation model climate feedbacks determined from perturbed sea surface temperature experiments. J. Geophys. Res. — Atmospheres, 102 (D16), 19383–19402. Emanuel, K., and Pierrehumbert, R. T., 1996: Microphysical and dynamical control of tropospheric water vapor. In Clouds, Chemistry and Climate, NATO ASI Series 35. Springer: Berlin, 260 pp. Frierson D. M. W., Held I. M., and Zurita-Gotor P., 2006: A gray-radiation aquaplanet moist GCM. Part I: Static Stability and Eddy Scales. J. Atmos. Sci., 63, 2548–2566. Galewsky J., Sobel A., and Held I., 2005: Diagnosis of subtropical humidity dynamics using tracers of last saturation. J. Atmos. Sci., 62, 3353–3367.

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184 | Pierrehumbert, Brogniez, Roca Hall A., Manabe S., 1999: The role of water vapor feedback in unperturbed climate variability and global warming. J. Climate, 12, 2327–2346. Haynes P., and Anglade J., 1997: The vertical-scale cascade in atmospheric tracers due to largescale differential advection. J. Atmos. Sci., 54, 1121–1136. Held I. M., and Soden B. J., 2000: Water vapor feedback and global warming. Annu. Rev. Energ. Env., 25, 441–475. Karatzas, I., and Shreve S. E., 1991: Brownian Motion and Stochastic Calculus, Springer: New York, 469 pp. Kelly K. K., Tuck A. F., and Davies T. 1991: Wintertime asymmetry of upper tropospheric water between the Northern and Southern Hemispheres. Nature, 353, 244–247. Kiehl J. T., Hack J. J., Bonan G. B., Boville B. A., Williamson D. L., and Rasch P. J., 1998: The National Center for Atmospheric Research Community Climate Model: CCM3. J. Climate, 11, 1131–1149. Legras B., Joseph B., and Lefevre F., 2003: Vertical diffusivity in the lower stratosphere from Lagrangian back-trajectory reconstructions of ozone profiles. J. Geophys. Res. — Atmospheres, 108 (D18), 4562, doi:10.1029/2002JD003045. Lesigne E., 2005: Heads or Tails: An Introduction to Limit Theorems in Probability (A. Pierrehumbert, translator), American Mathematical Society, 149 pp. Lindzen R. S., 1990: Some coolness concerning global warming. B. Am. Meteorol. Soc., 71, 288–299. Manabe S., and Wetherald R. R., 1967: Thermal equilibrium of atmosphere with a given distribution of relative humidity. J. Atmos. Sci., 24, 241–259. Minschwaner K., and Dessler A. E., 2004: Water vapor feedback in the tropical upper troposphere: Model results and observations. J. Climate, 17, 1272–1282. Pauluis O., and Held I. M., 2002: Entropy budget of an atmosphere in radiative-convective equilibrium. Part II: Latent heat transport and moist processes. J. Atmos. Sci., 59, 140–149. Petoukhov V., Ganopolski A., Brovkin V., Claussen M., Eliseev A., Kubatzki C., and Rahmstorf S., 2000: CLIMBER-2: A climate system model of intermediate complexity. Part I: Model description and performance for present climate. Climate Dynamics, 16, 1–17. Pierrehumbert R. T., 1998: Lateral mixing as a source of subtropical water vapor. Geophys. Res. Lett., 25, 151–154. Pierrehumbert R. T., 1999: Subtropical water vapor as a mediator of rapid global climate change. In Clark P. U., Webb R. S., and Keigwin L. D., eds., Mechanisms of Global Change at Millennial Time Scales. American Geophysical Union: Washington, D.C. Geophysical Monograph Series 112, 394 pp. Pierrehumbert R. T., 2002: The hydrologic cycle in deep time climate problems. Nature, 419, 191–198. Pierrehumbert, R. T., and Roca R., 1998: Evidence for control of Atlantic subtropical humidity by large scale advection. Geophys. Res. Lett., 25, 4537–4540. Roca R., Lafore J.-Ph., Piriou C., and Redelsperger J. L., 2005: Extra-tropical dry air intrusions into the West African Monsoon mid-troposphere: An important factor for the convective activity over the Sahel. J. Atmos. Sci., 62, 390–407. Salathé E. P., Jr., and Hartmann D. L., 1997: A trajectory analysis of tropical upper-tropospheric moisture and convection. J. Climate, 10, 2533–2547. Sherwood S. C., and Dessler A. E., 2000: On the control of stratospheric humidity. Geophys. Res. Lett., 27, 2513–2516.

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Relative Humidity of the Atmosphere | 185 Shine K. P., and Sinha A., 1991: Sensitivity of the Earths climate to height-dependent changes in the water-vapor mixing-ratio. Nature, 354 (6352), 382–384. Soden B. J., 1998: Tracking upper tropospheric water vapor. J. Geophys. Res., 103 (D14), 17069–17081. Soden B. J., and Bretherton F. P., 1993: Upper-tropospheric relative-humidity from the Goes 6.7µ channel—method and climatology for July 1987. J. Geophys. Res., 98 (D9), 16669–16688. Spencer R. W., and Braswell W. D., 1997: How dry is the tropical free troposphere? Implications for global warming theory. Bull. Amer. Meteorol. Soc., 78, 1097–1106. Tompkins A. M., and Emanuel K. A., 2000: The vertical resolution sensitivity of simulated equilibrium temperature and water-vapour profiles. Quart. J. Roy. Meteorol. Soc., 126B, 1219–1238. Weaver A. J., Eby M., Wiebe E. C., et al. 2001: The UVic Earth System Climate Model: Model description, climatology, and applications to past, present and future climates. AtmosphereOcean, 39, 361–428. Yang, H., and Pierrehumbert, R. T., 1994: Production of dry air by isentropic mixing. J. Atmos. Sci., 51, 3437–3454. Zhang C. D., Mapes B. E., and Soden B. J., 2003: Bimodality in tropical water vapour. Quart. J. Roy. Meteorol. Soc., 129 (594), 2847–2866. Kalnay, E., and coauthors, 1996: The NCEP/NCAR 40-year reanalysis project. Bull. Amer. Meteorol. Soc., 77, 437–471.

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Chapter 7

Quasi-Equilibrium Dynamics of the Tropical Atmosphere Kerry Emanuel

7.1. Introduction Compared to the extratropical atmosphere, the dynamics of the tropical atmosphere are poorly understood. The cornerstone of dynamical meteorology—quasigeostrophic theory and its contemporary encapsulation in potential-vorticity thinking—works well in middle and high latitudes, where the motion is quasi-balanced over a large range of scales and where diabatic and frictional effects are usually small and can often be neglected on short time scales. For this reason, the dynamics of fundamental processes such as Rossby wave propagation and baroclinic instability are well understood, and owing to the steep energy spectrum of quasigeostrophic turbulence, the evolution of the extratropical atmosphere can be predicted many days in advance. By contrast, much of what occurs in the tropical atmosphere is neither quasibalanced nor adiabatic, and thus the tools that have served middle- and high-latitude meteorology so well are poorly suited to understanding the Tropics. Moreover, most of the belt between the Tropics of Cancer and Capricorn is covered by ocean, and so before the advent of satellites, the atmosphere in this region was poorly observed. Even though there have been remarkable advances in satellite remote sensing, it is still difficult to produce reliable analyses of wind and water vapor. Advanced data assimilation techniques that work well outside the Tropics yield questionable results in tropical regions, not only because of the paucity of observations but because global atmospheric models are often inconsistent in rendering such basic tropical phenomena as the Madden-Julian Oscillation (Slingo et al. 1996). Poor models and sparse observations, together with the greater influence of convective and mesoscale phenomena, also make weather forecasting problematic. Under these circumstances, it is hardly surprising that tropical meteorology has advanced less rapidly than the meteorology of middle and high latitudes. The advent of better models and faster computers has led to the circumstance that numerical simulation of the tropical atmosphere, for all of its problems, has in many ways outpaced

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advances in conceptual understanding. It may be that ever-improving model physics and resolution will someday overcome present obstacles to improved analysis and prediction and thereby obviate any practical need for improved understanding. But history teaches us that simulation without understanding can be perilous, and is in any case intellectually barren. Yet there has been very substantial progress over the lasts few decades in understanding tropical phenomena. It is my purpose here to present an overview of the most important cornerstones of that progress, and to offer some speculations about future directions. I begin, in the next section, by positing radiative-moist convective equilibrium as the logical starting point for understanding the Tropics, in much the same way that zonal wind in thermal wind balance is an important starting point for understanding extratropical dynamics. Yet the radiative-convective equilibrium state is by no means trivial, and there are subtleties about it that continue to elude understanding. Because it is a statistical rather than an actual equilibrium, the classical analysis of the stability of such a state to small fluctuations is circumscribed by the requirement that the space and time scales of the fluctuations be sufficient to average over the random fluctuations of the equilibrium state, much as continuum mechanics is restricted to fluid motions on super-molecular scales. In section 7.3 I review the linear theory of small perturbations to radiative-convective equilibrium, focusing on the great simplifications that result from assuming that convection maintains moist adiabatic lapse rates. The phenomenon of moist convective damping is examined, and an overview is presented of possible energy sources for the perturbations, including surface fluxes and cloud-radiation interactions. The successes and failures of the linear theory to explain observed tropical phenomena are discussed. The theory of finite amplitude perturbations to radiative-convective equilibrium—in which deep convection is entirely suppressed in certain regions—is reviewed in section 7.4 and compared to observations of such phenomena as the Hadley and Walker circulations, monsoons, and tropical cyclones. Section 7.5 presents a few examples illustrating the application of quasi-equilibrium, and section 7.6 presents a summary.

7.2. Radiative-Convective Equilibrium: A Useful Starting Point In the absence of large-scale circulations, the tropical atmosphere would assume a state of radiative-moist convective equilibrium, in which the divergence of the net vertical radiative flux (shortwave and longwave) would be compensated by the convergence of the vertical flux of enthalpy in convective clouds, except for in a thin boundary layer next to the surface in which ordinary dry turbulence would carry the flux. There is, of course, no guarantee that such a radiative-convective state is stable to large-scale perturbations, and there is now considerable evidence that it is not. This does not eliminate its utility as an equilibrium state, any more than the instability of a zonal flow in thermal

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wind balance obviates its use as a basic state, though it may pose problems for its numerical simulation. The canonical problem of radiative-dry convective equilibrium was first developed by Prandtl (1942). In this problem, a semi-infinite but Boussinesq atmosphere subject to constant radiative cooling overlies a surface of fixed temperature. Assuming high Reynolds Number turbulence, there is only one external control parameter in the problem, the surface buoyancy flux F s ; this is also proportional to the longwave radiative flux at infinity. On dimensional grounds alone, one may deduce that the turbulence kinetic energy scales as (F s z)2/3, where z is the altitude above the surface, while the unstable stratification decreases as z −4/3. This fully turbulent convecting fluid does not support any waves. Later work on the finite version of this problem (Deardorff 1972) demonstrated that the turbulence kinetic energy throughout the layer scales as (F s h)2/3, where h is the layer depth. One easy variant of the classical Prandtl problem replaces the lower boundary by a water surface, but assumes that any condensed water remains suspended in the air, i.e., there is no precipitation. Under these circumstances, most of the air is water-saturated. As in the classical dry problem, the turbulent enthalpy flux is known at each altitude since it must balance the imposed radiative flux, but here the system specific enthalpy k is given by k = cpT + L ν q ,

[7.1]

where cp is the heat capacity at constant pressure, T the temperature, L ν the latent heat of vaporization, and q the mass concentration of water vapor. Since the entire system is water-saturated, q is given by its saturation value q ∗, which by the Clausius-Clapeyron equation is just a function of temperature and pressure. Thus the buoyancy flux has a direct relationship to the known enthalpy flux (as in the dry problem), and it can be shown that at each level the buoyancy flux is reduced from its dry counterpart by the factor
1 + (L νq ∗ RdT )
, 1 + (L 2νq ∗ c p RνT 2) where Rd and Rν are the gas constants for dry air and water vapor, respectively. This is just the ratio of the moist adiabatic and dry adiabatic lapse rates and is always less than unity, showing that for an imposed radiative flux, the buoyancy flux will be less than in the dry case. In our atmosphere, this factor can be as low as 0.3. But in other respects, the saturated Prandtl problem is nearly isomorphic to the classical problem, and once again the fluid is everywhere unstably stratified and does not support waves. The fun begins when one allows condensed water to precipitate. This deeply irreversible process depletes water from the ascending, cloudy currents, causing much of the descending air to be subsaturated and, owing to partial re-evaporation of the falling rain, driving strong downdrafts. As shown by Bjerknes (1938), the dry air between

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clouds is stably stratified and the asymmetry in stability to upward saturated displacements versus downward, unsaturated displacements causes a profound asymmetry in the fractional area occupied by updrafts and downdrafts. For Earth-like conditions, the updrafts occupy a very small fractional area, so that most of the fluid is unsaturated and stably stratified, thereby supporting buoyancy oscillations. Since air outside of clouds is generally subsaturated, the enthalpy flux (7.1) no longer specifies the buoyancy flux but instead acts as a constraint on the upward flux of water. As of this writing, there exists no generally accepted theory for the buoyancy flux in the precipitating Prandtl problem, though it can be shown to be necessarily less than that of the saturated Prandtl problem for the same imposed enthalpy flux. Although there is presently little understanding of precipitating radiativeconvective equilibrium, there have been a number of numerical simulations approximating this state in doubly periodic domains in which the troposphere is capped by a stable layer that represents the stratosphere. These simulations use non-hydrostatic models that explicitly, albeit crudely, simulate the convective cells themselves while at the same time having large enough domains to simulate many cells simultaneously. The surface temperature is generally fixed at a constant value, while internal radiative cooling is either calculated or imposed. Examples of three-dimensional simulations of radiativeconvective equilibrium include those of Islam et al. (1993), Robe and Emanuel (1996), and Pauluis and Held (2002). Among the important results of these simulations are: 1. Convection maintains a nearly moist adiabatic lapse rate from cloud base to the tropopause. 2. As expected, active updrafts occupy only a very small fraction of the domain at any one time. 3. In the absence of imposed shear, the convection is disorganized but more nearly regular than random; i.e., cells are less likely to be adjacent than if they were truly randomly distributed. But in at least some models, convection appears to self-aggregate even in the absence of shear (Tompkins 2001; Bretherton et al. 2005). 4. Imposing a background wind with vertical shear organizes convection in squall lines or arcs; these arcs may be parallel to the shear, perpendicular to it, or at an odd angle to it, depending on the shear profile and magnitude. 5. Momentum transport by the convection is broadly downgradient but highly nonlocal. 6. Increasing the forcing (i.e., the imposed rate of radiative cooling) results in increased spatial and temporal density of clouds but does not change the characteristic updraft velocities. This latter point is interesting, but so far lacks a theoretical explanation. Although explicit simulations of radiative-convective equilibrium have shed light on a number of important aspects of such states, they are very expensive and it is still not quite

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possible to perform such simulations in domains large enough to contain internal waves with wavelengths much greater than characteristic intercloud spacing. Thus it has not been possible, so far, to explicitly simulate in three dimensions the interaction between convection and circulations large enough in scale to consider the convection to be in statistical equilibrium with the larger scale.

7.3. Behavior of Small Perturbations to the Equilibrium State 7.3.1. Conditions for Statistical Equilibrium in Perturbations We now inquire about the disposition of small perturbations to radiative-convective equilibrium. But since the equilibrium is statistical, the nature of the problem depends very much on the scale of the perturbations. Convection itself is a chaotic process, and introducing perturbations on the scale of the convective clouds themselves can be expected to change the details of the state, but not its statistical properties. For example, an internal gravity wave introduced at a scale smaller than that of the characteristic intercloud spacing will rapidly lose its coherence amid the chaos of small-scale gravity waves excited by individual convective events. As with the continuum hypothesis in statistical mechanics, great simplifications can be made by considering perturbations large enough in space and long enough in time to average over many convective cells, so that the convection may be approximated as remaining in statistical equilibrium with the larger scales. This point was first emphasized by Arakawa and Schubert (1974). On the other hand, there is little appreciation of just how large perturbations need to be to meet the requirement of statistical equilibrium. Judging from the results of explicit simulations of radiative-convective equilibrium, where the intercloud spacing is on the order of several tens of kilometers, perturbations need to have scales of at least 100 km to “see” the convective state as a continuum. But when vertical shear is present, the spacing of squall lines in radiative-convective equilibrium simulations is on the order of several hundred kilometers, so in that case statistical equilibrium might only apply at scales more than 1000 km or so. This suggests that statistical equilibrium will never be as good an approximation, even for planetary scale phenomena, as the continuum hypothesis is for geophysical fluid flows. Certain macroscale atmospheric phenomena may prove as susceptible to the chaos of convection as a dust particle in Brownian motion is to random molecular fluctuations. In spite of these pessimistic considerations, much progress in tropical meteorology is arguably attributable to the success of statistical equilibrium theory. Virtually all global climate and weather–prediction models, and most regional models, rely on the parameterization of convection, which is as dependent on the notion of statistical equilibrium as the Navier-Stokes equations are on the continuum hypothesis. And although it is not generally recognized outside the tropical meteorology community,

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statistical equilibrium theory made several concrete predictions about tropical phenomena that were later verified by observations.

7.3.2. Formulation of Statistical Physics The cornerstone of the contemporary statistical equilibrium theory of convection was first stated by Arakawa and Schubert (1974), who postulated that moist convection consumes potential energy at the rate it is provided by larger-scale processes. (They called this the “quasi-equilibrium” hypothesis.) Thus potential energy, as quantified for example by the convective available potential energy (CAPE), does not accumulate in the atmosphere but is released by convection as fast as it is produced. This hypothesis, when coupled with a detailed model of how clouds redistribute enthalpy and water, provides a closed representation of convective heating and moistening in terms of large-scale variables. The Arakawa-Schubert formulation places no restrictions on the actual amount of CAPE available at any given time; nor does it require CAPE to be strictly invariant, any more than quasi-geostrophy requires flows to be steady. Quasiequilibrium is highly analogous to first-order closure in turbulence theory, which postulates that dissipation of turbulence kinetic energy locally balances generation. Energy (or buoyancy)-based closures have now largely replaced earlier, Kuo-type representations of moist convection based on the statistical equilibrium of water rather than energy. For simplicity and conceptual transparency, we here work with a somewhat more restrictive formulation of quasi-equilibrium which proves more amenable to analytical treatment. This formulation rests on two assumptions: 1. Moist convection always acts to maintain a moist adiabatic (virtual) temperature profile, and 2. Convection always acts to maintain the neutral buoyancy of air lifted from the subcloud layer to levels above cloud base. These general notions have been implemented in two ways. The first, and simplest, is to rigidly enforce both (1) and (2); this has sometimes been called “strict equilibrium.” The second involves relaxing the actual atmospheric state toward the satisfaction of (1) and (2) over some finite time scale; this is often referred to as “relaxed equilibrium.” The use of (1) has been advocated by Betts (1986), Emanuel (1987), Yano and Emanuel (1991), Neelin and Yu (1994), and Emanuel et al. (1994), among others. Assumption (2) was introduced by Raymond (1995) and applied by many others. In the spirit of simplicity, we will examine the behavior of small-amplitude perturbations to radiative-convective equilibrium states using approximations (1) and (2) above. By “small” we mean literally infinitesimal; at any rate, we do not here consider perturbations large enough to shut down deep convection anywhere. One working definition of “nonlinear” as applied to this problem is “sufficient to shut off deep

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convection somewhere in the perturbation.” As we shall see in section 7.4, the transition from “linear” to “nonlinear” thus defined may be fairly sharp.

7.3.3. Empirical Restrictions on the Validity of Statistical Equilibrium Before continuing, it is worthwhile to examine the theoretical and empirical bases for quasi-equilibrium in general and the restricted definition given by (1) and (2) above in particular. First, the neutral state for moist convection is one in which the temperature lies along a moist adiabat and for which boundary-layer air lifted vertically is precisely neutral. Convection is a comparatively fast process, and it is not unreasonable to assume that it is efficient in driving the observed state toward criticality, much as the dry convective boundary layer is always observed to have a dry adiabatic lapse rate, except very near the surface. But there are several potential problems with moist criticality. First, it may be objected that real convective clouds are observed to entrain environmental air, greatly reducing their average buoyancy; thus a state that is neutral to strictly adiabatic ascent would be quite stable to an entraining plume. Second, there is the nontrivial question of how to define a moist adiabat. The buoyancy of air lifted reversibly, for example, will be noticeably less than air lifted pseudo-adiabatically,1 in which the condensed water loading is absent. There is also the problem of how and if to include the ice phase, since supercooling of liquid water is common in clouds and thus phase equilibrium cannot be assumed. Yet observations of real convective clouds show that they are poorly modeled as classical, entraining plumes. In the latter, studied extensively in the laboratory (Morton et al. 1956), entrained fluid is assumed to be instantly homogenized across the plume, yielding a single buoyancy or a one-parameter buoyancy distribution. Real clouds are, however, highly inhomogeneous, containing a broad spectrum of mixtures, including some air that has ascended nearly adiabatically (Paluch 1979). This observation nicely explains the paradox identified by Warner (1970), that real clouds are both highly dilute and ascend nearly to levels predicted by adiabatic ascent of subcloud air. Tropical (and many extratropical) temperature soundings are, to a first approximation, moist adiabatic. Figure 7.1 is a buoyancy diagram, plotting the difference between the density temperature2 of a reversibly3 lifted parcel and that of its environment as a function of the level from which the parcel is lifted (abscissa) and the level to which it is lifted (ordinate). The ice phase has been ignored here, and the buoyancy has been averaged over an entire year (about 1400 soundings) from Kapingamarangi in the far western tropical Pacific. Note that the environment is precisely neutral (to within measurement error) to a parcel lifted reversibly from around 940 hPa. This seems to be a ubiquitous feature of the Tropics (Xu and Emanuel 1989), though one may question the relevance of the reversible adiabatic process. Figure 7.2 shows a time series from the same tropical station comparing the density temperature of a sample lifted reversibly from 950 hPa to the pressure-weighted vertical mean density temperature of its environment.

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Quasi-Equilibrium Dynamics | 193 Mean Reversible Density Temperature Difference (K ) −3−2

−1

−3−2 0

−1

−3

200

0

−2

1

−3

400

−1 −2

1 1

0

600

−1

Pressure to Which Parcel is Lifted (mb)

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800 0

−1 0

0 0

1000 1000

0

980

960

940

920

900

Parcel Origin Pressure (mb)

Figure 7.1. Contour plot of the difference between the environmental density temperature and the density temperature of a parcel lifted reversibly and adiabatically from the pressure level given on the abscissa to the pressure level on the ordinate. This quantity has been averaged over several thousand soundings taken at Kapingamarangi in the far western tropical Pacific.

A running average of 10 days has been applied to the lifted parcel density temperature, while the vertically averaged density temperature of the environment has been smoothed over 2 days.4 There is a reasonably good correspondence between fluctuations of the two quantities, with a correlation coefficient of 0.52, suggesting that on these time scales, convective criticality is a fairly good approximation. When the quantities are smoothed over shorter times, the correspondence is less good, though it is not clear whether this is more because criticality is a weaker approximation or because of measurement error. (The lifted parcel temperature is particularly sensitive to the water content of the sample, which suffers from relatively large measurement error. For example, a relative humidity error of 5% in the boundary layer yields a 2.5 K error in the temperature of a parcel lifted to 200 hPa. Thus to reduce the noise to a level comparable to the observed signal of order 0.5 K in the free tropospheric temperature, one needs to average over around 25 soundings, or 8 days.) Figure 7.3 compares the mean density temperatures of the lower and upper troposphere, smoothed over 2 days. Once again, there is good correspondence, with a correlation coefficient of 0.62, particularly of the low-frequency components, suggesting that assumption (1) above is reasonable. There is still considerable uncertainty about the space and time scales over which quasi-equilibrium may hold. Brown and Bretherton (1997) compared Microwave

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270

Density Temperature (K)

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268

Environment Parcel 267

0

100

200

300

400

Days Starting July 1

Figure 7.2. A 2-day running mean of the density temperature averaged between 900 and 200 hPa, and a 10-day running mean of the density temperature of a sample lifted reversibly from 940 hPa, also averaged between 900 and 200 hPa. The time series at Kapingamarangi runs from 1 July 1992 to 30 June 1993.

Sounding Unit (MSU) temperature observations to measurements of boundary layer entropy fluctuations and concluded that strict equilibrium can only be applied on time scales longer than about 40 days. Islam et al. (1993) analyzed space-time variability of precipitation in an explicit three-dimensional simulation of radiative-convective equilibrium and concluded that averaging over several hours and around 100 km is necessary to appreciably reduce the variance produced by the chaotic behavior of individual convective cells. To illustrate the process by which perturbations to moist equilibrium decay, we first ran a single-column model into a state of radiative-convective equilibrium and then introduced a small perturbation to the equilibrium state. The single column model uses a parameterization of moist convection developed by Emanuel and Živkovi´cRothman (1999) as well as a sophisticated radiative-transfer formulation, and applies a fixed ocean temperature and time-invariant clouds. Figure 7.4 shows the decay with time of a temperature perturbation of peak magnitude 3 K introduced into the equilibrium state at 450 hPa. The temperature perturbation spreads vertically while decaying over a time scale of roughly 12 hours, but it also clearly has an oscillatory component with a period of around 5 days. Note that this relaxation is purely owing to convection and radiation; as this is a single column model, there is no large-scale circulation. (In the real world, a spatially localized temperature perturbation would also produce internal waves that, likewise, would tend to remove the perturbation over time.)

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287 Density Temperature (K)

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285 Lower troposphere T Upper troposphere T + 34K 284

0

100

200

300

400

Days Starting July 1

Figure 7.3. A 2-day running mean of the density temperature averaged between 900 and 500 hPa (solid) and between 545 and 200 hPa (dashed); an offset of 34 K has been added to the latter. The time series at Kapingamarangi runs from 1 July 1992 to 30 June 1993.

Taken together, these findings suggest that the assumption of convective neutrality fails on time scales shorter than around 2 days and space scales less than around 100 km, but a more precise determination of these limiting scales awaits further research.

7.3.4. Implications of the Moist Adiabatic Lapse Rate for the Structure of Tropical Disturbances Now consider the structure of hydrostatic perturbations to radiative-convective equilibrium states in which moist adiabatic lapse rates are strictly enforced. We shall slightly simplify the definition of “moist adiabatic” to the state in which the saturation moist entropy, s ∗ , is invariant with altitude. This quantity is defined as the specific entropy air would have were it saturated with water vapor at the same temperature and pressure. It is given approximately by s ∗ = c p ln (T/T0) − Rd ln ( p/ po) +

Lν q∗ , T

[7.2]

where T0 is a reference temperature, po is a reference pressure, and the other symbols are as defined previously 5. We shall now show that the requirement that s ∗ remain constant with height places strong constraints on the structure of perturbations.

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100 200 300

Pressure (mb)

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400 500 600 700 800 900 1000

1

2

3

4

5

6

7

8

9

Time (days)

Figure 7.4. Evolution with time of the vertical profile of perturbation temperature in a singlecolumn model in which the initial temperature has been perturbed away from equilibrium by a maximum of 3 K, centered at 450 hPa. Solid lines indicate positive values, with a contour interval of 0.2 K, while dashed lines indicate negative values, with a contour interval of 0.05 K.

Denoting perturbations from radiative-convective equilibrium by primes, the hydrostatic equation in pressure coordinates may be written ∂φ  = −α , ∂p

[7.3]

where φ is the geopotential and α is the specific volume. Ignoring virtual temperature effects, we regard α as a function of the two state variables p and s ∗. Fluctuations of α at constant p can then be written     ∂α ∂T  ∗ α = s = s ∗, [7.4] ∂s ∗ p ∂p s∗ where in the second part we have made use of one of Maxwell’s relations (Emanuel 1994). The quantity in brackets on the far right side is just the moist adiabatic lapse rate. Substituting (7.4) into (7.3) and integrating in pressure gives   φ (x, y, p, t) = φb (x, y, t) + T (x, y, t) − T (x, y, p, t) s ∗(x, y, t), [7.5] where T is the pressure-weighted vertical mean temperature in the troposphere. We have chosen the integration constant so that φb is the pressure-weighted vertically averaged

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geopotential perturbation in the troposphere. Thus we can interpret φb as the barotropic component of the geopotential perturbation. This shows that in a moist adiabatic atmosphere, geopotential perturbations associated with hydrostatic motions consist of a barotropic part plus a “first baroclinic mode” 6 contribution with a single node in the troposphere. This latter has a vertical structure dictated by the shape of a moist adiabat. The restrictive vertical structure of geopotential perturbations indicated by (7.5) also constrains the structure of horizontal velocity. In particular, the linearized inviscid momentum equations show that both horizontal velocity components must have a structure identical in form to (7.5). Integrating the hydrostatic version of the mass continuity equation   ∂ω ∂u ∂ν =− + ∂p ∂x ∂y with height (pressure) gives  ω = ( po − p)

 ∗     p0 ∂u ∂ν ∗ ∂ub ∂νb + − ( po − p)T − + , [7.6] T d p ∂x ∂y ∂x ∂y p

where ω is the pressure velocity, po is the surface pressure, and we have represented the baroclinic parts of the horizontal velocities u and ν with asterisks. Thus the structure of the vertical velocity is the accumulated difference between the temperature along a moist adiabat and its vertical mean value. By the definition of T , the second term on the right of (7.6) vanishes when ω is evaluated at the tropopause, so  ∂ub ∂νb ωt = ( po − pt) + , ∂x ∂y 

[7.7]

where the subscript t denotes evaluation at the tropopause. Thus if we apply a rigid lid at the tropopause, we then require that the divergence of the barotropic component of the velocity vanish. In the linear undamped system, there is then no coupling between the baroclinic and barotropic motions, and the latter are independent of the former. We shall see later that the barotropic component must be present if we allow wave radiation into the stratosphere, and/or if the baroclinic and barotropic components are coupled through surface friction or nonlinearity. The observed tendency of the tropical atmosphere to maintain moist adiabatic lapse rates is consistent with the prominence of first baroclinic mode structure in tropical disturbances, as first noted by Reed and Recker (1971) and Madden and Julian (1971, 1972) among others, and collapses the linear inviscid primitive equations into shallow-water equations. What is left to determine is the feedback of perturbation motions on the temperature.

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7.3.5. Feedback of Air Motion on Temperature In the classical theory of dry wave motions in the tropical atmosphere, first formulated by Matsuno (1966), the wave motions influence temperature by adiabatic warming and cooling associated with vertical air motion. Here we regard as canonical the problem of the disposition of an infinitesimal internal wave launched into a state otherwise in radiative-convective equilibrium, with the restriction that we must consider waves whose space and time scales are large compared to characteristic scales associated with the convection, so that statistical equilibrium may be assumed. It is clear from the outset that such perturbations cannot even crudely be considered (dry) adiabatic. But for the sake of simplicity, in what follows we shall neglect all diabatic effects except for those associated with the phase changes of water. Convection only redistributes enthalpy and so cannot alter the mass-weighted vertical enthalpy given by (7.1). Thus the moist static energy h, given by the sum of the specific enthalpy and the specific potential energy, can only be changed by large-scale advection. In the absence of horizontal advection and perturbations to the surface enthalpy flux and the radiative cooling, the evolution of the vertically integrated enthalpy is given by   ∂ ∂h kd p = − ω dp, [7.8] ∂t ∂p where the integral is over the depth of the troposphere and ω is the wave-associated vertical motion. Using tropical sounding data, Yu et al. (1998) showed that an assumed
profile of ω together with the observed structure function of ∂h ∂p in the Tropics yield negative integrated enthalpy tendency when air is ascending. From (7.1), we have    ∂ ∂T ∂q [7.9] c p dp = k dp − L ν dp. ∂t ∂t ∂t One observes that during ascent, the troposphere generally moistens7, so that the right side of (7.9) is negative definite. This shows that temperature (and therefore s ∗) decreases during ascent and increases during descent, yielding what Neelin and Held (1987) referred to as “effective stratification.” It has since been shown by Back and Bretherton (2005) that observed profiles of ω and ∂h/∂p in certain regions, such as the intertropical convergence zone in the Pacific, yield negative values of the effective stability, but these usually cover small areas. There is an alternative way to look at this, perfectly consistent with the above, advocated by Yano and Emanuel (1991). Suppose, on the one hand, that the perturbations to the convection yield no perturbations to the surface precipitation. In that limit, there can be no perturbation to the vertically integrated convective heating, and thus the adiabatic temperature changes associated with vertical motion would be unopposed, and the wave would “feel” the dry stratification associated with an atmosphere with a moist adiabatic temperature profile. This stratification can be characterized by N 2,

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where N is the buoyancy frequency. At the opposite limit, suppose all the water that condenses in clouds falls to the surface as rain without any re-evaporation. In that case, there would be no downdrafts and no way to alter the specific entropy of subcloud-layer air. This, coupled with convective criticality, implies that there could be no change in free troposphere temperature. Thus the effective stratification would be zero, and no internal wave could exist. Yano and Emanuel (1991) proposed an interpolation between these limits, giving an effective stratification of 2 Neff = (1 − εp)N 2,

[7.10]

where ε p is the precipitation efficiency of the convective clouds. This quantity is unity if all the condensation in a column results in an increase in dry static energy and none is used to moisten the environment. It is generally thought to be sensitive to the relative humidity of the lower and middle troposphere. As will be shown explicitly in the next section, these results suggest that small perturbations to radiative-convective equilibrium for which the statistical equilibrium hypothesis is valid will behave like solutions to the shallow-water equations but with a reduced equivalent depth (Emanuel et al. 1994; Neelin and Yu 1994). (This also assumes that the tropopause can be treated as a rigid lid; this is in general not a good approximation, a point we shall return to later.) This prediction was, to a good approximation, verified in the observational analysis by Wheeler and Kiladis (1999), which revealed most of the low-frequency components of classical equatorial wave theory, but with substantially reduced phase speeds.

7.3.6. Quasi-Linear Equatorial System for the First Baroclinic Mode The preceding development can be codified in a relatively simple set of equations set on an equatorial β plane. The main approximations used here are that convection maintains a moist adiabatic lapse rate, so that (7.5) applies, that the tropopause can be treated as a rigid lid, and that friction acts linearly on the first baroclinic mode, so that we may neglect the barotropic component. This is likely not to be a good approximation, but we apply it here in the spirit of simplicity. Also, we neglect all advective nonlinearity, but retain nonlinearity in the surface fluxes, following the scaling and empirical analysis of Yano et al. (1995). The quasi-linear equations are ∂u ∂s ∗ = (Ts − T ) + βyν − r u, ∂t ∂x

[7.11]

∂ν ∂s ∗ = (Ts − T ) − βyu − r ν, ∂t ∂y

[7.12]

∂s ∗ d ˙ N2 = (εp M − w), Q r ad + ∂t m m

[7.13]

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h

∂s b = C k|V|(s 0∗ − s b) − (M − w)(s b − s m), ∂t

[7.14]

∂u ∂ν w + + = 0. ∂x ∂y H

[7.15]

The momentum equations (7.11) and (7.12), are derived using (7.5) and adding a linear damping of the first baroclinic mode. The thermodynamic equation for the free troposphere (7.13) uses the saturation entropy as the thermodynamic variable and includes the effects of radiative heating Q˙ r ad, vertical advection, and convective heating, given by εp M, where εp is the precipitation efficiency and M is the convective updraft mass flux. (In this equation, d and m are the dry and moist adiabatic lapse rates, and N is the (dry) buoyancy frequency.) It is important to note that the assumption of moist adiabatic lapse rates renders s ∗ not a function of height, so the momentum equations have the same mathematical form as the shallow-water equations. Equation (7.14) is an equation for the actual moist entropy of the subcloud layer, assumed to have a thickness h. The first term on the right is the surface entropy flux, where C k is the enthalpy exchange coefficient, |V| the magnitude of the surface wind, and s 0∗ the saturation entropy of the ocean surface. The second term on the right represents the import of low entropy into the subcloud layer by convective and nonconvective downdrafts; s m is a characteristic entropy of the middle troposphere. In the mass continuity equation (7.15), H is a characteristic half-depth of the troposphere. Note that the assumption that s ∗ is independent of height implies that the righthand side of (7.13) is also independent of height, constraining the vertical profile of εp M given a profile of the radiative cooling. At the same time, the convective mass flux that appears in (7.14) is conceptually evaluated at the top of the subcloud layer. For a fixed vertical profile of radiative cooling, and for constant precipitation efficiency, this would be some function of the mass flux that appears in (7.13); for simplicity, we represent both by a single function M. The set of equations (7.11)–(7.15) is augmented by making the assumption of convective neutrality: ∂s b ∂s ∗ = . ∂t ∂t

[7.16]

The system is now closed, except for the specification of Q˙ r ad, s 0∗, s m , and εp. Equating the left sides of (7.13) and (7.14) using (7.16) may be regarded as a closure for the convective updraft mass flux M. Raymond (1995) made a further simplification by neglecting h∂s b/∂t in (7.14), giving a simplified expression for the convective updraft mass flux: M = w + C k|V|

s 0∗ − s b . sb − sm

[7.17]

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Sobel and Bretherton (2000) make the additional approximation that ∂s ∗/∂t is small in (7.13) (the weak temperature gradient approximation, or WTG), and this effectively closes for both M and w, to wit s ∗−s

M=

C k |V| s b0−s mb + dNQ2r ad ˙

1 − εp

,

[7.18]

and s ∗−s

w=

εp C k |V| s b0−s mb + dNQ2r ad 1 − εp

˙

.

[7.19]

The evolution of the free tropospheric temperature, as represented by s ∗, is now governed by the momentum equations, with temperature slaved to vorticity, which is in turned slaved to w as determined by (7.19). The mean value of s ∗ can be chosen to ensure that the mass continuity equation is satisfied (Bretherton and Sobel 2002). Note that in this system, the convective mass flux is determined (through [7.18]) by the surface entropy flux, the radiative cooling of the troposphere, and the humidity of the free troposphere, as reflected in the value of s m , the middle troposphere entropy. As will be discussed in section 7.3.13, humidity variations probably also affect the precipitation efficiency ε p and thereby the convective mass flux in (7.18). The system given by (7.11)–(7.16) can be examined analytically by further linearization. An important aspect of this linearization is the treatment of the surface flux term in (7.14). One must account for the fact that many scales of motion contribute to the surface wind speed |V|. For example, a common linearization of this term about a mean zonal wind U is U |V| ∼ u, = 2 U + u2∗

[7.20]

where u is the perturbation zonal wind and u∗ is a measure of small-scale gustiness.

7.3.7. Slowly Varying Motions Forced by Sea Surface Temperature Anomalies An interesting application of quasi-equilibrium dynamics is to the steady response of the atmosphere to imposed sea surface temperature (SST) anomalies, as reflected by s 0∗. To illustrate this, we ignore friction, fluctuations of the radiative cooling, the precipitation efficiency, and the middle troposphere entropy, and we use the boundary-layer quasiequilibrium approximation (7.17) and linearize the surface fluxes, including the wind speed, according to (7.20). Nondimensionalizing all the variables gives a particularly simple set of equations. In the following, we have used (7.17) for the convective mass flux and (7.15) to relate vertical velocity to divergence: ∂s + yν = 0, ∂x

[7.21]

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∂s − yu = 0, ∂y

[7.22]

∂u ∂ν + + αu − χ s = −s 0, ∂x ∂y

[7.23]

where all the variables are nondimensional; α is a nondimensional parameter measuring the strength of the wind-dependence of the surface fluxes, and χ is a nondimensional parameter measuring the surface-flux damping of entropy perturbations. These can be combined into a single equation for the nondimensional saturation entropy: ∂s ∂s + αy − χ y 2s = −y 2s 0. ∂x ∂y

[7.24]

These are similar to the equations of Gill (1980), except that Gill formulated them in terms of the response of a shallow-water system to specified internal heating, whereas here we are only specifying the sea surface saturation entropy anomaly s 0. It is interesting that mathematically, s 0 plays exactly the same role here as specified heating does in Gill’s formulation, except that there is no equivalent of the α term in Gill; i.e., there is no equivalent of the wind effect on surface fluxes. Many solutions of (7.21)–(7.23) have been presented in the literature (without the α term); broadly, SST (and thus in s 0) anomalies on the equator give rise to vertical motion more or less coincident with the SST, with gyres to the north and south and flow perturbations to the east and west set up by, respectively, Kelvin and Rossby wave propagation. The effect of the α term is to offset the perturbation response upwind (with respect to the mean zonal wind) of the SST anomalies.

7.3.8. Transient Modes at Fixed Sea Surface Temperature If we make the same approximations and linearizations as in section 7.3.7 above but retain the time dependence and ignore the damping term χ and variability of the sea surface temperature, we obtain the set ∂u ∂s = + yν, ∂t ∂x ∂ν =δ ∂t



 ∂s − yu , ∂y

[7.25]

[7.26]

and ∂u ∂ν ∂s = + + αu, ∂t ∂x ∂y

[7.27]

where the new parameter δ measures departures from geostrophy of the zonal wind. If solutions that are periodic in x and t are assumed, the resulting equations can be combined into a single second-order equation in y whose solutions are in the

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form of parabolic cylinder functions provided an eigenvalue equation is satisfied. This equation gives the dispersion relation, whose solutions, in the absence of α, are exactly the equatorially trapped neutral modes found by Matsuno (1966). These consist of families of eastward propagating Kelvin waves, eastward and westward propagating mixed Rossby gravity waves, westward propagating Rossby modes, and eastward and westward propagating fast inertia-gravity modes. The WISHE term α destabilizes some of these modes, as found by Neelin and Held (1987) and Emanuel (1987) (see section 7.3.11).

7.3.9. Effect of Departures from Strict Equilibrium As shown in Fig. 7.4 and suggested by a range of studies, convection restores equilibrium on time scales that appear to lie between a few hours and a few days. The effects of this finite time scale were examined by Emanuel (1993) and Neelin and Yu (1994) and found to have a damping effect on wave motions, proportional to their frequency. This was called “moist convective damping” by Emanuel et al. (1994), who gave a heuristic description of the effect. For example, consider a moist Kelvin wave propagating eastward on the equator. In strict equilibrium (and neglecting diabatic effects other than convection), the convection is exactly in phase with the wave vertical motion and thus in quadrature with the wave temperature perturbations. Allowing the convection to respond over a finite time scale to the wave, the convection lags behind the vertical motion field, displacing it into the cold phase of the wave. The resulting negative correlation between wave perturbation temperature and perturbation convective heating then acts to damp the wave. The magnitude of the phase shift depends on the ratio of the convective response time to the wave period, so that highfrequency waves are preferentially damped. This may help explain the very red spectrum found in the Tropics (Wheeler and Kiladis 1999). Although the assumption of deep convective neutrality leads to a greatly simplified mathematical and conceptual description of tropical dynamics, there is increasing observational evidence that it is at least incomplete. A robust prediction of convective neutrality is the dominance of the first barotropic and first baroclinic structures (but see section 7.3.10 below, which describes an additional complication), yet observations suggest an important role for higher-order structure, particularly the second baroclinic mode structure (Wheeler et al. 2000; Straub and Kiladis 2003). This presents a competing potential explanation for the relative slow phase speeds observed in tropical waves, first advanced by Mapes (2000). This holds that the observed phase speeds are close to that of second baroclinic-mode structures, which excite primarily shallow convection with zero or small precipitation efficiency and thus have effectively dry dynamics. This hypothesis successfully addresses some of the problems that arise from strict neutrality physics, and at the time of this writing is receiving increasing attention in theoretical and observational analyses.

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7.3.10. Taking the Lid Off The tropopause is not a physical barrier but represents a rather large change from the small effective stability of the tropical troposphere to the large stability of the stratosphere. The classical theory of (dry) equatorial waves may be supposed to apply in the stratosphere, and much work has been done on this topic, owing partially to the importance of upward-propagating waves in driving the quasi-biennial oscillation of the equatorial stratosphere (Baldwin et al. 2001). In general, higher frequency waves also lose energy more rapidly to the stratosphere. Yano and Emanuel (1991) coupled a linear moist model of the tropical troposphere to a dry stratosphere, applying a wave radiation condition to the top, and showed that all but the lowest frequency moist waves of the troposphere are strongly damped by upward wave radiation. This is another effect that undoubtedly reddens the tropical tropospheric wave spectrum, but other than the above cited work, very little analytical research has been performed on this subject. 7.3.11. WISHE It is important to note that in the statistical equilibrium theory of convection reviewed here, the interaction between large-scale circulations and convection per se is a stable one, leading to neutral or damped oscillatory solutions of linear equations that, when a rigid lid is applied at the tropopause, are mathematically identical to the shallow-water equations. Observed circulations in the Tropics must therefore originate in stochastic forcing of the equatorial waveguide, in physical processes that serve to destabilize the oscillations, in externally forced gradients of the sea surface temperature, or in coupled instabilities of the ocean-atmosphere system. One mechanism for internally destabilizing tropical perturbations takes advantage of the large reservoir of potential energy inherent in the thermodynamic disequilibrium that normally exists between the tropical oceans and atmosphere. (This disequilibrium is responsible for hurricanes, for example.) On reasonably short time scales, the ocean temperature may be assumed fixed, and fluctuations in the surface enthalpy flux will then be largely owing to fluctuations in near-surface wind speed. Neelin and Held (1987) and Emanuel (1987) showed that such an interaction, in the context of strict statistical equilibrium, destabilizes many of the moist equatorial modes, though their growth rates maximize at small scales. (Neelin and Held referred to this destabilization as “wind-evaporation feedback,” while I have referred to it as “wind-induced surface heat exchange,” or WISHE.) Later work showed that either allowing wave radiation into the stratosphere (Yano and Emanuel 1991) or accounting for the finite time scale of convection (Emanuel 1993) damps the higher frequency modes, yielding maximum growth rates at synoptic to planetary scales. In particular, Emanuel (1993) showed that the surviving unstable modes are primarily planetary scale eastward-propagating Kelvin modes, eastward- and westward-propagating mixed Rossby-gravity modes, and westward-propagating Rossby modes. This is well in accord with the later findings of

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Wheeler and Kiladis (1999), analyzing space-time spectra of satellite-measured outgoing longwave radiation. Yet Madden-Julian Oscillation (MJO)-like modes, initially thought to be Kelvin-like but now believed to be separate phenomena, are not among those predicted by WISHE theory.

7.3.12. Cloud-Radiation and Water Vapor-Radiation Interactions Tropical weather systems produce very large perturbations of outgoing longwave radiation owing to the thick, high cloud usually associated with them. As is well known, on seasonal and longer time scales, there is a large cancellation between these longwave effects and the additional reflection of sunlight by such clouds, but on short time scales, the latter has little bearing owing to the thermal capacity of the upper ocean. There is also some evidence that on subseasonal time scales, the shortwave effects dominate (Lin and Mapes 2004). From a strictly thermodynamic point of view, the perturbations to column heating by clouds are at least as large as those owing to perturbations in surface enthalpy flux. Yet until quite recently there has been little attempt to understand how cloud radiative effects might affect intraseasonal variability, though to be sure, such effects are represented, one way or another, in most global models. Nilsson and Emanuel (1999) integrated a two-column model of the nonrotating tropical atmosphere using only clear-sky radiative transfer and a convective scheme with a hydrologic cycle. They found that the radiative-convective equilibrium spontaneously breaks down into a moist, ascending column and a dry, descending column, owing to differential absorption by water vapor. (Their model allowed the surface temperature of a slab ocean to vary in accord with the surface energy balance.) Raymond (2000) examined the stability of a zonally symmetric equatorial atmosphere overlying an ocean with fixed, constant sea surface temperature and found that cloud and water vapor interactions with radiation lead to a spontaneous Hadley-like circulation with ascent on one side of the equator and descent on the other. Fuchs and Raymond (2002) built a simple analytical model of a nonrotating atmosphere with cloud-radiation feedbacks and found that these destabilize a variety of modes and reduce the effective stratification for longwave disturbances. Bretherton and Khairoutdinov (2004) ran a cloud-resolving numerical model with cloud-radiation interactions included into a state of statistical radiative-convective equilibrium and found spontaneous aggregation of convection into O(100 km) mesoscale clumps owing specifically to cloudradiation interaction. Bony and Emanuel (2005) undertook a linear stability analysis of a nonrotating atmosphere to two-dimensional disturbances with parameterized cloud-radiation interactions and, similar to Fuchs and Raymond (2002), showed that these slowed down the propagation of longwaves, but also found that such interactions destabilize very short nonpropagating modes. Based on these studies, it is clear that cloud and water vapor interactions with radiation have a profound effect on intraseasonal variability in the Tropics.

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Unfortunately, our understanding of and ability to parameterize such effects are still poor, and representations used in global models vary widely, perhaps accounting for the very different intraseasonal variability observed in different global models (Slingo et al. 1996). Clearly this is an important and potentially fruitful area for further research.

7.3.13. Role of Humidity Fluctuations Above the Boundary Layer The quasi-equilibrium theory of tropical convection delineates a clear dependence of convection and large-scale ascent on the relative humidity of the lower and middle troposphere. Broadly, to balance a given surface enthalpy flux, there have to be more or stronger convective downdrafts if the entropy of the middle troposphere (the source region of the downdrafts) is higher, implying more convection. The WTG equations (7.18) and (7.19) make it clear that, all other things being equal, there will be more convection and more large-scale ascent in regions where the middle troposphere entropy s m is larger. Moreover, the precipitation efficiency (ε p) is almost certainly a function of the relative humidity of the troposphere, being larger in a more humid atmosphere where less evaporation of precipitation occurs, and this works in the same direction. On the other hand, it is observed that ascending regions are generally more humid than regions where air is descending on the large scale, implying a possible feedback between mid-level moisture and convection. This feedback has come to be known as “moistureconvection” feedback and was perhaps first detected in the two-dimensional cloudresolving simulations by Held et al. (1993). They found that when vertical wind shear is suppressed, moisture accumulates in one small region and the moist convection locks on to that region, being suppressed elsewhere. But modest amounts of shear disrupt the moisture anomaly and randomize the cloud field. Apparently, this feedback is not strong enough to organize moist convection in three dimensions. According to the work of Bretherton and Khairoutdinov (2004), self-aggregation does not occur in the absence of cloud radiative feedbacks. But Grabowski and Moncrieff (2004) find that that moistureconvection feedback is essential to obtain large-scale organization of convection in their aqua planet, constant sea surface temperature model. Since moisture-convection feedback is sensitive to small-scale processes such as turbulent entrainment and cloud microphysics, the magnitude of this effect is bound to depend on the nature of the model used. As in the case of cloud-radiation interaction, this is an area ripe for further research.

7.3.14. Coupling to the Ocean At time scales longer than a few days, the ocean cannot be considered to be an infinite heat capacitor and one must account for the energy budget of at least the ocean’s mixed layer. The tropical oceans exhibit substantial variability on intraseasonal time scales (Krishnamurti et al. 1988) and there is evidence that coupling to the ocean may be

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essential for explaining aspects of this variability (Sobel and Gildor 2003; Maloney and Sobel 2004). At even longer time scales, the dynamics of the ocean come into play, as for example with El Niño/Southern Oscillation.

7.4. Behavior of Finite Amplitude Perturbations The quasi-linear theory of small perturbations to convective atmospheres, based on quasi-equilibrium ideas, may be expected to be approximately valid as long as the perturbations do not become so strong that deep convection is completely annihilated in certain regions. Once this happens, the convection-free regions are not constrained to be convectively neutral, and the physics that limits the strength of the circulations changes. A simple model of the Walker circulation, based on quasi-equilibrium and WTG, suffices to illustrate some of the issues involved. Models like this have been developed by Pierrehumbert (1995), Miller (1997), Nilsson and Emanuel (1999), Larson et al. (1999), Clement and Seager (1999), Kelly and Randall (2001), and Shaevitz and Sobel (2004). This model represents the circulation in terms of two boxes, one over relatively warm ocean and the other over relatively cold ocean, as illustrated in Fig. 7.5. For simplicity, we take the boxes to have the same dimensions, and look for steady solutions to the quasi-equilibrium WTG equations. There are two possible regimes: one in which deep convection occurs in both boxes (Fig. 7.5a) and one in which deep convection is suppressed over the cold water (Fig. 7.5b). According to the postulate of convective neutrality, we insist on moist adiabatic temperature lapse rates in both boxes in the first regime and over the warmer water in the second regime. But WTG also insists that there be no temperature gradient in the free troposphere between the two boxes. Taken together, these two postulates mean that we can represent the temperature in both boxes, in both regimes, as a single value of the saturation entropy s ∗. Convective neutrality also stipulates the boundary layer entropy s b be equal to the saturation entropy of the free troposphere wherever there is deep convection. But in the second regime, the cold box does not contain deep convection and we allow s b there, s bc, to be lower than the free troposphere s ∗. For simplicity, we omit WISHE, cloud-radiation, and moisture-convection feedbacks simply by taking C D|V| = constant, d Q˙ r ad ≡ −R = constant, N2 εp = constant, and s b − s m ≡ s = constant.

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b

S*

S*

Sb

Sb S0w

S0c

S*

S*

Sb S0w

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Figure 7.5. Illustrating the two regimes of a two-box model of the Walker circulation. In (a), deep convection is assumed to occur in both boxes, while in (b) it is assumed to occur only over the warmer water, at left. The saturation entropy (i.e., the temperature) of the ocean surface is specified as s 0w for the warm column at left and s 0c for the cold column at right. The free troposphere temperature in the warm box is assumed to lie along a moist adiabat, so that the saturation entropy s ∗ is constant with height; the weak temperature gradient (WTG) approximation holds that the free troposphere temperature (and therefore s ∗) must be the same over the cold water. In the warm regime, convective neutrality specifies that the boundary-layer entropy s b must equal the saturation entropy of the overlying troposphere in both boxes, but in the cold regime the boundary-layer entropy s bc in the cold box is less than s ∗, signifying stability to deep moist convection.

We then apply the WTG equations (7.18) and (7.19) to each box of the first regime. Mass continuity demands that wwar m = −wc ol d, and this condition, together with (7.18) and (7.19), determines the system saturation entropy, s ∗. It is convenient to replace the variables in this problem by nondimensional counterparts according to w→

R w, 1 − εp

M→

R M, 1 − εp

s →

Rs s. C D|V|εp

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Then in the first regime, where deep convection occurs in both boxes, the solutions for the single value of the entropy (which is the saturation entropy of both boxes and the boundary-layer entropy of both boxes) and the vertical motion and convective mass fluxes in each box are 1 (s 0w + s 0c) − 1, 2 1 ww = −wc = (s 0w − s 0c), 2 1 1 (s 0w − s 0c), Mw = − 1 + εp 2εp s =

Mc =

1 1 −1− (s 0w − s 0c). εp 2εp

Thus the system entropy is just the average of the saturation entropy of the sea surface in both boxes, minus a constant related to the radiative cooling of the troposphere. The large-scale circulation increases linearly with the imposed ocean saturation entropy difference, while the mass flux over the cold water decreases linearly with same. The mass flux over the cold water will decrease to zero when the saturation entropy difference between the two boxes exceeds the critical value 2(1 − εp). At that point, the solutions above are no longer viable and we must confront a different balance over the cold water. When deep convection is absent over the cold water, subsidence warming of the free troposphere must balance radiative cooling. In dimensional terms, this means that wc = −R. At the same time, the surface entropy flux in the cold box must be balanced by the downward advection of low entropy from above the boundary D|v| layer, in dimensional terms, −wc = Cs (s 0c − s c). These two conditions determine the boundary-layer entropy of the cold box s c. Since s c now differs from s w, we have to allow for advection of entropy from the cold box to the warm box. At the same time, the original WTG equations must apply in the warm box, modified by the horizontal entropy advection and, of course, wwar m = −wc ol d. Using the same nondimensionalizations as before, the solution in this regime is given by wc = −ww = −(1 − εp), s c = s 0c − εp, sw = s ∗ = Mw =

s 0w + εp − 2 + χ (s 0c − εp) , 1+χ

2(1 − εp) , εp

Mc = 0,

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where χ ≡ R/C k|V|. (This measures the relative strength of horizontal entropy advection.) Clearly, the circulation no longer increases with increasing sea surface temperature difference in this regime: the magnitude of the circulation is absolutely limited by the radiative cooling over the cold water. Reducing the temperature of the cold water reduces both the system entropy and the boundary-layer entropy over the cold water, but does not affect the circulation strength. In the real word, though, the relative areas covered by the deep-convection and no-convection regimes are free to vary, and so there may be additional increases in the strength of the circulation with increasing SST gradient. Bretherton and Sobel (2002) took this into account in a similar model of the Walker circulation, but one that is horizontally continuous. They also showed that the interaction between high clouds and radiation is an important effect in such circulations when the sea surface temperature is specified, although the cancellation of the shortwave and longwave effects reduces its importance when the sea surface temperature is allowed to respond to the surface energy balance. (A review of these considerations may be found in Sobel et al. [2004].) Quantitatively, it takes little sea surface temperature gradient to shut off deep convection over the colder water. Once this happens, the magnitude of the circulation becomes rate-limited by the amount of radiational cooling that can occur in the dry regions. The fact that most tropical circulations are strong enough to shut off deep convection in various regions perhaps limits the utility of the theory of linear perturbations to radiative-convective equilibrium states, though it does not, in and of itself, invalidate the notion of quasi-equilibrium. But it does mean that the separation between moist ascending regions and dry subsiding regions, on a large scale, is an important and irreducibly nonlinear aspect of many tropical circulation systems. A possible exception to this conclusion is the case of tropical cyclones. The existing theory (see Emanuel [2003] for a review) shows that the intensity of the circulation is limited by surface fluxes in the high-wind core near the eyewall with no rate-limiting role played by the broad descent outside the core. This is no doubt owing to the circular geometry of these storms, which allows for broad outer regions of descent to compensate for a narrow, intense plume of ascent at the core.

7.5. Some Examples of the Application of Quasi-Equilibrium Closure To illustrate the power of quasi-equilibrium, I here show two rather different examples of the application of quasi-equilibrium dynamics to the tropical atmosphere. The first is a simple model of the equatorial beta plane on an aqua planet, and the second is the real-time prediction of hurricane intensity.

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7.5.1. A Simple Quasi-Equilibrium Model of the Equatorial Waveguide This model simply integrates the first baroclinic mode equations (7.11)–(7.15), ignoring any contribution from the barotropic mode. (For this reason, it effectively assumes a rigid lid and ignores the coupling of baroclinic and barotropic motions through surface friction or nonlinearity.) An equilibrium convective updraft mass flux is defined using boundary layer quasi-equilibrium, according to (7.17), and the actual convective updraft mass is relaxed toward this equilibrium value over a finite time to allow for the effects of nonzero convective response time, as discussed in section 7.3.9. Cloudradiation interactions are parameterized as a function of the middle troposphere relative humidity, following Bony and Emanuel (2005), and the precipitation efficiency is also a function of the middle troposphere relative humidity. The model is run in an equatorially centered channel, from 40o S to 40o N, with only 42 grid points in each direction. For the simulations presented here, the underlying sea surface temperature is specified as a zonally symmetric profile with a broad, flat peak at the equator. Figure 7.6 shows the equatorially symmetric and asymmetric power spectra of the vertical velocity averaged over 120 days. On these plots, positive zonal wavenumbers correspond to eastward propagation, while negative wavenumbers show westward propagation. The plot of the symmetric component (Fig. 7.6a) shows a prominent Kelvin wave signal corresponding to eastward propagation at about 20 m s−1, while the asymmetric component (Fig. 7.6b) shows a mixed Rossby-gravity signal. Power spectra of the asymmetric meridional wind (not shown) also display prominent planetary Rossby modes. These diagrams are quite similar to the power spectra of outgoing longwave radiation produced by Wheeler and Kiladis (1999), but the Madden-Julian Oscillation (MJO) is notably absent. The variability evident in Fig. 6 is caused in the model mostly by WISHE and cloud-radiation interactions.

7.5.2. Tropical Cyclones In some ways, the inner core of a tropical cyclone provides a severe test for quasiequilibrium.8 The intrinsic dynamical space and time scales of the inner core are related to the eye diameter and the rotational frequency νmax/r max, the ratio of the maximum tangential wind to the radius of maximum winds. These space and time scales are not very different from those associated with cumulonimbus clouds, and so one should be concerned whether the convection ever approaches statistical equilibrium with the cyclone. Nevertheless, a simple model based on quasi-equilibrium simulates actual tropical cyclones well enough that it is used as a forecasting aid by operational prediction centers. The model in question is called the Coupled Hurricane Intensity Prediction System (CHIPS) and is described in detail in Emanuel et al. (2004). Quasi-equilibrium

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Figure 7.6. Frequency-wavenumber power spectra of the (a) equatorially symmetric and (b) asymmetric parts of the vertical velocity averaged over a 120-day integration of an equatorial channel model in which the troposphere is assumed to have a moist adiabatic lapse rate, only the first baroclinic mode is retained, and convection is represented according to a relaxed version of boundary-layer quasi-equilibrium. Positive wavenumbers indicate eastward wave propagation.

is implemented in this model by assuming that the free tropospheric temperature lapse rate is always moist adiabatic on surfaces of constant absolute angular momentum about the storm’s central axis. This is tantamount to assuming that in a strongly baroclinic environment, such as the eyewall of a hurricane, slantwise (rather than upright) convective neutrality is maintained. No first-baroclinic-mode assumption is made; instead, angular momentum conservation is invoked at the tropopause as an additional constraint, and the partitioning between first baroclinic and barotropic modes is made naturally by the model itself. As in the equatorial waveguide model described in

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120 Maximum Surface Wind Speed (knots)

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Figure 7.7. Hindcast of the evolution with time of the maximum surface winds in Atlantic Hurricane Charley of 2004 (dashed) compared to estimates from observations (solid). The simulation was done using the Coupled Hurricane Intensity Prediction System (CHIPS), a model that uses a simple quasi-equilibrium convection scheme. The hindcast uses the actual track of the hurricane to estimate environmental conditions.

section 7.5.1, relaxed boundary-layer quasi-equilibrium is employed, although in this case it is necessary to account for horizontal entropy advection within the boundary layer. In the limiting case of a tropical cyclone reaching its potential intensity, the surface entropy flux is balanced by radial advection of entropy rather than by import of low entropy by convective downdrafts. Figure 7.7 shows a hindcast of a particular Atlantic hurricane. The hindcast has the advantage that the actual track is known, and the environmental shear is better known than in a forecast. The model does a good job simulating the intensity evolution of this and other events. This suggest that the assumption of quasi-equilibrium is, at any rate, not a terribly bad one even in a hurricane.

7.6. Summary and Concluding Thoughts Moist convection plays an important role in many and perhaps most tropical circulation systems. Convection itself is a deeply chaotic phenomenon with space scales of meters to kilometers and time scales of second to hours. On the other hand, the space and time scales of many tropical circulations systems are substantially larger, offering the hope that the fast, small-scale convective processes may be regarded as approximately

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in statistical equilibrium with the slower and bigger tropical circulations. In this review I discussed one popular rendition of statistical equilibrium in which it is assumed that convection maintains moist adiabatic lapse rates and the neutral stability of the tropical troposphere with respect to adiabatically lifted air from the subcloud layer. This is not the only form of statistical equilibrium that has been tried, nor is it necessarily the best, but it does seem to work well for a variety of phenomena, even including tropical cyclones. As pointed out by Arakawa and Schubert (1974), statistical equilibrium is a precondition for parameterization. Without some kind of equilibrium or near-equilibrium assumption, the collective effects of a small-scale process cannot be uniquely related to large-scale variables. Of course, convection in the real world may not be characterized as nearly in statistical equilibrium, at least in certain places at certain times. One of the supposed benefits of simulating the tropical atmosphere with convection-resolving models is the ability to do away with convective parameterization and all the baggage it carries, including the assumption of equilibrium. But it is important to be clear about this. To the extent that statistical equilibrium does not exist, then more than one realization of convection may occur for the same large-scale flow. If the feedback of convection on the large scale is important, this means that there may be more than one (and perhaps many) large-scale tendencies for exactly the same large-scale realization. Fundamentally, this means that the high-frequency chaos of small-scale convection is immediately felt at the large scale, and one must then confront a basic predictability problem. Explicitly resolving the unpredictable small-scale convection buys one nothing unless one is prepared to run large ensembles. Resolving the unpredictable is not obviously preferable to predicting the unresolvable. In its current state of development, the statistical equilibrium theory of moist convection makes certain testable predictions about tropical dynamics. First, it predicts that the available potential energy inherent in the vertically unstable stratification of the tropical atmosphere is not available to power large-scale disturbances. Indeed, the simple rendition of equilibrium as enforcing moist adiabatic lapse rates and boundarylayer quasi-equilibrium predicts that large-scale motions will be damped by their interaction with convection. Observed tropical atmospheric phenomena would then have to be powered by one or more of the following: • Sea surface temperature gradients • Energy propagation into the Tropics from higher latitudes • Wind-induced surface heat exchange (WISHE) • Cloud and moisture interactions with radiation • Feedbacks involving convection and middle-tropospheric moisture • Feedbacks involving the ocean and/or land surfaces

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Simple models based on statistical equilibrium postulates appear to be successful in explaining basic aspects of many tropical phenomena, ranging from tropical cyclones to the Walker circulation. On the other hand, no generally accepted paradigm for the Madden-Julian Oscillation has emerged, though there is no shortage of candidates. Among the many tropical processes that could profit from a better theoretical understanding are: • Coupling of the first baroclinic and barotropic modes, through surface friction

and through wave radiation into the stratosphere • The possible role of the second and higher baroclinic modes • The effects of convective momentum transfer • Cloud and moisture interactions with radiation • The effect of mid-level moisture on convection

As advancing computer power begins to allow us to resolve convective clouds and large-scale circulations simultaneously, it will become possible to rigorously test the quasi-equilibrium postulate, to refine our understanding of some of the processes mentioned above, and also to build better representations of those processes to run in faster and cheaper models.

Notes 1. In a pseudo-adiabatic process, condensed water is removed as soon as it forms. This affects both the density temperature (see note 2) and, because the heat capacity of the system is affected by condensed water, the actual temperature of the displaced sample. 2. The density temperature is defined so that multiplying by the gas constant for dry air and dividing by pressure gives the actual inverse density. Its relationship to temperature and water 
 substance is given by Tρ = T 1 + q ε − q t , where q is the mass concentration of water vapor, q t is the mass concentration of all water in the sample, and ε is the ratio of the molecular weight of water to that of dry air. 3. In a reversible displacement, entropy and total water mass are conserved, so there is no fall-out of condensed water. 4. The longer averaging of the lifted parcel temperature was found necessary owing to the larger noise associated with it; this is in turn due to the relatively large error in relative humidity measurement. 5. See Emanuel (1994) for an exact definition. 6. Although this has become a common description of this structure, it is not technically accurate as no actual normal modes are involved. 7. Observations suggest, however, that the maximum moistening precedes maximum precipitation and ascent in the Tropics (Sherwood and Wahrlich 1999). 8. On the other hand, convection is so strong in tropical cyclones that there is seldom any issue of convection shutting off, and shallow convection almost certainly plays less of a role here than elsewhere.

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References Arakawa, A. and W. H. Schubert (1974). Interaction of a cumulus cloud ensemble with the largescale environment, Part I. J. Atmos. Sci. 31, 674–701. Back, L. E., and C. S. Bretherton (2005). The relationship between wind speed and precipitation in the Pacific ITCZ. J. Climate 18, 4317–4328. Baldwin, M. P. and coauthors (2001). The quasi-biennial oscillation. Rev. Geophys. 39, 179–229. Betts, A. K. (1986). A new convective adjustment scheme. Part I: Observational and theoretical basis. Quart. J. Roy. Meteor. Soc. 112, 677–691. Bjerknes, J. (1938). Saturated-adiabatic ascent of air through dry-adiabatically descending environment. Quart. J. Roy. Meteor. Soc. 64, 325–330. Bony, S. and K. Emanuel (2005). On the role of moist processes in tropical intraseasonal variability: cloud-radiation and moisture-convection feedbacks. J. Atmos. Sci. 62, 2770–2789. Bretherton, C. S., P. N. Blossey and M. F. Khairoutdinov (2005). An energy-balance analysis of deep convective self-aggregation above uniform SST. J. Atmos. Sci. 62, 4273–4292. Bretherton, C. S. and M. F. Khairoutdinov (2004). Convective self-aggregation in large cloudresolving model simulations of radiative convective equilibrium. AMS Conference on Hurricanes and Tropical Meteorology, Miami, Amer. Meteor. Soc. Bretherton, C. S. and A. H. Sobel (2002). A simple model of a convectively coupled Walker Circulation using the weak temperature gradient approximation. J. Climate 15, 2907–2920. Brown, R. G. and C. S. Bretherton (1997). A test of the strict quasi-equilibrium theory on long time and space scales. J. Atmos.. Sci. 54, 624–638. Clement, A. C. and R. Seager (1999). Climate and the tropical oceans. J. Climate 12, 3383–3401. Deardorff, J. W. (1972). Numerical investigation of neutral and unstable planetary boundary layers. J. Atmos.. Sci. 29, 91–115. Emanuel, K. A. (1987). An air-sea interaction model of intraseasonal oscillations in the tropics. J. Atmos. Sci. 44, 2324–2340. Emanuel, K. A. (1993). The effect of convective response time on WISHE modes. J. Atmos. Sci. 50, 1763–1775. Emanuel, K. A. (1994). Atmospheric Convection, Oxford Univ. Press, New York. Emanuel, K. (2003). Tropical cyclones. Ann. Rev. Earth Plan. Sci. 31, 75–104. Emanuel, K., C. DesAutels, C. Holloway and R. Korty (2004). Environmental control of tropical cyclone intensity. J. Atmos. Sci. 61, 843–858. Emanuel, K. A., J. D. Neelin and C. S. Bretherton (1994). On large-scale circulations in convecting atmospheres. Quart. J. Roy. Meteor. Soc. 120, 1111–1143. Emanuel, K. A. and M. Živkovi´c-Rothman (1999). Development and evaluation of a convection scheme for use in climate models. J. Atmos. Sci. 56, 1766–1782. Fuchs Z., and D. J. Raymond (2002). Large-scale modes of a nonrotating atmosphere with water vapor and cloud–radiation feedbacks. J. Atmos. Sci. 59, 1669–1679. Gill, A. E. (1980). Some simple solutions for heat-induced tropical circulation. Quart. J. Roy. Meteor. Soc. 106, 447–462. Grabowski, W. W. and M. W. Moncrieff (2004). Moisture-convection feedback in the tropics. Quart. J. Roy. Meteor. Soc. 130, 3081–3104. Held, I. M., R. S. Hemler and V. Ramaswamy (1993).: Radiative-convective equilibrium with explicit two-dimensional moist convection. J. Atmos. Sci. 50, 3909–3927.

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Quasi-Equilibrium Dynamics | 217 Islam, S., R. L. Bras and K. Emanuel (1993). Predictability of mesoscale rainfall in the tropics. J. Appl. Meteor. 32, 297–310. Kelly, M. A. and D. A. Randall (2001). A two-box model of a zonal atmospheric circulation in the Tropics. J. Climate 14,(19) 3944–3964. Krishnamurti, T. N., D. K. Oosterhof and A. V. Mehta (1988). Air–sea interaction on the timescale of 30 to 50 days. J. Atmos. Sci. 45, 1304–1322. Larson, K., D. L. Hartmann and S. A. Klein (1999). The role of clouds, water vapor, circulation, and boundary layer structure in the sensitivity of the tropical climate. J. Climate 12,(8) 2359–2374. Lin, J.-L. and B. E. Mapes (2004). Radiation budget of the tropical intraseasonal oscillation. J. Atmos. Sci. 61, 2050–2062. Madden, R. and P. R. Julian (1971). Detection of a 40–50 day oscillation in the zonal wind in the tropical Pacific. J. Atmos. Sci. 28, 702–708. Madden, R. and P. R. Julian (1972). Description of global circulation cells in the tropics with a 40–50 day period. J. Atmos. Sci. 29, 1109–1123. Maloney, E. D. and A. H. Sobel (2004). Surface fluxes and ocean coupling in the tropical intraseasonal oscillation. J. Climate 17, 4368–4386. Mapes, B. E. (2000). Convective inhibition, subgrid-scale triggering energy, and stratiform instability in a toy tropical wave model. J. Atmos. Sci. 57, 1515–1535. Matsuno, T. (1966). Quasi-geostrophic motions in the equatorial area. J. Meteor. Soc. Japan 44, 25–42. Miller, R. L. (1997). Tropical thermostats and low cloud cover. J. Climate 10,(3) 409–440. Morton, B. R., G. I. Taylor and J. S. Turner (1956). Turbulent gravitational convection from maintained and instantaneous sources. Proc. Roy. Soc. London A234, 1–23. Neelin, J. D. and I. M. Held (1987). Modeling tropical convergence based on the moist static energy budget. Mon. Wea. Rev. 115, 3–12. Neelin, J. D., I. M. Held and K. H. Cook (1987). Evaporation-wind feedback and low-frequency variability in the tropical atmosphere. J. Atmos. Sci. 44, 2341–2348. Neelin, J. D. and J. Yu (1994). Modes of tropical variability under convective adjustment and the Madden-Julian oscillation. Part I: Analytical theory. J. Atmos. Sci. 51, 1876–1894. Nilsson, J. and K. A. Emanuel (1999). Equilibrium atmospheres of a two-column radiative convective model. Quart. J. Roy. Meteor. Soc. 125, 2239–2264. Paluch, I. R. (1979). The entrainment mechanism in Colorado cumuli. J. Atmos. Sci. 36, 2462–2478. Pauluis, O. and I. M. Held (2002). Entropy budget of an atmosphere in radiative-convective equilibrium. Part I: Maximum work and frictional dissipation. J. Atmos. Sci. 59, 125–139. Pierrehumbert, R. T. (1995). Thermostats, radiator fins, and the local runaway greenhouse. J. Atmos. Sci. 52, 1784–1806. Prandtl, L. (1942). Führer durch die Strömungslehre, Vieweg und Sohn, Braunschweig. Raymond, D. J. (1995). Regulation of moist convection over the west Pacific warm pool. J. Atmos. Sci. 52, 3945–3959. Raymond, D. J. (2000). The Hadley circulation as a radiative–convective instability. J. Atmos. Sci. 57, 1286–1297. Reed, R. J. and E. E. Recker (1971). Structure and properties of synoptic-scale wave disturbances in the equatorial western Pacific. J. Atmos. Sci. 28, 1117–1133. Robe, F. R. and K. Emanuel (1996). Dependence of tropical convection on radiative forcing. J. Atmos. Sci. 53, 3265–3275.

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218 | Kerry Emanuel Shaevitz, D. A. and A. H. Sobel (2004). Implementing the weak temperature gradient approximation with full vertical structure. Mon. Wea. Rev. 132, 662–669. Sherwood, S. C. and R. Wahrlich (1999). Observed evolution of tropical deep convective events and their environment. Mon. Wea. Rev. 127, 1777–1795. Slingo, J. M. and coauthors (1996). Intraseasonal oscillations in 15 atmospheric general circulation models: Results from an AMIP diagnostic subproject. Clim. Dyn. 12, 325–357. Sobel, A. H. and C. S. Bretherton (2000). Modeling tropical precipitation in a single column. J. of Climate 13, 4378–4392. Sobel, A. H., C. S. Bretherton, H. Gildor and M. E. Peters (2004). Convection, cloud-radiative feedbacks and thermodynamic ocean coupling in simple models of the Walker circulation. In Earth’s Climate: The Ocean-Atmosphere Interaction, ed. C. Wang, S.-P. Xie and J. A. Carton, Amer. Geophys. Union. 147, 393–405. Sobel, A. H. and H. Gildor (2003). A simple time-dependent model of SST hot spots. J. Climate 16, 3978–3992. Straub, K. H. and G. N. Kiladis (2003). The observed structure of convectively coupled Kelvin waves: Comparison with simple models of coupled wave instability. J. Atmos. Sci. 60, 1655–1668. Tompkins, A. M. (2001). Organization of tropical convection in low vertical wind shears: The role of water vapor. J. Atmos. Sci. 58, 529–545. Warner, J. (1970). On steady state one-dimensional models of cumulus convection. J. Atmos. Sci. 27, 1035–1040. Wheeler, M. and G. N. Kiladis (1999). Convectively coupled equatorial waves: Analysis of clouds and temperature in the wavenumber-frequency domain. J. Atmos. Sci. 56, 374–399. Wheeler, M., G. N. Kiladis and P. J. Webster (2000). Large-scale dynamical fields associated with convectively coupled equatorial waves. J. Atmos. Sci. 57, 613–640. Xu, K.-M. and K. A. Emanuel (1989). Is the tropical atmosphere conditionally unstable? Mon. Wea. Rev. 117, 1471–1479. Yano, J.-I. and K. A. Emanuel (1991). An improved WISHE model of the equatorial atmosphere and its coupling with the stratosphere. J. Atmos. Sci. 48, 377–389. Yano, J.-I., J. C. McWilliams, M. W. Moncrieff and K. Emanuel (1995). Hierarchical tropical cloud systems in an analog shallow-water model. J. Atmos. Sci. 52, 1723–1742. Yu, J. Y., C. Chou and J. D. Neelin (1998). Estimating the gross moist stability of the tropical atmosphere. J. Atmos. Sci. 55, 1354–1372.

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Chapter 8

Simple Models of Ensemble-Averaged Tropical Precipitation and Surface Wind, Given the Sea Surface Temperature Adam H. Sobel

8.1. Introduction 8.1.1. Definition of the Problem In this chapter we will address the following problem. Consider the sea surface temperature (SST) field as given, and find the quasi-steady component of the tropical surface wind and precipitation field over the tropical oceans. We do not explicitly address precipitation over land. By “quasi-steady” we mean, in principle, ensemble-averaged over all possible realizations associated with that SST, taking deep convection to have a stochastic component. In practice, monthly climatologies, or perhaps individual monthly means, will have to be good enough. We are interested in the total fields, not just deviations from the time or zonal means. We further specify that the problem must be solved without running a general circulation model (GCM). The model must be considerably simpler than that. More specifically, all the models we will discuss here have severely truncated vertical structure, to just one, two, or at most three vertical modes or layers. Their physical parameterizations are also all greatly reduced in complexity from what would appear in a GCM. Our goals are to review the various sorts of models of this type that have been constructed, and to assess both the degree to which these models are successful and to which they are similar to or different from one another. Models constructed on different principles seem to produce qualitatively similar results; we will try to explain this and to ascertain where differences in the results do exist that might be used to test the theoretical ideas underlying the models. The figures of merit here will be the precipitation and surface wind. The surface wind is important particularly because of its influence on the ocean. Much effort on models of the sort discussed here has been expended in the context of the prediction of the El Niño-Southern Oscillation (ENSO) phenomenon. In such models the

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precipitation, being proportional to the vertical integral of the convective heating, is of interest primarily through the influence of that heating on the winds. In the present discussion we are also interested in precipitation for its own sake. The questions of to what degree the heating associated with precipitation drives the surface winds, and to what degree convergence of the surface winds drives precipitating convection, will be closely considered. The discussion will focus in particular on what controls the strength and position of intertropical convergence zones (ITCZs). The limitation to the surface wind means that we do not need to find the free-tropospheric circulation, except to the extent that it may be necessary in order to find the surface circulation. This means that we will avoid questions regarding angular momentum conservation, which are discussed in detail in chapter 9 in this volume.

8.1.2. Observations We first review the salient features of the observations. Figure 8.1 shows climatological maps of SST, precipitation, and temperature at 500 hPa for January at tropical latitudes. Figure 8.2 shows surface wind and divergence (only negative values, i.e., convergence, are shown) from the same season for the tropical Pacific only. We see that to a crude first approximation, maxima in rainfall and SST coincide, and little or no rain falls over low SST. Temperature gradients at 500 hPa are small compared to those at the surface, so the horizontal structure of the difference in temperature between the sea surface and 500 hPa, which we might take as a crude measure of stability of the atmosphere to deep convection, is governed mainly by the horizontal structure of the SST. Focusing more closely on the Pacific, we see that in this season, surface convergence is larger in the eastern Pacific ITCZ than in the south Pacific convergence zone, but rainfall is greater in the latter.

8.1.3. Theories Theories for surface wind and precipitation are to some degree distinct, although in many cases, as is natural, one theory predicts both. We will consider them separately, although they are obviously related and the relationship is of interest. In both cases we have two broad classes of theories. In broad terms, for both surface wind and precipitation the theories can be categorized into those that view the surface wind (strictly, its convergence) as causing the precipitation, and those that view the precipitation as causing the surface wind. By “causing the precipitation,” we really mean here “causing large-scale, large-amplitude horizontal variations in precipitation.” In radiative-convective equilibrium (see chapter 7 in this volume) precipitation will be equal to the local surface evaporation, which itself has some horizontal structure.

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Figure 8.1. Top: climatological SST (◦C, contour interval 2◦C below 26◦C, 1◦C above) from the Reynolds and Smith (1994) dataset. Middle: precipitation ( mm d−1, contour interval 2 mm d−1 below 4 mm d−1, 4 mm d−1 above) from the CMAP dataset (Xie and Arkin 1997). Bottom: 500 hPa temperature (◦C, contour interval 2◦C) for January from the NCEP/NCAR reanalysis (Kalnay et al. 1996). Adapted from Sobel (2002). (Reproduced with permission from the c 2002.) American Institute of Physics


In practice, however, horizontal variations in precipitation greatly exceed those in evaporation, and are thus balanced in the moisture budget by the variations in the horizontal divergence of the vertically integrated moisture flux due to the largescale circulation (which when negative is called “moisture convergence” for short). Regarding what causes precipitation, one class of theories tends to focus on moisture convergence, treating it as an external causal factor influencing precipitation. The other class tends to focus on the vertical profiles of temperature and humidity in a column as the important causal factors. The discussion here has considerable projection on a broader debate, occurring over the last decade or so, regarding the parameterization of cumulus convection and the meaning or validity of “conditional instability of the second kind” (CISK). We will avoid dealing with these issues in their most general form, partly because a number of good reviews address them (Emanuel et al. 1994; Stevens et al. 1997; Smith

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10 m s–1

Figure 8.2. Climatological (1949–1992) surface wind ( m s−1) and divergence (s−1, positive values not shown, contour interval 5 × 10−6) for January, from the Comprehensive OceanAtmosphere Data Set (COADS). Adapted from Stevens et al. (2002). (Reproduced with c 2002.) permission from the American Meteorological Society


1997; Arakawa 2004) and partly because the issues are somewhat different when we focus on the quasi-steady circulation, as opposed to the broader question of convective parameterization in models that aim to simulate variability on a broad range of time scales. Inasmuch as averages such as those shown in Figs. 8.1 and 8.2 show much clearer relationships between SST and precipitation (for example) than would be evident in individual daily maps, tropical climate should be easier to understand than tropical weather, and it behooves us to attempt to view the climate problem separately. For example, convective parameterizations whose closures are based on moisture convergence have problems in both physical justification and behavior that make them unsuited for GCMs. One reason for this is that moisture convergence is not sufficiently external to deep convection to be viewed as an environmental forcing. However, when we are dealing with the quasi-steady circulation only, one may argue that large-scale factors associated with the boundary conditions, such as the SST gradient, induce moisture convergence in a way that may to some extent be viewed as external to the convection, so that a moisture convergence closure may be more defensible. We at least ought to reconsider our views on convective parameterization with an open mind when we are dealing with the quasi-steady problem in isolation. At the same time, inasmuch as there are convective parameterization issues in both GCMs and the simple models we will discuss here, the differences in the nature of the parameterization problems in the two sorts of models—most importantly, the need to simulate transients in one but not the other—mean that we have to be careful in extrapolating any conclusions we may draw from the simple models to the GCM context. This caveat is important, given that much motivation for considering the problems addressed here comes from the problem of understanding tropical biases in GCMs (see chapter 11 in this volume).

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8.2. Surface Wind 8.2.1. Theory 8.2.1.1. Matsuno-Webster-Gill Models A number of models, based on early work by Webster (1972) and Gill (1980), and following still earlier fundamental work on linear equatorial wave theory (Matsuno 1966; Lindzen 1967), assume that the surface winds are driven by heating associated with the condensation of water vapor in deep convection, and that linear dynamics are adequate to understand the surface-wind response to this heating. For brevity and consistency with convention, I refer to these as “Gill models” hereafter, while recognizing the contributions of earlier workers. In these models, one assumes a spatial distribution of heating and solves a forced, steady linear wave problem to find the winds. This could be done in three spatial dimensions, but these studies all first assume that the vertical structure is separable and known and has a “first baroclinic mode” form (see chapter 7 in this volume), so that only a set of shallow-water equations is actually solved. True vertical-mode solutions only exist for an artificial rigid-lid upper-boundary condition. While this prevents some leakage of wave energy to the stratosphere that would otherwise occur, for steady linear flow in the presence of some dissipation the error associated with the rigid lid is not too large (Geisler and Stevens 1982).1 A Gill model requires prior knowledge of the heating field, essentially equivalent to knowledge of the precipitation, and thus is not a complete theory for the surface wind, given the SST. A number of models have been constructed that remedy this by developing recipes of varying degrees of complexity for finding the heating, but avoiding convective parameterization in the sense of actual cloud models and the like. Such models include those of Webster (1980), Zebiak (1982, 1986), Weare (1986), Davey and Gill (1987), Seager (1991), Kleeman (1991), and Wang and Li (1993). Some of these, developed for the purpose of seasonal prediction, are anomaly models linearized about a seasonally varying climatology. Gill models are subject to a number of criticisms. To give reasonable results, they tend to require damping coefficients (mechanical, thermal, or both) considerably larger than can be justified based on physical processes actually occurring in the free troposphere. It may be argued that since the desired output is the surface wind, and the free troposphere is not of interest, such strong damping is justified. This is closer to an engineering than a physical argument. The neglect of the barotropic mode may be a large part of the reason such strong damping is needed on the momentum field. When a barotropic mode is included, as in a true two-layer model or the “quasi-equilibrium tropical circulation model” (QTCM) of Neelin and Zeng (2000), weak surface winds tend to be attained by cancellation of the barotropic and baroclinic modes at the surface (see, e.g., Burns et al. [2006] for a detailed discussion of this in the context of an axisymmetric Hadley circulation). Accomplishing the same thing by damping a pure

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single baroclinic mode will certainly lead to large errors in the upper troposphere (where the two modes, if both present, will tend to add constructively rather than cancel), but may perhaps cause other problems even for the surface solution. Another criticism is that when the three-dimensional wave-propagation problem is considered for a heat source that has significant amplitude only above the planetary boundary layer (PBL), as may be claimed for deep convective heating, even small damping prevents waves from reaching the surface due to the small vertical group velocities of the waves of interest, and thus the long time available for damping to act before the signal reaches the surface (Wu et al. 1999). This implies that the resulting surface winds should be much smaller than the first baroclinic mode structure suggests. However, convective heating does in fact reach the surface via downdrafts (though these cool rather than heat). Also, even if the PBL heating is zero, and the direct-wave response is totally dissipated before reaching the surface so that significant winds are produced directly by heating only above the PBL, turbulent entrainment of momentum into the PBL can bring the heating-induced wind signal down to the surface (Chiang et al. 2001; Stevens et al. 2002). 8.2.1.2. Lindzen and Nigam Model Lindzen and Nigam (1987) introduced an apparently entirely different theory for the surface winds. Strictly, they applied their theory only to the zonally asymmetric component of the wind, but showed that it has some relevance for the zonally symmetric component as well, and we will consider its potential relevance to the total wind here. Rather than viewing the winds as a response to deep convective heating, Lindzen and Nigam posited that the winds could be considered to be driven by baroclinic pressure gradients imposed directly on the PBL by turbulent fluxes, which act to effect an adjustment of the PBL temperature toward the underlying SST. The surface pressure in their model is, with a simple linear relationship, low over warm SST and high over cool SST. The pressure gradient at the top of the PBL, which would in general be added to the local SST-related pressure gradient to produce the surface pressure gradient, and which is presumably determined largely by deep convective heating, is assumed negligible. Rather than solving a forced-wave problem, the simplest version of the Lindzen-Nigam model consists of linear, damped, shallow-water momentum equations with the pressure field given. This simplest version also assumes that the PBL top is at a fixed height, implying vertical mass fluxes through the PBL top to balance horizontal mass divergence. The Lindzen-Nigam model assumes that deep convection occurs in regions of convergence, venting mass rapidly from the PBL so as to keep its depth from increasing. To obtain reasonable simulations, Lindzen and Nigam found that they had to go beyond the simplest version of their model, and added a “back-pressure” effect in which the PBL top was allowed to rise slightly, raising the surface pressure, in regions of PBL convergence. This effectively amounts to a linear damping on the perturbation

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pressure (as in [8.3] below). The back-pressure effect amounts to an assumption that mass venting by deep convection is not infinitely fast. Emanuel et al. (1994) provided an interpretation of the back-pressure effect in terms of downdrafts. Conceptually, an attractive feature of the Lindzen-Nigam model is that it provides a very simple recipe for obtaining the surface winds from the SST without finding the precipitation. The precipitation can be diagnosed if desired. The claim implicit in the model is that the surface winds are determined independently of the freetropospheric convective heating. The PBL momentum budget, constrained only by the SST, determines the mass convergence in the PBL. With a simple assumption about the moisture content of the PBL, such as fixed relative humidity, this in turn determines the low-level moisture convergence. Requiring the steady-state diagnostic moisture budget to be satisfied, given some reasonable parameterization of surface evaporation, then determines the precipitation. In the Lindzen-Nigam model, the PBL pressure perturbation is proportional to the assumed PBL depth. Lindzen and Nigam assumed this depth to be 300 hPa, rather thick compared to typical observations of trade-wind boundary layers. It might be argued that the model falsely extends the free-tropospheric pressure gradient (which is explicitly ignored) into the PBL by making the PBL artificially thick (Chiang et al. 2001). Another criticism is that the model ignores the mass of the free troposphere, using the full value of gravity where it should use a reduced gravity for purposes of computing the back-pressure, but this is easily addressed by a straightforward modification of the model (Battisti et al. 1999). Another criticism of the Lindzen-Nigam model, as well as other models with equal drag coefficients for zonal and meridional momentum, is that it ought to include the stress on the PBL due to entrainment of momentum from the free troposphere (Chiang and Zebiak 2000; Stevens et al. 2002; McGauley et al. 2004). This entrainment acts differently than does Rayleigh drag with a constant coefficient. This also can be dealt with by straightforward modification of the model, most simply by allowing different drag coefficients for u and v.

8.2.1.3. The Relationship Between the Lindzen-Nigam and Gill Models The mechanisms involved in these two arguments, those of Lindzen and Nigam and those based on Matsuno-Webster-Gill-type models, are entirely different. In one view the surface wind is determined by convective heating, while in the other convective heating is irrelevant to the surface wind. Nonetheless, taking the coarse-grained, qualitative perspective from which one typically evaluates simple theoretical models, the two views have many important similarities, both in their formal structure and in the predictions to which they lead. Because both assume fixed vertical structure, they can both be reduced to a set of linear, damped, steady, thermally forced shallowwater equations (Neelin 1989). Let’s write these on an equatorial β plane, in somewhat

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general form: −βyv = −φx − uu,

[8.1]

βyu = −φ y − vv,

[8.2]

∇ · u = Q −  pφ,

[8.3]

where  and φ are the mean and perturbation geopotential, Q represents a fixed component of the heating (mainly convective heating, but possibly including a radiative component), and other notation is standard. Q here has units of geopotential per time, and is related to the heating in standard units (e.g., degrees per day) by hydrostatic balance. What vertical structure the prognostic variables are assumed to have varies from model to model. In Lindzen-Nigam, φ can be thought of as geopotential on a pressure surface near the surface of the Earth. It is taken proportional, with a negative coefficient, to a PBL temperature, which in turn is assumed to be simply related to the SST. In Gill, a deep layer is represented and φ is proportional to a deep tropospheric temperature. I have assumed Rayleigh-type damping in all three equations, but allowed the three to have different rates, u, v, p. One concrete difference between our two views lies in how these rates are chosen. A second, perhaps more important difference lies in how Q, the convective heating, is determined. Yet there as well, the difference in practice may not be entirely obvious.

8.2.2. Surface Wind: Evidence A number of recent studies have shed new light on the evidence for one or the other of the two views described above regarding the question of how the surface winds are determined. We are particularly interested in narrow ITCZ regions, where SST gradients are high and we expect Lindzen-Nigam arguments to be most applicable. We do this despite that the flow in such regions projects substantially on the zonal mean, to which Lindzen and Nigam themselves did not claim their arguments were primarily relevant. Much of the debate here can be boiled down to the question of how much of the pressure gradient at the surface is imprinted from above by the free troposphere vs. the hydrostatic result of temperature gradients in the PBL. For the zonal-mean zonal wind, this argument is irrelevant because there is no zonal pressure gradient in the zonal mean. Thus this debate applies to the meridional wind (both in the zonal mean and deviations from it) and to the departures from the zonal-mean zonal wind. A key assumption of the Lindzen-Nigam model is that the pressure gradient at the top of the PBL is small enough to be unimportant compared to the component of the pressure gradient imposed directly on the PBL from below by the SST. Observations from the East Pacific Investigation of Climate (EPIC) experiment have been used to test this central assumption of the Lindzen-Nigam model directly (McGauley et al. 2004; Raymond et al. 2004). These observations show that when the time-averaged fields

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are considered, this assumption holds fairly well for the meridional component of the pressure gradient. The picture is quite different for the synoptic to intraseasonal time scale transients, which have a large component of their pressure gradients imposed from the free troposphere. Most precipitation appears to be associated with these transients, so the relationship of the steady Lindzen-Nigam component of the meridional wind (which presumably accounts for most of the convergence) to the steady component of the rainfall is unclear. Bacmeister and Suarez (2002) did a similar analysis for the zonal component of the pressure gradient in both a GCM and the NCEP/NCAR reanalysis dataset. In both cases they found that, for the most part, the free-tropospheric component of the pressure gradient dominated the SST-controlled component. They did not examine the meridional component of the pressure gradient. Chiang et al. (2001) addressed this issue using a dry linear GCM. They incorporated a PBL of Lindzen-Nigam type, but also a deep heating whose spatial structure was derived from observations. They varied a number of parameters, including PBL depth and damping coefficients, in a systematic way to determine which set gave the best fit of the resulting surface winds to observations. They found that, for the best-fit parameters, the free-tropospheric pressure gradient (associated with the deep heating) tended to dominate the zonal component while the SST-controlled component tended to dominate the meridional component. The agreement among these studies seems to justify a fairly high degree of confidence in the conclusion that the Lindzen-Nigam picture explains much of the meridional wind, particularly in regions of large meridional SST gradient, while freetropospheric pressure gradients produced by deep convective heating explain most of the zonal wind’s departures from its zonal mean.

8.3. Precipitation 8.3.1. Theory In one view, PBL momentum dynamics drives deep convection by controlling moisture convergence in the PBL. The Lindzen-Nigam model is an example of this view, but there are other examples that allow factors excluded from the Lindzen-Nigam model to influence the PBL momentum budget. Such factors are free-tropospheric pressure gradients and nonlinearity. In the other view, deep convection is thermodynamically controlled. By this we mean that precipitation is determined by the local profiles of temperature and humidity, the momentum budget is essentially irrelevant to precipitation, and moisture convergence, while diagnostically closely related to precipitation, is not an external factor that can be viewed as causing precipitation. Again, there are multiple ways, differing in detail, in which this view can be expressed in an explicit theory.

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As in the case of the surface winds, these two classes of ideas appear totally different. One requires only local information, while horizontal structure is essential to the other. One assumes that precipitation can be found independently of the momentum budget, while the momentum budget of the PBL is essential to the other. I will first describe some of the ideas that fall into these classes in more detail, then consider some evidence that may help us distinguish between them. In the process we will examine whether there are any points of connection between these two apparently very different views.

8.3.1.1. PBL Momentum Control The notion of control of precipitation by the momentum budget in the context of the quasi-steady circulation has roots in an early body of literature (well reviewed by Waliser and Somerville [1994] and Gu and Zhang [2001]) that addresses what sets the latitude of the ITCZ. While in general the ITCZ lies close to an SST maximum, rendering some form of direct control of precipitation by local SST perhaps the most obvious explanation (Pike 1971), a number of studies proposed mechanisms that could place the ITCZ off the equator even if the SST were uniform or had a maximum on the equator. Some of these studies invoked transients (Holton et al. 1971; Lindzen 1974) while others did not (Charney 1971). All of these studies, as well as Waliser and Somerville (1994), make what are essentially CISK arguments—to the effect that, for one reason or another, heating at some particular finite latitude is particularly effective at inducing PBL convergence, which then provides moisture to “fuel” the deep convection that produces the heating. Since these ideas assume a given heating, and then ask how this heating can best induce PBL convergence, they may be classified as momentumcontrol ideas in that they take the primary bottleneck controlling the occurrence or intensity of deep convection to lie in the PBL momentum budget. In the Lindzen-Nigam view described above, the SST induces PBL pressure gradients directly, and these drive the PBL flow by linear dynamics. At places where this flow converges, convection occurs to vent mass from the PBL. Several other theories invoke PBL momentum dynamics as an agent influencing deep convection, but emphasize other aspects of PBL momentum dynamics. Tomas and Webster (1997) and Tomas et al. (1999) took a longitudinallyindependent perspective (appropriate to long, thin, east-west oriented ITCZs) focused on the meridional flow in regions of large cross-equatorial pressure gradient. They took the pressure gradient as given, whether induced directly by the Lindzen-Nigam mechanism or imposed at the top of the PBL. (To the extent that the latter [freetropospheric] component is important, one expects it to be strongly influenced by the location and intensity of deep convection. Since the theory aims to predict the distribution of convection, the theory is to some extent diagnostic, rather than fully predictive. Tomas and Webster (1997) argued that the preexisting pressure gradient

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was induced on “larger scales” by land-sea contrasts or SST gradients, which might be taken to imply that the theory aims to predict perturbations to the ITCZ intensity and location that are in some sense “small scale”.) If this pressure gradient is strong enough, the cross-equatorial flow that it drives in the PBL is able to displace the zero absolute-vorticity contour significantly off the equator. By a mechanism that involves the meridional advection of relative vorticity, and is thus nonlinear, and which has some common elements with symmetric instability (though the flow itself is not linearly unstable in the presence of substantial friction, as appropriate for the PBL), this leads to strong convergence at a location on the high-SST side of the equator, but, in general, equatorward of the SST maximum, assuming an SST distribution like that commonly observed in the eastern Pacific. The authors argued that this would both strengthen the ITCZ precipitation and shift it equatorward of the location where it would otherwise be. Pauluis (2004) derived another idea from an analysis of axisymmetric GCM simulations. This idea applies to situations in which the cross-equatorial SST gradients are small, as opposed to large in the case of Tomas and Webster (1997) and Tomas et al. (1999). In Pauluis’s simulations, the SST has a single positive peak north of the equator and is uniform elsewhere, with the SST gradient on the equator being close to zero. In this case, if the PBL is also sufficiently shallow (remember that the pressure gradient induced by the Lindzen-Nigam method increases with PBL depth, for fixed SST gradient), the pressure-gradient force is unable to balance friction for any significant meridional velocity on the equator. Further poleward, on both sides of the equator, southerly flow (towards the SST maximum) does occur, with the drag being balanced by the Coriolis force on the zonal flow. Since the flow in the Southern Hemisphere is towards the equator but then stops at the equator, convergence occurs at or just south of the equator; the flow must ascend and cross the equator above the PBL, where there is little friction and so less pressure gradient is required. This implies a secondary maximum in precipitation (the primary maximum being over the SST maximum), and Pauluis argued that this could be a mechanism for the formation of a double ITCZ. While these various views differ in the PBL momentum balances they invoke in order to generate convergence, they all wind up generating convergence one way or another through the momentum balance, and then assume that this convergence produces convection. In Lindzen and Nigam (1987), nothing about convection need be known in order to find the PBL convergence, so that the theory truly predicts precipitation. In the others, to one degree or another knowledge of the occurrence and intensity of deep convection must be assumed to find the PBL convergence, but the ability to produce PBL convergence, given the constraint imposed by the PBL momentum budget, is nonetheless considered the limiting factor controlling the convection. The overall direction of causality is thus the same in each case, as far as the present argument is concerned.

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8.3.1.2. Thermodynamic Control There are at least two entirely distinct sorts of ideas by which thermodynamic factors can be hypothesized to control convection. Stability. The first set of ideas relies on parcel stability considerations of one form or another. In one form of this view, precipitation will occur where positive convective available potential energy (CAPE) exists, or would exist if precipitation were not occurring (since the occurrence of deep convection will tend to squelch CAPE). To evaluate the properties of the hypothetical nonprecipitating state, we can (for example) use a well-understood one-dimensional model of the nonprecipitating trade-wind boundary layer (e.g., Betts and Ridgway 1989). The requirement of positive CAPE can be replaced by one that CAPE be above some finite threshold. We can also stipulate further that the rain rate will be some monotonic function of the CAPE in the nominal nonprecipitating state. CAPE increases as the moist static energy (or moist entropy) of the near-surface air increases and as the free-tropospheric temperature decreases. The warmer and moister the air would be in the nominal nonprecipitating state, or, the colder the free-tropospheric sounding, the greater the CAPE, and thus the greater the rain rate we expect to actually occur. Free-tropospheric temperature is constrained to be close to horizontally uniform in the deep Tropics, by dynamical adjustment under strong (dry) stratification and small Coriolis parameter. We assume it is uniform enough that we can neglect its variations relative to those of the PBL moist static energy in computing CAPE (the “weak temperature gradient” [WTG] approximation; e.g., Sobel and Bretherton [2000]; Sobel et al. [2001]). Its vertical stratification can be assumed to be moist adiabatic (see chapter 7 in this volume). The rain rate is thus a function only of the PBL moist static energy in the hypothetical nonprecipitating equilibrium. We further assume that the PBL moist static energy in this state is a function only of the SST. This is fairly reasonable assumption for qualitative purposes. Thus, the rain rate depends only on the SST in this model. Since deep convection consumes CAPE rapidly, it may make more sense to relate precipitation to the rate of CAPE production (as opposed to CAPE itself), which suggests a focus on surface fluxes. This in turn suggests that surface wind speed as well as SST should be an important factor, as is true in hurricanes and as has been argued for intraseasonal variability (Emanuel 1987; Neelin et al. 1987). Here we must be careful not to generalize from the uncoupled to the coupled problem, since high wind speed will also reduce SST on seasonal time scales, and surface fluxes are limited on these time scales by surface radiation and ocean heat transport regardless of wind speed (e.g., Waliser and Graham 1993; Sobel 2003; Seager et al. 2003). We can make this argument more sophisticated in a number of additional ways. One is to include some dependence on free-tropospheric relative humidity, which appears in observations to exert some control over deep convection (e.g., Sherwood

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1999; Parsons et al. 2000), presumably through its effect on entrainment and downdrafts. Free-tropospheric moisture does not enter standard—non-entraining, pseudoadiabatic—definitions of CAPE (e.g., Emanuel 1994; Curry and Webster 1999), and so is considered a separate influence according to standard categorizations.2 Another is to relate precipitation to other indices of parcel stability instead of or in addition to CAPE, such as convective inhibition (CIN) as argued by Mapes (2000) in the context of transient variability, or other measures of disequilibrium between PBL and free-tropospheric parcels, as in “boundary layer quasi-equilibrium” (Raymond 1995; Raymond et al. 2003). All these measures can generally be expected to vary in the same way with SST; high SST regions tend in general to have high tropospheric moisture, low CIN, etc. Because of this, and because none of these ideas requires any information about the PBL momentum budget, for our purpose here it is not important to distinguish between them. These thermodynamic ideas are manifest in a number of models for the steady circulation that use so-called quasi-equilibrium convective closures (e.g., Seager 1991; Kleeman 1991). The control of precipitation by SST according to these ideas is perhaps most transparently seen in Sobel and Bretherton (2000), who use single-column models (with essentially CAPE-based convective closures) to predict precipitation, with the freetropospheric temperature profile held fixed (WTG) and SST the only input parameter that is allowed to vary from one point to the next. Conserved Variable Budgets. An entirely distinct view, originating with Neelin and Held (1987), invokes the moist static energy budget and ignores CAPE or any other measure of parcel buoyancy entirely. I give an exposition of this view here, drawing directly on the treatment by Neelin (1997). We stick with moist static energy although the same arguments can be phrased in terms of moist entropy or equivalent potential temperature with no important changes. In pressure coordinates, the equation for the moist static energy, h = c pT + L vq + g z, with c p the heat capacity at constant pressure, T temperature, L v latent heat of vaporization, q specific humidity, g gravitational acceleration, and z geometric height, is ∂th + u · ∇h + ω∂ p h = −R,

[8.4]

where −R is the radiative cooling (R assumed positive), and we have not yet performed any averaging in time or space. Now, taking a time average over some interval long enough that the average tendency becomes negligible, and then vertically integrating from the surface ( p = ps ) to a nominal tropopause ( p = pt) at which the vertical velocity ω and all turbulent fluxes are assumed to vanish, leads to u · ∇h + ω∂ ph = F net,

[8.5]

F net = E + H − R

[8.6]

where

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is the total moist static energy input to the atmospheric column, E is the surface latent heat flux, H is the surface sensible heat flux, the overbar represents the time average, and the angle brackets represent a vertical average:
ps −1 X = ( ps − pt) Xd p. pt

Now let us neglect horizontal advection. This is a major assumption, not necessarily justified. Horizontal temperature advection can be generally assumed small in the deep Tropics, but horizontal moisture advection cannot, and this implies that significant horizontal advection of moist static energy can occur. Nonetheless, we make this assumption here for the sake of argument, and return to it later. We then also neglect transients, assuming ω∂ ph ≈ ω∂ ph. This may not necessarily be justified either, and needs to be reconsidered later together with the neglect of horizontal advection. We now make one additional key assumption. We assume ω to have separable horizontal and vertical dependence: ˆ ω(x, y, p) = ( p)ω(x, y),

[8.7]

with the dimensionless function ( p) known. We assume that  is of order unity and positive; assuming the vertical velocity to have a deep, single-signed vertical structure, as normally associated with the first baroclinic mode, ωˆ can be thought of as the value of ω at a midtropospheric level, say 500 hPa. The theory can be generalized to allow ( p) to be a function of horizontal position as long as it is a slowly varying function compared ˆ Then, we can take ωˆ out of the integral to obtain to ω. ˆ = −ω−∂ ˆ −ωM ph = F net.

[8.8]

We now have a predictive theory for the horizontal structure of the vertical velocity, if we know three quantities: the surface fluxes (of which the latent heat flux is by far the dominant contributor over the tropical oceans; sensible heat flux can be neglected there to a good approximation), the radiative cooling, and the gross moist stability M: M ≡ −∂ ph.

[8.9]

M is not directly related to any measure of local stability at a point (e.g., the buoyancy frequency), nor to any measure of column stability derived from buoyancy considerations, such as CAPE. (Note that here M has the units of ∂ p h, that is, J kg−1 hPa−1. This is different than the convention used, for example, by Neelin and Zeng (2000), though the difference is only multiplication by a dimensional constant.) To illustrate the dependence of M and thus ωˆ on the vertical structure ( p), consider a simple example illustrated by Fig. 8.3. The figure shows a typical tropical

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Figure 8.3. Moist static energy profile, showing level of concentrated inflow pi n (where moist static energy is h i n) and level of concentrated outflow pout (where moist static energy is h out) for a hypothetical region of idealized ascent, uniform for pout < p < pi n , zero elsewhere.

moist static energy profile from a deep convective region (in this case a mean over slightly less than two months at Kwajalein, Marshall Islands, during the KWAJEX field experiment [Sobel et al. 2004; Yuter et al. 2005]). Imagine a two-dimensional control volume with horizontal flow into it localized as a delta function in the vertical at a pressure pi n, and a similarly localized outflow at pout . The horizontal divergence is thus ∇ · u = C [δ( p − pout) − δ( p − pi n)], where C is a constant with dimensions of hPa s−1. The vertical velocity is thus constant, call it ω0, for pout < p < pi n , and zero elsewhere, and the vertical structure function (0( p), say) is reasonably chosen to be unity for pout < p < pi n , and zero elsewhere. Using (8.9), the gross moist stability is M=

h out − h i n , p

[8.10]

the moist static energy difference between the outflow and inflow levels divided by the pressure depth of the troposphere p. We will have M > 0 as long as pout is low enough (the altitude of outflow high enough) that h out > h i n . From the figure it is evident that we can change the magnitude and even the sign of M quite easily by making only modest changes in pi n or pout . For fixed pi n below the h minimum (around 600 hPa in this profile), and pout above the minimum, M increases as pout decreases. In reality, of course, horizontal divergence is continuous in the vertical, and the integral in (8.8) is

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not so transparently evaluated, but it remains generally true that deeper ascent generally leads to larger M, and shallower ascent to smaller or even negative M. ˆ we can find the precipitation by applying the same steps and Knowing ω, approximations that led to (8.5) to the equation for dry static energy, s = c pT + g z, yielding ˆ s = P − R, ω∂ ps  = −ωM

[8.11]

where we have defined the gross dry stability, Ms = −∂ ps , and P is the precipitation expressed in energy units. Eliminating ωˆ between (8.8) and (8.11) yields P=

Ms F net + R. M

[8.12]

The second term on the right-hand side accounts for the precipitation that would occur in radiative-convective equilibrium, while the first term represents variations associated with the large-scale circulation. Keep in mind that while surface fluxes contribute importantly to F net, this first term does not represent rainfall balanced directly by surface evaporation (that is accounted for by the second term, though it isn’t immediately obvious because (8.12) is most directly a heat budget rather than a moisture budget), but rather that component of rainfall that is balanced by large-scale moisture convergence. Through the decomposition of the energy transport into a mass transport and a gross moist stability, the net vertical energy input into the atmosphere, F net , which includes surface fluxes, appears as a forcing that drives the circulation that accomplishes that convergence. It is worth pausing for a moment to examine the logical structure of this argument because the result appears somewhat miraculous: we have obtained a theory for the time-mean precipitation without either using any convective parameterization or considering the horizontal structure of any field. Up to (8.7), we had done nothing but construct a budget, do some averaging, and neglect certain terms. While the neglect of these terms is debatable, we can at least find some situations in which it is valid (for example, at local maxima and minima in h, horizontal advection must vanish). Even the assumption of separability and known vertical structure, (8.7), does not at first glance seem all that restrictive; consider that a model with fixed vertical structure can still have a broad range of convective parameterizations (e.g., Wang and Li 1993). There are several problems, however. The first, as mentioned above, is the neglect of horizontal advection. Analysis of reanalysis datasets (L. Back and C. Bretherton, personal communication) yields the result that in many convective zones, horizontal advection of moist static energy is competitive with or even dominates vertical advection in the vertical average, making neglect of the former a poor approximation. Even if

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Figure 8.4. Climatological January surface evaporation (mm d−1) from the COADS dataset.

we ignore this, there are more subtle difficulties related to our assumptions of what is known and what must be predicted. Equation (8.12) predicts P if we know F net and Ms /M. While the surface fluxes can ultimately not be taken as an external forcing, being part of the solution themselves, it is nonetheless reasonable to consider the problem with fixed-flux boundary conditions as a first step. Since departures from radiative-convective equilibrium are generally large, the spatial structure of precipitation is accounted for more by variations in moisture convergence than those in surface evaporation, the latter of which are considerably smaller and do not even have the same spatial structure as those in precipitation (see Fig. 8.4, and compare to precipitation shown in Fig. 8.1); thus, taking the surface fluxes as given does not make the problem trivial. Obtaining R is a job for a radiative transfer model, which can be considered more or less independent of the rest of the problem. Radiative cooling depends strongly on cloudiness and humidity, but these are closely related to precipitation; so a simple parameterization by which R is made a function of P , linear in the simplest case R = R0 − r P , with r a positive dimensionless constant (expressing P and R in the same units, say W m−2) apparently in the range 0.1–0.2 (e.g., Su and Neelin 2002; Bretherton and Sobel 2002; Lin et al. 2004), captures the essence of the physics without adding any fundamental complexity to the problem. The quantity Ms /M is the main difficulty. It is not at all clear that this quantity can be considered known or fixed. Neelin and Held (1987) hypothesized that spatial variations in M exert a primary control on spatial variations in ωˆ (and thus implicitly P ), but, consistent with the assumption of known vertical structure (in their case, a two-layer model), took those M variations to be determined by variations in ∂h/∂ p. However, M is very sensitive to changes in  because ∂h/∂ p changes sign in the troposphere while ω often does not (or at least does not do in a way that resembles ∂h/∂ p), so there is generally a lot of cancellation in the integral in (8.9). In the toy case above, for example, we can plausibly choose pi n and pout such that h i n = h out , so M = 0; then increasing or decreasing pout by a tiny amount leads to M of opposite sign. Since ∂s /∂ p does not change sign in the troposphere, Ms is much less sensitive, so the ratio Ms /M retains the sensitivity.

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Yu et al. (1998) computed M using ∂ ph computed from a large-scale meteorological analysis and from tropical soundings. They did not use meteorological analyses directly to estimate , but took the latter to have the structure associated with a first baroclinic mode, an approximately half-sine-type structure between ps and pt, and implemented a similarity theory, allowing pt to vary depending on the depth of convection diagnosed from CAPE. Using this together with observed ∂h/∂ p from observations, they found that variations in the depth of convection tended to be comparable, but of opposite sign to those due to variations in ∂h/∂ p, so that M tended to remain roughly constant. However, to the extent that vertical profiles of vertical velocity obtained from assimilation datasets are any guide, the assumption of a first baroclinic mode, even a stretchable one with varying pt , is too restrictive for the purpose of this calculation. When typical variations in both h and ω from such datasets are considered, one comes to the conclusion that variations in ω generally dominate variations in the gross moist stability. In a model with parameterized (or resolved) convection, the vertical structure of the vertical velocity will be strongly influenced by the convection, since in the deep Tropics we expect WTG balance to hold, ω

∂s ≈ Q, ∂p

[8.13]

where Q is the total heating and the dry stability ∂s /∂ p varies little. Thus the sensitivity of M to the vertical structure of  hides a dependence of this theory on convective parameterization, which determines the vertical structure as well as the magnitude of Q. Finally, perhaps the most important weakness of the moist static energy budget argument is that it does not by itself predict the occurrence of significant deep convection or large-scale ascent (the two being inseparable on time scales of a day or greater), only their intensities at those locations where they are already known by other means to be occurring. This is clear from the study of Yu et al. (1998), for example, who only plot M in regions of significant precipitation. Looking at (8.8) and assuming M to be positive, it may be tempting to think that we can associate mean ascent with positive net mean forcing on the column moist static energy budget, F net > 0, corresponding in steady state to net export of moist static energy from the column. This is not correct because many tropical and subtropical regions of mean descent also export moist static energy. This is apparent from the analysis of Trenberth et al. (2001), and is also illustrated here in Fig. 8.5. The figure shows a climatology for September of precipitation and F net from the second NCEP reanalysis. (One may be legitimately concerned about estimating F net from assimilation datasets in which the energy fluxes are entirely model products; all we can say is that we have made similar figures for several such datasets as well as general circulation models, and the features of interest here are remarkably consistent across all of these.) While there is arguably some correspondence between F net and precipitation, particularly

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20N

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Figure 8.5. September climatology of precipitation (contours; contour interval 4 mm d−1, minimum contour 2 mm d−1) and total moist static energy forcing F net (positive values shaded, contour interval 50 W m−2, negative values white) computed from the second NCEP reanalysis. (Figure courtesy of Michela Biasutti.)

over land, we see clearly that F net over the southern tropical Atlantic is as large as in the ITCZ to the north, but precipitation in the south is less than 2 mm d−1. The associated convective heating is well below the radiative cooling in this region, so this is a region of descent. The positive F net implies moist static energy export nonetheless. Some of this export may be due to transients, but it is entirely possible that a large fraction of it is explainable within the confines of the argument above, that is, by a local argument in which steady vertical advection of moist static energy is the key factor. In descent regions, the free troposphere tends to be dry, so the minimum in the moist static energy profile tends to be relatively pronounced. The PBL, on the other hand, still has high relative humidity, and particularly if the SST is not too low, may still have fairly large moist static energy. All we need in order to have net moist static energy export via descent, according to the argument above, is an  profile that is not too top-heavy. Figure 8.6 illustrates this, using the same sounding as in Fig. 8.3 but with choices of pi n and pout such that M < 0, though we should not really call this “gross moist instability” as we would if we had M < 0 in an ascent region, because in this case we are still exporting energy, just doing it through descent rather than ascent.

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Figure 8.6. As in Fig. 8.3, but for a descent region.

To the extent that the time-mean vertical velocity profile in descent regions can in fact configure itself in such a way as to accomplish moist static energy export, F net does not determine the sign of the large-scale vertical velocity even in a steady, horizontally uniform atmosphere. What does? Presumably we must fall back on arguments having to do with CAPE, or some other similar measure of convective stability (and thus ultimately relatable to the SST) or to PBL momentum considerations. While such considerations may not contradict the moist static energy budget arguments, they are outside the scope of the latter, and the need for these extra considerations demonstrates the incompleteness of the moist static energy budget arguments.

8.3.2. Precipitation: Evidence As described above, we have two classes of theories for the precipitation. The mechanisms involved in the two classes are almost entirely unrelated. One focuses on local thermodynamics, the other on boundary-layer momentum dynamics. Each class contains multiple specific theories that are at least somewhat distinct. In the case of thermodynamic control in particular, the ideas based on stability and on conserved variable budgets are quite different. There are many possible ways one could go about testing these ideas. Many of them would be particular to specific theories rather than to the whole class to which the theory belongs. I am most interested in distinguishing between the two classes of ideas.

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In this light, we first consider recent observational evidence from the East Pacific Investigation of Climate (EPIC) field experiment (Raymond et al. 2004), which contradicts the notion that PBL convergence necessarily implies deep convection. This notion is essential to all PBL momentum control ideas in their strongest forms. Weaker forms, in which PBL momentum control modulates deep convection to some degree, changing the distribution of precipitation quantitatively from what it would otherwise be based on thermodynamics alone, are still tenable in the face of this evidence. The EPIC studies mentioned above in the context of the surface wind (Zhang et al. 2004; McGauley et al. 2004; Raymond et al. 2004) find that when the SSTinduced pressure gradient is strong, and produces convergence, there need not be deep ascent or rain associated with that convergence. Rather, there can be shallow ascent and divergence just above the boundary layer, with a shallow return flow there rather than a deep Hadley circulation. This is just what we should expect if convection were thermodynamically controlled. If the thermodynamic conditions are not right for deep convection, but the SST gradient drives low-level convergence nonetheless, the only way to satisfy mass conservation (without the deep ascent that would necessarily come with deep convection) is to have a shallow circulation of this type. The observations confirm this theoretical expectation. Presumably this situation also occurs in the second, southern ITCZ during the large fraction of the year when it exists in surface convergence but not precipitation (Liu and Xie 2002). Again, this evidence from EPIC does not prove that PBL convergence has no influence on deep convection whatsoever, only that it is not sufficient to induce deep convection if the thermodynamic conditions are not right. It is still possible that under some thermodynamic conditions, dynamically-induced PBL convergence can either induce deep convection or increase its intensity. More broadly, we would like to determine whether PBL momentum control or thermodynamic control ideas are more correct by using each one to produce a simulation of the precipitation, and comparing to observations. This turns out not to give an unambiguous result. Simulations using simple models based on both theories, as well as combinations of the two (e.g., Wang and Li 1993), have been done, and all reproduce the observations at comparable levels of accuracy—with significant errors, as one expects from simple models, but with some fidelity to the main gross features of the climatological precipitation distribution (or in some studies, the deviations from the time or zonal mean). A very simple model problem gives some idea of why the two sets of ideas do not lead to dramatically different predictions. Consider an ocean slab-symmetric in one horizontal direction (call it y, so the SST varies only in x) with no planetary rotation. In that case, (8.1) becomes p x + uu = 0,

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which, solving for u and taking minus the partial derivative with respect to x to find the convergence, gives p xx . −u x = u Assuming rainfall to be proportional to convergence and p to be proportional to minus the SST, both being the case in the Lindzen-Nigam model, we find that the rainfall is proportional to minus the curvature of the SST, P ∼−

∂2 (SST). ∂ x2

[8.14]

The stability ideas described in section 8.3.1.2, on the other hand, suggests that the rainfall will be related to the local value of the SST, relative to some background value, of which the tropical mean is presumably approximately representative. If the SST is high compared to the tropical mean, we expect significant precipitation, but not otherwise. For small perturbations, we may assume the relation to be linear above a threshold, P ∼ H(SST − SSTmi n)(SST − SSTmi n),

[8.15]

where H is the Heaviside function and SSTmi n , the minimum SST at which convection can occur, is a function of the mean tropical climate, for example presumably related to the mean SST over the Tropics (e.g., Sobel et al. 2002). Comparing (8.14) and (8.15), we see that even at the very crude level of argument used to obtain these relations, the fundamental difference between the two models is clear, with one involving the local value of SST and the other involving the spatial structure. Nonetheless, it is also apparent why it is not so simple to use observations to determine which is correct. If we assume that spatial and temporal variations in SST have characteristic magnitude and spatial scale that are known, then the two models will not disagree greatly since both predict rainfall to maximize at SST maxima. The actual predicted SST-rainfall relationships will, of course, depend quantitatively on the parameters in the models, which have been ignored in the arguments here, but if at all possible it is desirable to resolve theoretical arguments without having to debate too closely what these parameters should be. At the same time, this example does suggest one situation that may be used to distinguish between the two views: a local maximum in SST, even a very sharp one, that is not a global maximum. In that case, the SST curvature can still be as large, so the PBL convergence and thus rainfall according to a Lindzen-Nigam-type view will be large, while the thermodynamic-control view will predict less or zero precipitation, depending on just how large or small the local SST is compared to values elsewhere in the Tropics. The southern ITCZ in the eastern Pacific Ocean, which has significant precipitation only in northern spring in reality but year-round in many GCMs, may be viewed

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Figure 8.7. June climatologies of SST (top) and precipitation (bottom), using the same datasets and plotting conventions as in Fig. 8.1.

as just such a case. Although it does not contain a true local maximum in SST, it is a local maximum in the meridional direction, in which gradients are sharpest. This is enough to produce precipitation by the Lindzen-Nigam mechanism. At the same time, during most of the year the local SST is lower than that in the northeast Pacific, or the western Pacific, or much of the rest of the tropical oceans, so by thermodynamic-control arguments we expect little precipitation. As an example, see Fig. 8.7. The same arguments, to a lesser degree, may also apply to the northeast Pacific ITCZ. The SST there is high enough to induce convection by thermodynamic-control arguments (assuming the absence of very strong dry air advection or some other anomalous suppressing effect) but, most of the time, not quite as high as in the west Pacific warm pool. The meridional SST gradients are as sharp as anywhere in the Tropics, however, so we might expect both models to predict precipitation, but boundary-layer momentum-control arguments, all else equal (in some admittedly imprecise sense), to predict greater precipitation. Simulations with simple models appear to bear this out. Models based on Gill formulations, with no explicit boundary layer, tend not to have a spurious southern ITCZ. At the same time, these models tend to underestimate the precipitation in the northern ITCZ. This difficulty with a Gill model was discussed by Seager (1991). Figure 8.8 shows results from the quasi-equilibrium tropical circulation model

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242 | Adam H. Sobel January QTCM1-V2.3 Model Climatology Monthly Mean Precipitation (1982–98) (10 member ensemble) 40N 20N EQ 20S 40S 0

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(Neelin and Zeng 2000; Zeng et al. 2000; chapter 10, this volume). This model is more complex in several respects than other models discussed in this chapter, but it lacks a boundary layer (its vertical structure consists of two modes, barotropic and first baroclinic). The figure shows essentially zero precipitation in the northeast Pacific ITCZ in January, where the Xie-Arkin dataset shows a fairly strong, narrow ITCZ in that season, as shown in the top panel of Fig. 8.1. Models with a boundary layer can do much better in this regard, though sensitivity to the convective parameterization is large. Figure 8.9, from Wang and Li (1993), shows results from their model, which contains a boundary layer as well as a first baroclinic mode. The first scheme, called “linear heating,” is a moisture convergence-based scheme, similar to those of Kuo (1974) and to those used in simple models by Webster (1972), Zebiak (1986), and others, but is truly linear in that negative precipitation is allowed. The second is essentially the same scheme, except that negative precipitation is not allowed to occur. The resulting precipitation field is not exactly the same as would be obtained from setting all negatives in the first panel to zero, because the heating has feedbacks with the rest of the model dynamics. The third panel uses a similar scheme, but one in which precipitation is not allowed to occur unless SST is greater than a threshold value. In the fourth panel, precipitation does not occur for SST below the threshold, is linearly proportional to SST above that threshold, as in (8.15), and then has no SST dependence above a second, higher threshold; in addition to the SST dependence, the precipitation retains the moisture-convergence dependence of the preceding schemes. The schemes used in the third and fourth panels, through their explicit SST dependences, bring the model closer to thermodynamic-control ideas. We can see that in these two panels the southern ITCZ in the eastern Pacific is nonexistent, but the northern one is also weak or nonexistent. The two upper panels, using models closer to momentum-control ideas, both have strong northern ITCZs. In the linear scheme, this is accompanied by negative precipitation south of the equator. The conditional-heating scheme is perhaps the best, giving a strong northern ITCZ

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(a) LINEAR HEATING 20N 10N

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accompanied by a modest but clear southern ITCZ, for a double ITCZ bias comparable to what one might find in a GCM.

8.4. Discussion 8.4.1. Some Open Questions We are left with the impression that boundary-layer momentum dynamics does play some role in producing the sharp, intense northeast Pacific ITCZ that is observed, as concluded by Wang and Li (1993). At the same time, too much emphasis on PBL momentum control, or too little thermodynamic control, can lead to a double ITCZ bias. In this light, the double ITCZ bias in GCMs, discussed in more detail in chapter 11 in this volume, seems particularly paradoxical. Apparently, the GCMs are responding too strongly to the boundary-layer momentum forcing and producing deep convection in the Southern Hemisphere, where instead they should produce a shallow circulation of the sort found at certain times in the north during EPIC (McGauley et al. 2004; Raymond et al. 2004). Why should the GCMs do this when nearly all modern convection schemes used today are essentially CAPE adjustment schemes of one sort or another, which is to say they are constructed on thermodynamic-control principles? Few if any current models actually build in an explicit tendency towards deep convection where there is low-level convergence (in the way that older models using, for example, the Kuo scheme and its relatives did). Based on the simple models discussed above, one might expect that many GCMs would have no southern ITCZ, but might rather err on the side of too weak a northern ITCZ. The former is clearly not true, and the latter does not seem to be systematically true either. As discussed in section 8.1, we cannot directly apply conclusions from the simple-model context to the GCMs, but the question here suggests that it may be fruitful to focus attention precisely on the different roles of the convective parameterizations in the two sorts of models. The need to simulate transients in GCMs but not the simple models is the most obvious difference, and indeed some recent work suggests that transients may be playing a large role in the double ITCZ bias in at least one GCM (Bacmeister et al. 2006). This leads to another, closely related and more fundamental question: how does PBL convergence induced by momentum forcing actually influence convection? Simply stating, as some models built on these principles do, that deep convection must occur so that the mass and moisture can be vented from the boundary layer is not satisfactory because the mass and moisture can also be vented by a shallow circulation with little or no precipitation. Additionally, at least the most basic tenet of thermodynamic-control thinking must be satisfied: the atmosphere must be in some appropriate sense unstable in order for deep convection to occur. Our understanding should include a grasp of how the PBL convergence influences the stability of the atmosphere so as to make deep convection more probable.

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Several mechanisms exist by which PBL convergence can destabilize the atmosphere: • An influence on CAPE. If the ascent induced by the PBL convergence reaches a

significant height, it can have a significant influence on CAPE because the ascent will adiabatically cool the troposphere. However, a shallow circulation will have little influence on CAPE, as most CAPE resides in the difference between the moist static energy of a near-surface parcel and the saturation moist static energy of the environment at relatively high levels (McBride and Frank 1999). • An influence on convective inhibition (CIN). This is more likely in the case of a shallow circulation, as only ascent at the top of the PBL is needed to induce cooling there, reducing the negative buoyancy of near-surface parcels that have ascended to that level and making it easier for them to break through to their levels of free convection. • An influence on free-tropospheric moisture. Shallow ascent will advectively moisten the lower free troposphere, reducing the negative contribution to buoyancy that entrainment imparts to updrafts, and reducing the tendency of downdrafts to stabilize the sounding. At least the latter two mechanisms are quite plausible, but it would be very valuable, not least from the point of view of convective parameterization and model development, to determine quantitatively which is more important, or whether some other factor, not named above, is responsible for the enhancement or strengthening of deep convection in the ITCZ by PBL momentum-driven convergence.

8.4.2. Moist Static Energy Argument Despite having earlier offered a critique of moist static energy arguments, we can offer a hypothesis here in terms of such arguments. We may be motivated to do this in order to provide an alternative perspective on the questions above, or perhaps just because analyzing a budget of the most conserved variable in the system is such a fundamental principle in physics that we should not abandon it easily. Some aspects of this hypothesis are testable. This hypothesis is relevant to regions where the SST is low enough that the persistent absence of deep convection is possible, but not so low that deep convection is nearly impossible, as is the case in the eastern Pacific cold tongue, for example. In such regions, in the absence of PBL momentum-driven convergence, deep convection will be weak or absent. In its absence, the troposphere will cool radiatively and this will induce subsidence. Assuming the subsidence profile is not too top-heavy, it is possible for the steady divergent circulation to export moist static energy. We assume that in our

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regions of interest the total column moist static energy forcing (surface fluxes minus radiative cooling) is positive, and to make the argument simple, we assume that it takes a constant value, whether deep convection is occurring or not. Our descending circulation configures itself so as to export moist static energy at the rate necessary to balance the forcing. Up to this point our argument has been thermodynamic, considering only the local SST value relative to some large-scale tropical mean, except that by assuming the absence of momentum-driven convergence, we implicitly assume that the horizontal structure of the SST field is not such as to generate such convergence. Now consider the case in which such convergence does occur, via Lindzen-Nigam dynamics for example, because our location is a local SST maximum (though not a global one). The environment being stable to deep convection, this convergence will generate a shallow circulation with ascent only near PBL top and a divergent return flow shortly above, with the vertical velocity becoming downward again well below (say) the freezing level. Taken on its own, this shallow circulation has negative gross moist stability because ∂h/∂ p is positive (h decreasing with height) over most or all of the shallow region of ascent. (Even if the ascent extends somewhat past the moist static energy minimum, we can still have M < 0 as long as it does not go too far past that point.) We can view the resulting import of moist static energy as an additional column forcing on top of F net in equations (8.5) or (8.12). We can also say that the means of exporting moist static energy by descent has been cut off by the imposed near-surface circulation, since the configuration of ω and ∂h/∂ p over the whole column that allowed net export is no longer dynamically possible. The result is that deep convection must occur in order to export moist static energy at the required rate. This argument is made quantitative by the model of Sobel and Neelin (2006), which couples a free-tropospheric model with vertical structure similar to that of Neelin and Zeng (2000) with an explicit PBL in which Lindzen-Nigam effects (among other mechanisms) can operate.

8.5. Conclusions There are two classes of theories for the quasi-steady component of tropical precipitation and surface wind, given the SST. In one, the SST determines the winds directly, and the convergence of the winds at low levels determines the location and intensity of precipitation. In the other, precipitation is determined locally by the SST and related thermodynamic factors, and the surface winds are then induced by the heating associated with the precipitation. Available evidence suggests that deep convective heating is largely responsible for determining the surface zonal wind, but that particularly in regions of strong meridional SST gradients, those gradients to a significant extent induce the meridional wind directly, essentially as in the model of Lindzen and Nigam (1987). While the picture

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is less clear for precipitation, there is also some evidence to suggest that in these same regions—the narrow ITCZs—the convergences associated with these meridional winds, induced through the PBL momentum budget independently of deep convection, play a role in modulating the precipitation, though thermodynamic factors must play a significant role as well. We have argued that the interplay of PBL momentum dynamics and freetropospheric thermodynamics may possibly be understood by considering the moist static energy transport by the flow deduced from the PBL momentum budget forced directly by SST. When the latter is divergent and the SST not too high, we expect descent, and this can still be associated with positive F net and moist static energy export. When it is convergent, we can consider the moist static energy transport directly associated with this PBL-induced flow as another external forcing on the moist static energy budget of the deep troposphere, which will add constructively to F net. The descending export mode is forbidden by the PBL momentum budget, so deep ascent is the only means for the necessary moist static energy export to be accomplished, and this implies precipitation. Formulating a simple model for the quasi-steady component of the precipitation and surface wind over the tropical oceans is obviously a more limited task than formulating a GCM, which must simulate a much broader range of phenomena. Finding the best idealized model and understanding its properties would not necessarily lead immediately to improved GCM simulations of tropical climate. Nonetheless, if we do not understand—in a way that we can express through the formulation of a simple model in which cause and effect are clear—the basic factors controlling the location, spatial extent, and intensity of the tropical precipitation zones, our efforts to improve their simulation in GCMs can amount to little more than groping in the dark. Further effort with models at the simpler end of the hierarchy seems warranted at this moment.

Acknowledgments This chapter draws on conversations with many people, including Julio Bacmeister, Michela Biasutti, Chris Bretherton, Mark Cane, John Chiang, Kerry Emanuel, Isaac Held, Ron Miller, David Neelin, Richard Seager, and others I have no doubt forgotten to mention, whom I ask to forgive me. Michela Biasutti kindly provided Fig. 8.5, Joyce Meyerson and David Neelin kindly provided Fig. 8.8, and Bin Wang and Qing Bao kindly provided Fig. 8.9. The moist static energy plot in Figs. 8.3 and 8.6 was kindly provided by Sandra Yuter. I am indebted to Michela Biasutti, Ray Pierrehumbert, and Tapio Schneider for thoughtful and insightful reviews that led to a number of improvements.

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Notes 1. This error increases with frequency and is a major issue for high-frequency transients. 2. Free-tropospheric moisture does enter the non-entraining CAPE through its influence on the virtual temperature of the free troposphere, but that effect is both small in magnitude and of opposite sign to the effect of the same moisture on the likelihood and intensity of deep convection via entrainment and downdraft effects.

References Arakawa, A., 2004: The cumulus parameterization problem: Past, present, and future. J. Climate, 17, 2493–2525. Bacmeister, J. T., and M. J. Suarez, 2002: Wind stress simulations and the equatorial momentum budget in an AGCM. J. Atmos. Sci., 59, 3051–3073. Bacmeister, J. T., M. J. Suarez, and F. R. Robertson, 2006: Rain re-evaporation, boundary-layer/ convection interactions, and Pacific rainfall patterns in an AGCM. J. Atmos. Sci., 63 3383–3403. Battisti, D. S., E. S. Sarachik, and A. C. Hirst, 1999: A consistent model for the large scale steady surface atmospheric circulation in the Tropics. J. Climate, 12, 2956–2964. Betts, A. K., and W. Ridgway, 1989: Climatic equilibrium of the atmospheric convective boundary layer over a tropical ocean. J. Atmos. Sci., 46, 2621–2641. Bretherton, C. S., and A. H. Sobel, 2002: A simple model of a convectively coupled Walker circulation using the weak temperature gradient approximation. J. Climate, 15, 2907–2920. Burns, S. P., A. H. Sobel, and L. M. Polvani, 2006: Asymptotic solutions to the moist axisymmetric Hadley circulation. Theor. Comp. Fluid Dyn., 20, 443–467. Charney, J. G., 1971: Tropical cyclogenesis and the formation of the Intertropical Convergence Zone. Mathematical Problems in Geophysical Fluid Dynamics, W. H. Reid, Ed., Lectures in Applied Mathematics, 13, Amer. Math. Soc., 355–368. Chiang, J. C. H., and S. E. Zebiak, 2000: Surface wind over tropical oceans: Diagnosis of momentum balance, and modeling the linear friction coefficient. J. Climate, 13, 1733–1747. Chiang, J. C. H., S. E. Zebiak, and M. A. Cane, 2001: Relative roles of elevated heating and surface temperature gradients in driving anomalous surface winds over tropical oceans. J. Atmos. Sci., 58, 1371–1394. Curry, J. A., and P. J. Webster, 1999: Thermodynamics of Atmospheres & Oceans. Academic Press, 471 pp. Davey, M. K., and A. E. Gill, 1987: Experiments on tropical circulation with a simple moist model. Quart. J. Roy. Meteorol. Soc., 113, 1237–1269. Emanuel, K. A., 1987: An air-sea interaction model of intraseasonal oscillations in the Tropics. J. Atmos. Sci., 44, 2324–2340. Emanuel, K. A., 1994: Atmospheric Convection. Oxford University Press, 580 pp. Emanuel, K. A., J. D. Neelin, and C. S. Bretherton, 1994: On large-scale circulations in convecting atmospheres. Quart. J. Roy. Meteor. Soc., 120, 1111–1143. Geisler, J. E., and D. E. Stevens, 1982: On the vertical structure of damped steady circulation in the Tropics. Quart. J. Roy. Meteor. Soc., 108, 87–93. Gill, A. E., 1980: Some simple solutions for heat-induced tropical circulation. Quart. J. Roy. Meteorol. Soc., 106, 447–462.

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Tropical Precipitation and Wind | 249 Gu, G., and C. Zhang, 2001: A spectrum analysis of synoptic-scale disturbances in the ITCZ. J. Climate, 14, 2725–2739. Holton, J. R., J. M. Wallace, and J. M. Young, 1971: On boundary layer dynamics and the ITCZ. J. Atmos. Sci., 28, 275–280. Kalnay, E., and coauthors, 1996: The NMC/NCAR 40-year reanalysis project. Bull. Amer. Meteor. Soc., 77, 437–471. Kleeman, R., 1991: A simple model of the atmospheric response to ENSO sea surface temperature anomalies. J. Atmos. Sci., 48, 3–18. Kuo, H.-L., 1974: Further studies of the parameterization of the influence of cumulus convection on the large-scale flow. J. Atmos. Sci., 31, 1232–1240. Lin, J., Mapes, B. E., and M. Zhang, 2004: Radiation budget of the tropical intraseasonal oscillation. J. Atmos. Sci., in press. Lindzen, R. S., 1967: Planetary waves on beta-planes. Mon. Wea. Rev., 95, 441–451. Lindzen, R. S., 1974: Wave-CISK in the Tropics. J. Atmos. Sci., 31, 156–179. Lindzen, R. S., and S. Nigam, 1987: On the role of sea surface temperature gradients in forcing low-level winds and convergence in the Tropics. J. Atmos. Sci., 44, 2418–2436. Liu, W. T., and X. Xie, 2002: Double intertropical convergence zone—a new look using scatterometer. Geophys. Res. Lett., 29, 222072, doi:10.1029/2002GL015431. Mapes, B. E., 2000: Convective inhibition, subgrid scale triggering, and stratification instability in a toy tropical wave model. J. Atomos. Sci., 57, 1515–1535. Matsuno, T., 1966: Quasi-geostrophic motions in the equatorial area. J. Meteor. Soc. Japan, 44, 25–43. McBride, J. L., and W. M. Frank, 1999: Relationships between stability and monsoon convection. J. Atmos. Sci., 56, 24–36. McGauley, M., C. Zhang, and N. A. Bond, 2004: Large-scale characteristics of the atmospheric boundary layer in the eastern Pacific cold tongue/ITCZ region. J. Climate, 17, 3907–3920. Neelin, J. D., 1989: On the interpretation of the Gill model. J. Atmos. Sci., 46, 2466–2468. Neelin, J. D., 1997: Implications of convective quasi-equilibria for the large-scale flow. The physics and parameterization of moist atmospheric convection, R. K. Smith, Ed., 413–446, Elsevier. Neelin, J. D., and I. M. Held, 1987: Modeling tropical convergence based on the moist static energy budget. Mon. Wea. Rev., 115, 3–12. Neelin, J. D., I. M. Held, and K. H. Cook, 1987: Evaporation-wind feedback and low frequency variability in the tropical atmosphere. J. Atmos. Sci., 44, 2341–2348. Neelin, J. D., and N. Zeng, 2000: A quasi-equilibrium tropical circulation model—formulation. J. Atmos. Sci., 57, 1741–1766. Parsons, D., K. Yoneyama, and J.-L. Redelsperger, 2000: The evolution of the tropical western Pacific atmosphere-ocean system following the arrival of a dry intrusion. Quart. J. Roy. Meteor. Soc., 126, 517–548. Pauluis, O., 2004: Boundary layer dynamics and cross-equatorial Hadley circulation. J. Atmos. Sci., 61, 1161–1173. Pike, A. C., 1971: The inter-tropical convergence zone studied with an interacting atmosphere and ocean model. Mon. Wea. Rev., 99, 469–477. Raymond, D. J., 1995: Regulation of moist convection over the west Pacific warm pool. J. Atmos. Sci., 52, 3945–3959. Raymond, D. J., S. K. Esbensen, C. Paulson, M. Gregg, C. S. Bretherton, W. A. Petersen, R. Cifelli, L. K. Shay, C. Ohlmann, and P. Zuidema, 2004: EPIC2001 and the coupled ocean-atmosphere system of the tropical east Pacific. Bull. of the Amer. Meteor. Soc., 85, 1341–1354.

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250 | Adam H. Sobel Raymond, D. J., G. B. Raga, C. S. Bretherton, J. Molinari, C. Lopez-Carrillo, and Z. Fuchs, 2003: Convective forcing in the intertropical convergence zone of the eastern Pacific. J. Atmos. Sci., 60, 2064–2082. Reynolds, R. W., and T. M. Smith, 1994: Improved global sea surface temperature analyses. J. Climate, 7, 929–948. Seager, R., 1991: A simple model of the climatology and variability of the low-level wind field in the Tropics. J. Climate, 4, 164–179. Seager, R., R. Murtugudde, A. Clement, and C. Herweijer, 2003: Why is there an evaporation minimum at the equator? J. Climate, 16, 3792–3801. Sherwood, S. C., 1999: Convective precursors and predictability in the tropical western Pacific. Mon. Wea. Rev., 127, 2977–2991. Smith, R. K., 1997: On the theory of CISK. Quart. J. Roy. Met. Soc., 123, 407–418. Sobel, A. H., 2002: Water vapor as an active scalar in tropical atmosphere dynamics. Chaos, 12, 451–459. Sobel, A. H., 2003: On the coexistence of an evaporation minimum and precipitation maximum in the warm pool. J. Climate, 16, 1003–1009. Sobel, A. H., and C. S. Bretherton, 2000: Modeling tropical precipitation in a single column. J. Climate, 13, 4378–4392. Sobel, A. H., I. M. Held, and C. S. Bretherton, 2002: The ENSO signal in tropical tropospheric temperature. J. Climate, 15, 2702–2706. Sobel, A. H., and J. D. Neelin, 2006: The boundary layer contribution to intertropical convergence zones in the quasi-equilibrium tropical circulation model framework. Theor. Comp. Fluid Dyn., 20, 323–350. Sobel, A. H., J. Nilsson, and L. M. Polvani, 2001: The weak temperature gradient approximation and balanced tropical moisture waves. J. Atmos. Sci., 58, 3650–3665. Sobel, A. H., S. E. Yuter, C. S. Bretherton, and G. N. Kiladis, 2004: Large-scale meteorology and deep convection during TRMM KWAJEX. Mon. Wea. Rev., 132, 422–444. Stevens, B., J. Duan, J. C. McWilliams, M. Munnich, and J. D. Neelin, 2002: Entrainment, Rayleigh friction, and boundary layer winds over the tropical Pacific. J. Climate, 15, 30–44. Stevens, B., D. A. Randall, X. Lin, and M. T. Montgomery, 1997: A comment on: On largescale circulations in convecting atmospheres by Emanuel, Neelin and Bretherton. Quart. J. Roy. Meteor. Soc., 123, 1771–1778. Su, H., and J. D. Neelin, 2002: Teleconnection mechanisms for tropical Pacific descent anomalies during El Niño. J. Atmos. Sci., 59, 2694–2712. Tomas, R. A., and P. J. Webster, 1997: The role of inertial instability in determining the location and strength of near-equatorial convection. Quart. J. Roy. Meteorol. Soc., 123, 1445–1482. Tomas, R. A., J. R. Holton, and P. J. Webster, 1999: The influence of cross-equatorial pressure gradients on the location of near-equatorial convection. Quart. J. Roy. Meteorol. Soc., 125, 1107–1127. Trenberth, K. E., J. M. Caron, and D. P. Stepaniak, 2001: The atmospheric energy budget and implications for surface fluxes and ocean heat transports. Clim. Dyn., 17, 259–276. Waliser, D. E., and N. E. Graham, 1993: Convective cloud systems and warm-pool surface temperatures: coupled interactions and self-regulation. J. Geophys. Res., 98, 12881–12893. Waliser, D. E., and R. C. J. Somerville, 1994: Preferred latitudes of the intertropical convergence zone. J. Atmos. Sci., 51, 1619–1639. Wang, B., and T. Li, 1993: A simple tropical atmosphere model of relevance to short-term climate variations. J. Atmos. Sci., 50, 260–284.

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Tropical Precipitation and Wind | 251 Weare, B. C., 1986: A simple model of the tropical atmosphere with circulation dependent heating and specific humidity. J. Atmos. Sci., 43, 2001–2016. Webster, P. J., 1972: Response of the tropical atmosphere to local, steady forcing. Mon. Wea. Rev., 100, 518–541. Webster, P. J., 1980: Mechanisms determining the atmospheric response to sea surface temperature anomalies. J. Atmos. Sci., 38, 554–571. Wu, Z., D. S. Battisti, and E. S. Sarachik, 1999: Rayleigh friction, Newtonian cooling, and the linear response to steady tropical heating. J. Atmos. Sci., 57, 1937–1957. Xie, P., and P. A. Arkin, 1997: Global precipitation: A 17-year monthly analysis based on gauge observations, satellite estimates, and numerical model outputs. Bull. Amer. Meteor. Soc., 79, 2539–2558. Yu, J.-Y., C. Chou, and J. D. Neelin, 1998: Estimating the gross moist stability of the tropical atmosphere. J. Atmos. Sci., 55, 1354–1372. Yuter, S. E., R. A. Houze Jr., E. A. Smith, T. T. Wilheit, and E. Zipser, 2005: Physical characterization of tropical oceanic convection observed in KWAJEX. J. Appl. Meteor., 44, 385–415. Zebiak, S. E., 1982: A simple atmospheric model of relevance to El Niño. J. Atmos. Sci., 39, 2017–2027. Zebiak, S. E., 1986: Atmospheric convergence feedback in a simple model for El Niño. Mon. Wea. Rev., 114, 1263–1271. Zeng, N., J. D. Neelin, and C. Chou, 2000: A quasi-equilibrium tropical circulation model— implementation and simulation. J. Atmos. Sci., 57, 1767–1796. Zhang, C., M. McGauley, and N. A. Bond, 2004: Shallow meridional circulation in the tropical eastern Pacific. J. Climate, 17, 133–139.

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Chapter 9

Dynamical Constraints on Monsoon Circulations R. Alan Plumb

9.1. Introduction While much has been written on monsoon circulations, a comprehensive theory of their dynamics is still lacking. It is clear that, given the distribution of precipitation, many of the observed characteristics of monsoon flows can be understood on the basis of the linear response to diabatic heating (e.g., Gill 1980; Hoskins and Rodwell 1995). Nevertheless, such solutions fall short of a complete theory. For one thing, in general the distribution of precipitation is not determined a priori by the externals of the problem but is in itself a part of the atmosphere’s response, and this is especially true in the presence of the heterogeneous surface conditions typical of monsoon situations. Explaining the distribution of precipitation over land is a key part of the monsoon problem. A second issue not resolved by linear theory involves placing the monsoon problem into the broader context of the theory of tropical dynamics. The widely accepted paradigm for the tropical Hadley circulation, the theory of Held and Hou (1980), hinges on nonlinear angular momentum advection in the upper troposphere, assumed inviscid, creating uniform absolute angular momentum right across the Hadley cell. To some extent, monsoon circulations form a part of the seasonal Hadley circulation. This is especially true in northern summer, when the zonally averaged upward motion in the Tropics is dominated by the contribution over south Asia, so that in this season the Hadley circulation—if by that one means the zonally averaged meridional overturning—is essentially the zonal average of the south Asian monsoon. Placed in that context, it is difficult to reconcile linear theories of monsoon flows with nonlinear theories for the Hadley cell. In fact, applying the principles of the Held-Hou theory to zonally symmetric cases with applied forcing off the equator leads to the conclusion that a threshold forcing must be exceeded before radiative-convective equilibrium is replaced by a deep (nonviscous) meridional circulation (Plumb and Hou 1992; Emanuel 1995), although under

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C

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Figure 9.1. Schematic of a divergent monsoonal flow. The focus of discussion in the text is on the upper-level potential vorticity budget within the isentropic layer enclosed by the contour C.

moist dynamics the relevant criterion involves the distribution of boundary-layer moist entropy and is more diagnostic than predictive. In principle, extending the zonally symmetric constraint of uniform angular momentum to a more general requirement of zero potential vorticity (PV) in the absence of zonal symmetry is straightforward (Schneider 1987). However, a patch of zero PV localized off the equator may become unstable; during northern summer, this is manifested in the shedding of eddies from the Tibetan anticyclone (Hsu and Plumb 2000; Popovic and Plumb 2001). A distinct, better known, instability of the low-level easterly jet that in simple models is ubiquitous in monsoon circulations (and is in fact strongest prior to monsoon onset) further complicates the picture. Nevertheless, the low-level moist entropy distribution seems to hold the key to the existence of strong monsoonal flows, and to the distribution of continental precipitation. These issues will be discussed in greater detail in this chapter. There will be no attempt here to give a comprehensive account of monsoon dynamics (but see chapter 10 in this volume). Rather, what follows will be a personal view of the “circulation constraint” on divergent tropical flows, the way in which this constraint is violated, and its implications for monsoon dynamics.

9.2. Sustained Divergent Flows: The Circulation Constraint in the Upper Troposphere Consider a divergent, monsoonal flow such as that depicted schematically in Fig. 9.1. We focus, following the arguments of Schneider (1987), on the upper-level outflow, where the flow might reasonably be assumed to be inviscid and adiabatic.

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P1 P

a

P2

b Figure 9.2. Schematic of the implications of equation (9.1). In each case, the shaded ellipse represents a region of upper-level divergence; the contours represent PV on an uppertropospheric isentropic surface. In (a), closed PV contours enclose the divergent region; in (b), the PV contours span the globe.

The inviscid equation of motion can be written ∂t u + ζa × u = −∇ B, where ζa is absolute vorticity and B is the Bernoulli function. If  = circulation around any fixed closed contour C, then  ∂t  + ζa × u · dl = 0.




C u · dl

is the

C

Now if the contour lies within an isentropic surface and the flow at the level of interest is locally adiabatic, then   ζa × u · dl = Pρθ u · n dl , C

where ρθ = −g −1∂θ p is the isentropic density, P = −g ζa ∂ p θ is the PV, and n is the outward unit normal. If, furthermore, we choose the contour C to be one of constant P , we finally have, for steady flow (∂t  = 0),  P ρθ u · n dl = 0. [9.1] In inviscid, adiabatic, steady flow, therefore, there can be no net divergent flow across any contour of constant PV on an isentropic surface unless the value of PV on that contour is zero. On the face of it, this statement represents a major constraint on divergent tropical circulations in the upper troposphere, where one might regard the flow as nearly inviscid. Figure 9.2 illustrates the point. Consider a localized region of upper-level divergence, represented by the shaded regions. In the first case, (a), it is envisaged that the region of divergence lies within a closed P contour, in which case a steady conservative flow of this kind can exist, with nonzero net divergence within the contour, only if P = 0 on all closed contours enclosing the region of divergence, and thus on all such contours except those that enclose all the compensating convergence as well as the divergence. If the PV contours span all longitudes, as in case (b), then if the region between the two contours P = P1 and P = P2 contains the region of divergence

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Figure 9.3. July 1987–90 mean PV (upper panel; units are 10−6 K m2 s−1 kg−1) and Montgomery potential M = c pT + g z (lower panel; units are m2 s−2) at 370 K according to NCEP reanalysis data. From Popovic and Plumb (2001). (Reproduced with permission from c 2001.) the American Meteorological Society


(and not all of the compensating convergence), there must be net divergent flow across at least one of those contours, in which case P must be zero on that contour. This constraint is well known in the theory of the zonally averaged Hadley circulation (Held and Hou 1980; Lindzen and Hou 1988). Under zonal symmetry,  −1 ζa = − a 2 cos φ ∂φm, where m = a 2 cos2 ϕ + ua cos ϕ is the absolute angular momentum per unit mass, and so vanishing of upper-tropospheric absolute vorticity over the region of the divergent Hadley circulation amounts to uniform absolute angular momentum there. In reality, absolute vorticity does not vanish across the tropical upper troposphere, although in much of the region relative vorticity is anticyclonic, making the absolute vorticity weaker than the planetary term alone. Thus, while the theory goes some way toward explaining the observed circulation, it is clear that angular momentum is not well conserved in reality: air is permitted to cross angular momentum surfaces through the agency of eddy momentum transport, whose effects are neglected in the foregoing theory. The important role of eddy momentum transport in the Hadley circulation has recently been highlighted by Walker and Schneider (2006). Monsoon circulations comprise a major component of the divergent tropical flow, especially in northern summer when the circulation is dominated by the flow of the South Asian monsoon. Four-year means of upper-tropospheric potential vorticity and Montgomery potential M for July are shown in Fig. 9.3. (Note that M/ f , where f is the Coriolis parameter, is the streamfunction for the geostrophic flow within the isentropic surface.) The huge Tibetan anticyclone that so dominates M is also evident in the absolute vorticity distribution; the core of the anticyclone is a local minimum

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in ζa (which coincides with the elevated tropopause over the region). Nevertheless, summertime-mean ζa is significantly different from zero across south Asia, especially between the monsoon region and the equator. How, then, does the extensive, crossequatorial divergent circulation satisfy the circulation constraint? Vorticity budget analysis in the region of the Tibetan anticyclone shows that, as in the zonally averaged picture, transient eddy fluxes play a role at leading order. These do not, however, appear to be eddies of midlatitude origin crashing into the tropical upper troposphere; rather, both models and observations indicate that the anticyclone itself is the source of these eddies.

9.3. Criteria for Sustained Divergent Circulations The circulation constraint states that, if the upper-level PV remains cyclonic, there can be no sustained upper-level divergence, assuming the flow there to be steady and inviscid. Plumb and Hou (1992) argued, on the basis of a dry, zonally symmetric model with diabatic heating and cooling represented by Newtonian relaxation to an equilibrium temperature Te possessing a localized subtropical maximum, that this implies that a threshold of the magnitude of the Te maximum must be exceeded before a divergent circulation can be sustained. This follows from consideration of the radiative equilibrium state, when T = Te and the zonal wind u (assumed zero at the ground) is in gradient wind balance with the temperature field. It is straightforward to show that absolute vorticity is cyclonic at the tropopause (z = D), and therefore that the radiative equilibrium solution is regular, provided  3  g D sin φ cos φ ˆ − ∂φ ∂φT e < f 2a 2, Tr cos3 φ sin φ

[9.2]

where f = 2 sin φ is the Coriolis parameter, Tˆ e is the vertically averaged radiative equilibrium temperature, and Tr a reference temperature. When (9.2) is satisfied, no upper-level divergent flow is permitted and numerical simulations show only weak (viscously driven) meridional circulations. On the other hand, when the forcing is strong enough for (9.2) to be violated, the equilibrium solution becomes irregular, and numerical solutions show a strong meridional circulation spanning an extensive region of near-zero upper-level PV. Emanuel (1995) extended the argument to a moist atmosphere, in which case the radiative-convective equilibrium state is assumed to be moist adiabatic, and thus dictated by the distribution of boundary-layer moist entropy; then (9.2) is replaced by  3  sin φ cos φ − 3 ∂φ T ∂φs b < f 2a 2, cos φ sin φ

[9.3]

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where T is the temperature difference between the surface and tropopause and s b the boundary-layer moist entropy. The relevance of (9.3) has been demonstrated in zonally symmetric moist models (Emanuel 1995; Zheng 1998). It has even been found to be relevant in three-dimensional models where, moreover, the location of the s b maximum appears to determine the poleward limit of the divergent circulation, and thus the maximum inland penetration of the monsoon (Privé and Plumb 2007a, 2007b), consistent with the findings of Chou and Neelin (2003). Thus, the essential dynamical constraint of the upper-level circulation budget manifested in (9.2) does seem to extend into the moist, three-dimensional case, though with some caveats. The first caveat is that in the moist case, the threshold criterion is less useful as a predictor than in the dry case, simply because s b is not always a simple function of the external boundary conditions. Advection by even a weak divergent circulation in the boundary layer (which is always present, even in subcritical cases) can have a substantial impact on the distribution of s b near a coast. A second caveat is that as we have seen, time-averaged upper-level PV does not vanish over extensive regions (though it may become weak). Three-dimensional dynamics—i.e., the presence of eddies—allows the assumptions leading to (9.1), and thence to (9.2) and (9.3), to be violated. These eddy processes, which appear to be inevitable in a monsoon circulation, will be discussed in the following sections. Their existence means that in realistic situations (9.3) can only be regarded as approximate.

9.4. Violation of the Circulation Constraint in the Upper Troposphere How the steady circulation constraint can be broken may be understood from simple arguments, and is revealed by simple models. Let us represent the divergent uppertropospheric flow as the outflow from a localized mass source in a single layer on a beta-plane, as depicted in Fig. 9.4. As zero-PV air spreads out from the source region, it does so asymmetrically, spreading preferentially westward as a “beta-plume” (Rhines 1983). As the plume extends westward, it becomes unstable, as expected for an elliptical vortex when its aspect ratio exceeds 3; a wave forms along its edges, rolls up as an anticyclonic eddy, and is shed westward from the now-diminished source vortex. An example of this occurring in a shallow-water model is shown in Fig. 9.5. In such simple cases, the behavior is very periodic, and it suffices to show a single period. Low-PV air emanating from the source is shed from the source region within the detached eddy. Thus, the source flow diverges away from the source region into an environment of nonzero PV, violating the circulation constraint by virtue of the sustained timedependence of the flow. In fact, by conducting the circulation analysis with respect to a time-dependent PV contour (Sobel and Plumb 1999), it can be shown that viscous effects, acting on the small-scale filamentary structures accompanying the detachment, are key to the existence of nonzero divergence within the contour. Thus, from a timeaveraged perspective, net divergent flow occurs through the agency of transient eddy PV

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Figure 9.4. Schematic of shallow-water flow away from a localized source (marked by the heavy dot) on a beta-plane. North is at the top.

fluxes, the eddies produced, inevitably, as a consequence of the instability of the vorticity distribution created by the divergent flow itself. This kind of shedding behavior is indeed evident in the upper troposphere, at least in the vicinity of the Tibetan anticyclone in northern summer. Figure 9.6 shows one example of this, an event that took place during 11–13 July 1990. Over the 54-hr period shown, the main anticyclone elongated westward and split, shedding a large anticyclone as far as 30◦ E. Subsequently (not shown), the detached eddy becomes caught up in the westerly jet immediately to its north, and is sheared out rapidly eastward into a filament that appears, to some extent, to re-merge with the main anticyclone. This kind of event is not an isolated occurrence; typically, three or four such events can be identified over the course of a summer (Popovic and Plumb 2001). This behavior makes it apparent that, in reality, the circulation constraint can be violated in the upper troposphere, and the divergent flow can thus extend across finite regions of nonzero PV; the dynamics of the divergent flow itself allows this to happen (rather than, say, incidental transport from eddies of extraneous origin). Does this upper-tropospheric behavior influence the overall characteristics of the monsoon flow? The direct signal of anomalies in PV and M extends down only as far as 340 K (about 400 hPa) (Popovic and Plumb 2001), so there is no obvious feedback onto the lower-tropospheric flow. Nevertheless, Randel and Park (2006) have identified a coherent relationship between anomalies in upper-level PV and in outgoing longwave radiation (OLR), suggesting a relationship with deep convection; however, since the

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t=0

t = 1/4

t = 1/2

t = 3/4

Figure 9.5. Periodic shedding of an anticyclone from a mass source on a shallow-water betaplane in the Northern Hemisphere. Grayscale shading represents absolute vorticity (medium gray is near zero, lightest gray intermediate, and darkest gray the most positive values); contours are free surface height; arrows show velocity. One period of the cycle is shown; times are in fractions of a period. Adapted from Hsu and Plumb (2000). (Reproduced with permission c 2000.) from the American Meteorological Society


OLR signal appears to lead the PV signal, it is not clear whether this implies an impact of upper-tropospheric behavior onto the convection. As an aside (since our main focus here is on the dynamics of the monsoon), the upper-tropospheric activity we have noted may be relevant to issues of transport in the vicinity of the tropical tropopause. Note from Fig. 9.5 that in concert with the shedding of the anticyclone, cyclonic air is entrained equatorward immediately to the east of the

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Figure 9.6. Eddy shedding at 370 K (just below the tropopause). Time sequence of PV field (shading) and the geopotential height at 200 hPa (contours) over the Asian summer monsoon at successive 18-hour time intervals from 0000 UTC 11 July to 0600 UTC 13 July 1990. The grayscale contour interval is 0.05 PVU. The black contour is 25 m. Adapted from Hsu and Plumb (2000). (Reproduced with permission from the American Meteorological Society c 2000.)


source region and rolls up cyclonically to the west. If this occurs in reality (and there are hints of it in Fig. 9.6), this is tantamount to entrainment of stratospheric air into the tropical upper troposphere. Such transport would be complementary to the poleward transport of tropical tropopause air into the stratosphere noted by Dethof et al. (1999). Randel and Park (2006) have recently discussed the observed characteristics of tracer transport into and in the vicinity of the Tibetan anticyclone.

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Figure 9.7. Schematic showing the middle- or upper-level absolute-vorticity distribution with latitude for a sequence of cases with a localized s b maximum of increasing magnitude (corresponding to decreasing minima of ζa at y = φ/φ0 = 1). The sign of the absolute-vorticity gradient changes sign at lower magnitude than the sign of absolute vorticity itself.

9.5. Lower Tropospheric Eddies and Their Impact Eddy formation in the lower troposphere, quite distinct from what may happen near the tropopause, appears to be ubiquitous in summertime situations where land lies poleward of an equatorial ocean. In such situations, which are not restricted to those with an active monsoon, the easterly jet associated with the landward temperature gradient becomes unstable, generating easterly waves (e.g., Rennick 1976; Thorncroft and Hoskins 1994). In fact, in a seasonally varying model with a simple continent, Xie and Saiki (1999) found monsoon onset to occur just after the appearance of such waves, and ascribed the onset to the influence of these eddies. Xie and Saiki’s remark gives grounds for doubting the relevance of the steadycirculation constraint in the non-axisymmetric case. Consider the schematic Fig. 9.7. Suppose that s b has a localized maximum at latitude φ0; the distribution of ζa in the middle or upper troposphere is shown schematically for four different magnitudes of this maximum (including zero, the straight line). The absolute vorticity ζa first reaches zero in case 3, but it is evident that its gradient changes sign, thus raising the possibility of instability, long before this stage is reached. If the waves generated by the instability are responsible for monsoon onset, then this may occur at lower forcings than (9.3) would suggest. The role of these waves, amongst other things, was investigated by Privé and Plumb (2007a, 2007b) in two- and three-dimensional models of the response of a moist

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Figure 9.8. A particularly simple example of a westward-propagating eddy produced by instability of a coastal easterly jet. A continent with prescribed surface heat flux occupies the region everywhere north of 16◦ N; to the south is ocean with prescribed SST. This case is slightly subcritical in the sense that the continental heat flux is just too weak to produce a monsoon. The lower frame shows rainfall at the coastal grid point vs. longitude and time; the upper frame shows rainfall and 1000-hPa flow, composited with respect to the moving disturbance. After Privé and Plumb (2007b).

atmosphere to the presence of continents with simple geometries. In the case shown in Fig. 9.8, a single continent was placed everywhere north of 16◦N, with prescribed SST (with an equatorial maximum) south of the coast. The total surface heat flux on the continent was prescribed, with a maximum near the coast and decreasing monotonically inland, and the model run (for several hundred days) to an equilibrated state. For sufficiently large continental heat flux, a monsoonal state was produced. For

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weaker (subcritical) forcing, the continent was found to be very arid and hot almost everywhere (except very close to the coast) with relatively shallow dry convection; the corresponding mid-level easterly jet along the coast became unstable and generated eastward propagating disturbances. The case of Fig. 9.8 is a particularly simple example of this, with a single eddy propagating westward along the coast. The steadily translating cyclonic eddy has a rainfall maximum on its eastern flank, where moisture is advected onto the land from the ocean, and a very dry region to its west, where the low-level flow is divergent (as a consequence of Rossby-wave-induced subsidence—the “RodwellHoskins” effect) and dry air with relatively low s b is being advected off the interior of the continent. While in such cases the eddies are obviously influential in organizing the rainfall pattern, do they in fact facilitate monsoon development? With this configuration of a zonally uniform continent, it is relatively easy to address this question directly by comparing the results of the three-dimensional model with those of a zonally symmetric two-dimensional model, in which of course the eddies and their effects are suppressed. Making such a comparison, Privé and Plumb (2007b) found the impact of eddies to be surprisingly modest and, in fact, to be such as to inhibit monsoon development— in the sense that creation of a monsoon requires slightly larger surface heat fluxes over the land in the presence of eddies—rather than to encourage it. The reason for this is straightforward when framed in terms of boundary-layer moist entropy: air advected from the ocean onto the land by the eddies, while moist, has lower entropy than that in the continental boundary layer. Thus, the effect of the eddies is to weaken the land surface maximum in s b , and thereby to suppress monsoonal circulations. While the generation of westward-propagating disturbances weakens once a monsoonal circulation appears (when the low-level temperature gradient decreases), they do not entirely disappear. Figure 9.9 shows results from a case with a Northern Hemisphere continent occupying the region north of 16◦ N, and extending 0◦–180◦ in longitude. Like the previous case, the surface heat flux is specified (as a function of latitude only) at the land surface; SSTs are specified, with a maximum at 8◦ N. Significant time-averaged precipitation on the continent is confined to the extreme southeast corner, where it almost merges with the midlatitude rainband associated with the oceanic storm track. What little precipitation there is along the central and western parts of the south coast is associated with weak westward-propagating disturbances, as shown in the bottom frame of the figure. (In fact, as also found by Cook and Gnanadesikan (1991), continental aridity appears difficult to avoid in models with such simplified geography.) The aridity of all but the far-eastern portion of the continent is not primarily a Rossby wave effect (it extends too far for that) but a consequence of the low-level advection of low-s b air eastward from the cool midlatitude ocean. Like Chou et al. (2001) and Chou and Neelin (2003), we find that the extent of the monsoonal rainfall—in longitude as well as in latitude—is very much restricted by the requirement of high s b. In fact, if advection of low-entropy air onto the continent

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Figure 9.9. A case with a continent lying north of 16◦ N and between 0◦ and 180◦ in longitude, with prescribed SST peaking at 8◦ N. Top: Time-averaged 1000-hPa winds and rainfall; bottom: rainfall at 16◦ N vs. longitude and time, showing westward-propagating disturbances.

is suppressed by the addition into the model of impermeable walls (extending up to 700 hPa) at the eastern and western coasts, intense rain occurs over the land all along the south coast.

9.6. Conclusions While the circulation constraint yields, for dry dynamics under zonal symmetry, a simple predictor for the existence of divergent circulations in terms of the external forcing (the radiative equilibrium temperature distribution), it becomes progressively less direct when moisture and three-dimensionality are added. While the constraint still exists, and can be framed in terms of boundary-layer moist entropy for steady flow, it is

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no longer a simple statement of external conditions and is at best approximate because of the presence of eddies that arise from instability of either the monsoon flow itself (in the upper troposphere) or the low-level easterly jet that is a consequence of the very same land-sea contrast that produces the monsoon. The upper-level instability ensures that the zero PV limit is never reached (on a time-average; instantaneously, analyses suggest the frequent occurrence of zero or near-zero PV). Whether the quasi-periodic shedding of eddies in the upper troposphere has any significant manifestations in the “big picture” of the monsoon or is merely an upper-level curiosity is not clear, though suggestions of a relationship between fluctuations of upper-level PV and outgoing longwave radiation, reported by and Randel and Park (2006), are intriguing. The low-level disturbances seen in models with simple continents (Privé and Plumb 2007b), which appear to be representative of easterly waves in pre-monsoon conditions and perhaps of monsoon depressions in an active monsoon, are very effective at organizing rainfall, and indeed in producing rainfall in otherwise arid regions. Nevertheless, they appear to play a rather modest role in monsoon onset, delaying it slightly by advecting low-entropy air from the ocean onto the land.

Acknowledgments I thank Nikki Privé for many discussions and for providing some of the figures, and Bill Randel for sharing his results of upper-tropospheric transport in advance of publication. This work was supported by the National Science Foundation.

References Chou, C., and J. D. Neelin, 2003: Mechanisms limiting the northward extent of the northern summer convection zones. J. Climate, 16, 406–425. Chou, C., J. D. Neelin, and H. Hsu, 2001: Ocean-atmosphere-land feedbacks in an idealized monsoon. Quart. J. R. Meteor. Soc., 127, 1869–1891. Cook, K. H., and A. Gnanadesikan, 1991: Effects of saturated and dry land surfaces on the tropical circulation and precipitation in a general circulation model. J. Climate, 4, 873–889. Dethof, A., A. O’Neill, J. M. Slingo, and H. G. J. Schmidt, 1999: A mechanism for moistening the lower stratosphere involving the Asian summer monsoon. Quart. J. R. Meteor. Soc., 125, 1079–1106. Emanuel, K. A., 1995: On thermally direct circulations in moist atmospheres. J. Atmos. Sci., 52, 1529–1534. Gill, A. E., 1980: Some simple solutions for heat-induced tropical circulation. Quart. J. R. Meteor. Soc., 106, 447–462. Held, I. M., and A. Y. Hou, 1980: Nonlinear axially symmetric circulations in a nearly inviscid atmosphere. J. Atmos. Sci., 37, 515–533. Hoskins, B. J., and M. J. Rodwell, 1995: A model of the Asian summer monsoon. Part I: The global scale. J. Atmos. Sci., 52, 1329–1340.

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266 | R. Alan Plumb Hsu, C. J., and R. A. Plumb, 2000: Non-axisymmetric thermally driven circulations and upper tropospheric monsoon dynamics. J. Atmos. Sci., 57, 1254–1276. Lindzen, R. S., and A. Y. Hou, 1988: Hadley circulations for zonally averaged heating centered off the equator. J. Atmos. Sci., 45, 2416–2427. Plumb, R. A., and A. Y. Hou, 1992: The response of a zonally-symmetric atmosphere to subtropical thermal forcing. J. Atmos. Sci., 49, 1790–1799. Popovic, J. M., and R. A. Plumb, 2001: Eddy shedding from the upper tropospheric Asian monsoon anticyclone. J. Atmos. Sci., 58, 93–104. Privé, N. C., and R. A. Plumb, 2007a: Monsoon dynamics with interactive forcing. Part I: Axisymmetric studies. J. Atmos. Sci., 64, 1417–1430. Privé, N. C., and R. A. Plumb, 2007b: Monsoon dynamics with interactive forcing. Part II: Impact of eddies and asymmetric geometries. J. Atmos. Sci., 64, 1431–1442. Randel, W. J., and M. Park, 2006: Deep convective influence on the Asian summer monsoon anticyclone and associated tracer variability observed with UARS. J. Geophys. Res., 111, D12314, doi:10.1029/2005JD006490. Rhines, P., 1983: Lectures on geophysical fluid dynamics. Lect. Appl. Math., 20, 1–58. Rennick, M. A., 1976: The generation of African waves. J. Atmos. Sci., 33, 1955–1969. Rodwell, M. R., and B. J. Hoskins, 1996: Monsoons and the dynamics of deserts. Quart. J. R. Meteor. Soc., 122, 1385–1404. Schneider, E. K., 1987: A simplified model of the modified Hadley circulation. J. Atmos. Sci., 44, 3311–3328. Sobel, A. H., and R. A. Plumb, 1999: Quantitative diagnostics of mixing in a shallow-water model of the stratosphere. J. Atmos. Sci., 56, 2811–2829. Thorncroft, C. D., and B. J. Hoskins, 1994: An idealized study of African easterly waves. I: A linear view. Quart. J. R. Meteor. Soc., 120, 953–982. Walker, C. C., and T. Schneider, 2006: Eddy influences on Hadley circulations: Simulations with an idealized GCM. J. Atmos. Sci., 63, 3333–3350. Xie, S.-P., and N. Saiki, 1999: Abrupt onset and slow seasonal evolution of summer monsoon in an idealized GCM simulation. J. Meteor. Soc. Japan, 77, 949–968. Zheng, X., 1998: The response of a moist zonally symmetric atmosphere to subtropical surface temperature perturbation. Quart. J. R. Meteor. Soc., 124, 1209–1226.

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Chapter 10

Moist Dynamics of Tropical Convection Zones in Monsoons, Teleconnections, and Global Warming J. David Neelin

10.1. Introduction 10.1.1. Approach In tropical dynamics a leading feature is the interaction of the large-scale circulation with moist processes arising at small scales. This chapter summarizes a recent avenue of tropical theory for this interaction, in which constraints from the moist dynamics can actually simplify large-scale theory under certain conditions. Important tools in this approach include convective quasi-equilibrium (QE) and the moist static energy (MSE) budget. Convective QE (Arakawa and Schubert 1974; Emanuel et al. 1994) is the postulate, used in various forms in a number of convective parameterizations, that moist convection tends to alter the large-scale temperature and moisture vertical profiles in a manner that reduces the buoyancy available to small-scale overturning motions. A further summary of QE may be found in chapter 7 in this volume. The aspects required here are the tendency of moist convection to constrain the temperature profile and to establish a relationship between temperature and moisture. In outlining the approach to moist dynamics advocated here, three applications are considered: monsoons, tropical teleconnections, and tropical precipitation changes under global warming. Background for these is provided in the remainder of this section, aiming also to convey the challenges of analyzing large-scale flow interacting with moist convection. With this motivation, section 10.2 outlines the MSE budget and a useful quantity that arises from it under QE approximations, the gross moist stability. A numerical model, the Quasi-Equilibrium Tropical Circulation Model (QTCM), whose formulation has been designed to mesh with this theoretical framework, is sketched in section 10.3. This model is used in simulating the phenomena considered, but its particular value is the relative simplicity of analysis—either in pointing to solution features to analyzes in observations or more complex models, or generating simpler models from its equations.

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A consequence of viewing the thermodynamics via the MSE budget is the importance of the net energy flux into the atmospheric column and implications for land-sea contrast, as outlined in section 10.4. This ties directly to the first application, mechanisms setting monsoon poleward boundaries, in section 10.5. Theory for how precipitation anomalies arise in these moist dynamical interactions, along with treatment of cloud feedbacks and land-versus-ocean cases in this framework, is summarized in sections 10.6 and 10.7. How such diagnostics apply to moist teleconnection theory (section 10.9) and tropical precipitation anomalies under global warming (section 10.10) is then addressed. We know there are limits of validity to the approximations used here, particularly as one moves to the small scales. The boundaries of the regime of validity have not yet been well determined, and this is part of the current direction of this research area. Statements of where limitations or modifications are believed by this author (in discussion with several colleagues noted in the acknowledgments) to be likely to arise are included in the form of postulates, denoted open question or postulate as appropriate. The hope is to summarize phenomena and scales where this particular approach shows promise, while tempting the reader with aspects that are as yet unresolved.

10.1.2. Monsoon Background There is a rich literature on various monsoon systems: Asian (Ramage 1971; Riehl 1979; Luo and Yanai 1984; Murakami 1987; Ding 1992; Yasunari and Seki 1992; Ding 1994; Lau and Yang 1996; Li and Yanai 1996; Webster et al. 1998; Annamalai et al. 1999; and references therein), North American (Douglas et al. 1993; Higgins et al. 1997; Barlow et al. 1998; Yu and Wallace 2000), African (Palmer 1986; Xue and Shukla 1993; Lare and Nicholson 1994; Rowell et al. 1995; Eltahir and Gong 1996; Cook 1997; Semazzi and Sun 1997; Janicot et al. 1998), and South American (Lenters and Cook 1997; Zhou and Lau 1998, Nogues-Paegle et al. 2002). It is not possible here to review this vast body of work (see Webster [1987] and Young [1987] for an introduction). Rather, the aim here is to add a particular moist dynamical perspective on the large-scale aspects of these systems. Chapter 9 in this volume provides a complementary perspective on monsoon dynamics. The term “monsoon” has migrated from older definitions (Ramage 1971) to recent usage emphasizing regions with “pronounced summertime precipitation maxima” (Yu and Wallace 2000), sometimes including ocean regions adjacent to continental monsoon regions (Chao 2000). Many aspects of the circulation such as an upperlevel anticyclone and energetics commonly associated with monsoons (Webster 1987; Douglas et al. 1993; Zhou and Lau 1998) are intimately linked to this large-scale seasonal movement of the tropical convergence zones. For reference, and to illustrate issues that arise in a precipitation-centered view of the monsoons, Fig. 10.1 shows climatological rainfall (from the Xie and Arkin

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[1997] dataset, 1979–1998 average) for December–February (DJF) and June–August (JJA) processed in two ways: as a percent of annual rainfall and as the seasonal excess relative to the annual mean. Both are measures of the monsoon characteristic of high summer rainfall relative to the annual mean. From Fig. 10.1 (a) and (b), 40% of annual

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rainfall occurring in JJA would be a plausible monsoon indicator (50% would exclude some traditional monsoon regions). But by this measure, monsoon regions would extend far poleward in continental interiors in the summer hemisphere. Discussing the “Canadian Prairie monsoon” or the “Siberian monsoon” might be stretching the terminology, although by this measure they clearly exceed the “Arizona monsoon.” To have an impact on the circulation, the monsoon deep convective diabatic heating should be sufficiently large. From this point of view, Fig. 10.1 (c) and (d) provides a better measure. A threshold as simple as 2 mm/day summer-season precipitation excess (above the annual average) yields regions that correspond fairly well to current notions of the large-scale monsoons. The 2 mm/day precipitation contour also delineates the poleward boundary reasonably well in most locations, and this will be used later in model evaluation. In terms of what maintains summer monsoons, land-sea contrast has long been considered fundamental (Webster 1987; Young 1987). One might conjecture that the thermodynamics of ocean-atmosphere-land interaction could be the controlling factor in the forcing of monsoons. However, in examining the relation between solar heating of the continent and the associated monsoon rainfall, the rain zone does not extend as far poleward as the maximum heating would seem to indicate. This suggests that other mechanisms besides the land-sea heating contrast determine the northward extent of the summer monsoon. Many studies (Lofgren 1995; Meehl 1994a, 1994b; Xue and Shukla 1993; Yang and Lau 1998; Nicholson 2000) discuss the importance of land processes, such as soil moisture and surface albedo, in affecting the magnitude and position of the monsoon. Topography, such as the Tibetan Plateau, also affects the monsoon circulation (Flohn 1957; Murakami 1987; Meehl 1992; Yanai and Li 1994; Wu and Zhang 1998). The approach presented here follows Chou et al. (2001) and Chou and Neelin (2001, 2003— CN03 hereafter) in showing how the land-sea contrast can be viewed in terms of the MSE budget and how large-scale dynamical mechanisms can mediate land-sea contrast to determine the poleward extent of the monsoon.

10.1.3. Moist Teleconnection/Tropical Precipitation in Global Warming Background To motivate development of theory for precipitation changes in tropical teleconnections and global warming, Fig. 10.2 shows a composite El Niño-Southern Oscillation (ENSO) precipitation pattern from observations and three models forced by observed SST. Two are general circulation models (GCMs) used in numerical weather prediction from the European Centre for Medium-Range Weather Forecasts (ECMWF) and the National Center for Environmental Prediction (NCEP), and one is the intermediate complexity climate model QTCM summarized in section 10.3. The composite is for warm events (average of 1982–83, 1986–87, 1991–92, 1994–95) minus cold events (average of 1983–84, 1988–89, 1995–96).

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b

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Figure 10.2. Composite precipitation anomaly (mm/day) for ENSO events during DJF for observations and three models forced by observed SST (no data assimilation). (a) Observed (Xie-Arkin dataset); (b) ECMWF model; (c) NCEP model; (d) QTCM1 v2.3. ECMWF and NCEP datasets are from the Atmospheric Model Intercomparison Project (AMIP2).

Without belaboring the obvious, there is no way to construe the comparison of the GCM response to observations in Fig. 10.2 as satisfactory. This poor simulation occurs despite the fact that these are well-respected models on which the most widely used reanalysis datasets are based. The point this underlines is that the simulation of teleconnected precipitation anomalies is a very challenging problem. This difficulty not only motivates the effort to understand the physical processes. Furthermore, the sensitivity to slightly different model representations appears consistent with the several mechanisms that are argued to be responsible for such precipitation anomalies in section 10.9, building on the theory from previous sections. Regarding the simulation by the QTCM in Fig. 10.2(d), the agreement with observations is qualitative in terms of the region of negative anomalies, although the magnitude is reasonable. The negative anomalies over equatorial South America occur along the margin of the model simulated climatology of the convection zone, which is shifted with respect to observed. The fact that the QTCM fares slightly better than the GCMs in Fig. 10.2(b) and (c) likely has to do with the fact that ENSO precipitation anomalies were examined among other factors, such as simulation of the climatology, during model development and revision (Zeng et al. 2000; Su et al. 2001) because

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Figure 10.3. Schematic of the mechanisms at work in moist teleconnections, including wave dynamics spreading tropospheric warming, interactions with convection and lower tropospheric moisture, cloud radiative feedbacks, land feedbacks, and climatological mean flow acting on the induced moisture and temperature gradients, as postulated in 1998.

the model was designed with teleconnection theory in mind. The simulation makes it plausible that mechanisms diagnosed from the QTCM make a reasonable starting point, subject to suitable caveats, for taking apart the complex processes of moist teleconnections. Figure 10.3 is a schematic based on analysis of the QTCM that was used in a series of talks trying to promote interest in moist teleconnection mechanisms during QTCM development in the late 1990s. One natural reaction to the diagram is that it appears complex. The processes, including the spreading of the warming by wave dynamics, the interaction of this with convection through QE considerations, cloud radiative feedbacks, and the role of mean flow acting on gradients of temperature and moisture, are consistent with present understanding that section 10.9 attempts to summarize. These processes roughly form the basis of such studies as Chiang and Sobel (2002), Su and Neelin (2002), Neelin et al. (2003, NCS03 hereafter) Neelin and Su (2005, NS05 hereafter), Chiang and Lintner (2005), Lintner and Chiang (2005) which have helped to systematize and somewhat condense the processes. Nonetheless, the fact remains that even in intermediate complexity models there is a fairly complex set of processes that can occur in moist teleconnections. This helps to explain the GCM difficulties in capturing these noted in Fig. 10.2, and raises the hope that improved understanding will help guide improved simulation. In global warming simulations, large tropical precipitation changes occur (e.g., Boer et al. 2000; Hu et al. 2000; Dai et al. 2001; Douville et al. 2002; Meehl et al. 2000; Roeckner et al. 1999; Yonetani and Gordon 2001; Williams et al. 2001; Allen and Ingram 2002; Johns et al. 2003; Manabe et al. 2004), including regions of substantial negative anomalies. The agreement on the regional distribution of these is poor (Houghton et al. 2001; NCS03), akin to the agreement between models seen in the ENSO case in Fig. 10.2. Model evidence that a subset of the mechanisms at work in ENSO teleconnections creates the anomalies in global warming simulations is reviewed in section 10.10.

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10.2. Moist Static Energy Analysis and Gross Moist Stability The moist static energy (MSE) equation (10.3) sidesteps some of the large but canceling terms that occur in the individual moisture and temperature equations. One can then exploit the tendency of convective QE to constrain temperature vertical structure through a deep convective layer, and to tie together moisture and temperature equations (section 10.2.5). If temperature and moisture in a column are not independent variables due to the QE constraints, then the MSE budget can determine the thermodynamics at leading order, under certain approximations. Even when this does not fully apply, analysis of the MSE budget can guide hypotheses regarding mechanisms. An effective static stability for deep convective motions, the gross moist stability (section 10.2.3), also arises under certain conditions and greatly simplifies understanding of phenomena to which it applies. In presenting these, it has proved tempting to include comments regarding extensions or ongoing work (sections 10.2.2 and 10.2.4) that are aimed at those familiar with this area.

10.2.1. Moist Static Energy Budget The vertically integrated thermodynamic equation and moisture equation from the primitive equations are ∂t
T  +
v·∇T  +
ω∂ ps  =
Q c + S net + R net + H

[10.1]

∂t
q  +
v·∇q  +
ω∂ pq  =
Q q + E ,

[10.2]

where the pressure p is used as the vertical coordinate. The dry static energy is s = T + φ, with φ the geopotential. Temperature T and specific humidity q are in J kg−1, absorbing heat capacity at constant pressure and latent heat of condensation, respectively, for compactness. The sum of convective heating and moistening, Q c and Q q, must cancel when vertically integrated, since horizontal transport by smallscale convective motions is negligible. Precipitation is given by the vertical integral of either of these terms, P =
Q c = −
Q q. Vertical integrals
·  through the depth
of the atmosphere are here mass integrals · d p/g so all terms are in W/m2, following NS05. When the vertically integrated thermodynamic equation (10.1) and moisture equation (10.2) are added, using cancellation of the convective heating and moisture sink terms, the MSE equation is ∂t(
T  +
q ) +
v·∇(T + q ) +
ω∂ ph = F net , where the MSE is h = s + q .

[10.3]

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The net flux into the atmospheric column F net, signed positive when heating the atmosphere, is F net = F tnet − F snet,

[10.4]

where the net flux at the top of the atmosphere is F tnet = S ↓t − S ↑t − R ↑t

[10.5]

and the net surface flux, signed positive when heating the ocean, is F snet = S ↓s − S ↑s + R ↓s − R ↑s − E − H .

[10.6]

Signs on individual flux terms are chosen such that they are typically positive in the climatology. Surface evaporation and sensible heat are denoted E and H. On the solar radiation terms S and longwave radiative terms R, arrows denote direction of the flux in a two-stream radiation treatment, subscripts s and t denote surface and model top, and net heating of the atmospheric column terms are S net = S ↓t − S ↑t − S ↓s + S ↑s and R net = −R ↑t + R ↑s − R ↓s , with R ↓t ≈ 0. In GCMs and in QTCM there are also horizontal diffusion terms in (10.3) that are small but not negligible in large-scale budgets. We omit them for presentation purposes but include them in the discussion. Note that (10.3) involves neglect of time variation of surface pressure and is interpreted in the approximation that tropospheric transports dominate (or that vertical integrals are over the troposphere, with F tnet at tropopause level). 10.2.2. Some Remarks on Format and vψ versus vχ Note that the
ω∂ pφ contribution to the
ω∂ ps  term in (10.1), and implicity in (10.3) in the
ω∂ ph term, is associated with conversions to the kinetic energy equation, so the MSE equation is not a standard conservation equation; h is not conserved over the domain;
T + q  is conserved by horizontal motions in absence of sources and sinks. The MSE equation is the thermodynamic contribution to the primitive total energy equation, in which
T + q + K  is conserved integrated over a closed domain when the kinetic energy K equation is added. This is perhaps more clearly seen from the flux form of the MSE equation, here given for clarity in the approximation that ω terms vanish at the upper and lower limits of vertical integration. Using the continuity equation (10.8) and integration by parts, (10.3) becomes ∂t(
T  +
q ) +
∇ · (hv) −
v·∇φ = F net .

[10.7]

The
v·∇φ term is the conversion to kinetic energy. The local time change term is the tendency of
T + q , but this will vanish when time averages are taken. The flux term is a flux of h, so the term MSE equation is still appropriate, despite caveats on conservation properties.

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There is a history of using an MSE equation that conserves h but is itself an approximation. Typically this is derived by assuming that a −αd p term in the thermodynamic equation can simply be replaced by dφ. Because the d p is following the parcel, this assumption is inaccurate in presence of horizontal gradients. Betts (1974) gives an insightful discussion of this approximate equation. The implications of using this approximation at large scales seem not to have been evaluated. The interpretation in which the MSE equation (10.3) is directly from the primitive equations but h is not conserved is preferred here. The usefulness of (10.3) or (10.7) is due to the removal of the large but canceling convective-heating and moisture-sink terms to permit analysis of balances among terms that might seem small in the individual T and q equations but are important to the dynamics. For instance, the horizontal advection terms v·∇(T, q ) will prove important in some of the mechanisms discussed below. In the horizontal advection, it can be useful to distinguish between the irrotational (purely divergent) component of the flow vχ and the nondivergent (purely rotational) component of the flow vψ, where χ and ψ denote the associated velocity potential and streamfunction, respectively. The continuity equation ∂ p ω = −∇ · v ≡ −∇ · vχ

[10.8]

links the vertical velocity directly to vχ. Thus this term is best conceptually combined with the vertical advection term
ω∂ ph. Furthermore, global conservation properties are maintained in experiments that alter the vψ contribution because vψ·∇(T + q ) ≡ ∇ · [vψ(T + q )] vanishes in a global integral. The properties of vψ and vχ differ considerably in their effect on an advected scalar field, in particular moisture. The nondivergent vψ cannot raise moisture above its upstream value. In a continent, when surface evaporation is secondary and no precipitation occurs, vψ simply carries the inflow oceanic air mass around within the continent. On the other hand, for a field like moisture where other physics maintains a decrease of moisture with height, vχ and the implied vertical velocity acts to increase/decrease moisture when vχ is convergent/divergent at low levels. Postulates: In GCMs, commonly v·∇q and v·∇T terms are not explicit but are evaluated in flux form. The following recommendations have not been tested and should be regarded as postulates. Recommendation 1: Evaluate and save the vχ and vψ contributions of flux form transports separately in GCM runs. The MSE transport terms of (10.3) can then be written
v·∇(T + q ) +
ω∂ ph =
∇ · (vψ(T + q ) + [
∇ · (vχ(T + q )) +
ω∂ p φ]. [10.9]

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The term in square brackets relates directly to convergent motions and vertical velocity and thus with the work required to raise large-scale air masses, the essence of the moist stability discussed in the following section, while the vψ term is associated with advecting air masses horizontally without convergence. Where strong gradients occur, the term in square brackets may make more sense to interpret than ω and vχ terms separately. Recommendation 2: In GCMs, terrain-following “sigma” coordinates are often used, but output is often interpolated to pressure levels. Sometimes variables on a staggered grid in the horizontal are also interpolated onto a common grid. The MSE budget, because of the strong cancellation between large terms, can be very sensitive to interpolation. Computing the quantities above in the native coordinates at run time and saving the vertical averages would add little to the storage requirements and much to the accuracy of MSE diagnostics.

10.2.3. Gross Moist Stability Suppose for some set of phenomena of interest, associated with deep convection, one can define a dominant vertical structure of vertical velocity 1( p) such that ω(x, y, p, t) ≈ −1( p)∇ · v1(x, y, t), where v1 is the projection coefficient of a wind vertical structure V1 and  ps V1d p. 1 = −

[10.10]

[10.11]

p

Signs are chosen so that 1 is positive when V1 has the sign of upper-level flow for a first baroclinic structure. One important case where this arises is given in the next section. Then the
ω∂ ph term in (10.3) is replaced by
ω∂ ph ≈ M∇ · v1,

[10.12]

where M is the gross moist stability (GMS) M = −
1∂ ph.

[10.13]

This is an effective static stability for moist motions that includes the partial cancellation of adiabatic cooling by latent heating (Neelin and Held 1987; Neelin and Yu 1994). Here M is defined in units of J m2 following NS05, i.e., absorbing ( p T/g ) relative to Yu et al. (1998), where p T is the pressure depth of the integration through the troposphere. The GMS gives a measure of the work that must be done to raise a large-scale air mass with a vertical velocity of the given vertical profile. The measure above gives the MSE transport by motions converging at low levels, rising and diverging aloft, under conditions where horizontal gradients are small, as is conventional to assume for dry static stability measures.

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10.2.4. The Gross Moist Stability in Flux Form While the GMS has most commonly been discussed with respect to vertical motions, it applies to flux form equations as well. Integrating (10.13) by parts, and using  = 0 at top and bottom, yields M =
V1h.

[10.14]

For convenience, define components of this as M = MT + Mφ − Mq,

(MT, Mφ, Mq) =
V1(T, φ, −q ).

[10.15]

The MSE equation in flux form (10.7), assuming that limits of vertical integration have negligible gradients, that v ≈ V1( p)v1, and taking a time average ( ) , becomes ∇ · (Mv1) − v1·∇ Mφ = F net,

[10.16]

where the second term, involving Mφ, is the conversion to kinetic energy and is presumably locally smaller than the MSE convergence, ∇ · (Mv1). The latter can be divided into rotational and divergent components using v1 = v1ψ + v1χ. The MSE convergence is then ∇ · (Mv1ψ) + ∇ · (Mv1χ)

[10.17]

with the second term being due to all the effects of the divergent flow and associated vertical velocity and the first term acting only by vψ·∇ advection terms. Postulates: When strong horizontal gradients of temperature and moisture occur, then vχ·∇q and vχ·∇T terms can be large. The flux form of M (10.14), with (10.16), appears appropriate to these situations. In GCM diagnostics, the term in square brackets in (10.9) may provide an easier means of evaluating the MSE convergence associated with the GMS. In defining V1 for (10.14), it may be best to use the typical vertical structure of vχ for the region and phenomenon, since the convergent motion is so important to the energetics but tends to be smaller than the purely rotational component. Evaluation of V1χ( p) may also be aided by the tendency of χ = ∇ −2(∇ · v) to have a smoothing effect that emphasizes larger spatial scales. Open question: If more than one vertical structure is necessary to describe the vertical velocity field for the set of phenomena of interest, then
ω∂ ph has contributions from more than one GMS (Neelin and Zeng 2000). The flux form of M makes the extension to this case appear straightforward. Using a sum of a few velocity vertical  structures, v ≈ i Vi( p)vi(x, y, t), then the MSE convergence term becomes  ∇ · (Mivi), with Mi =
Vih. [10.18] i

Exploration of the implications of this has only recently begun, and it will be interesting to see under what conditions it leads to simple interpretations. Cases that treat slow

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spatial variation of 1 and M may be found in Yu and Neelin (1997) and Yu et al. (1998). Moskowitz and Bretherton (2000) and Sobel and Neelin (2006) find nontrivial contributions from impacts of atmospheric boundary-layer convergence on vertical structure, bringing into a QE context effects previously noted in a large literature summarized in chapter 8 in this volume. The likelihood that changes in vertical structures affect M in narrow intertropical convergence zones (ITCZs) is inferred from reanalysis budgets in Back and Bretherton (2006). 10.2.5. Relationship to Quasi-Equilibrium An important case where the GMS arises is when convective QE constrains temperature (Emanuel et al. 1994) and thus baroclinic pressure gradients, and where frictional effects may be considered secondary. Then the baroclinic wind coefficient v1 appears in the divergence, giving the horizontal variations of the vertical motions, whose vertical structure has been obtained by hydrostatic and continuity constraints. A sketch of the derivation follows (for details, see Neelin [1997] and Neelin and Zeng [2000]). Consider the momentum equations combined with the hydrostatic equation in an approximation where friction is neglected and only the barotropic wind v0 is retained as the advecting velocity:  pr s (∂t + v0 · ∇)v + f k × v = −∇ κ T dln p − ∇φs , [10.19] p

where φs is the geopotential at the surface reference pressure level pr s and κ = R/c p , where R is the gas constant for air and appears in the baroclinic pressure-gradient term since T has absorbed c p . If QE tends to move the column toward a preferred temperature structure such that ∇T ≈ a1( p)∇T1(x, y, t),

[10.20]

then (10.19) can be separated to yield an equation for the barotropic velocity component v0 with constant vertical structure and an equation for a baroclinic velocity component v1 driven by the baroclinic pressure gradients associated with T1. The associated baroclinic vertical structure is V1( p) = a1+ − a1+, [10.21]
p where a1+ = p r s κa1 d ln p, and () denotes a vertical average. From (10.21), 1( p) can be derived by (10.11). The gross moist stability (10.13) then arises naturally for motions obeying QE. Open questions: This derivation will not work uniformly for all motions. At small scales, departures from QE temperature structure will occur. Yu and Neelin (1994) have noted that QE slowly gives way to other balances as one moves from large spatial scales to scales much smaller than the time scale on which convection establishes QE times the

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gravity wave speed. In other situations, baroclinic pressure gradients may not dominate friction or baroclinic advection. For instance, Back and Bretherton (2006) have noted that in the zonally elongated ITCZs, MSE balances occur in which ω∂ ph is not the dominant export of MSE, and strong low-level convergence occurs. Postulate: In ITCZs the meridional velocity is primarily balanced by friction at low levels and by baroclinic advection at upper levels, not by meridional pressure gradients. The vertical structure of the mean meridional wind is then not given by (10.21). However, a large-scale wave disturbance passing through the same ITCZ whose convergent velocity does obey these balances will experience a QE-based GMS.

10.3. Quasi-Equilibrium Tropical Circulation Model The QTCM, introduced in Neelin and Zeng (2000) and Zeng et al. (2000), projects the primitive equations onto vertical basis functions that are derived under QE approximations as in the previous sections. The resulting equations are not necessarily in or near QE unless convection happens to be very strong at a particular location, but the vertical structures are well suited to QE conditions. The MSE budget for the QTCM is a close counterpart to the primitive equations case (10.3). The considerations of the previous section lead to the simpler form for the MSE divergence term (10.12), with M given by (10.13). Here the anomaly case is presented for use in discussion of teleconnection t or global-warming anomalies. For time averages over some period of interest X c t c as departure from a climatology X , the anomaly is X  = X − X . Neglecting time derivative terms, the anomaly MSE equation becomes (M∇ · v1)  +
v·∇T  +
v·∇q   = F tnet  − F snet ,

[10.22]

where the vertical integrals are evaluated on the retained vertical structures. The time averages can also be extended to include ensemble averaging from a set of numerical experiments. Anomalies in nonlinear terms such as
v·∇q  contain changes in the contributions of transients. For instance, if double primes denote departures from the t c time average in which they occur (X or X ), there are transient terms such as
ω∂ ph  as discussed in Su and Neelin (2002) implicity present in (M∇ · v1). The transient terms will be implicitly retained, although discussion will center on terms like M∇ · v1, which also result from expanding terms like (M∇ · v1) but don’t explicitly address the role of transients. Postulate: The role of terms due to transients is likely to be scale dependent, since at the scale of individual convective elements, transients must dominate. Evaluation of (10.9) with some spatial averaging scale may be a route to resolving this. While the transient terms can be important in the climatology, for climate anomalies their role is not clear. In QTCM simulations (which have substantial explicit diffusion presumed

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to stand in for some aspects of transients), the role of transients is often, though not always, secondary. This appears not to have been well examined in observations.

10.4. Some Implications of the MSE Budget and QE 10.4.1. Implications for Land Convective Zones Because of the low heat capacity over land, the net surface energy flux on time scales longer than a day obeys F snet = 0,

[10.23]

so F snet drops out of (10.3) over land. Under conditions where the MSE budget is the leading thermodynamic balance, this provides considerable simplifications for treatment of land surface feedbacks within convection zones (Zeng and Neelin 1999; Neelin and Zeng 2000). Only top-of-atmosphere radiative forcing F tnet and transport processes, i.e., advection of temperature and moisture, come into play in balancing the MSE divergence. To a first approximation, i.e., to the extent that convective QE holds and the MSE budget alone controls the dynamics in the convective zones, the atmospheric dynamics can be solved without any reference to Ts. The partition between E  and other fluxes does have some effect, especially when departures from QE must be considered. Nonetheless, this is a powerful leading-order simplification. The physics can be understood as follows: in a deep convective region, moisture and temperature equations are being strongly linked by convection through the entire troposphere; the partition between latent and other forms of heat release at the surface is thus less important at leading order than the flux of all forms of energy into the tropospheric column. 10.4.2. Land-Ocean Contrast and the Net Flux into the Atmosphere The consequences of the zero net flux condition (10.23) over land for land-ocean contrast are striking. Figure 10.4 shows an estimate of the net flux into the atmospheric column from observations (Chou and Neelin 2001; CN03). The top-of-atmosphere net flux is from the Earth Radiation Budget Experiment. Over land, this is the only flux required, due to (10.23). Over ocean there is a strong net surface flux varying with season, balanced by ocean heat storage and transport. While estimates of the net flux at the ocean surface differ in detail among datasets (here from the Comprehensive OceanAtmosphere Data Set and Darnell et al. [1992] shortwave), the land-ocean contrast is large enough to be a robust feature. Seasonally, at midlatitudes it is on the order of 50–100 W/m2 difference from land to ocean. In the Tropics, differences between ocean regions with large heat export and continents have similar magnitude. In the view argued here, this difference in net flux is the leading cause of land-ocean contrast,

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Figure 10.4. Net flux into the atmospheric column of longwave, shortwave, sensible, and latent heat through the surface and top of the atmosphere, estimated from observations for (a) January (from Chou and Neelin 2001) and (b) July (from Chou and Neelin 2003). c 2001 and from the (Reproduced with permission from the American Geophysical Union
c 2003.) American Meteorological Society


not surface temperature. Consequences of this difference for summer monsoons are elaborated in the following section.

10.5. Dynamical Mechanisms in Monsoon Poleward Boundaries Comparing the poleward boundary of strong seasonal precipitation in Fig. 10.1 (c) and (d) with the poleward extent of positive net flux into the atmospheric column over summer continents in Fig. 10.4 raises a question posed in Chou et al. (2001), Chou and Neelin (2001), and CN03. If the leading balance typical of tropical dynamics applied, positive net flux into the column would lead to rising motion and precipitation, with the associated low-level convergence supplying sufficient moisture to meet the convective threshold (a substantial fraction of the time). Since the boundary of the convection zone, as measured by substantial precipitation, clearly does not extend nearly as far poleward

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as the region of positive net flux, there must be a transition in terms of the dynamics of the system that sets this boundary. What are the dynamics setting the poleward boundary? Two main dynamics mechanisms were found. 1. The ventilation mechanism, defined as the import into the continents of low MSE air from ocean regions (or export of high MSE from continents). Advection terms vψ(T + q ) are important to this import, which breaks the simplest balance introduced above and prevents convective criteria from being reached in large parts of the continent. Ocean seasonal heat storage and transport keep the oceanic MSE low relative to what would be required to meet convective criteria in a warm continent. 2. The “interactive Rodwell-Hoskins” (IRH) mechanism due interaction of Rossby-wave-induced subsidence to the west of monsoon heating with the convergence zone. Rodwell and Hoskins (1996, 2001) examined baroclinic Rossby wave subsidence to the west of a specified heating. The adjective “interactive” in IRH is intended to underline that the monsoon heating and its spatial pattern are themselves determined interactively with the subsidence pattern. To test the ventilation mechanism, CN03 suppressed the vψ·∇(q , T ) terms in the model moisture and temperature equation over target regions. Note that conservation relations are maintained within the target region because vψ is divergence-free (see section 10.2.1 or CN03 for details). They found that when ventilation is suppressed, monsoon rainfall indeed extends far poleward of the normal boundary for North American and Asian monsoons, while similar experiments in Chou and Neelin (2001) showed the same for the South American case. The exception is the northern African summer monsoon, where albedo plays a lead role in current climatology. This may be anticipated from Fig. 10.4 because over northern Africa, there is not a substantial net flux of energy into the atmospheric column. In the model, if albedo is artificially set to constant over northern Africa, then the dynamical mechanisms become important. Implications of the ventilation mechanism for the mid-Holocene “green Sahara” problem are explored in Su and Neelin (2005) and Hales et al. (2006), including interaction with vegetation feedback. An example of the effects of the ventilation mechanism is seen in Fig. 10.5. Rather than showing climatologies with and without ventilation, it shows a succession of days during the transition from an experiment with ventilation suppressed to an experiment with standard-model dynamics. The initial and final contour roughly typify the climatology without and with ventilation, respectively. There is transient synoptic variability present as well, but the change over the continent is representative of the change in the climatology (CN03). The 2 mm/day precipitation contour is used as a

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Tropical Convection Zones | 283 May 30 (No Ventilation) June 1 (Ventilation On) June 2 June 8

Figure 10.5. Boundary of the monsoon as measured by the 2 mm/day precipitation contour for an experiment where the ventilation mechanism is switched on May 30, beginning from a prior experiment where the ventilation was artificially suppressed. The precipitation thus initially extends artificially far poleward in the continent (ghost-shaded contour labeled May 30). Once the ventilation mechanism is switched on, low MSE oceanic air properties are advected in from the west. The retreat of the monsoon boundary over the subsequent week is shown by a succession of contours labeled by date. The full shaded area shows the region with precipitation exceeding 2 mm/day on June 8, which is typical of the model climatology. Data from experiments conducted for Chou and Neelin (2003).

measure of the edge of the convection zone (monsoon boundary). In the climatology, the model maximum precipitation over Mexico reaches 10 mm/day. While the ventilation is off, the precipitation extends far poleward over the continent. A dry region persists in the southwestern United States due to the IRH mechanism. In general both ventilation and IRH mechanisms contribute to east-west asymmetry in the continent and maintenance of southwestern deserts. Suppressing the beta effect has been used to demonstrate the role of IRH internal Rossby wave dynamics in this. When the ventilation is restored, the oceanic air mass sweeps into the continent over a period of a week, carrying modest moisture values insufficient to maintain sustained convection over most of the continent. Open questions: Such experiments have not been conducted on full GCMs, so there is as yet no corroboration of the mechanisms analyzed in the QTCM. Analysis of output might yield some insight, although challenges exist because of the number of features acting at once. For instance, an oceanic air mass carried into a dry region created by subsidence tends to raise moisture, but when carried into a region that has enough moisture to meet convective criteria over a warm continent, it tends to reduce moisture. Postulate: A potentially useful (but untested) diagnostic: find the low-level moisture or MSE value that would be required to initiate convection at every point in the model climatology. This would be done by approximating the value typically associated with the onset of strong convection for a given free-tropospheric temperature. Tracing

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back along streamlines of vψ, these values could be compared to the inflow values of moisture and potential temperature at the oceanic boundaries to address the importance of ventilation.

10.6. Implications of the MSE Budget and QE: Anomaly Case 10.6.1. Implications for Precipitation and the GMS Multiplier Effect Following NS05, the moisture budget (10.2) gives the precipitation anomalies (vertically integrated convective heating) as P  = (Mq∇ · v1) −
v · ∇q  + E . Using the MSE equation (10.22) for the divergence yields

 Mq  Mq   net  net  −
v·∇T  + F t − F s P = − + 1
v·∇q  M M

Mq  + − M + Mq ∇ · v¯ 1 + E  M

[10.24]

[10.25]

The nonlinear terms of the form
v·∇q  must be expanded into v and q  contributions (and an additional term for changes in transients). Each term in (10.25) is typically associated with some particular pathway for precipitation anomaly impacts, as elaborated in sections 10.9 and 10.10. The factor (M q/M) is a measure of the moisture convergence relative to the MSE divergence. This factor tends to be several times larger than unity (Yu et al. 1998; Zeng and Neelin 1999). NS05 refers to the effects summarized in (M q/M) as the GMS multiplier effect. It boosts a small signal in the MSE equation to a convective heating and precipitation impact several times larger. This effect is basically what has earlier been referred to as a “convergence feedback” (Webster 1972; Zebiak 1986) recast in terms of moist thermodynamics. In the observation-based estimates of Yu et al. (1998), this would be roughly 4 or 5 in the deep convective zones. Postulate: Even for GCMs where M has not been estimated or where it is challenging to do so, the principle of this effect may be evaluated: large vertical velocity and moisture convergence, with large consequences for precipitation, may occur with little work being done in the MSE equation. 10.6.2. Implications for Anomalies Over Ocean Consider an ocean surface layer governed by c s∂tTs + Ds = F snet ,

[10.26]

where Ts is SST and we have assumed the vertical integral over the ocean surface layer
  s ρc wT dz ≈ c sTs , so c s is an effective heat capacity of the layer; Ds is the divergence

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of the ocean heat transport in the layer, including turbulent and advective fluxes from below; and primes denote anomalies. A simple case is a passive ocean surface layer, such as a fixed-depth mixed layer, simply reacting to atmospheric heat flux anomalies with Ds = 0. The surface-layer energy budget (10.26) then implies that the surface flux vanishes when the layer equilibrates and c s∂tTs → 0. Thus despite nonzero SST anomalies, the F snet  term drops out of the perturbation MSE budget (10.22) and hence out of the precipitation anomaly equation (10.25). While the ocean is in disequilibrium with the teleconnected forcing, the SST anomaly can contribute to driving MSE divergence. Once the SST anomaly reaches approximate equilibrium with the atmosphere, then the SST is approximately a by-product in convective regions, much like the land surface temperature. This has considerable implications for global warming cases, where nearequilibrium conditions often apply. For regions where net surface flux is small compared with other terms in the anomaly MSE budget, the SST anomaly should be regarded as a by-product in analyzing precipitation changes. Open questions: To what extent do disequilibrium surface fluxes contribute to precipitation anomalies in interannual teleconnections? If anomalies in winds or surface fluxes create changes in ocean transports, then the conditions above do not hold. Nonzero net surface flux can then continue to impact the atmospheric MSE budget and drive sustained anomalies. For which phenomena is this important?

10.7. Cloud Radiative Feedbacks as a Modification to the Effective Static Stability A manipulation of cloud radiative effects that clarifies their nature as a feedback was presented in Zeng and Neelin (1999) for the land case (under different notation) and in Su and Neelin (2002) for the ocean case. Here, these cases are summarized and contrasted under uniform notation following NS05. Versions have been used in Bretherton and Sobel (2002) and Sobel and Gildor (2003). Both solar and longwave feedbacks involving deep clouds and the associated cirrostratus/cirrocumulus (CsCc) have a strong linkage to convection and thus to precipitation. For land regions, only the top-of-atmosphere radiative fluxes count in the MSE budget, while over the ocean, in presence of nonzero net surface flux, the net forcing at top-of-atmosphere minus surface is important. The cloud radiative forcing (CRF) at top-of-atmosphere, CRFt, and the net CRF on the tropospheric column CRFnet  can be approximated as feedback terms proportional to precipitation, CRFt ≈ C t P 

land,

CRFnet  ≈ C net P 

ocean.

[10.27]

The coefficients of these linearizations depend on basic state and thus on location; for instance, mean cloudiness and surface albedo are important to the shortwave

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contribution. For the parameters in QTCM v2.2, for example, over equatorial South America, C t ≈ −0.05, and over the equatorial Atlantic, C net ≈ 0.12. Positive C net indicates CRF tending to warm the atmospheric column where there is a positive precipitation anomaly, while the small negative C t corresponds to deep cloud feedbacks tending to slightly cool the whole column including the surface (when the surface is equilibrated). Because C t results from cancellation of large individual longwave and shortwave terms, it is smaller and more sensitive than C net. The value that applies over ocean on short time scales, C net, is larger because surface fluxes are lost (at least temporarily) into the ocean. Longwave heating of the column with increased cloud dominates this term. An approximate MSE budget that takes the CRF into account can thus be written (using the moisture equation [10.2] to rewrite P  in (10.27) as moisture convergence and other terms) as net  (Meff∇ · v1) +
v·∇T  + (1 + C )
v·∇q  = F eff ,

[10.28]

where an effective moist stability Meff that includes cloud radiative feedbacks has been defined as Meff = M − MCRF = M − C Mq, C = C t,

land; C = C net,

[10.29] ocean.

[10.30]

The effective flux forcing occurring in the tropospheric MSE budget is approximately (neglecting nonlinear cross-terms, notably between cloud and moisture effects) net  F eff = − T tT1 + q tq 1 + TstTs,

land

= − T T1 + qq 1 + TsTs + (1+C )E  + H ,

ocean,

[10.31]

where T = T t + T s is the longwave cooling rate of the troposphere per unit temperature, given by top-of-atmosphere and surface contributions T t, T s ; similarly for moisture longwave heating coefficients, q = q t + q s ; Ts is the coefficient for tropospheric absorption of longwave radiation due to surface temperature anomalies Ts; Tst is the (small) coefficient of top-of-atmosphere contribution of Ts to outgoing longwave radiation (OLR); and T1, q 1 are projections of temperature and moisture on their respective typical vertical profiles (Zeng et al. 2000). When evaporation is linearized, and QE is applied to link q  to T  (and land surface feedbacks are incorporated in the land case), this effective flux has the form net  net  F eff = − eff T1,

[10.32]

net where eff changes from land to ocean, being roughly twice as large over oceans due to downward loss to the surface, while over land there is only loss to space due to the zero net surface flux condition.

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Figure 10.6. Composite tropospheric temperature anomaly (200–850 mb average) (◦C) for ENSO events during DJF for reanalysis data and three models forced by observed SST as in Fig. 10.2. (a) NCEP reanalysis (assimilated observations); (b) ECMWF model; (c) NCEP model; (d) QTCM v 2.3.

In summary, the zero net surface flux condition over land allows land surface feedbacks and cloud radiative feedbacks to be combined into a modification of the GMS. This modification is small over land due to approximate cancellation of longwave and shortwave feedbacks. Over oceans, on time scales shorter than equilibration of the ocean surface layer, the effective moist stability can be reduced by more than 25% due to cloud longwave feedbacks. This can enhance vertical velocity anomalies and thus precipitation anomalies.

10.8. The Kelvinoid Solution as an Example of Moist Wave Dynamics The remote impacts, for instance of ENSO Pacific SST variations, must be mediated by wave dynamics, and tropospheric temperature is a good indicator of the baroclinic contribution to this. Figure 10.6 presents the ENSO composite corresponding to Fig. 10.2 but for vertically averaged temperature above the boundary layer. A reanalysis product (which should be a reliable proxy for observations because satellite tropospheric temperature estimates are assimilated) and the same three models are shown. The comparison between models and observations is considerably better than in precipitation, at least in terms of the general features of the tropical warming signal, and the models agree reasonably well over the equatorial Pacific and South America. Among the qualitative features are: warming is confined within on the order of ◦ 20 of latitude of the equator; maxima just off the equator in the Pacific consistent with Rossby wave dynamics; and an equatorial eastward extension of the warming that one might hypothesize to be in some way related to Kelvin wave dynamics, albeit with some more complicated features that vary among models. If the pattern over Africa is related to this eastward extension, then there is a poorly explained increase relative to the Atlantic. In both observations and models, there is a substantial gradient of temperature

0

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both westward of the main ENSO region and in the eastward warming over South America. This gradient changes as a function of lag (e.g., Chiang and Sobel 2002). To provide an example of how such gradients might be affected by moist dynamics, the “Kelvinoid” solution of NS05 is useful. It includes momentum damping, but the simplest instance occurs when this is neglected. A simple case of the momentum equation for baroclinic mode zonal wind anomalies u1 has the form ∂tu1 + u¯ u∂xu1 + κ∂xT1 = 0

[10.33]

for a solution with negligible v  and damping, where u¯ u is a suitable projection of the mean wind, assumed constant. In steady state, the balance in (10.33) is between momentum advection and the baroclinic pressure gradient term resulting in a relationship (−¯uu)u1 = κ T1, for ∂x(u, T ) = 0, between anomalies of wind and temperature that depends on the mean wind. Geostrophic balance of u1 with ∂ yT1 immediately yields Gaussian y-dependence with length scale [(−¯uu)/β]1/2. While this is the same as a stationary Rossby wave scale, the balances here differ; it is simply the only available scale because the y-structure can be obtained from the vorticity equation alone with no reference to the x-structure. In more general cases, for instance if a term representing surface drag/convective momentum transport is included, the radius of deformation also enters. Combining the steady form of (10.33) with a similar form of (10.28) and using the QE temperature-moisture relation yields a moist wave equation of the form 2 net  (c eff − u¯ uu¯ h)∂xT1 = u¯ u eff T1,

[10.34]

where c eff is an effective moist phase speed (see also chapter 7 in this volume) that is 1/2 proportional to Meff ; u¯ h is the mean wind with a vertical projection suitable to the MSE equation (10.28). net 2 The solution decays eastward at a rate that depends on u¯ u , c eff , and eff . Compared net 2 to land, ocean regions have smaller c eff due to cloud feedbacks (10.29), and larger eff due to surface fluxes (10.32). This switch in the magnitude of the parameters of (10.34) creates changes in divergence, vertical velocity, and precipitation just associated only with land-ocean contrast. Furthermore, the v · ∇T and v · ∇q terms of (10.28) give rise to the u¯ h term of (10.34). It may be seen that the gradient of temperature, and thus divergence, depends on the interplay of these with the work done against the effective moist stability. This can be interpreted as the basic-state velocity opposing effects associated with an eastward phase speed, permitting the flux terms to have more effect per unit distance. Postulates: Inclusion of varying basic state in (10.33) has impacts in addition to the basic-state variations discussed above. Ongoing work to quantify this may help shed light on the temperature variations seen in Fig. 10.6. The wave equation (10.34)

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30N 15N EQ 15S 30S 40S

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Atmospheric wave dynamics tends to spread warming, reducing pressure gradients, creating non-local T´,V´ 180

120W

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Figure 10.7. Schematic modified from Su and Neelin (2002) of a basic principle postulated to occur in moist teleconnections, here shown for the case of a warm ENSO event in the Pacific. Wave dynamics acting to reduce baroclinic pressure gradients spreads the warming from the Pacific. The teleconnected warming and associated circulation anomalies then interact with convection in a manner that depends on basic-state circulation, moisture, temperature, and convective thresholds. In particular subregions, conditions are satisfied for strong local descent and precipitation anomalies. A number of moist thermodynamic balances can apply in these regions, as elaborated in Table 10.1.

does not include moisture gradients due to exit from convection zones, essential to one mechanism discussed below that applies on margins of convection zones. Extensions to this case are feasible.

10.9. Moist Teleconnection Mechanisms As summarized in section 10.1.3, teleconnections in the Tropics—including the important impacts on land precipitation—are complicated by interactions with convective heating, shortwave and longwave cloud radiative feedbacks, and land surface feedbacks. Traditional notions of descent anomalies being balanced by radiative cooling, such as are implicit in Gill (1980), are argued here and in Su and Neelin (2002) and NS05 not to be leading balances in the regions of strongest precipitation and descent anomalies. Using the MSE budget and QE considerations outlined above aids understanding of the contributing mechanisms. The main principles common to all mechanisms are schematized in Fig. 10.7, adapted from Su and Neelin (2002): 1. Wave dynamics acts to reduce baroclinic pressure gradients that would otherwise be created by the local heating by surface fluxes in the source region (e.g., during El Niño), spreading warm temperatures.

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2. The direct effects of the wave dynamics are expressed by tropospheric temperature anomalies T  and by the associated wind anomalies v. Convective QE links moisture anomalies q  to T  within convection zones (QE mediation). The teleconnected T , v, and the QE-mediated q  impact various terms in the MSE budget in particular regions determined by basicstate conditions. 3. Anomalies in the vertical motion determined by interplay with other terms in the MSE equation imply large effects on precipitation because anomalous moisture convergence tends to be much larger than the associated MSE divergence (the GMS multiplier effect of section 10.6.1). In analyzing these teleconnections, we have simplifications outlined in previous sections that hold under specified approximations resulting from: (i) The GMS, M (section 10.2.3), which accounts for the partial cancellation of adiabatic cooling by convective heating as a reduced static stability; (ii) the zero net surface flux condition over land (section 10.4.1) within convecting regions (where moisture equations and dry thermodynamic equations are linked by convection); (iii) cloud radiative feedbacks as a change in M (section 10.7). These simplifications help to handle some of the feedbacks, while analysis of the MSE budget shows which terms balance anomalous descent. This analysis yields a small zoo of mechanisms, summarized in Table 10.1, many of which have overlapping steps in their pathways. Mechanisms are stated here for an El Niño warming case but the mechanisms also tend to work for a La Niña case with signs reversed, e.g., cool troposphere is in QE with reduced low-level moisture. These can be tested numerically in experiments that artificially suppress terms involved in the mechanisms. Different mechanisms hold in particular subregions within the area where wave dynamics has affected tropospheric temperature, wind, and moisture. Note that the mechanisms are not additive and some have common elements. Regarding terminology, C. Bretherton has suggested that “effect” might sound better than “mechanism.” Definitions—effect: “a real phenomenon, usually named after its discoverer,” and mechanism: “the means by which an effect is produced” (per Webster’s dictionary)—allow for either. Here the term originally appearing in the literature is used for simplicity and readers may substitute “effect,” as in “upped-ante effect,” if they prefer.

10.9.1. The Upped-Ante Mechanism Figure 10.8 shows a schematic of the upped-ante mechanism. In convective regions, an increase in tropospheric temperature would tend to decrease parcel buoyancy unless low-level MSE increases to compensate. In NCS03 terminology motivated by a poker analogy, this “ups the ante” for the amount of lower-tropospheric moisture a region must have to convect, in competition with neighboring regions for the available

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Table 10.1. Moist teleconnection mechanisms, with emphasis on those leading to precipitation anomalies. Note that not all are independent. “Common” denotes effects common to several or all mechanisms. Right arrows indicate causal pathways between mnemonic symbols indicating the terms involved. The gross moist stability (GMS) is M, v is horizontal velocity, ∇ · v is a measure of the divergence and vertical velocity anomaly, T  is free tropospheric temperature anomaly, q  is low-level moisture anomaly; 2 c eff is proportional to Meff and measures wave propagation propagation relative to mean flow projections u¯ h, u¯ u; F snet  is net surface flux anomaly, Ss  is net surface shortwave anomaly and Ts is SST.

Mechanism GMS multiplier effect (common) QE mediation (common) Cloud radiative feedback (common) Upped-ante mechanism Moist wave decay mechanisms (easterly flow effects, radiative damping, surface drag) M  mechanism Anomaly wind (v) mechanisms Troposphere/SST disequilibrium (surface flux) mechanisms (CRF, QE-mediated E , v effects on E )

Pathway or Mediating Term MSE equation ⇒ ∇ · v ⇒ P  ≈ (Mq/M)∇ · v Convection links low-level q  with free tropospheric T  CRF ≈ MCRF∇ · v acts like modification to GMS; Meff = (M − MCRF) differs between land and ocean Teleconnected T  ⇒ q  ⇒ v¯ · ∇q  v·∇T  due to (v · ∇v), combined with surface drag and thermal damping; 2 e.g. (c eff − u¯ hu¯ u) vs. damping  T ⇒ q  ⇒ Mq , M ; M ∇ · v¯ ⇒ M∇ · v Teleconnected v ⇒ v·∇ T¯ , v·∇ q¯ , v effects on E  T  ⇒ q  ⇒ E , v ⇒ E , Ss  yield F snet  only while ∂tTs = 0

moisture supply. In regions where there is no other impact on the moisture or energy budget, the boundary-layer moisture simply increases and precipitation continues. The moisture supply required for the increase is small and can be met by a temporary decrease in precipitation that does not persist after the moisture returns to convective QE with the warmer troposphere. However, because moisture is not being increased in neighboring non-convective regions, moisture gradients are created. In regions where there is low-level inflow of air from a non-convective region, a v·∇q  moisture advection term creates a drying effect in the margins of the convective zone. This impacts vertical motion via the GMS multiplier effect, leading to a substantial precipitation reduction from a modest v·∇q . NS05 found the upped-ante mechanism to be the leading

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292 | J. David Neelin Margin of convective zone with ν¯ inward

El Niño heating

Wave dynamics

T´ > 0 Convective adjustment (fast)

q´ > 0 Moistens to meet new convective threshold

Clim. descent region

Reduced convection

ν¯ Drying by ν¯ · q´

Moistens less

Figure 10.8. Schematic of the upped-ante mechanism. The free-tropospheric temperature is affected by remote heating via wave dynamics. To maintain convection into a warmer free troposphere (T  > 0), the lower-tropospheric moisture must increase (q  > 0), creating moisture gradients relative to neighboring non-convective regions. Advection by low-level inflow tends to oppose the moisture increase, leading to reduced precipitation on the margin of the convective zone. Adapted from NCS03. (Reproduced with modifications with permission c 2003.) from the American Geophysical Union


contributor to the precipitation reduction over eastern equatorial South America and a major contributor to anomalies over the Atlantic ITCZ during ENSO.

10.9.2. Moist Wave Decay Mechanisms Section 10.8 provides an example of a moist Kelvinoid solution, a steady, eastward decaying solution that would remain after a time-dependent set-up by a Kelvin wave packet interacting with moist convection, the mean flow, and Rossby contributions. Some features of the interaction would also apply to decaying Rossby-wave-like solutions. In deep convection zones, the GMS implies greater vertical motion anomalies and slower phase speed than would occur for a dry wave of the same vertical structure, due to the partial cancellation of adiabatic cooling/warming by diabatic heating anomalies. The mean wind in which the wave propagates can play an important role because momentum advection terms are the main terms balancing baroclinic pressure gradients; and the slow moist wave speed, on the order of 10–15 m s−1, implies that the zonal wind can be comparable to the phase speed. The cancellation between u¯ terms, such as u¯ ∂xT , and the moist stability terms that give the phase-speed term can increase gradients in the anomalies, i.e., faster zonal decay in T  and greater descent and precipitation anomalies. This eastward decay involved in the u¯ ∂xT  term can be balanced in part by radiative cooling to space, or over oceans by surface flux damping terms. Traditional notions of the balance would have suggested that radiative damping would be the leading term in driving anomalous descent. Radiative cooling associated with temperature is only

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a few W m−2 for even a large ENSO event, considerably smaller than other terms, so would not drive much descent if a dry static stability were the main balance. However, the moist processes and advection can conspire to reduce the energy required to induce descent anomalies. Over ocean, the radiative damping effect is roughly doubled because loss occurs (temporarily) to the surface as well as to space. Surface drag communicated upward in the advection terms (as in Bacmeister and Suarez 2002), though also small, can likewise contribute to the wave decay for analogous reasons.

10.9.3. Anomalous Wind Mechanisms Anomalous wind mechanisms occur via teleconnected v contributions to moisture and temperature advection, v · ∇(q¯ , T¯ ), and to surface fluxes. The surface flux case is listed under troposphere/SST disequilibrium mechanisms below. Su and Neelin (2002) found these are leading contributors to precipitation anomalies for ENSO impacts in the Pacific.

10.9.4. Troposphere/SST Disequilibrium (Surface Flux) Mechanisms Mechanisms associated with surface fluxes have often been considered as a means of teleconnected warming of SST (e.g., Enfield and Mayer 1997; Klein et al. 1999; Lau and Nath 2001; Chiang and Sobel 2002). The MSE budget makes clear that the flux warming the ocean is lost from the troposphere. In deep convective regions this tends to be balanced by descent anomalies that reduce precipitation. If ocean transport divergence anomalies are negligible, then effects of surface fluxes disappear in the MSE equation once the ocean surface layer equilibrates with the troposphere. Thus SST is simply adjusted to cancel heat flux contributions by wind or tropospheric temperature plus convection—i.e., SST can be viewed as a by-product of surface heat-flux equilibrium much as in the land surface case. The proposed terminology, troposphere/SST disequilibrium, serves as a reminder that surface flux effects are temporary (unless supported by ocean heat transport) and that the free troposphere is involved as well as the atmospheric boundary layer. For the time scale of the onsetting El Niño, surface fluxes can be important. The GMS multiplier effect acts on the net surface flux anomaly to yield a substantial negative precipitation contribution. There are several mechanisms active in producing the net surface flux anomalies. As deep convection acts to bring the boundary layer and free troposphere into QE, surface fluxes act to bring the atmospheric boundary layer and ocean surface layer into equilibrium (Chiang and Sobel 2002). Cloud radiative feedbacks, especially shortwave, yield a substantial surface-warming tendency, while longwave radiative fluxes associated with temperature and moisture contribute a weaker tendency toward equilibrium with the teleconnected tropospheric warming. Additionally, teleconnections via wind create a contribution to surface flux anomalies.

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10.10. Mechanisms for Tropical Regional Precipitation Anomalies in Global Warming As discussed in section 10.1.2, tropical regional precipitation anomalies simulated in global warming scenarios can be substantial. The root-mean-square change over the Tropics can be significantly larger than the tropical average (Neelin et al. 2006). Three basic mechanisms have been identified based on diagnostics of the type discussed above, and in the QTCM have been verified by numerical experiments that suppress them. One mechanism is due to changes in ocean transport, which drive the atmospheric response via surface fluxes. The primary example of this occurs in the equatorial Pacific in an El Niño-like response. The other two mechanisms occur even in an equilibrated mixedlayer ocean with negligible surface fluxes. For these, SST is not useful as an indicator, since it is simply equilibrated toward a vanishing surface flux condition.

10.10.1. Upped-Ante Mechanism The upped-ante mechanism in global warming works much as in the teleconnection case above (Fig. 10.8), but the tropospheric warming is due to absorption of longwave radiation by increased greenhouse gases. The temperature equilibrates toward canceling the top-of-atmosphere net radiative flux so there is little driving of circulation by fluxes. Instead, the large-scale warming induces differential moistening between convection zones and non-convective regions. At the inflow margins of convection zones the uppedante mechanism then induces regional negative precipitation anomalies (NCS03, Chou and Neelin 2004). This occurs in the QTCM, and there is evidence supporting it in at least one coupled ocean-atmosphere GCM (Chou et al. 2006). The signatures to be sought in GCMs are differential moistening between convection zones and their surroundings, and regions where v¯ ·∇q  has a drying tendency coinciding with negative precipitation anomalies.

10.10.2. M  or Rich-Get-Richer Mechanism Figure 10.9 shows a schematic of the M  (anomalous gross moist stability) mechanism, also known as the rich-get-richer mechanism because it favors precipitation increases in regions of large climatological precipitation and decreases in dry regions as the troposphere warms. Moisture increases in convective regions, as in the closely related upped-ante mechanism, to maintain QE balance with warm tropospheric temperatures. This tends to increase Mq, thus reducing M. In the anomaly MSE equation (10.22), the balance M∇ · v1 = −M ∇ · v1 [the two nonzero components of (M∇ · v1)] yields a low-level convergence anomaly where there is mean convergence and a low-level divergence in the climatological descent regions where ∇ · v1 has opposite sign. There is

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Greenhouse warming T´ > 0

Convective adjustment (fast) q´ > 0

Moistens to meet new convective threshold Ts´ > 0

Center of convergence zone: incr. moisture lower gross moist stability incr. convergence Figure 10.9. Schematic of the M  or rich-get-richer mechanism, shown for the case of tropospheric warming T  due to greenhouse gas increase. The increase of moisture with T  creates increased moisture convergence and lower gross moist stability (GMS). The latter is balanced in the moist static energy (MSE) budget by additional convergence. Both aspects increase precipitation in the convergence zones and decrease it in the descent regions.

also a contribution by mean divergence acting moisture. Together these  on the increased Mq    lead to the contribution to P in (10.25), − M M + Mq ∇ · v¯ 1. This tends to increase precipitation in climatological convergence zones while decreasing it in subtropical descent regions. Chou and Neelin (2004) find this to be important in QTCM globalwarming experiments. Chou et al. (2006) find evidence in a GCM of the signature ¯ pq   driving positive precipitation anomalies in the center of convection zones, of
ω∂ enhanced by
ω∂ pq¯  feedback.

10.11. Final Remarks A theme that recurs in both the ventilation mechanism, applied to the poleward extent of monsoons, and in the upped-ante mechanism, applied to precipitation changes under ENSO teleconnections and global warming, is the role of inflow of air into a region that might potentially convect. The inflow air properties must be considered relative to a critical value for convection to onset under the QE postulate—for instance, low-level moisture must exceed a threshold relative to free-tropospheric temperature. Between this and the MSE and moisture budgets, the transition from convecting to non-convecting region is set (for the climatology) or shifted (for anomalies). The upped-ante mechanism and ventilation mechanisms are thus essentially variants on a theme.

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The mediating terms for these mechanisms can be thought of as the advection terms, such as v · ∇q . However, there appears likely to be an advantage to thinking in terms of advection by the purely rotational flow vψ and the horizontally convergent flow vχ separately. The latter can be wrapped neatly with the effects of convergence/vertical velocity, including in the gross moist stability. The sum of these effects tends to concentrate column moisture in regions of low-level convergence. The vψ term, on the other hand, simply advects the air mass. Often the contribution of vψ in the moisture budget is substantial, tending to oppose convection where it flows into a convection zone (in realistic cases; two-dimensional analogs, such as Hadley-cell cases, typically have only vχ crossing moisture contours by construction, and thus should be used with caution). This is postulated to be implicated in the concentration of droughts in particular regions along the margins of the convective zones in global warming, and perhaps may be relevant to the sharp equatorial “notches” in the convective zones such as over northeastern Brazil. Theory for the GMS is still at a rudimentary stage. The clearest case applies for inviscid motions when temperature structure is in convective QE and the baroclinic pressure gradient is a leading term in the momentum equation. While there may be many motions in the atmosphere for which these are reasonable assumptions, there are also phenomena for which modification of these is likely required. For example, a zonally symmetric ITCZ does not have baroclinic pressure gradients balancing the convergent component of the wind, and does depend strongly on a frictional boundary layer. Zonally elongated ITCZs are then likely to require different approximations. The approach advocated here should be viewed as one that is “in progress,” with several useful results and tools, but where a number of model-based postulates must be further assessed in data, and where progress so far suggests many aspects that await refinement. It is hoped that the combination of summary and postulated directions here will help stimulate this process.

Acknowledgments This work was supported in part by National Science Foundation grant ATM0082529 and National Oceanographic and Atmospheric Administration grants NA05OAR4311134 and NA04OAR4310013. Graphical work by J. E. Meyerson is gratefully acknowledged. C. Chou, C. Holloway, and A. Sobel provided helpful comments on the manuscript. It has been a pleasure to interact with C. Bretherton, J. Chiang, C. Chou, K. Emanuel, I. Held, B. Lintner, M. Majda, M. Munnich, A. Sobel, H. Su, and others on developments in this area. Recent discussions with them and with L. Back and M. Peters have helped fuel the open question/postulate paragraphs.

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References | 297

References Allen, M. R. and W. J. Ingram, 2002: Constraints on future changes in climate and the hydrologic cycle. Nature, 419, 224–232. Annamalai, H., J. M. Slingo, K. R. Sperber, and K. Hodges, 1999: The mean evolution and variability of the Asian summer monsoon: Comparison of ECMWF and NCEP-NCAR reanalyses. Mon. Wea. Rev., 127, 1157–1186. Arakawa, A. and W. H. Schubert, 1974: Interaction of a cumulus cloud ensemble with the largescale environment, Part I. J. Atmos. Sci., 31, 674–701. Bacmeister, J. T. and M. J. Suarez, 2002: Wind stress simulations and the equatorial momentum budget in an AGCM. J. Atmos. Sci., 59, 3051–3073. Back, L. E. and C. S. Bretherton, 2006: Geographic variability in the export of moist static energy and vertical motion profiles in the tropical Pacific. Geophys. Res. Lett., 33, L17810, doi: 10.1029/2006GLO26672. Barlow, M., S. Nigam, and E. H. Berbery, 1998: Evolution of the North American monsoon system. J. Climate, 11, 2238–2257. Betts, A. K., 1974: Further comments on “A comparison of the equivalent potential temperature and the static energy.” J. Atmos. Sci., 31, 1713–1715. Boer, G. J., G. Flato, and D. Ramsden, 2000: A transient climate change simulation with greenhouse gas and aerosol forcing: Projected climate to the twenty-first century. Clim. Dyn., 16, 427–450. Bretherton, C. S. and A. H. Sobel, 2002: A simple model of a convectively-coupled Walker circulation using the weak temperature gradient approximation. J. Climate, 15, 2907–2920. Chao, W. 2000: Multiple quasi-equilibria of the ITCZ and the origin of monsoon onset. J. Atmos. Sci., 52, 641–651. Chiang, J. C. H. and B. R. Lintner, 2005: Mechanisms of remote tropical surface warming during El Niño. J. Climate, 18, 4130–4149, doi:10.1175/JCLI3529.1. Chiang, J. C. H. and A. H. Sobel, 2002: Tropical tropospheric temperature variations caused by ENSO and their influence on the remote tropical climate. J. Climate, 15, 2616–2631. Chou, C. and J. D. Neelin, 2001: Mechanisms limiting the southward extent of the South American summer monsoon. Geophys. Res. Lett., 28, 2433–2436. Chou, C., J. D. Neelin, and H. Su, 2001: Ocean-atmosphere-land feedbacks in an idealized monsoon, Quart. J. Roy. Meteor. Soc., 127, 1869–1891. Chou, C. and J. D. Neelin, 2003: Mechanisms limiting the northward extent of the northern summer monsoons over North America, Asia and Africa. J. Climate, 16, 406–425. Chou, C. and J. D. Neelin, 2004: Mechanisms of global warming impacts on regional tropical precipitation. J. Climate, 17, 2688–2701. Chou, C., J. D. Neelin, J.-Y. Tu, and C.-T. Chen, 2006: Regional tropical precipitation change mechanisms in ECHAM4/OPYC3 under global warming. J. Climate, 19, 4207–4223. Cook, K. H., 1997: Large-scale atmospheric dynamics and Sahelian precipitation. J. Climate, 10, 1137–1152. Dai, A., G. A. Meehl, W. M. Washington, T. M. L. Wigley, and J. M. Arblaster, 2001: Ensemble simulation of twenty-first century climate changes: Business-as-usual versus CO2 stabilization. Bull. Amer. Meteor. Soc., 82, 2377–2388. Darnell, W. L., W. F. Staylor, S. K. Gupta, N. A. Ritchey, and A. C. Wilber, 1992: Seasonal variation of surface radiation budget derived from international satellite cloud climatology project C1 data. J. Geophys. Res., 97, 15741–15760.

June 12, 2007

Time:

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298 | References Ding, Y.-H., 1992: Summer monsoon rainfalls in China. J. Meteor. Soc. Japan, 70, 373–396. Ding, Y.-H., 1994: Monsoons over China. Kluwer Academic Publishers, Dordrecht. 419 pp. Douglas, M. W., R. A. Maddox, and K. Howard, 1993: The Mexican monsoon. J. Climate, 6, 1665–1678. Douville, H., F. Chauvin, S. Planton, J.-F. Royer, D. Salas-Mélia, and S. Tyteca, 2002: Sensitivity of the hydrological cycle to increasing amounts of greenhouse gases and aerosol. Clim. Dyn., 20, 45–68. Eltahir, E. A. B. and C. Gong, 1996: Dynamics of wet and dry years in West Africa. J. Climate, 9, 1030–1042. Emanuel, K. A., J. D. Neelin, and C. S. Bretherton, 1994: On large-scale circulations in convecting atmospheres. Quart. J. Roy. Meteor. Soc., 120, 1111–1143. Enfield, D. B., and D. A. Mayer, 1997: Tropical Atlantic sea surface temperature variability and its relation to El Niño-Southern Oscillation. J. Geophys. Res., 102, 929–945. Flohn, H., 1957: Large-scale aspects of the “summer monsoon” in south and east Asia. J. Meteor. Soc. Japan, 35, 180–186. Gill, A. E., 1980: Some simple solutions for heat induced tropical circulation. Quart. J. Roy. Met. Soc., 106, 447–462. Hales, K., J. D. Neelin, and N. Zeng, 2006: Interaction of vegetation and atmospheric dynamical mechanisms in the mid-Holocene African monsoon. J. Climate, in press. Higgins, R. W., Y. Yao, and X. L. Wang, 1997: Influence of the North American monsoon system on the U.S. summer precipitation regime. J. Climate, 10 2600–2622. Houghton, J. T., et al. eds. 2001: Climate Change: The Scientific Basis: Contribution of Working Group I to the Third Assessment Report of the Intergovernmental Panel on Climate Change Scientific Assessment. Cambridge University Press, Cambridge, U.K., New York. 881 pp. Hu, Z.-Z., M. Latif, E. Roeckner, and L. Bengtsson, 2000: Intensified Asian summer monsoon and its variability in a coupled model forced by increasing greenhouse gas concentrations. Geophys. Res. Lett., 27, 2681–2684. Janicot, S., A. Harzallah, B. Fontaine, and V. Moron, 1998: West African monsoon dynamics and eastern equatorial Atlantic and Pacific SST anomalies (1970–88). J. Climate, 11, 1874–1882. Johns, T. C. and coauthors, 2003: Anthropogenic climate change for 1860 to 2100 simulated with the HadCM3 model under updated emissions scenarios. Clim. Dyn., 20, 583–612. Klein, S.A., B. J. Soden, and N.-C. Lau, 1999: Remote sea surface temperature variations during ENSO: Evidence for a tropical atmospheric bridge. J. Climate, 12, 917–932. Lare, A. R. and S. E. Nicholson, 1994: Contrasting conditions of surface water balance in wet years and dry years as a possible land surface-atmosphere feedback mechanism in the West African Sahel. J. Climate, 7, 653–668. Lau, N.-C. and M. J. Nath, 2001: Impact of ENSO on SST variability in the north Pacific and north Atlantic: Seasonal dependence and role of extratropical sea-air coupling. J. Climate, 14, 2846–2866. Lau, K.-M. and S. Yang, 1996: The Asian monsoon and predictability of the tropical oceanatmosphere system. Quart. J. Roy. Meteor. Soc., 122, 945–957. Lenters, J. D. and K. H. Cook, 1997: On the origin of the Bolivian high and related circulation features of the South American climate. J. Atmos. Sci., 54, 656–677. Li, C. and M. Yanai, 1996: The onset and interannual variability of the Asian summer monsoon in relation to land-sea thermal contrast. J. Climate, 9, 358–375. Lintner, B. R. and J. C. H. Chiang, 2005: Reorganization of tropical climate during El Niño: A weak temperature gradient approach. J. Climate, 18, 5312–5329, doi:10.1175/JCLI3580.1.

June 12, 2007

Time:

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References | 299 Lofgren, B. M., 1995: Surface albedo-climate feedback simulated using two-way coupling. J. Climate, 8, 2543–2562. Luo, H. and M. Yanai, 1984: The large-scale circulation and heat sources over the Tibetan Plateau and surrounding areas during the early summer of 1979. Part II: Heat and moisture budjets. Mon. Wea. Rev., 112, 966–989. Manabe, S., P. C. D. Milly, and R. Wetherald, 2004: Simulated long-term changes in river discharge and soil moisture due to global warming. Hydrolog. Sci. Jour., 49, 625–642. Meehl, G. A., 1992: Effect of tropical topography on global climate. Annu. Rev. Earth Planet. Sci. 20, 85–112. Meehl, G. A., 1994a: Coupled land-ocean-atmosphere processes and South Asian monsoon variability. Science, 266, 263–267. Meehl, G. A., 1994b: Influence of the land surface in the Asian summer monsoon: External conditions versus internal feedbacks. J. Climate, 7, 1033–1049. Meehl, G. A., W. D. Collins, B. A. Boville, J. T. Kiehl, T. M. L. Wigley, and J. M. Arblaster, 2000: Response of the NCAR climate system model to increased CO2 and the role of physical processes. J. Climate, 13, 1879–1898. Moskowitz, B. M. and C. S. Bretherton, 2000: An analysis of frictional feedback on a moist equatorial kelvin mode. J. Atmos. Sci., 57, 2188–2206. Murakami, T., 1987: Orography and Monsoons. In Monsoons, ed. J. S. Fein and P. L. Stephens, John Wiley, New York, 331–364. Neelin, J. D., 1997: Implications of convective quasi-equilibrium for the large-scale flow. In The Physics and Parameterization of Moist Atmospheric Convection, ed. R. K. Smith, Kluwer Academic Publishers, Dortrecht, 413–446. Neelin, J. D., C. Chou, and H. Su, 2003: Tropical drought regions in global warming and El Niño teleconnections. Geophys. Res. Lett., 30(24), 2275, doi:10.1029/2003GL018625. Neelin, J. D. and I. M. Held, 1987: Modelling tropical convergence based on the moist static energy budget. Mon. Wea. Rev., 115, 3–12. Neelin, J. D., M. Munnich, H. Su, J. E. Meyerson, and C. Holloway, 2006: Tropical drying trends in global warming models and observations. Proc. Nat. Acad. Sci., 103, 6110–6115. Neelin, J. D. and H. Su, 2005: Moist teleconnection mechanisms for the tropical South American and Atlantic sector. J. Climate, 18, 3928–3950. Neelin, J. D. and J.-Y. Yu, 1994: Modes of tropical variability under convective adjustment and the Madden-Julian oscillation. Part I: Analytical results. J. Atmos. Sci., 51, 1876–1894. Neelin, J. D. and N. Zeng, 2000: A quasi-equilibrium tropical circulation model—formulation. J. Atmos. Sci., 57, 1741–1766. Nicholson, S., 2000: Land surface processes and Sahel climate. Rev. Geophys., 38, 117–139. Nogues-Paegle, J., C. R. Mechoso, and coauthors, 2002: Progress in Pan American CLIVAR research: Understanding the South American monsoon. Meteorologica, 27 3–32. Palmer, T. N., 1986: Influence of the Atlantic, Pacific and Indian Oceans on Sahel rainfall. Nature, 322, 251–253. Ramage, C. S., 1971: Monsoon Meteorology. Academic, New York, 296 pp. Riehl, H., 1979: Climate and Weather in the Tropics. Academic, New York, 611 pp. Rodwell, M. J. and B. J. Hoskins, 1996: Monsoons and the dynamics of deserts. Quart. J. Roy. Meteor. Soc., 122, 1385–1404. Rodwell, M. J. and B. J. Hoskins, 2001: Subtropical anticyclones and summer monsoons. J. Climate, 14, 3192–3211.

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Time:

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300 | References Roeckner, E., L. Bengtsson, J. Feichter, J. Lelieveld, and H. Rodhe, 1999: Transient climate change simulations with a coupled atmosphere-ocean GCM including the tropospheric sulfur cycle. J. Clim., 12, 3004–3032. Rowell, D. P., C. K. Folland, K. Maskell, and M. N. Ward, 1995: Variability of summer rainfall over tropical north Africa (1906–92): Observations and modelling. Quart. J. Roy. Meteor. Soc., 121, Part A, 669–704. Semazzi, F. H. M. and L. Sun, 1997: The role of orography in determining the Sahelian climate. Intl. J. Clim., 17, 581–596. Sobel, A. H. and H. Gildor, 2003: A simple time-dependent model of SST hot spots. J. Climate, 16, 3978–3992. Sobel, A. H., I. M. Held, and C. S. Bretherton, 2002: The ENSO signal in tropical tropospheric temperature. J. Climate, 155, 2702–2706. Sobel, A. H. and J. D. Neelin, 2006: The boundary layer contribution to intertropical convergence zones in the quasi-equilibrium tropical circulation model framework. Theor. Comp. Fluid Dyn., doi:10.1007/s00162-006-0033-7. Su, H., J. D. Neelin, and C. Chou, 2001: Tropical teleconnection and local response to SST anomalies during the 1997–1998 El Niño. J. Geophys. Res., 106, D17, 20,025–20,043. Su, H. and J. D. Neelin, 2002: Teleconnection mechanisms for tropical Pacific descent anomalies during El Niño. J. Atmos. Sci., 59, 2694–2712. Su, H. and J. D. Neelin, 2005: Dynamical mechanisms for African monsoon changes during the mid-Holocene. J. Geophys. Res., 110, D19105, doi:10.1029/2005JD005806. Webster, P. J., 1972: Response of the tropical atmosphere to local, steady forcing. Mon. Wea. Rev., 100, 518–541. Webster, P. J., 1987: The elementary monsoon. In Monsoons, ed. J. S. Fein and P. L. Stephens, John Wiley, New York, 3–32. Webster, P. J., V. O. Magaña, T. N. Palmer, J. Shukla, R. A. Tomas, M. Yanai, and T. Yasunari, 1998: Monsoons: Processes, predictability, and the prospects for prediction. J. Geophys. Res., 103, 14451–14510. Williams, K. D., C. A. Senior, and J. F. B. Mitchell, 2001: Transient climate change in the Hadley Centre models: The role of physical processes. J. Clim., 14, 2659–2674. Wu, G. and Y. Zhang, 1998: Tibetan Plateau forcing and the timing of the monsoon onset over South Asia and the South China Sea. Mon. Wea. Rev., 126, 913–927. Xie, P. and P. A. Arkin, 1997: Global precipitation: A 17-year monthly analysis based on gauge observations, satellite estimates, and numerical model outputs. Bull. Amer. Meteorol. Soc., 78, 2539–2558. Xue, Y. and J. Shukla, 1993: The influence of land surface properties on Sahel climate. Part I: Desertification. J. Climate, 6, 2232–2245. Yanai, M. and C. Li, 1994: Mechanism of heating and the boundary layer over the Tibetan Plateau. Mon. Wea. Rev., 122, 305–323. Yang, S. and K.-M. Lau, 1998: Influences of sea surface temperature and ground wetness on Asian summer monsoon. J. Climate, 11, 3230–3246. Yasunari, T. and Y. Seki, 1992: Role of Asian monsoon on the interannual variability of the global climate system. J. Met. Soc. Japan, 70, 177–189. Yonetani, T. and H. B. Gordon, 2001: Simulated changes in the frequency of extremes and regional features of seasonal/annual temperature and precipitation when atmospheric CO2 is doubled. J. Clim., 14, 1765–1779.

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References | 301 Young, J. A., 1987: Physics of monsoons: The current view. In Monsoons, ed. J. S. Fein and P. L. Stephens, John Wiley, New York, 211–243. Yu, B. and J. M. Wallace, 2000: The principal mode of interannual variability of the North American monsoon system. J. Climate, 13, 2794–2800. Yu, J.-Y., C. Chou, and J. D. Neelin, 1998: Estimating the gross moist stability of the tropical atmosphere. J. Atmos. Sci., 55, 1354–1372. Yu, J.-Y. and J. D. Neelin, 1997: Analytic approximations for moist convectively adjusted regions. J. Atmos. Sci., 54, 1054–1063. Yu, J.-Y. and J. D. Neelin, 1994: Modes of tropical variability under convective adjustment and the Madden-Julian oscillation. Part II: Numerical results. J. Atmos. Sci., 51, 1895–1914. Zebiak, S. E., 1986: Atmospheric convergence feedback in a simple model for El Niño. Mon. Wea. Rev., 114, 1263–1271. Zeng, N. and J. D. Neelin, 1999: A land-atmosphere interaction theory for the tropical deforestation problem. J. Climate, 12, 857–872. Zeng, N., J. D. Neelin, and C. Chou, 2000: A quasi-equilibrium tropical circulation model— implementation and simulation. J. Atmos. Sci., 57, 1767–1796. Zhou, J. and K.-M. Lau, 1998: Does a monsoon climate exist over South America? J. Climate, 11, 1020–1040.

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Chapter 11

Challenges in Numerical Modeling of Tropical Circulations Christopher S. Bretherton

11.1. Introduction The large-scale dynamics of the tropical atmosphere are inextricably linked to the small-scale turbulent motions of moist convection, ranging from cumulonimbus cloud systems to shallow, nonprecipitating cumuli to stratocumulus cloud sheets. Cumulonimbus cloud systems are the primary heat engine of tropical circulations and keep the temperature profile nearly moist-adiabatic throughout low latitudes. Moist convection also controls both the distribution of humidity and radiative transfer through the tropical atmosphere. Moist convection is a sub-gridscale process in current atmospheric and coupled ocean-atmosphere general circulation models (AGCMs and CGCMs). It involves parameterizations for convective transport, cloud microphysics, subgrid inhomogeneity of cloud condensate, and boundary-layer processes that must work seamlessly together. This has proven a thorny challenge, not satisfactorily solved despite 40 years of sustained research efforts. However, improvements in cloud-resolving models (CRMs), increased computational power, and a dramatic increase in satellite data on low-latitude clouds, winds, and water vapor are improving our fundamental understanding of how to parameterize low-latitude moist convection. At the same time, the parameterization challenges are changing as increasingly fine space scales are explicitly resolved within global models. The character of moist convection is tightly linked to the underlying land or ocean surface. The surface radiation balance and winds are in turn affected by the convection and associated clouds. Current realistically initialized CGCMs rapidly develop important biases in the seasonal distribution of sea and land surface temperatures, precipitation, and El Niño-Southern Oscillation (ENSO) within months to a few years. Errors in parameterizing moist convective processes are widely thought to play an important (though not exclusive) role in these biases.

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Challenges in Numerical Modeling | 303

Section 11.2 documents some key low-latitude biases in CGCMs and AGCMs that are likely associated with moist convection. Section 11.3 will discuss insights derived from using CRMs to resolve some of these biases. Section 11.4 discusses some new global climate modeling techniques using CRMs and their potential for reducing tropical biases. Section 11.5 discusses another modeling challenge, uncertainties in the feedback of low-latitude clouds on global warming.

11.2. Tropical Biases in Climate Simulations In this section, we will illustrate five long-standing model biases that plague most current CGCMs and AGCMs. These are: 1. 2. 3. 4. 5.

A warm eastern ocean/double intertropical convergence zone (ITCZ) bias. ENSOs with periods that are too short (2–3 years). A Madden-Julian Oscillation (MJO) that is too weak and too fast. A diurnal cycle of convective cloud and rainfall and that is several hours early. Biases in cloud vertical distribution and optical thickness.

This chapter was partially inspired by a May 2003 workshop on tropical biases in Princeton, New Jersey, organized by the author and Dr. Ping Chang, which focused on the first two of these biases.

11.2.1. The Warm Eastern Ocean/Double ITCZ Bias The warm eastern ocean/double ITCZ bias is a persistent problem for almost all current CGCMs, as documented by intercomparisons by Mechoso et al. (1996), Davey et al. (2002), and others. Here we illustrate this bias using two figures derived from a CGCM/AGCM comparison study performed by Sperber et al. (2003) for the aforementioned 2003 tropical biases workshop. Figure 11.1 shows the remarkably similar character of this bias in three representative leading CGCMs, the Geophysical Fluid Dynamics Laboratory (GFDL) model (Knutson and Manabe 1998), the National Center for Atmospheric Research (NCAR) Community Climate System Model (CCSM) version 2 (Kiehl and Gent 2004), and the Hadley Centre Coupled Model version 3 (HadCM3, Gordon et al. 2000). The common bias pattern is characterized by annualmean sea surface temperatures (SSTs) that are too warm in the cool-SST regions off the east coasts of South and North America and South Africa. In the central Pacific, warm off-equatorial SST anomalies combine with an overly pronounced equatorial cold tongue to support a “double ITCZ” structure with excessive rainfall south of the equator. Over the continents, Atlantic and Indian Oceans, and midlatitudes, biases are much more model-dependent, though in some CGCMs they are also very significant.

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304 | Christopher S. Bretherton GFDL (2003)

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Figure 11.1. Biases in annual-mean climatological SST (shading) and rainfall (1 mm d−1 contours; negative contours dashed) in three leading near-current CGCMs. Biases are computed relative to observed SST (Reynolds and Smith 1994) and satellite-derived rainfall (Xie and Arkin 1997), respectively shown in lower right panel with 27◦, 28◦, and 29◦C SST contours and shading for rainfall (mm d−1). Adapted from Sperber et al. (2003).

There is controversy about how much of this bias is due to parameterization of atmospheric moist processes. Figure 11.2 shows the rainfall simulated by the component AGCMs of the three coupled models shown in Fig. 11.1, but now forced with the observed SST distribution for the 15-year period 1979–1993. All three AGCMs have a double ITCZ rainfall bias, but not nearly as strong as in the corresponding CGCMs. Two other AGCMs in the Sperber et al. study (ECHAM4 and GFDL-1996) had no double ITCZ bias. However, these models are not plotted here because the corresponding CGCMs used flux correction, which artificially suppresses double ITCZ biases by keeping the CGCM-simulated SST climatology congruent with observations. Oceanic explanations of aspects of this bias have been proposed. Current CGCM experience is that they can reduce parts of the bias pattern but not the double ITCZ. A treatment of the penetration of insolation into the ocean dependent on productivity and wavelength, as well as diurnally-resolved coupling of the ocean to the atmosphere, both help reduce the equatorial cold SST biases (Danabasoglu et al. 2006). Unresolved tropical instability waves in the upper ocean just north of the equator also produce significant vertical and horizontal oceanic and surface heat flux divergence within the

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Figure 11.2. Climatological annual rainfall bias simulated by the component AGCMs of the three CGCMs in Fig. 11.1 when forced with the observed SST distribution for 1979–1993. The lower-right panel repeats the observational estimate of Fig. 11.1. Contour interval and shading are as in Fig. 11.1. Adapted from Sperber et al. (2003).

cold tongue, though a high-resolution ocean model suggests these effects compensate so as to have little net impact on cold tongue SST (Jochum et al. 2005). Large and Danabasoglu (2006) found that overwriting the ocean state in a CGCM with observed climatology (or even just specifying the observed wind stress) in the easternmost 250 km of the southeast Pacific and southeast Atlantic reduces SST and rainfall biases throughout neighboring region of the tropical oceans. The ocean model may contribute to biases, but is probably not their primary cause. Atmospheric moist process errors can distort rainfall patterns, profoundly affecting the entire tropical circulation. Even if the simulated rainfall distribution is reasonable, faulty parameterizations can induce large biases in the surface wind stress and net surface turbulent or radiative heat fluxes. The response of surface winds and SST in a CGCM to such errors can be highly nonlocal. For instance, Kiehl (1998) showed that the coupled response to an excessive heat flux into the west Pacific warm pool was to amplify equatorial easterlies that increased latent heat flux out of the warm pool, counterintuitively cooling its SST. Yu and Mechoso (2001) showed that the response of their CGCM to enhancement of southeast Pacific stratocumulus cloud cover was to change the rainfall patterns and winds over the entire east and central Pacific and remove a double ITCZ otherwise present in their simulations.

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11.2.2. ENSO Biases ENSO is a challenging simulation problem for CGCMs. ENSO intercomparison studies by Latif et al. (2001) and AchutaRao and Sperber (2002), as well as studies of individual CGCMs, show that many CGCMs simulate ENSOs of reasonable but often somewhat too-small amplitude (as measured using eastern equatorial Pacific SST). In most (though not all) CGCMs, simulated ENSOs have unrealistically short periods of 2–3 years and tend to have SST, rainfall, and cloud anomalies concentrated in too narrow a latitude band around the equator. Many models displace the maximum SST anomalies too far to the west. A variety of explanations exist for these biases, and the reasons may partly be model-specific. Guilyardi et al. (2004), in a study comparing different pairings of AGCMs and ocean GCMs, concluded that the ENSO amplitude and period are more sensitive to the atmospheric model than the ocean model. Deser et al. (2006) documented the aforementioned biases in the NCAR CCSM3. They found that if the atmospheric component of that model was forced by the correct time-varying SST, the wind stress and sea-level pressure signature are too equatorially localized even though the ENSO rainfall anomalies have a fairly reasonable geographical structure and amplitude. Idealized simple coupled-model studies (e.g., Kirtman 1997) explain the consequences of wind stress biases in terms of the delayed-oscillator view of ENSO (Schopf and Suarez 1988). Kirtman found that if the wind stress anomaly response to an ENSO SST anomaly is more equatorially localized, less of the Rossby wave response is carried to the western boundary by slow off-equatorial Rossby wave modes. This offequatorial response slows the ENSO period by delaying the growth of a reflected Kelvin wave response that can reverse the east Pacific SST anomaly. Other related mechanisms have been proposed that have a similar effect on ENSO period. Regardless of the details of the ocean response, ENSO theory implies that more accurate simulation of the nearsurface atmospheric wind anomaly field in response to ENSO-like SST anomalies is required to improve CGCM simulations of ENSO. The surface winds are highly coupled to the overall tropical circulation, which inextricably combines many feedbacks, but they are certainly influenced both by planetary boundary layer (PBL) dynamics and by vertical momentum exchange in cumulus clouds. The latter process is neglected in many CGCMs (including CCSM3) and treated crudely in some others, likely contributing significantly to the surface wind biases.

11.2.3. MJO Biases The Madden-Julian Oscillation (MJO) (Madden and Julian 1971, 1972, 1994) is a longstanding simulation challenge for most weather forecast models and GCMs. The MJO is the dominant mode of 30–90-day rainfall variability over much of the Tropics, and

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forces midlatitude Rossby wave trains that affect extratropical weather (e.g., Weickmann 1983, Kiladis and Weickmann 1992). The MJO is a quasi-periodic disturbance of quite variable amplitude and 40–50-day period. It consists of a broad-scale band of enhanced convective rainfall and baroclinic zonal wind anomalies that develops over the Indian Ocean then propagates slowly eastward at 5–10 m s−1 into the west Pacific warm pool (e.g., Hendon and Salby 1994). The rainfall anomaly dissipates near the dateline, but the upper-tropospheric wind anomalies propagate around the globe at a phase speed of roughly 30 m s−1, possibly helping to stimulate the next cycle of MJO activity. An interesting feature of the MJO (and other convectively coupled waves) is that the deep convection tends to be preceded by a period of lower tropospheric moist anomalies capped by a dry, warm upper troposphere that has been interpreted as a form of “preconditioning” for deep convection (Maloney and Hartmann 1998; Wang and Schlesinger 1999; Wheeler et al. 2000). MJO simulation has mainly been studied in the AGCM context of prescribed SST, and has been the subject of countless papers and several intercomparison studies (e.g., Slingo et al. [1996]; see also the report by Waliser et al. [2003]). It is quite sensitive to the treatment of cumulus parameterization. AGCMs with cumulus parameterizations based on moisture-convergence closures often simulated a strong MJO-like disturbance (e.g., Hayashi and Sumi 1986), usually with too short a period, that was explained as a form of wave-CISK (conditional instability of the second kind) instability. After such closures fell from favor to be replaced by CAPE (convective available potential energy)regulating closures, investigators such as Wang and Schlesinger (1999) concluded that a more vigorous and realistic MJO could be obtained by supplementing CAPE closure with a limiter that only allows deep convection when the PBL and/or lower troposphere are sufficiently moist. Inness and Gregory (1997) and others have found that parameterized cumulus momentum transport affects the MJO, mainly as a form of damping. Parameter and physics choices within individual AGCMs are made mainly to optimize the mean tropical climate, and in practice these have often not led to the best possible MJO simulations. Thus, MJO intercomparisons discussed by Waliser et al. (2003) and Slingo et al. (1996) show that most current AGCMs still have too little intra-seasonal outgoing longwave radiation (OLR) variability in the Tropics, especially associated with coherent eastward-propagating disturbances. Comparisons of CGCMs with fixed SST simulations with the same AGCM indicate that ocean coupling tends to somewhat amplify and slow down the simulated MJO, and favor eastward propagation, but not enough to substantially reduce the model biases (Waliser et al. 2003).

11.2.4. Diurnal-Cycle Biases The diurnal cycle of low-latitude rainfall tends to peak several hours earlier in the day in AGCMs than in observations (Yang and Slingo 2001). Over land, the simulated rainfall often peaks at or before noon, while observations suggest a late afternoon or evening

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peak; over ocean the diurnal cycle is weaker and the bias is less consistent, but also tends to be advanced in AGCMs compared to observations.

11.2.5. Cloud Biases Climate model developers have long been aware of the importance of clouds on the radiation balance at the top of the atmosphere (TOA) and the surface. For 15 years, modelers have used satellite observations of the space-time distribution of TOA shortwave and longwave cloud radiative forcing from the Earth Radiation Budget Experiment (ERBE) and its successor, Clouds and the Earth’s Radiant Energy System (CERES), as a test of the fidelity of cloud-radiation interactions in AGCMs. Cloud radiative forcing is defined as the difference between the net incoming radiative flux, given the observed clouds, and the corresponding clear-sky net radiative flux; in the current climate its global average is negative, corresponding to a net cooling effect of clouds on the Earth system. TOA cloud radiative forcing is a widely used measure of the cloud effect on climate because it has a natural physical interpretation and can be accurately estimated from satellite observations (Ramanathan et al. 1989). Cloud radiative forcing is partitioned between longwave cloud forcing (LWCF), which is usually positive due to the greenhouse effect of clouds, and shortwave cloud forcing (SWCF), which is usually negative because clouds reflect additional sunlight compared to the cloud-free atmosphere. The boundary-layer clouds of subsidence regions have slight LWCF since they radiate at a temperature similar to the underlying surface, and they have a range of SWCF depending on cloud fraction and thickness. Deep-convective cloud systems in ascent regions have large positive LWCF associated with their extensive cold-topped cirrus clouds, and roughly compensating negative SWCF due to both the cirrus and underlying cumulus clouds. This discussion cannot do justice to the large literature on the AGCM cloudbias problem, but I will briefly illustrate some issues using results from a recent paper by Wyant et al. (2006a, hereafter W06). W06 compared low-latitude cloud biases in three leading U.S. AGCMs. They used a method proposed by Bony et al. (2004) in which a monthly climatology is constructed over each AGCM grid column in 30◦ S– 30◦ N, and column-months are binned into dynamical regimes using their 500 mb mean vertical pressure velocity ω500. Ascent regimes (ω500 < 0) are rainy and have frequent mesoscale convective systems with large cirrus anvils. Subsidence regimes (ω500 > 0) favor boundary-layer cumulus and stratocumulus cloud over the oceans and clear skies over land. W06 showed that current flagship U.S. AGCMs are fairly skillful at reproducing strong observed correlations over the tropical oceans between more ascent and rainfall, more longwave cloud radiative forcing (due particularly to the greenhouse effect of cirrus anvils), and more negative shortwave radiative forcing (enhanced albedo due both to cirrus and underlying cloud).

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Figure 11.3. Frequency of clouds in all grid columns in 30◦ S–30◦ N and all months of the year, binned by radiatively-inferred cloud-top pressure and 500 hPa monthly mean vertical pressure velocity ω500. In each column the top three panels correspond to thin, medium, and thick cloud optical depth ranges, and the bottom panel is their sum. Right column: ISCCP observations. Left and middle columns: NCAR CAM3 and GFDL AM2.12 AGCMs. Adapted from Wyant c et al. (2006a). (Reproduced with permission from Springer Science and Business Media
2006.)

However, these same models have serious systematic biases in the cloud height and thickness distributions that create the observed cloud radiative forcing compared to data from the International Satellite Cloud Climatology Project (ISCCP). This data can be compared with the output of an “ISCCP simulator” (Klein and Jakob 1999) applied to clouds simulated in an AGCM. The right column of Fig. 11.3, adapted from W06, plots the ISCCP-derived frequency of clouds in all grid columns in 30◦ S–30◦ N and all months of the year, binned by radiatively-inferred cloud-top pressure and monthly mean ω500. The four panels in the column correspond to three cloud optical depth ranges (thin, medium, and thick clouds), and their sum. The left two columns show ISCCP-simulator output from the NCAR CAM3 and GFDL AM2.12 AGCMs, averaged and binned in the same manner. In ascent (deep convective) regions, the AGCMs tend to have inadequate midlevel clouds, inadequate clouds of intermediate optical depth, and a compensating excess

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of high, thick cloud. This same error pattern has been exposed in studies comparing CRMs with single-column models (SCMs) (e.g., Wu and Moncrieff 2001). In subsidence regions (ω500 > 0) both AGCMs have too much thick, low-topped cloud, but too little thin- and medium-thickness cloud. There are some imperfections in this comparison, e.g. associated with ISCCP non-detection of sub-pixel clouds, but these are likely small compared to the model-observation differences. Similar biases were found across a broader set of AGCMs in another recent intercomparison study (Zhang et al. 2005) that used latitudinal rather than ω500 binning. Low-latitude cloud biases in AGCMs in both ascent and subsidence regimes involve a combination of interacting parameterizations. Coarse resolution in space and time also contributes to these biases. For example, many deep and shallow cumulus parameterizations tend to pulse on and off given the long time step (30 min or more) of many GCMs, creating short lived but thick clouds. The coarse vertical resolution of GCMs also favors the production of excessively deep, optically thick cloud layers. For some processes, such as ice microphysics in tropical cirrus anvils, fundamental issues, such as what determines the number and size distribution of ice particles, are still quite poorly understood. Other parameterization issues involve economical representation of known physics. For instance, the joint horizontal and vertical subgrid variability of water vapor, cloud cover, and condensate in convective cloud systems can be fairly well simulated in CRM simulations of convective cloud systems, but schemes for representing this using simplified classes of probability density functions and vertical overlap schemes are difficult to make general, accurate, and numerically tractable. A third type of parameterization problem involves interaction of parameterizations. For instance, in the CAM3 AGCM, we have found grid columns in the subtropical trade-wind regimes that in almost one fifth of the time steps throughout the entire year, there is simulated rainfall but no cloud liquid water. This involves inconsistent handling of interactions between the cumulus and prognostic liquid water parameterizations. The cumulus parameterization is active and produces precipitation. The parameterized cumulus convection contributes to the assumed fractional cloud cover used in the radiation scheme. However, the calculation of cloud liquid water content is not coordinated with the calculation of cloud fraction. Instead, it inconsistently makes the assumption that the cumulus updrafts themselves have negligible areal coverage. The cumulus parameterization contributes to the prognosed within-cloud liquid water content only indirectly, by detraining moist, condensate-filled air. This air is assumed to homogeneously mix with the ambient air before the resolved-scale liquid water content is computed. If the environment in the detrainment zone is sufficiently dry, the mixing process will evaporate all the detrained condensate and the model predicts there is no resolved-scale liquid water in its clouds. Such inconsistencies are hard to avoid when multiple parameterizations are interacting on the sub-gridscale. It can be quite awkward to design resolved-scale

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cloud fraction and microphysical schemes that compensate for these biases. The simulated excess of cirrus, for instance, may be a result of tuning the AGCM to create sufficient longwave and shortwave radiative forcing to match ERBE/CERES using only resolved-scale clouds that do not precipitate very much, because the condensate within parameterized, heavily precipitating cumulonimbi is not adequately radiatively accounted for. W06 and Zhang et al. (2005) concluded that AGCMs are achieving reasonable global means and broad-scale geographic distributions of LWCF and SWCF by compensating errors. LWCF and SWCF are fields widely used by model developers to assess model changes. Both studies suggest that these fields have evolved into agreement with the observations more by a “tuning” process than due to correct representation of cloud processes.

11.3. Using CRMs to Understand Tropical Biases Deficiencies in the parameterization of cumulus convection and associated cloud processes likely play an important role in tropical biases. In this section, we will focus on one such deficiency, the inadequate sensitivity of most current cumulus parameterizations to ambient lower- and mid-tropospheric relative humidity. Many observational studies have shown that tropical oceanic convection can be suppressed by mid-level dry advection, even when undilute boundary-layer air would have substantial CAPE (e.g., Numaguti et al. 1995). This reflects the importance of entrainment of dry mid-level air into rising moist thermals, which evaporatively cools them and robs them of buoyancy. All cumulus parameterizations account in some way for entrainment of surrounding air through the cloud sides and top, but if the parameterized entrainment rates are too small, the cumulus parameterization will be insensitive to the properties of lower- and mid-tropospheric air. Recent CRM-SCM intercomparison studies suggest that this is indeed the case. Derbyshire et al. (2004) simulated cumulus convection in a suite of soundings (maintained by nudging) with different values of cloud-layer relative humidity (RH), but all having the same temperature profile and subcloud properties. Results from six SCMs were compared with two CRM simulations. Fig. 11.4a shows that the United Kingdom Meteorological Office (UKMO) CRM is highly sensitive to environmental RH. It plots the vertical profile of cumulus mass flux Mc(z), calculated by summing the product of vertical velocity and density at all cloudy gridpoints at each model level and dividing by the total number of gridpoints at that model level. For the driest case of RH = 25%, no convective mass flux penetrates above 2 km. For moist RH, vigorous convective mass flux extends throughout the troposphere. A second CRM gave similar results.

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Fig. 11.4b shows the parameterized cumulus mass-flux profile for the UKMO SCM, which performed better in this simulation than most of the other SCMs, but nevertheless had much weaker sensitivity to RH than the CRM. 11.3.1. Causes and Solutions of the Problem Why are SCMs systematically biased relative to the CRMs? Dimensional arguments and laboratory studies, summarized in chapter 2 of Emanuel (1994), suggest that the relative rate at which a dry buoyant thermal entrains mass is inversely proportional to the radius of the thermal; thus smaller cumuli are particularly prone to entrainment drying. Most parameterizations of precipitating cumulus convection have been designed primarily to represent deep, relatively large, cumulonimbus clouds that entrain slowly and precipitate much of their condensate out before it has a chance to evaporate due to entrainment mixing. These schemes (even spectral schemes in which mass flux is partitioned across a population of cumuli spanning a large spectrum of entrainment rates, e.g., Arakawa and Schubert [1974]; Moorthi and Suarez [1992]; Zhang and McFarlane [1995]) appear to underestimate the contribution of the population of smaller clouds to the overall upward transports. Kuang and Bretherton (2006) proposed an approach for diagnosing the partitioning of mass flux in a CRM between effective lateral entrainment rates. Their approach was based on use of moist static energy as a tracer that is adiabatically approximately conserved but which is sensitive to lateral mixing. They diagnosed the partitioning of mass flux between entrainment rates during a high-resolution CRM simulation of an idealized transition from nonprecipitating oceanic trade cumulus convection to congestus, then cumulonimbus convection. A striking feature of their results, seen throughout the transition, was how much of the mass flux at all levels was in updrafts so diluted by entrainment as to be only weakly buoyant. Their approach can be directly compared with predictions of cumulus parameterizations, and may perhaps provide a firmer basis for partitioning mass flux into more and less entrainment-diluted drafts. 11.3.2. Relation to Diurnal Cycle Rainfall Biases over Warm Continents The GCM diurnal cycle biases discussed in section 11.2.4 may also reflect inadequate sensitivity of deep cumulus development to relative humidity. Grabowski et al. (2006) presented a CRM-SCM intercomparison of the daytime development of deep convection over Amazonia. It was initialized with a typical pre-dawn thermodynamic profile for this region. Specified time-evolving heat and moisture fluxes were applied to force moist convection. Figure 11.5 shows the evolution of typical CRM and SCM cumulus mass flux profiles over the first six hours of simulation. The CRMs evolved a shallow cumulus population within two hours, but then took an additional three hours to develop deep convection with significant rainfall. During this three hours, the shallow

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Figure 11.4. Sensitivity of cumulus mass flux profile to specified environmental relative humidity: (a) UKMO cloud-resolving model; (b) UKMO single-column model. Extracted from Fig. 9 of Derbyshire et al. (2004). (Reproduced with permission from the c 2004.) Royal Meteorological Society


convection was steadily moistening or “preconditioning” the lower free troposphere, allowing cumuli to deepen more and more as entrainment became less effective in drying and cooling their updrafts. In contrast, the SCMs developed deep convection after 2–3 hours, 3 hours in advance of the CRMs, and exactly in the sense of the AGCM diurnal cycle bias over warm continents.

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Figure 11.5. Hourly evolution of cumulus mass flux profile during the morning development of cumulus convection over Amazonia. (a) NCAR cloud-resolving model. (b) UKMO singlecolumn model. Adapted from Grabowski et al. (2006). (Reproduced with permission from the c 2006.) Royal Meteorological Society


11.4. Application of CRMs to Global Climate Dynamics In the previous section, we saw that CRMs can address several important tropical biases seen in AGCMs, including problems with the diurnal cycle and MJO. Recently, this potential has inspired some exciting new strategies for using CRMs in global simulations of the atmosphere and coupled ocean-atmosphere system. Using the Japanese Earth Simulator supercomputer, Tomita et al. (2005) have run aqua-planet simulations with a fully global CRM at horizontal resolutions down to 3.5 km for periods of a few days or weeks. These impressive computations may portend the future of climate modeling, but are still too short and idealized for assessment of mean tropical biases in a global CRM simulation. For the next few years, practical global climate applications of CRMs still require shortcuts. Two such shortcuts are superparameterization and Diabatic Acceleration and REscaling (DARE). Superparameterization, also known as cloud-resolving cumulus parameterization or multiscale modeling, was pioneered by Grabowski (2001) and was first embedded in a realistic AGCM by Khairoutdinov and Randall (2001). The physical parameterizations

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within each grid column of a GCM are replaced by a small CRM with the same vertical resolution and approximately 50–100 horizontal columns, which simulates the cumulus convection and associated subgrid cloud processes and radiation within that grid column. The CRMs are coupled through the large-scale advection processes simulated by the AGCM, which are applied as horizontally homogeneous time-dependent forcings on the individual CRMs. Results presented by Khairoutdinov et al. (2005), embedding their CRM in the CAM3 AGCM to get what we will call the CAM3-SP, represent the current state of this rapidly evolving methodology. Superparameterization involves approximately 100 times more computations than a conventional GCM to simulate the same length of time, so is unlikely to replace conventional AGCMs. At present, this has limited superparameterization to a small number of specified-SST simulations less than twenty years long. There are important questions about the adequacy of the CRM vertical and horizontal resolution, domain size, and dimensionality used in current implementations of superparameterization. Because superparameterization avoids the need for cumulus parameterization, it sidesteps the apparent parameterization problems with RH-convection feedback and may shed light on how AGCM biases might change if RH-convection feedback were better represented. Khairoutdinov et al. (2005) found the simulated diurnal cycle of tropical rainfall, like the MJO, was much improved in CAM3-SP compared to its conventional counterpart. This suggests that premature development of parameterized deep convection in conventional AGCMs plays a central role in this bias. Grabowski (2001), Khairoutdinov and Randall (2001), and Khairoutdinov et al. (2005) found that superparameterization robustly produces a vigorous MJO with realistic vertical structure and eastward propagation. The MJO in Grabowski’s study is probably distorted and amplified by unrealistic cumulus momentum flux feedbacks associated with a two-dimensional CRM geometry. More diagnostic analysis is required to fully understand the dynamics of Khairoutdinov and Randall’s superparameterized MJO. Wyant et al. (2006b) used a monthly-mean ω500-binned ISCCP-simulator analysis to examine regime-sorted biases in CAM3-SP low-latitude cloud climatology, as in Fig. 11.3. They found CAM3-SP had biases in the joint distribution of cloud optical thickness and cloud-top pressure that are significantly smaller than (though in the same sense as) CAM3. Interestingly, these biases are reduced not only in the ascent (deep convection) regions, but also in subsidence regions in which one might expect poorer performance from superparameterization, since it under-resolves the boundary-layer cloud-forming eddies. Superparameterization currently produces mean tropical rainfall biases comparable to a conventional AGCM. For instance, a particular problem found by Khairoutdinov et al. (2005) was excessively strong rainfall during boreal summer at the northern edge of the west Pacific warm pool. Because current superparameterizations do not resolve the turbulent circulations that produce boundary-layer cumulus and stratocumulus, they

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are just as prone to parameterization error over the subsidence regions of the Tropics as conventional AGCMs. Their mean rainfall biases may partly reflect nonlocal responses to errors in the subsidence regions. However, the superparameterization does seem to correctly predict the east margin of the rainfall in the southwest and northwest tropical Pacific, correcting the bias we noted in CAM3. Thus, superparameterization qualitatively improves all the major biases that we suggested might arise from imperfectly parameterized RH-convection feedback, without yet translating this improvement into a tropical mean rainfall distribution in an AGCM. Future refinements of superparameterization may further improve simulations of tropical cloud-turbulence-climate interactions. An embedded ultra-high-resolution, large-eddy simulation model inserted at grid columns with only boundary-layer cloud and no deep cumulus convection could allow superparameterization to avoid the need for boundary-layer turbulence and shallow cumulus convection parameterizations to accurately simulate boundary-layer cloud regimes. The cloud-microphysical parameterizations in CRMs are arguably more physically based than in GCMs because they can work on air parcels that are realistically circulating through clouds instead of working only on a statistical representation of a cloud ensemble. However, the microphysical parameterizations in present CRMs have not been widely tested against TOA and surface-radiation measurements, and will need to be refined to reduce such radiative biases. Blossey et al. (2007) show an example of such a test against a 52-day dataset from a tropical convection field experiment in the central Pacific ITCZ, including sensitivity of the CRM simulations to various changes in the CRM microphysical and radiative parameterizations. The CRM is found to have particular difficulty simulating sufficient anvil cirrus during periods of isolated deep convection producing only scattered showers, even though it can simulate the TOA radiative fluxes fairly well during periods of more intense rainfall. Superparameterization still introduces an artificial scale gap between the global and grid column scales because CRMs in different columns interact only through their domain-mean prognosed variables. Another approach, DARE (Diabatic Acceleration and REscaling; Kuang et al. 2005), involves a global or near-global CRM simulation of a rescaled miniature planet that is a large factor γ ≈ 10 smaller than the real Earth, rotates a factor γ faster, and in which all diabatic processes such as radiative and surface turbulent heating, rainfall, and autoconversion of cloud to rain/snow are also increased by a factor γ . This produces weather and climate that are surprisingly similar to the real Earth (see Kuang et al. 2005), but allows a multiyear climate simulation with a CRM of 2–4 km resolution on a 16-node Linux cluster in a few days. Rather than introducing an artificial scale gap, DARE brings the global scale a factor γ closer to the convective scale. This allows DARE, unlike superparameterization, to simulate such intermediatescale weather systems as hurricanes. However, DARE does appear to somewhat distort the cloud cover and condensate, and, like superparameterization, does not adequately resolve boundary-layer cloud circulations. Furthermore, it is not yet clear if DARE with

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a horizontal grid spacing of x actually provides benefits over a non-hydrostatic global CRM run without a cumulus parameterization on our true planet at a resolution γ x (Pauluis et al. 2006). Despite their challenges, superparameterization, DARE, and full global CRMs, tested and improved using the fire hose of new satellite and field observations, should play a major role in furthering our understanding of the feedbacks between cloud systems and large-scale circulations in the Tropics over the next few years.

11.5. Low-Latitude Cloud-Climate Feedback Uncertainties Our society looks to climate models for reliable prediction of the climate response to increased greenhouse gases. This is often quantified in terms of the “climate sensitivity” of a GCM, defined as the steady-state globally averaged surface air temperature increase due to CO2 doubling. The climate sensitivity of current CGCMs varies by a rather large factor of 2–3 (IPCC 2001). A dominant uncertainty in such predictions is cloud radiative feedback on climate change (Cess et al. 1989; IPCC 2001). This feedback is often cast in terms of TOA cloud radiative forcing. If global-mean cloud radiative forcing becomes less negative in a warmer climate, this is usually interpreted as a positive cloud feedback on the warming, subject to caveats discussed by Soden et al. (2004). One of the most pressing challenges in climate modeling is to reduce this cloudfeedback uncertainty. In low latitudes, this challenge is magnified because most clouds are formed via sub-gridscale dynamical processes such as cumulus convection that involve multiple physical parameterizations. The cloud-feedback problem is related to mean tropical-bias problems in that both involve physical parameterizations in similar ways, and because it makes little sense to analyze the climate sensitivity of a simulated cloud climatology that does not resemble reality. However, the cloud-feedback problem involves some important additional considerations. We might hope that if we reduce mean tropical cloud and circulation biases, this is an indication that our GCM is physically realistic and will also respond correctly to climate perturbations. However, we do not yet have observations that reliably document the cloud changes that might accompany greenhouseinduced global warming and must test our models using the cloud responses to other climate perturbations of different space-time structure, such as ENSO and the seasonal cycle. Furthermore, GCMs will probably always have uncertain parameters with values tuned to best match the model simulations to current climate. Given such tuning, a good simulation of the current climate is no guarantee of an equally good simulation of a perturbed climate. On the other hand, climate sensitivity responds mainly to cloud radiative feedbacks important on global scales, while tropical rainfall and circulation biases can also strongly respond to geographically localized errors.

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In this section, we compare the tropical cloud responses of some AGCMs and some CRM-based models to CO2 doubling. These responses include changes in cloud fractional coverage, thickness, and altitude in a perturbed climate.

11.5.1. AGCM Cloud Response to Climate Warming We start with the NCAR and GFDL AGCMs discussed earlier in section 11.2.5. The climate sensitivity of both AGCMs coupled to slab-ocean models and run to equilibrium climates is between 2.5 and 3 K. However, this response involves a cancellation of substantially different cloud feedbacks in the two models. Figure 11.6 shows the change in low-latitude ω500-binned longwave and shortwave cloud forcing (LWCF, SWCF) for the two models associated with CO2 doubling. The two models show some qualitative similarities, including decreased LWCF (less anvil cloud) in strong ascent regions and more negative SWCF (more low cloud) in subsidence regions. However, the balance of LWCF and SWCF changes is different between the two models, such that CO2 doubling produces slightly negative net tropical cloud forcing in the NCAR model (a negative cloud feedback on greenhouse warming) but positive net tropical cloud forcing in the GFDL model. To complicate matters further, extratropical cloud radiative feedbacks cancel some of these differences in cloud response. Bony and DuFresne (2005) analyzed a large set of simulations of the climate effects of a transient CO2 doubling done with many CGCMs for the 2007 IPCC climate assessment. They found that intermodel differences in predicted SWCF change at CO2 doubling tend to be larger than those in LWCF change, and that the subsidence regimes especially contribute to the model spread. They concluded that the parameterization of the marine boundary-layer clouds that dominate the subsidence regimes is the most important uncertainty about modeling tropical cloud feedbacks on climate sensitivity.

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11.5.2. Cloud Feedbacks in a CRM-Simulated Mock-Walker Circulation CRM simulations of idealized tropical circulations are computationally easy compared to global CRM simulations. They can give hints about mechanisms of cloud feedback on climate sensitivity that may be relevant across the global Tropics, while avoiding the complex moist parameterizations in AGCMs. The idea is to force a plausible organized large-scale circulation internal to the CRM somewhat analogous to the tropical divergent circulation, which therefore may have similar modes of cloudradiation-convection-circulation feedback. Here we describe a few results, some a bit surprising, from CRM simulations of the climate sensitivity of an idealized two-dimensional Walker circulation above a mixed layer or “slab” ocean. This is previously unpublished joint work of the author and Peter Blossey. It was originally inspired by Grabowski et al.’s (2000) CRM simulations of an idealized Walker simulation forced by a sinusoidally varying SST. Our Walker circulation model is patterned on the idealized studies of Sobel et al. (2004) and Peters and Bretherton (2005). It should be loosely regarded as an east-west section along the equator, so atmospheric meridional heat transports and circulations are not simulated. In the model, a specified horizontally varying heat flux Soc n(x) out of the ocean forces the Walker circulation and corresponding horizontal SST gradients. The heat flux Soc n(x) is a loose proxy for the combined effects of all ocean dynamics and the meridional atmospheric heat flux divergence. To obtain realistic SST with realistic insolation, the domain averaged Soc n(x) should be negative to stand in for the 50–60 W m−2 heat export out of 15◦ S–15◦ N to higher latitudes that occurs on Earth (Trenberth and Caron 2001). We use a two-dimensional version of the SAM6.1 CRM (Khairoutdinov and Randall 2003) on a periodic domain of 1024 km (“small”) or 4096 km (“large”) horizontal extent L , with 2 km horizontal resolution and 64 vertical layers extending up into a damping region from 20–27 km in the stratosphere. We use the CAM3 radiation parameterization (interactive with clouds and water vapor) with time-invariant, horizontally uniform insolation of 425 W m−2. The Coriolis parameter is set to zero in the examples shown. The slab ocean is assumed to have a uniform depth D (20 m by default) and a vertically uniform temperature equal to the SST. At each x, the SST tendency is determined by the imbalance between the net surface-energy input and the ocean heat flux divergence, chosen to be Soc n = 35 + 50|x − L /2| W m−2. Thus, the “cold pool” will be centered on the edges of the domain and the “warm pool” will be centered at the domain center. The simulation with a 20 m-deep slab ocean converges to a statistical steady state within 5 years, and was run out 5 more years to look at the climatological equilibrium solution. Parallel integrations were performed with a default “1×” CO2 concentration of 365 ppm and a “2×” concentration of 730 ppm. We plan to soon write a more detailed article describing the various simulations that we have done; this chapter will only show preliminary highlights that demonstrate some challenges in

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simulating cloud feedbacks on circulation and climate sensitivity with a CRM, even in an idealized modeling framework. Figure 11.7 shows the 5–10-year mean small-domain horizontal distributions of the SST, precipitation, column relative humidity (CRH, defined as the ratio of the precipitable water in the atmospheric column to that which would saturate the column with respect to condensation of liquid water), and cloud liquid and ice water paths. In the 1× run, the cold pool SST is 298 K and the warm pool is 5 K warmer due to the 50 W m−2 reduction in ocean heat loss. Rainfall and both lower-tropospheric (liquid) and upper-tropospheric (ice) cloud condensate are localized in the central fifth of the domain. As predicted by idealized models (Sobel 2003; Sobel et al. 2004; Peters and Bretherton 2005), the SST is nearly uniform across this region. Surprisingly, but as also found by Sobel (2003), the maximum SST is at the edges of the rainy region rather than the domain center. This is due to the powerful role of cloud shading in the rainy region in regulating SST by reducing the net radiative flux into the ocean surface (Ramanathan and Collins 1991). Outside the precipitating region, the CRH decreases rapidly from 0.8 over the warmest SST to less than 0.1 over the coldest SST. This provides a very strong water vapor feedback reinforcing the horizontal SST gradients in this region. In the 2× simulation, there is a nearly horizontally uniform 1.9 K increase in SST, which is rather similar to the 1.7 K increase one would expect from an atmosphere with no clouds, a moist-adiabatic lapse rate, and no impact of CO2 on relative humidity.

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This is due to a near-cancellation of both positive and negative feedbacks. The region of rainfall and clouds slightly narrows and intensifies in 2×, causing a slight drying of CRH on the wings of the convection and more condensate within the rainy region (which enhances albedo in this region). These are negative feedbacks on climate sensitivity. The drying of CRH occurs because when the convective region narrows, its edge shrinks back away from any fixed grid column, which allows for more subsidence to dry out the air as it traverses between the convective edge (where CRH is regulated by deep convection and is rather insensitive to climate change) and the column of interest. This is a weak version of Lindzen et al.’s (2001) water vapor iris effect (to be distinguished from their controversial cirrus cloud iris effect). Lindzen et al.’s analysis is generally regarded as flawed and exaggerated (e.g., Hartmann and Michelsen 2002), but their core notion that there may be significant feedbacks between fractional area of tropical deep convection and climate is borne out by these Walker simulations. However, there are other feedbacks associated with the vertical structure of the convective clouds. Figure 11.8 shows vertical distributions of domain-mean equilibrium temperature, relative humidity, cloud fraction, and nonprecipitating cloud condensate for these two simulations. The temperature profile shows “lapse-rate feedback”—larger 1×-2× changes in the upper troposphere, roughly following a warmer moist adiabat. The relative humidity, cloud fraction, and condensate profiles all show an upward offset from 1× to 2× that follows the rise of the mean isotherms, a generalized manifestation of the fixed anvil temperature hypothesis of Hartmann and Larson (2002). They show little other change except for the slight drying and reduction in cloud fraction in the lower troposphere and more upper-tropospheric ice in cirrus cloud. The net effect of the vertical structure change from 1× to 2 × is a weakly positive climate feedback—the isothermal rise of cloud tops in 2×, which increases the cloud greenhouse effect—that counteracts the weak negative feedbacks from the narrowing of the convective region. Otherwise this 1×-2× simulation pair matches the default clear-sky paradigm rather well and has a similar climate sensitivity. The overturning mass transport in the Walker circulation also slightly decreases from 1× to 2× in accordance that this transport scales with the ratio of net tropospheric radiative cooling to tropospheric column water vapor (Knutson and Manabe 1995). Clausius-Clapeyron arguments suggests that water vapor rises rapidly in a warmer climate, but net atmospheric radiative cooling is constrained by global energy balance to rise much less rapidly. This simulation pair has little boundary-layer cloud over the cooler SSTs, but simulations with large domains, higher vertical and horizontal resolution in the lower troposphere, and/or plausible changes to the subgrid turbulence scheme that inhibits vertical mixing have considerably more shallow cumulus cloud in the cool-SST regions. For example, Fig. 11.9 shows 5–10-year mean vertical section of the streamfunction and condensate distributions for 1024 and 4096 km-wide domains, showing much more extensive shallow cumuli and a broader ascent region in the latter run than in the former.

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We find that more extensive simulated boundary-layer cloud often reduces the simulated climate sensitivity dramatically because the boundary-layer cloud tends to become more extensive as the climate warms, as suggested by Miller (1997) and others. For instance, the climate sensitivity of the 4096 km simulation is 0.96 K, only half as large as for the 1024 km run, which has much less boundary-layer cloud. Thus, details of parameterized physics that are poorly resolved, inappropriately idealized, or truly uncertain can strongly affect predictions of climate sensitivities even in this highly controlled CRM framework, just as with CGCM simulations of our real planet. Unexpectedly, even the depth of the slab ocean can affect the simulated climate sensitivity. In equilibrium, the SST distribution should satisfy a balance between ocean surface energy gain and the specified ocean heat removal. The slab ocean depth does not enter this balance, and hence should not affect the equilibrium SST. Hence, we tried simulations with shallower slab oceans to speed up the approach to equilibrium. These simulations were carried out with an older model configuration with a different radiation scheme (including different specifications of effective radii for cloud water and ice), and used slightly different domain-mean heat fluxes out of the ocean so that they maintain realistic domain-mean SSTs, but we do not believe this has an important effect on the results shown in Fig. 11.10. In Fig. 11.10a, the simulation pair with

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D = 20 m, the SSTs smoothly asymptote in 5 years to their equilibrium values 1.3 K apart. In Fig. 11.10c, a simulation pair with the slab ocean depth D = 2 m, significant variability of SST and circulation on 50–100-day periods can be seen due to the rapid response of SST to cloud and circulation changes. Thus, although equilibrium is indeed more quickly attained, a lengthy averaging period of order 5–10 years is necessary to accurately measure the mean SST change. However, the 2 m simulation does appear to have the same climate sensitivity of 1.3 K as the 20 m simulation. Fig. 11.10b shows an intermediate simulation pair with a 5 m-deep slab ocean. These simulations have even larger low-frequency coupled oscillations in the SST and width of the rainy region that rectify onto their mean climates. Even after time-averaging, the implied climate sensitivity for this case was much lower—only 0.4 K! Thus, even the transient behavior of the system—which is dependent on the assumed domain geometry, method of forcing, lack of seasonal cycle or extratropical interactions, etc.—can profoundly affect the climate sensitivity in these simulations.

11.5.3. DARE Aqua-Planet Climate Sensitivity Over a Slab Ocean As a last perspective on tropical cloud feedbacks, this time applying global CRMbased methods, some DARE climate sensitivity results are presented. This previously unpublished joint work of Matt Peters and the author again uses the SAM6.1 CRM. This CRM does not support a full global grid, orography, or a realistic land-surface parameterization. Hence, we use a pair of aqua-planet simulations coupled to a 20 mdeep slab ocean (with no ocean heat transport) on a beta-plane between 50◦ S–50◦ N with a DARE factor γ = 20 run with 1× and 2× CO2 concentrations. Both simulations are run for 5 years. They take around 3 years to equilibrate, and we use averages over years 3.5–5 in the figures and for climate sensitivity calculations. Because the simulations are not truly global and have no potential for snow/ice-albedo feedback or polar amplification of surface temperature changes, we should expect the domain-mean temperature change to be smaller than if a truly global domain were simulated. Figure 11.11 shows zonally-averaged SST and rainfall, and TOA LWCF and SWCF for the two simulations. They show a weak climate sensitivity to CO2 doubling of 0.6 K in domain-mean SST, driven by strong negative cloud radiative feedbacks, especially in SWCF in the subsidence regions. These are associated with more condensate but less cloud fraction in the warmer climate. As in the idealized Walker circulation simulations, ITCZ rainfall narrows and intensifies in the warmer climate, but this effect is very slight. In both simulations, SWCF is much stronger than LWCF, even in the ITCZ, in contrast with the real Earth. Experience with this CRM suggests that the radiation parameterization, the microphysical parameterizations, and to a lesser extent the DARE acceleration may all contribute to this apparent bias, which somewhat diminishes our trust in the simulated CRM-based cloud feedbacks.

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More simulations of this type with different global CRM-based models will help us understand which features of the cloud response are most robust. The Japanese global CRM group has assessed climate sensitivity and cloud feedbacks using a pair of global 30-day aqua-planet CRM simulations with specified SSTs differing by 2 K everywhere (Miura et al. 2005). As in our DARE simulation, they find pronounced negative cloud feedbacks on climate sensitivity, driven strongly by extratropical SWCF changes. However, they simulate a broadening of the mean ITCZ rainfall belt in a warmer climate, in contrast to our results. A similar study using 3.5-year simulations with the superparameterized CAM3-SP and realistic Earth geography also finds strongly negative cloud feedbacks in both low and high latitudes, dominated by SWCF changes,

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but little broadening of the tropical rainfall belts (Wyant et al. 2006b). The common thread in all these CRM-based simulations is negative climate feedbacks from the SWCF associated with thicker, more extensive low-altitude clouds, causing all three models to have climate sensitivities at the bottom edge of the range found in current AGCMs (Wyant et al. 2006b). Over the short run, even global CRMs will not allow us to more confidently predict climate sensitivity, just as they do not fully solve tropical rainfall bias problems. They are better regarded as a different modeling perspective on the problem, with formidable strengths but some weaknesses compared to conventional AGCMs.

11.6. Summary Simulations of tropical climate by AGCMs and CGCMs are improving, albeit rather slowly. Ever-finer model grids are allowing more realistic parameterization approaches to be developed for all types of cloud systems. Cumulus and cloud microphysical parameterizations in GCMs continue to benefit from well-designed comparisons with CRMs and ongoing field studies. Progress has proved challenging and often slow over the last two decades, but enough smart minds working together on improving these parameterizations in a coupled-model setting should be able to diminish tropical bias problems in AGCMs and CGCMs considerably over the next decade, leading to better simulations of the mean seasonal cycle, ENSO, MJO, and other tropical circulations. CRMs and strategies for applying them to global-scale problems promise to put the simulation of clouds on a physically sounder footing and push the parameterization problems down to the air-parcel scale. CRM-based global simulation strategies such as superparameterization, DARE, and even full global CRMs are becoming more affordable, and already show promise for such simulation challenges as the diurnal cycle of continental convection and the MJO. Current CRM microphysical parameterizations need considerable further refinement to skillfully predict the radiative properties and feedbacks of the full global suite of cloud systems. Hence, CRM-based simulation of climate sensitivity to greenhouse gases is still quite sensitive to the parameterizations, idealizations, and resolution employed, and will ensure that tropical cloud systems will remain a stimulating challenge for the next generation of numerical modelers.

Acknowledgments Peter Blossey, Matt Wyant, and Matt Peters kindly provided new figures and research results for this chapter. Marat Khairoutdinov developed and provided the SAM CRM

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used for CRM simulations presented in this chapter. Funding from NASA, NSF, and NOAA supported previously unpublished research presented in this chapter. Adam Sobel and an anonymous reviewer provided valuable feedback on a draft of this chapter.

References AchutaRao, K., and K. R. Sperber, 2002: Simulation of the El Niño Southern Oscillation: Results from the Coupled Model Intercomparison Project. Climate Dyn., 19, 191–209. Arakawa, A., and W. H. Schubert, 1974: Interaction of a cumulus cloud ensemble with the largescale environment: Part I. J. Atmos. Sci., 31, 674–701. Blossey, P. N., C. S. Bretherton, J. Cetrone, and M. Khairoutdinov, 2007: Cloud-resolving model simulations of KWAJEX: Model sensitivities and comparisons with satellite and radar observations. J. Atmos. Sci., 64, 1488–1508. Bony, S., J.-L. Dufresne, H. Le Treut, J.-J. Morcrette, and C. Senior, 2004: On dynamic and thermodynamic components of cloud changes. Climate Dyn., 22, 71–86. Bony, S., and J.-L. Dufresne, 2005: Marine boundary layer clouds at the heart of cloud feedback uncertainties in climate models. Geophys. Res. Lett., 32, L20806, doi:10.1029/GL2005023851. Cess, R. D., and coauthors, 1989: Interpretation of cloud-climate feedback as produced by 14 atmospheric general circulation models. Science, 245, 513–516. Danabasoglu, G., and coauthors, 2006: Diurnal ocean-atmosphere coupling. J. Climate, submitted. Davey, M., and coauthors, 2002: STOIC: A study of coupled model climatology and variability in tropical ocean regions. Climate Dyn., 18, 403–420. Derbyshire, S. H., I. Beau, P. Bechtold, J.-Y. Grandpeix, J.-M. Piriou, J.-L. Redelsperger, and P. M. M. Soares, 2004: Sensitivity of moist convection to environmental humidity. Quart. J. Roy. Meteor. Soc., 130, 3055–3079. Deser, C., A. Capotondi, R. Saravanan, and A. S. Phillips, 2006: Tropical Pacific and Atlantic climate variability in CCSM3. J. Climate, submitted. Emanuel, K. A., 1994: Atmospheric Convection. New York: Oxford University Press, 580 pp. Gordon, C., and coauthors, 2000: The simulation of SST, sea ice extent, and ocean heat transport in a version of the Hadley Centre coupled model without flux adjustments. Climate Dyn., 16, 147–168. Grabowski, W., 2001: Coupling cloud processes with the large-scale dynamics using the cloudresolving convection parameterization (CRCP). J. Atmos. Sci., 58, 978–997. Grabowski, W. W., J.-I. Yano, and M. W. Moncrieff, 2000: Cloud-resolving modeling of tropical circulations driven by a large-scale SST gradient. J. Atmos. Sci., 57, 2022–2040. Grabowski, W. W., and coauthors, 2006: Daytime convective development over land: A model intercomparison based on LBA observations. Quart. J. Roy. Meteor. Soc., 28, 317–344. Guilyardi, E. P., and coauthors, 2004: Representing ENSO in coupled models: The dominant role of the atmospheric component. J. Climate, 17, 4623–4629. Hartmann, D. L., and K. Larson, 2002: An important constraint on tropical cloud-climate feedback. Geophys. Res. Lett., 29, 1951–1954. Hartmann, D. L., and M. L. Michelsen, 2002: No evidence for iris. Bull. Amer. Meteor. Soc., 83, 249–254.

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Time:

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328 | Christopher S. Bretherton Hayashi, Y.-Y., and A. Sumi, 1986: The 30-40 day oscillation simulated in an “aqua-planet” model. J. Meteor. Soc. Japan, 64, 451–467. Hendon, H. H., and M. L. Salby, 1994: The life cycle of the Madden-Julian oscillation. J. Atmos. Sci., 51, 2225–2237. Inness, P. M., and D. Gregory, 1997: Aspects of the intraseasonal oscillation simulated by the Hadley Centre Atmospheric Model. Climate Dyn., 13, 441–458. IPCC (Intergovernmental Panel on Climate Change), 2001: Climate Change 2001: The Scientific Basis, J. T. Houghton and co-eds., Cambridge University Press, 944 pp. Jochum, M., R. Murtugudde, R. Ferrari, and P. Malanotte-Rizzoli, 2005: The impact of horizontal resolution on the tropical heat budget in an Atlantic Ocean model. J. Phys. Oceanogr., 18, 841–851. Khairoutdinov, M. F., C. DeMott, and D. A. Randall, 2005: Simulation of the atmospheric general circulation using a cloud-resolving model as a super-parameterization of physical processes. J. Atmos. Sci., 62, 2136–2154. Khairoutdinov, M. F., and D. A. Randall, 2001: A cloud-resolving model as a cloud parameterization in the NCAR Community Climate System Model: Preliminary results. Geophys. Res. Lett., 28, 3617–3620. Khairoutdinov, M. F., and D. A. Randall, 2003: Cloud resolving modeling of the ARM Summer 1997 IOP: Model formulation, results, uncertainties, and sensitivities. J. Atmos. Sci., 60, 607–625. Kiehl, J. T., 1998: Simulation of the tropical Pacific warm pool with the NCAR Climate System Model. J. Climate, 11, 1342–1355. Kiehl, J. T., and P. R. Gent, 2004: The Community Climate System Model, Version Two. J. Climate, 17, 3666–3682. Kiladis, G. N., and K. M. Weickmann, 1992: Circulation anomalies associated with tropical convection during northern winter. Mon. Wea. Rev., 120, 1900–1921. Kirtman, B. P., 1997: Oceanic Rossby wave dynamics and the ENSO period in a coupled model. J. Climate, 10, 1690–1704. Klein, S. A., and C. Jakob, 1999: Validation and sensitivities of frontal clouds simulated by the ECMWF model. Mon. Wea. Rev., 127, 2514–2531. Knutson, T. R., and S. Manabe, 1995: Time-mean response over the tropical Pacific to increased CO2 in a coupled ocean-atmosphere model. J. Climate, 8, 2181–2199. Knutson, T. R., and S. Manabe, 1998: Model assessment of decadal variability and trends in the tropical Pacific ocean. J. Climate, 11, 2273–2296. Kuang, Z., P. N. Blossey, and C. S. Bretherton, 2005: A DARE approach for 3D cloud resolving simulations of large scale atmospheric circulation. Geophys. Res. Lett., 32, L02809, doi: 10.1029/2004GL021024. Kuang, Z., and C. S. Bretherton, 2006: A mass flux scheme view of a high-resolution simulation of transition from shallow to deep cumulus convection. J. Atmos. Sci., accepted. Large, W. G., and G. Danabasoglu, 2006: Attribution and impacts of upper-ocean biases in CCSM3. J. Climate, submitted. Latif, M., and coauthors, 2001: ENSIP—The El Niño Simulation Intercomparison Project. Climate Dyn., 18, 255–276. Lindzen, R. S., M.-D. Chou, and A. Y. Hou, 2001: Does the earth have an adaptive infrared iris? Bull. Amer. Meteor. Soc., 82, 417–432. Madden, R. A., and P. R. Julian, 1971: Detection of a 40–50 day oscillation in the zonal wind in the tropical Pacific. J. Atmos. Sci., 28, 702–708.

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Challenges in Numerical Modeling | 329 Madden, R. A., and P. R. Julian, 1972: Description of global-scale circulation cells in the Tropics with a 40–50 day period. J. Atmos. Sci., 29, 1109–1123. Madden, R. A., and P. R. Julian, 1994: Observations of the 40–50 day tropical oscillation—A review. Mon. Wea. Rev., 122, 814–837. Maloney, E. D., and D. L. Hartmann, 1998: Frictional moisture convergence in a composite life cycle of the Madden-Julian oscillation. J. Climate, 11, 2387–2403. Mechoso, C. R., and 19 coauthors, 1995: The seasonal cycle over the tropical Pacific in coupled ocean-atmosphere general circulation models. Mon. Wea. Rev., 123, 2825–2838. Miller, R. L., 1997: Tropical thermostats and low cloud cover. J. Climate, 10, 409–440. Miura, H., H. Tomita, T. Nasuno, S. Iga, M. Satoh, and T. Matsuno, 2005: A climate sensitivity test using a global cloud resolving model under an aquaplanet condition. Geophys. Res. Lett., 32, L19717, doi:10.1029/2005GL023672. Moorthi, S., and M. J. Suarez, 1992: Relaxed Arakawa-Schubert: A parameterization of moist convection for general circulation models. Mon. Wea. Rev., 120, 978–1002. Numaguti, A., R. Oki, K. Nakamura, K. Tsuboki, N. Misawa, T. Aisai, and Y.-M. Kodama, 1995: 4–5 day-period variation and low-level dry air observed in the equatorial western Pacific during the TOGA-COARE IOP. J. Meteor. Soc. Japan, 73, 267–290. Pauluis, O., D. Frierson, S. Garner, I. Held, and G. Vallis, 2006: The hypo-hydrostatic rescaling and its impacts on atmospheric convection. Theor. Comp. Fluid Dyn., submitted. Peters, M. E., and C. S. Bretherton, 2005: A simplified model of the Walker circulation with an interactive ocean mixed layer and cloud-radiative feedbacks. J. Climate, 18, 4216–4234. Ramanathan, V., R. D. Cess, E. F. Harrison, P. Minnis, B. R. Barkstrom, E. Ahmad, and D. Hartmann, 1989: Cloud-radiative forcing and climate: Results from the Earth Radiation Budget Experiment. Science, 243, 57–63. Ramanathan, V., and W. Collins, 1991: Thermodynamic regulation of ocean warming by cirrus clouds deduced from observations of the 1987 El Niño. Nature, 351, 27–32. Reynolds, R. W., and T. M. Smith, 1994: Improved global sea surface temperature analyses using optimum interpolation. J. Climate, 7, 929–948. Schopf, P. S., and M. J. Suarez, 1988: Vacillations in a coupled ocean-atmosphere model. J. Atmos. Sci., 45, 549–566. Slingo, J. M., and coauthors, 1996: Intraseasonal oscillations in 15 atmospheric general circulation models: Results from an AMIP diagnostic subproject. Climate Dyn., 12, 325–357. Sobel, A. H., 2003: On the coexistence of an evaporation minimum and a precipitation maximum in the warm pool. J. Climate, 16, 1003–1009. Sobel, A. H., C. S. Bretherton, H. Gildor, and M. Peters, 2004: Convection, cloud-radiative feedbacks and thermodynamic ocean coupling in simple models of the Walker circulation. In Ocean-atmosphere interaction and climate variability, ed., Chunzai Wang, Shang-Ping Xie, and Jim Carton, Amer. Geophys. Union Geophysical Monograph 147, 393–405. Soden, B. J., A. J. Broccoli, and R. S. Hemler, 2004: On the use of cloud forcing to estimate cloud feedback. J. Climate, 17, 3661–3665. Sperber, K., and coauthors, 2003: Tropical rainfall biases in coupled models. Lawrence Livermore National Laboratory Report UCRL-MI-153206, 41 pp. Tomita, H., H. Miura, S. Iga, T. Nasumo, and M. Satoh, 2005: A global cloud-resolving simulation: Preliminary results from an aquaplanet experiment. Geophys. Res. Lett., 32, L08805, doi:10.1029/2005GL022459. Trenberth, K. E., and J. M. Caron, 2001: Estimates of meridional atmosphere and ocean transports. J. Climate, 14, 3433–3443.

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330 | Christopher S. Bretherton Waliser, D. E., S. Schubert, A. Kumar, K. Weickmann, and R. Dole, 2003: Modeling, Simulation and Forecasting of Subseasonal Variability. Technical Report Series on Global Modeling and Data Assimilation. NASA/CP-2003-104606, Vol. 25. Wang, W., and M. E. Schlesinger, 1999: The dependence on convective parameterization of the tropical intraseasonal oscillation simulated by the UIUC 11-layer atmospheric GCM. J. Climate, 12, 1423–1457. Weickmann, K. M., 1983: Intraseasonal circulation and outgoing longwave radiation modes during Northern Hemisphere winter. Mon. Wea. Rev., 111, 1838–1858. Wheeler, M., G. N. Kiladis, and P. J. Webster, 2000: Large-scale dynamical fields associated with convectively-coupled equatorial waves. J. Atmos. Sci., 57, 613–640. Wu, X., and M. W. Moncrieff, 2001: Long-term behavior of cloud systems in TOGA COARE and their interactions with radiative and surface processes. Part III: Effects on the energy budget and SST. J. Atmos. Sci., 58, 1155–1168. Wyant, M. E., C. S. Bretherton, J. T. Bacmeister, J. T. Kiehl, I. M. Held, M. Zhao, S. A. Klein, and B. A. Soden, 2006a: A comparison of tropical cloud properties and responses in GCMs using mid-tropospheric vertical velocity. Climate Dyn., in press. Wyant, M. E., M. F. Khairoutdinov, and C. S. Bretherton, 2006b: Climate sensitivity and cloud response of a GCM with a superparameterization. Geophys. Res. Lett., in press. Xie, P., and P. A. Arkin, 1997: Global precipitation: A 17-year monthly analysis based on gauge observations, satellite estimates, and numerical model outputs. Bull. Amer. Meteor. Soc., 78, 2539–2558. Yang, G.-Y., and J. Slingo, 2001: The diurnal cycle in the Tropics. Mon. Wea. Rev., 129, 784–801. Yu, J.-Y., and C. R. Mechoso, 2001: A coupled atmosphere-ocean GCM study of the ENSO cycle. J. Climate, 14, 2329–2350. Zhang, G. J., and N. A. McFarlane, 1995: Sensitivity of climate simulations to the parameterizations of cumulus convection in the Canadian Climate Centre general circulation model. Atmos.Ocean, 33, 407–446. Zhang, M., and coauthors, 2005: Comparing clouds and their seasonal variations in 10 atmospheric general circulation models with satellite measurements. J. Geophys. Res., 110, D15S02, doi:10.1029/2004JD005021.

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Chapter 12

Challenges to Our Understanding of the General Circulation: Abrupt Climate Change Richard Seager and David S. Battisti

12.1. Introduction About 14,700 years ago (14.7 kyr BP), toward the end of the last ice age, the climate warmed dramatically and abruptly around the North Atlantic—by as much as the difference between full glacial and interglacial conditions—in no more than a decade or two. This is all the more remarkable because it occurred in the presence of massive ice sheets and continuation of the albedo forcing that presumably had been helping maintain glacial conditions up to that point. But it was not to last. Sometime just after 13 kyr BP this Bølling-Allerød warm period ended as climate first cooled, and then abruptly cooled, into the so-called Younger Dryas. As near-glacial conditions returned, glaciers advanced in Europe, and the forests that had established themselves in the preceding warm epoch died. The Younger Dryas ended with a second abrupt warming that occurred over only a decade or so and that shifted temperatures back to those of the Holocene and of today. The idea that the climate system goes through such abrupt shifts did not take the climate research community by storm but dribbled into acceptance in the 1980s and the early 1990s. Only when duplicate ice cores said the same thing and the evidence was found in multiple indicators within the ice—oxygen isotopes, dust concentrations, snow accumulation, and so on—and could be correlated with terrestrial and marine records did acceptance that abrupt climate change was a reality sink in. This gradual acceptance is telling. When Hays et al. (1976) showed just how well climate records from deep-sea cores could be matched to orbital cycles, it was deeply satisfying: the gradual waxing and waning of the great ice sheets could be explained by equally gradual changes in the distribution of delivery of solar radiation to the Earth’s surface. Insolation over high northern latitudes was deemed to be particularly important, with reduction in summer leading to retention of winter snow and ice sheet growth. All that remained was to show exactly how the climate system accomplished the necessary links.

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Three decades later we are still far from understanding how orbital changes are converted into ice-sheet growth and decay. While this is testimony enough to our limited understanding of the climate system and general circulation, abrupt climate change is now the star witness. In this case, the climate changes occurred not only abruptly but, apparently, in the absence of any external forcing. The lack of any theory for how such changes could occur helps explain the slow acceptance of what the data were saying. In the two decades since the discovery of abrupt change, two advances have been made. First, the spatial pattern of abrupt climate change has been better delimited, and it is now known that these events occurred essentially synchronously across much of the Northern Hemisphere (including the northern Tropics) within the atmosphere, the surface ocean, and the deep ocean. Abrupt changes are not found in the ice records from Antarctica, and the Southern Hemisphere remains a question because of limited data. These spatial patterns place some severe constraints on proposed mechanisms of abrupt climate change. Second, mechanisms have been advanced that revolve around the thermohaline circulation (THC). Broecker et al. (1985) were perhaps the first to suggest that rapid warmings and coolings of climate around the North Atlantic were caused by rapid switchings on and off of North Atlantic Deep Water formation with “on” states being associated with transport of warm waters into the subpolar North Atlantic. Despite difficulties explaining the paleoclimate record of abrupt changes with the THC theory, no competing idea has yet been offered. The paleoclimate record poses many challenges to our understanding of the general circulation of the atmosphere and ocean, of which explaining abrupt change is just one. How orbital changes cause ice-sheet growth and decay remains a major unsolved problem but will probably be solved as it becomes computationally feasible to integrate coupled atmosphere-ocean general circulation models (GCMs) through orbital cycles. It is only in recent years that time snapshot simulations of the Last Glacial Maximum (LGM) with coupled GCMs have become commonplace (Shin et al. 2003; Hewitt et al. 2003). As suggested by Ruddiman and McIntyre (1981), it is probably a solar radiation re-distribution that allows increased winter export of tropical moisture into higher latitudes, there to fall as snow, and reduced summer insolation at high latitudes, which allows the snow to be retained until next winter, causing ice sheets to grow. Further back in time, evidence of equable climates poses an enormous challenge to our understanding of the general circulation and the climate system. A particularly interesting example is the Eocene, when temperatures of high-latitude northern continental interiors remained above freezing in winter, allowing crocodilians to survive in subpolar regions. More recently, the greening of the Sahara in the mid-Holocene, when the worlds most impressive desert essentially became a moist savanna, remains a fascinating unexplained phenomenon. Certainly it was triggered by orbital changes that increase summer insolation over the Northern Hemisphere, but the apparently abrupt onset and demise of the African Humid Period (deMenocal et al. 2000), and the fact that

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other Northern Hemisphere monsoon regions show less impressive changes, suggest a nonlocal coupling between deserts and monsoons on paleoclimate time scales that is waiting to be elucidated. Baffling though these problems are, the focus in this chapter will be on abrupt climate changes in glacial times. We will advance a case for an important role for the Tropics in climate change. To begin, we will review the evidence for abrupt climate change and summarize the current knowledge of the spatial footprint. We will also review evidence for the relationship between abrupt changes in surface climate and deep ocean circulation. This will form the basis for a critique of the THC theory of abrupt change before we advance a case for a mechanism that involves global atmosphere-ocean coupling and an active role for the Tropics. This mechanism will be as sketchy as that of Broecker et al. (1985) but hopefully will inspire some future investigations.

12.2. Abrupt Climate Change in Polar Ice Cores The characteristics of abrupt climate change as recorded in polar ice cores from Greenland and Antarctica have been well described by Wunsch (2003). The problem can be seen in Fig. 12.1, which is reproduced in Wunsch’s paper from the original of Blunier and Brook (2001). The δ 18O content of ice in the Greenland core shows frequent rapid drops and even more dramatic increases throughout the last glaciation. The Younger Dryas appears to have been the most recent of these. The δ 18 O content of ice is supposed to be a proxy for the local temperature when the snow fell. Cuffey et al. (1995) used borehole temperature measurements from the Greenland ice core, and models of heat and ice flow, to infer that during the glacial period an 0.33 increase in δ 18O corresponded to a 1◦C increase in temperature. The δ 18O content in the ice core is, however, also influenced by the isotopic composition of the water that was evaporated and the change in composition due to evaporation/condensation cycling during transport to the ice core. Either way, the record shows dramatic climate changes as much as two-thirds of the size of the difference between full glacial and interglacial conditions. Using the Cuffey et al. relation, the inferred temperature changes are as large as 15◦–20◦C. These changes occurred in decades. The Antarctic cores do not show any such rapid climate changes. Wunsch (2003) concluded that the two are uncorrelated on the millennial time scale (but correlated on the glacial-interglacial time scale), although others have suggested more complex relationships between the two (Roe and Steig 2004). The time scale of the Antarctic core was adjusted to bring the methane records, also shown in Fig. 12.1, into agreement on the basis that methane is a well-mixed gas. The fact that this can be done indicates that the source regions for methane, especially in tropical wetlands, were affected by the rapid climate changes (Brook et al. 1999) and that the changes were not simply regional North Atlantic events.

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Statistical analysis presented by Wunsch shows to be true what the eye perceives, that the Greenland climate is bimodal, switching between two preferred states. The histograms of δ 18O in two Greenland cores (GRIP and GISP2) show bimodality, whereas the Antarctic core (Byrd) is unimodal, though with a long tail. This is striking evidence of nonlinearity and threshold behavior in the climate system, at least in Greenland. Wunsch suggests that this arises from switching between two different evaporative source regions for the water that falls as snow onto the Greenland core location, and can reproduce similar behavior with a two-source model, some noise, and some simple rules for transitioning between sources. If the two-source explanation is true, it implies that the atmospheric circulation is capable of switching between circulation regimes with different trajectories of atmospheric water vapor and condensate between the oceans and Greenland. However, it does appear that the original interpretation of the δ 18O in terms of temperature has some validity. Using an entirely independent methodology based on the temperature dependence of the diffusion of gases within the ice core, Severinghaus and Brook (1999) and Severinghaus et al. (1998) deduced a warming at the ice surface of as much as 9◦C

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in a few decades at the transition into the Bølling-Allerød warm period, an amount consistent with Cuffey et al.’s oxygen isotope paleothermometry. The kind of climate change seen in the Greenland ice cores almost perfectly fits Lorenz’s description of an “almost intransitive” system: “a particular solution extending over an infinite time interval will possess successive very long periods with markedly different sets of statistics” (Lorenz 1968). Lorenz developed this concept of climate change, which seems more relevant today than ever, even before the ice-core data was available! In summary, abrupt climate changes, consisting of coolings followed by rapid coolings and then abrupt warmings, punctuated the entire glacial period at Greenland, but no such thing happened in Antarctica. At Greenland, there is enticing evidence of nonlinearity of the climate system with thresholds and switches. The Younger Dryas is the most recent of these abrupt changes. Much of the discussion to follow on the spatial and temporal structure of abrupt changes will concern the Younger Dryas (which we have labeled in Figure 12.1) because this is the best observed, but it is anticipated that the descriptions are generally valid for all the abrupt changes.

12.3. Abrupt Climate Change in the Surface Atlantic Ocean Were abrupt climate change limited to Greenland, it would not pose too much of a challenge to our understanding of the general circulation. It would be easy to imagine some alteration of the circulation and movement of heat by stationary and transient eddies that could accomplish the observed effects. However, as time has gone by, changes elsewhere in the global Tropics and Northern Hemisphere have come to light, and these changes appear to be synchronous with those in Greenland. The first evidence was from proxies for sea surface temperature (SST) preserved in ocean bottom sediment cores. Bond et al. (1993) claimed that when Greenland was cold, the subpolar North Atlantic SSTs were low, although low sedimentation rates made the time resolution of the core too imprecise for easy cross-comparison with the Greenland record. Sachs and Lehman (1999) presented records from the Bermuda Rise, a region of high sedimentation that allows for good temporal resolution. They derived an SST from the alkenone unsaturation ratio measured in the sediments.1 The agreement between the alkenone-derived SST and the Greenland ice core record through numerous abrupt climate changes during the last glacial period is startling (Fig. 12.2): whenever Greenland is cold, the subtropical North Atlantic Ocean is cold too. While it is true that this conclusion partly depends on the fact that the age control on the Bermuda Rise record was fitted in order to maximize the correspondence between the two records, the fact that it is possible to get such a high correspondence testifies to a real link between climate changes in these two locations. The typical warming here from stadial to interstadial was about 3◦C. It should be noted that the high sedimentation rate on the Bermuda Rise

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Figure 12.2. Reconstructed sea-surface temperatures based on alkenone unsaturation ratios in sediments on the Bermuda Rise for the period from 30,000 to 60,000 years ago (gray line, left axis), plotted together with Greenland δ 18O (black line, right axis) on the GISP2 ice-core timescale. From Sachs and Lehman (1999). (Reproduced with permission from the American c 1999.) Association for the Advancement of Science


occurs because of transport of material—and hence alkenones—to the site, which can effect interpretation of the record (Ohkouchi et al. 2002). The Cariaco Basin just north of Venezuela is an invaluable treasure trove of climate records. The waters above are anoxic, which prohibits mixing of the sediments by organisms living within. This, and very high sedimentation rates because of the proximity of the continent, has created a record with close to annual resolution extending back into the last ice age. Lea et al. (2003) used Mg/Ca ratios in Cariaco Basin sediments to reconstruct SSTs during the last deglaciation. They found an abrupt warming at the transition into the Bølling-Allerød warm period, an abrupt cooling of as much as 4◦C at the beginning of the Younger Dryas, and an abrupt warming at its termination. This result is consistent with Sachs and Lehman and Bond and, despite necessary concern about the proxy indicators, improves confidence that the entire North Atlantic surface ocean cooled dramatically, by a few to several degrees Celsius during Greenland stadials.

12.4. Abrupt Climate Change Away from the North Atlantic 12.4.1. The Caribbean, Tropical Atlantic and Africa Using other proxies, the Cariaco Basin record off Venezuela has also been used to infer climate changes other than in surface ocean temperature. Hughen et al. (1996, 1998)

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showed that the sediments reveal transitions into and out of the Younger Dryas as abrupt as those observed in Greenland and interpreted them as abrupt changes in trade wind strength. Peterson et al. (2000) showed that almost all of the abrupt jumps seen in Greenland between 60 kyr BP and 25 kyr BP (and numbered in Fig. 12.1) are also seen in the Cariaco Basin record as shifts in sediment reflectance and major-element chemistry. These sediment characteristics record changes in the rate of riverine influx from South America north of the Amazon Basin, i.e., the balance of precipitation and evaporation in that region, suggesting that during Greenland cold stadials, northern regions of South America received less rain than now. The most obvious way in which this could happen is through a southward shift of the Intertropical Convergence Zone (ITCZ). Recent results from speleothems (calcium carbonate deposits in caves, more commonly known as stalactites and stalagmites) have indeed indicated that at times when northern South America was drier, northeast Brazil was wetter (Wang et al. 2004).2 These records make clear that abrupt climate changes impacted the tropical Atlantic region with a southward-shifted ITCZ during cold stadials in Greenland. There are numerous records that record the glacial and Holocene hydroclimate of tropical Africa. These come either from sediments in African lakes or from sediments offshore that record the rate of input of terrigenous material from the continent, assumed to be a proxy for precipitation over the land. Gasse (2000) reviews many of these records from the last glacial maximum and the deglaciation, and found rapid increases in lake levels in tropical West Africa at the onset of the Bølling-Allerød warm interval and at the end of the Younger Dryas. This is consistent with results obtained from sediments offshore by the mouth of the Niger River that show a dry Younger Dryas across West Africa (Lezine et al. 2005). Adegbie et al. (2003) examined a sediment core off Cameroon in the Gulf of Guinea and deduced not just a weaker West African monsoon during the Younger Dryas but dry conditions correlating with Greenland stadials throughout the last 30,000 years of the glacial period.

12.4.2. Northern Extratropics The Younger Dryas abrupt climate change was first identified in pollen records in northern Europe and is named after a cold-tolerant plant that reestablished itself during this time. Throughout Europe mountain glaciers advanced, as in Scandinavia and the Alps, or they reformed, as in the British Isles (see references in Denton et al. [2005]). Glacial advances require adequate snow in winter and cool enough summers for accumulation to exceed ablation. In North America, the record is mixed. Shuman et al. (2002) show that across eastern North America vegetation changed dramatically at the beginning and end of the Younger Dryas, but that these cannot simply be accounted for by a cooling, in part because summers may have been warmer as a consequence of increased insolation. In western North America, alpine glaciers advanced during

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the Younger Dryas but with notable exceptions in the Sierra Nevada and around Mt. Rainier (Licciardi et al. 2004). Since precipitation patterns are highly spatially variable, the simplest explanation for glacial advances in Europe and North America would be a colder climate, including summers. Together with the Greenland record this suggests that the Younger Dryas abrupt climate change involved cooling across much of the northern extratropics. Where longer records exist, there is obvious correlation with all of the abrupt changes in Greenland during the last glacial period. For example, these events extended to the northeast Pacific Ocean margin, appearing as changes in ocean oxygenation in the Santa Barbara basin and Gulf of California (Behl and Kennett 1996), although the interpretation of these records in terms of climate has proven difficult.

12.4.3. The Tropics and the Southern Hemisphere In the Atlantic sector, ice cores from Andean glaciers in Bolivia and Peru, both south of the equator, show evidence of the Younger Dryas abrupt climate change (Thompson et al. 1998). On the mountain of Sajama in Bolivia, the abrupt shifts at the transition into the Bølling-Allerød warm period and then into the Younger Dryas were as large as the shift between the depths of the LGM and the current climate. Younger Dryas-era glacial advances have also been reported in the Andes but dating is uncertain. These results probably indicate that the cooling encompassed not just the Northern Hemisphere but also the Tropics. Further afield, abrupt climate changes throughout the last glacial cycle, and including the Younger Dryas, appear in speleothems from China (Yuan et al. 2004). Two speleothems from caves 1000 km apart record remarkably similar changes of δ 18O within them. According to the authors, the isotope content here does not reflect temperature changes but indicates reduced precipitation during times when Greenland, and much of the Northern Hemisphere, was cold. The China monsoon record is consistent with that derived from ocean sediments in the Arabian Sea that record changes in biological productivity, with higher productivity presumed to be caused by stronger monsoons and upwelling. Schulz et al. (1998) found that over the last 65,000 years there was an impressive correlation, independently dated in each record, of weaker monsoons accompanying low temperatures in Greenland (see also Altabet et al. [2002]). Taken with the China record, this suggests that cold stadials in the Northern Hemisphere were associated with a weaker monsoon across all of Asia. The changes in monsoon strength on the time scales of abrupt climate changes are as large as the changes between glacial and interglacial climates. Thus, abrupt climate changes were as strong here as around the North Atlantic. An important record comes from the δ 18O content of surface-dwelling foraminifera recorded in sediments on the eastern edge of the Indonesian archipelago and in the west Pacific warm pool, with higher values interpreted to indicate increased

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salinity (Stott et al. 2002). The tropical Pacific salinity record does not clearly show the abrupt changes seen in Greenland, but there is a visible correlation between increased salinity in the far west of the warm pool and stadial conditions in Greenland. This suggests reduced precipitation in this area during Greenland stadials. The monsoon records and those from the Cariaco Basin, the Bermuda Rise, tropical Africa, the Santa Barbara Basin and the west Pacific warm pool make clear that abrupt climate changes with impacts across the Northern Hemisphere and throughout the global Tropics occurred during the last deglaciation and throughout the last glacial period. The Younger Dryas is just the best documented of these events. Further south, the trail begins to run dry. However, a recent record developed by Farmer et al. (2005) on the basis of the Mg/Ca ratios within planktonic foraminifera in a well-dated core off the Namibian coast of southwest Africa shows a potent Younger Dryas cooling of 2◦–3◦C. The SST evolution in this core during the Bølling-AllerødYounger Dryas period correlates impressively with the Cariaco Basin and Greenland records. This is the most important record to date documenting a Younger Dryas event in the southern extratropics. A report of a Younger Dryas glacial advance in New Zealand (Denton and Hendy 1994) has recently been claimed to be older by several hundred years (Broecker 2003). Both here and in the southern Andes much work remains to be done to identify and date late glacial advances.

12.4.4. The Global Mean Climate It is not yet clear whether the planet as a whole warmed and cooled during abrupt climate changes. If it did not, then it makes sense to look for causes purely in changes in atmosphere and ocean circulations and heat transports. However, if the global mean temperature also changed, then the circulation must have interacted with water vapor and/or clouds such that changes in greenhouse trapping and/or albedo allowed the planet to equilibrate with the incoming solar radiation at a different temperature. Changes in sea ice can also change the global mean temperature but have a lesser impact on planetary albedo than clouds because sea ice is most prevalent in locations and seasons with little solar radiation to reflect. Only improved temperature reconstructions from the Tropics and southern midlatitudes during times of abrupt changes will allow us to know if these were associated with planetary warming and cooling.

12.5. Abrupt Climate Change and the Deep Ocean Circulation Even before there was a reliable means of determining past changes in ocean circulation, Broecker et al. (1985) presented the still-reigning paradigm of abrupt climate change: switches on and off of deep-water formation in the North Atlantic Ocean and associated changes in the thermohaline circulation (THC). When deep-water formation does not

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occur, the subpolar branch of the THC does not operate and surface currents do not bring warm, salty waters northward into the Nordic Seas, there to lose heat to the atmosphere. Consequently the climate around the North Atlantic cools. The most compelling evidence to date of THC changes during abrupt climate changes is that of McManus et al. (2004), who showed that the THC was very weak between the LGM and the transition into the Bølling-Allerød warm period when it abruptly “turned on.” It then reduced to half strength, quite sharply, during the Younger Dryas and then gradually returned to Holocene strength (Fig. 12.3). As the THC weakens and strengthens, the transport of salty water from the subtropical Atlantic to the subpolar Atlantic would be expected to reduce and increase. Consequently, during times of a weak THC, the salinity of the subtropical North Atlantic would be expected to increase. Recently Schmidt et al. (2004) reported there was indeed an inverse relationship between Caribbean Sea salinity and the THC that reinforces the evidence for THC changes. Together with earlier data (Hughen et al. 2000; Bond et al. 1997), these data indicate that abrupt climate changes can involve not just surface climate but also the deep ocean circulation or THC. It also indicates that the THC is capable of rapidly “turning on” and that it can rapidly reduce in strength.

12.6. Seasonality of Abrupt Climate Change Around the North Atlantic Ocean Atkinson et al. (1987), using a method based on the current climate tolerances of hundreds of species of carnivorous beetles and the distribution of fossil remains of those species, concluded that over the British Isles the rapid warming at the beginning of the Bølling-Allerød and the rapid changes of the Younger Dryas involved only modest summer temperature changes but enormous changes in winter temperature. The implied changes in seasonality were enough to convert the British Isles from a climate after the LGM (and after deglaciation of the Isles) that was similar to that of current day northeastern Russia to one during the Bolling-Allerod that was similar to that of today, then back again to one of great seasonality during the Younger Dryas, and then finally back to a modern-day climate. Each transition was accomplished in decades. Denton et al. (2005) also show that in around Greenland the Younger Dryas glacier advance is sufficiently limited that it can only be consistent with modest summer cooling. If the annual mean temperature reconstructions from ice cores are correct, then the winter cooling in Greenland must have been around 20◦C—comparable to that in the British Isles. As they point out, all the paleoclimate data from around the North Atlantic Ocean, including Norway (Dahl and Nesje 1992), indicates that Younger Dryas cooling was largely contained within the winter, while summers cooled more modestly. Consequently, the climate flip-flops measured in the ice cores—which do seem representative of the climate of the region—represent rapid transitions between two

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Figure 12.3. Records of ocean circulation and climate in the North Atlantic region during the last deglaciation from 20,000 to 10,000 years ago. (A) shows ice-rafted debris from sediments in the subpolar North Atlantic, (B) shows the benthic δ 13C, which is impacted by the strength of deep water formation, (C) shows the GISP2 δ 18O record, (D) shows reconstructed subpolar SSTs from a variety of sources, and (E) shows the ratio 231Pa/230Th from the Bermuda Rise, a measure of the strength of meridional overturning. The ratio 231Pa/230Th reaches 0.093 for a total cessation of the THC. From McManus et al. (2004). (Reproduced with permission from c 2004.) Macmillan Publishers Ltd.


climate regimes. One is a maritime climate akin to the modern one and the other is a climate of marked seasonality in which winters are not tempered by the ameliorating effects of release of heat from the ocean or via atmospheric heat flux convergence. The most common explanation for how winters around the North Atlantic could become so severe is that sea ice expanded southward to the latitude of southern Britain. If winters around the North Atlantic got as cold as the reconstructions suggest they did during stadials and the Younger Dryas, then sea ice would almost certainly encroach this far. How this could ever happen is a subject we will shortly turn to.

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12.7. The Lack of Modern Analogues of Past Abrupt Climate Changes The period of widespread instrumental observations of weather and climate, beginning in the middle-nineteenth century when measurements from ships were begun, until the present contains many climate changes but none that appear to be analogues, even in weakened form, of past abrupt climate changes. The North Atlantic region has experienced climate changes on decadal time scales. These have primarily been associated with changes in the atmosphere circulation. For example the trend from a low-index to a high-index state of the Northern Annular Mode (NAM) between the 1960s and late 1990s brought striking changes of climate to western Europe, including drought in Spain and milder, snowier winters and advancing glaciers in western Norway. The NAM trended in the opposite direction from the 1920s through the 1960s. The more recent upward trend has been explained as a consequence of rising greenhouse gases (Shindell et al. 1999) or as a response to Indian Ocean warming (Hoerling et al. 2004). Both explanations make the earlier opposite trend hard to explain. Although basin-wide SST anomalies, which could be related to THC variations, do appear in the North Atlantic, they do not seem to explain the NAM behavior. The NAM behavior has been gradual, free of abrupt shifts. The most celebrated climate shift in the instrumental record is that of 1976/77. This winter ushered in an extended period in which the tropical Pacific Ocean was warmer than normal, as were the waters along the west coast of the Americas, while the central North Pacific Ocean was cold (Zhang et al. 1997; Mantua et al. 1997). In the record of tropical tropospheric temperatures, the transition does appear quite abrupt (Seager et al. 2004). More generally it appears as the dividing point between a period of strong ENSO events that began with the 1976/77 winter and a period of weaker ENSO activity in the decades before. This climate shift, and perhaps an opposite shift after the 1997/98 El Niño, does testify to the ability of the Tropics to organize midlatitude climate on multi-decadal time scales (Hoerling et al. 2004; Huang et al. 2005), but falls short of being directly relevant to past climate changes. The only other striking climate transition that has occurred in the instrumental record is the shift to a drier climate in the Sahel region of West Africa in the early and mid 1970s. This appears to be related to changes in tropical SSTs, including those in the Indian Ocean (Giannini et al. 2003), although the mechanisms remain obscure. Other monsoon regions have not gone through such clear transitions. The Sahel drying could be relevant to abrupt transitions into and out of the African Humid Period of the mid-Holocene, a time when the region of the current Sahara Desert was grassland (deMenocal et al. 2000). Neither indices of tropical climate variability (e.g., NINO3) nor extratropical circulation variability show evidence of bimodality such as that seen in the Greenland ice-core data (Wunsch 2003), and efforts to locate regime-like behavior in the climate system have so far failed (Stephenson et al. 2004). Since regime-like behavior did occur

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in climates of the past, this emphasizes that past abrupt climate changes are different, not just quantitatively, but also qualitatively, from those in the instrumental record.

12.8. Examining Proposed Causes of Abrupt Climate Change Next we turn our attention to the possible causes of abrupt climate change. First we will summarize what needs to be explained.

12.8.1. The Spatial Character of Abrupt Climate Changes According to the studies described in the previous sections, the climate of the North Atlantic region during glacial times moved between two different states of operation. The transitions in between occurred rapidly, especially for the warming. These rapid climate changes involved striking temperature changes across western Europe and eastern North America, enormous in winter but modest in summer. Consequently there were abrupt changes in the degree of seasonality. During North Atlantic cold stadials, the surface ocean in the subtropical North Atlantic cooled, the ITCZ over South America shifted south, the tropical Americas and the South Atlantic Ocean off Africa cooled, and the Asian monsoon weakened. In the tropical regions the transitions were as large as the difference between full LGM and modern states. Hence, the abrupt climate-change signal does not appear to become more muted with distance from the North Atlantic. Any proposed mechanism must be able to explain these observed climate changes.

12.8.2. Strengths and Weaknesses of the THC-Driving Theory 12.8.2.1. Role of the THC in Today’s Climate Because of the North Atlantic branch of the THC, with deep sinking at high latitudes compensated by northward flow at the surface and southward flow at depth, the North Atlantic Ocean moves heat northward at all latitudes. The following discussion is based on Seager et al. (2002). The North Atlantic Ocean moves about 0.8 PW across 35◦N (Trenberth and Caron 2001), enough to warm the area north of 35◦N by 3◦–4◦C. This pales in comparison to a warming of 27◦C due to the vastly greater atmosphere heat transport across 35◦N and a warming of another 27◦C in winter due to seasonal release of heat stored since the last winter. Not all of this ocean heat transport (OHT) is due to the THC, as the subtropical and subpolar gyres and wind-driven overturning also contribute. Those circulations persist even when the THC is weakened or shuts down, leaving a poleward OHT, though one that is greatly reduced. The heat transported by the North Atlantic Ocean is released to the atmosphere primarily in two regions. The first is in the Gulf Stream region east of North America

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where, during winter, cold, dry air from the continent flows over the warm waters offshore extracting up to 400 W m−2, in the seasonal mean, from the ocean. A sizable portion of this heat is picked up by transient eddies—atmospheric storms—and converges over eastern North America, ameliorating winters there. The transient eddy heat flux, as always, is acting diffusively to oppose temperature contrasts (Lau 1979). The other region where the North Atlantic OHT is preferentially released is in the Norwegian Sea, keeping it ice-free in winter and warming the coast of Norway. Because of this geography of heat release, the North Atlantic OHT warms the climate on both sides of the ocean, leaving the bulk of the temperature contrast of about 15◦C to be explained by the more basic continental-maritime climate distinction and by stationary waves, especially those forced by flow over the Rocky Mountains. 12.8.2.2. The Spatial Pattern of THC-Induced Climate Change What is the global signature of a sudden THC shutdown or onset? Many modeling groups have performed experiments in which the THC in a coupled ocean-atmosphere GCM is forced to shut down, usually by addition of a massive amount of freshwater to the surface of the subpolar North Atlantic Ocean (e.g., Manabe and Stouffer 1997; Rind et al. 2001; Vellinga et al. 2002; Vellinga and Wood 2002; Zhang and Delworth 2005). Freshwater dumping is usually justified in the paleoclimate context as an idealized representation of a meltwater discharge from glacial dammed lakes.3 All these experiments agree that once the water columns of the subpolar North Atlantic have been stabilized by the addition of low-density freshwater, deep-water formation is reduced or ceases, poleward flow into the region is weakened, the heat transport reduces, and the sea-ice extent increases. They also all agree that the North Atlantic region cools by as much as several degrees Celsius in the region of Iceland and that the cooling extends into Europe and over Greenland, although at reduced strength. Figure 12.4 shows the change in annual mean air temperature caused by a THC shutdown in the Geophysical Fluid Dynamics Laboratory (GFDL) coupled-model experiment performed by Zhang and Delworth (2005). The annual-mean cooling over Greenland is only a few degrees Celsius (Manabe and Stouffer 1997; Zhang and Delworth 2005), much smaller than that observed for abrupt climate changes (Severinghaus and Brook 1999). The modeled cooling over western Europe is also at most a few degrees Celsius, much less than that reconstructed from beetles (Atkinson et al. 1987) or periglacial evidence (see Denton et al. 2005). All models agree that the North Atlantic cooling extends down into the subtropics and perhaps as far as the equator, but that south of 45◦N the cooling is only of the order of 1◦C, less than that reproduced by Sachs and Lehman (1999). The models also do not produce a strong tropical cooling over South America, in conflict with ice core records there for the Younger Dryas (Thompson et al. 1998). Further, as shown in Figure 12.4, a THC shutdown causes warming in the South Atlantic Ocean that is in contrast to the cooling there during the Younger Dryas reported by Farmer et al. (2005).

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Figure 12.4. The change in annual mean surface air temperature between two states of a coupled GCM, one with a collapsed thermohaline circulation and one with an active thermohaline circulation. The model was the latest version of the GFDL coupled GCM. Shading indicates the change is significant at the 5% level. From Zhang and Delworth (2005). c 2005.) (Reproduced with permission from the American Meteorological Society


Rind et al. (2001), Vellinga and Wood (2002), and Zhang and Delworth (2005), using different coupled GCMs, all show that for a THC shutdown the ITCZ moves south in the tropical Atlantic, broadly consistent with cooling of the North Atlantic Ocean that occurred in the models, and with the interpretation of the Cariaco Basin record in terms of a reduction in precipitation over northern South America and with the speleothem record of increased precipitation just south of the Equator. The precipitation changes reach up to a few millimeters per day in some regions. The models of Vellinga and Wood (2002) and Zhang and Delworth (2005, see Fig. 12.5) have reduced precipitation in the Asian monsoon region, while the Rind et al. (2001) model has a very modest weakening of the Indian monsoon and an equally modest strengthening of the East Asian monsoon. Figures 12.4 and 12.5 show that in the GFDL model the impacts of a THC shutdown extend into the northern and tropical Pacific Ocean where the pattern of temperature change is oddly similar to that in the Atlantic, i.e., significant cooling in midlatitudes but warming immediately south of the equator. It is yet to be determined why this occurs. The Hadley Centre model of Vellinga and Wood (2002) has a similar response. In the GFDL model, tropical Pacific precipitation changes as expected given the change in surface temperature with a southward shift of the ITCZ.

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Figure 12.5. The change in annual mean precipitation (m/year) between two states of a coupled GCM, one with a collapsed thermohaline circulation and one with an active thermohaline circulation. The model was the latest version of the GFDL coupled GCM. Shading indicates the change is significant at the 95% level. From Zhang and Delworth (2005). c 2005.) (Reproduced with permission from the American Meteorological Society


In the North Atlantic region, therefore, there is sufficient agreement between models and paleoclimate data that changes in the THC were likely involved in abrupt climate changes. This said, changes in the THC, even shutdowns—at least as represented in GCMs—cannot explain the magnitude of the cooling around the North Atlantic, despite large increases in sea ice cover in the surrounding seas (Manabe and Stouffer 1997; Rind et al. 2001), or in the subtropics (Sachs and Lehman 1999). Therefore, even on its own home turf, the THC theory falls short of being able to offer a complete explanation of abrupt climate changes, unless all existing coupled GCMs are significantly in error (which they may be). Outside of the North Atlantic region, the THC theory of abrupt climate change cannot entirely explain the paleoclimate data. According to GCMs, the impact of the THC on temperature and precipitation over equatorial and southern South America is too weak to explain the impressive documentation of the Younger Dryas in tropical ice cores. Changes in the THC also cannot fully explain the equally impressive reduction in monsoon strength, coherent across the Asian monsoon, that occurred during cold stadial events when the THC was weaker. They also cannot explain cooling in the

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southeast subtropical Atlantic Ocean. On the other hand, the impacts of a THC shutdown do extend across the entire Northern Hemisphere and into the Tropics. 12.8.2.3. The Temporal Behavior of the THC McManus et al. (2004) showed, using radiochemical data, that the North Atlantic THC was operating during the LGM but was in a “drop dead” state during the beginning of the last deglaciation between the LGM and about 14.7 kyr BP. Then it abruptly increased to near-modern strength, apparently in decades, coinciding with the dramatic warming at the start of the Bølling-Allerød. At the beginning of the Younger Dryas, the THC rapidly weakened but remained active, and at the end it more gradually recovered to Holocene values. This remarkable result confirmed what had been suggested for a while: the THC is not a sluggish part of the climate system but can shift between modes of operation in years to decades. What can cause such changes? Manabe and Stouffer (1988) showed that a coupled GCM (the GFDL one) had two stable modes of operation, one with an active North Atlantic THC and one with a “drop dead” THC that could be induced by a massive addition of freshwater into the subpolar North Atlantic Ocean. This work inspired a generation of similar experiments and led to the development of the dominant paradigm of abrupt climate change: releases of glacial meltwater, whether it be from ice-dammed lakes or ice surges, caps the North Atlantic Ocean with fresh surface water, shutting down deep-water formation and turning off the flow of warm water from the subtropical North Atlantic into the subpolar Atlantic, thus cooling regional climate, which is amplified by an increase in sea-ice extent. Rind et al. (2001), using a coupled GCM with a free-surface ocean model that allows for an increase in river flow at any chosen location to be directly specified, were able to induce a North Atlantic THC shutdown within a decade or two when the flow through the St. Lawrence was increased. However, to do this required an inflow of 20 Sv for five years. Licciardi et al. (1999) and Clark et al. (2001) have attempted to reconstruct the history of freshwater flux from North America into the Atlantic Ocean during the last deglaciation. They show jumps of a fraction of a Sverdrup in combined St. Lawrence and Hudson River runoff occurring as a result of reroutings of continental drainage and from sudden emptying of ice-dammed lakes (such as Lake Agassiz). For changes in freshwater flux to the North Atlantic of this magnitude, coupled GCMs agree that the THC would weaken by only a modest amount and only very gradually, say, over hundreds of years (Manabe and Stouffer 1997; Rind et al. 2001). If the coupled GCMs are correct, then realistic freshwater influxes cannot explain the rapid decreases in the North Atlantic THC that, almost certainly, seem to have occurred in the past. Further, the paleoclimate record shows that in the North Atlantic region, it was actually the warmings, and THC resumptions, rather than the coolings, and THC slowdowns or shutdowns, that were the most abrupt. Coupled GCMs can produce rapid cessations of deep-water formation if given apparently unrealistic perturbations, but none has yet produced an abrupt resumption of deep-water formation and the

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THC. Instead, after a period of a weak or nonexistent THC, the THC gradually returns over a hundred years or so (Vellinga et al. 2002). It is also not clear that all of the abrupt changes seen in Fig. 12.1 were the result of freshwater perturbations. Further, no geological evidence has yet been found of the catastrophic flood from Lake Agassiz that was meant to trigger the Younger Dryas (Broecker 2006). This suggests there may be other mechanisms whereby the THC can be turned on and off. Simple climate models (often called “intermediate” models) that contain more than one stable mode of operation of the THC can reproduce many aspects of observed climate records, including rapid THC resumptions and warmings (Ganopolski and Rahmstorf 2001). The finding of multiple equilibrium states in the GFDL coupled model lent credibility to such models. However, the key physics acting in the intermediate climate models that have produced regime-like behavior in the THC include ocean convection, advection, and diffusion. The realism with which these processes are represented is affected by the very coarse horizontal and vertical resolution in the intermediate models. Intermediate models have non trivial problems in simulating the effects of topography on circulation, which strongly influences where ocean convection takes place and its stability of location and strength. The ocean components of the simple models will also have difficulty representing the real processes by which the ocean transports heat, a transport that, as shown by Boccaletti et al. (2005), largely contained within shallow, surface intensified circulations and not within deep overturning circulations. Similarly, the atmospheric components of the intermediate models are, essentially, extended energy-balance models. It is not clear that regime-like behavior would occur if the same ocean models were coupled to more realistic atmospheric models (such as those that allow for weather). In general, it needs to be demonstrated that multiple regimes can be supported by models with higher resolution that adequately resolve the physics responsible for the multiple regimes in intermediate models. To date, it is not even clear whether coupled GCMs other than the old Manabe and Stouffer GFDL model have multiple equilibrium states. Vellinga et al. (2002) say they have not found this behavior in the Hadley Centre coupled GCM. In general, the GCM model results are at sufficient variance with the paleoclimate record to raise several questions that will be addressed shortly.

12.8.3. Summary There is no doubt that abrupt climate changes, such as the stadial-interstadial transitions of the last ice age and the warming at the start of the Bølling-Allerød and Younger Dryas of the last deglaciation, involve changes in the North Atlantic THC, and probably as an active component. However, changes in the THC alone cannot explain the known global climate changes associated with these events, especially in the Tropics, even accounting for its impacts on sea ice. Further, no state-of-the-art climate model has shown that regime-like behavior of the THC with rapid transitions between states is possible.

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On this basis we suggest that there is a lot more to the puzzle of abrupt climate change than changes of the THC.

12.9. Atmospheric Circulation Regimes and Global Atmosphere-Ocean Coupling as Possible Causes of Abrupt Climate Change The discussions of the spatial extent of abrupt climate changes in glacial times and during the last deglaciation should make it clear that the causes must be found in changes in the general circulations of the global, as opposed to regional, atmosphere and ocean circulation. The idea that the THC changes and directly impacts a small area of the globe, and that somehow most of the rest of the world piggy-backs along in a rather systematic and reliable way seems dubious.

12.9.1. The Problem of North Atlantic Climate Change Consider the changes in seasonality documented around the North Atlantic Ocean, going from the LGM through the abrupt warm transition at the start of the BøllingAllerød and then cooling into the Younger Dryas, followed by the second abrupt warm transition that ended it. A wide collection of evidence indicates that winter temperatures in, for example, the British Isles changed up and down by about 20◦–30◦C during these transitions and summer temperatures by 4◦–6◦C (e.g., Atkinson et al. 1987). This could be accomplished if the sea ice edge extended south of the British Isles during winter and then retreated far north in summer. Imposing this seasonal cycle of sea ice cover under an atmosphere GCM, Renssen et al. (2001) reproduced, approximately, the observed different deglacial climates in a set of time snapshot experiments. When sea ice extends that far south, and if it is thick enough with few gaps, the surface temperature of the ice and the air above drops dramatically in winter as the atmosphere becomes insulated from the ocean below. It is likely that the sea ice extension, rather than the presence of ice sheets (which existed throughout the Bølling-Allerød and Younger Dryas over North America and Scandinavia), caused winter cooling in the North Atlantic sector. But sea ice never extends as far south as the British Isles in coupled GCMs when the THC is forced to shut down. This is consistent with the fact that in the North Pacific and Southern Ocean, sea ice typically does not extend equatorward of 60◦ even though, unlike the current North Atlantic Ocean but similar to one with a THC shutdown, there is close to zero poleward ocean heat transport there. Sea ice is melted from below and so its equatorward extent depends in part on the disposition of the ocean heat transport, but the placement of the ocean heat transport depends on the wind stress. In THC shutdown experiments, the sea ice expansion is restricted by the atmospheric circulation.

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Assuming that by some means, sea ice did extend as far south as the British Isles during the winters of stadials, the relative summer warmth then requires it to retreat far to the north and for the ocean to warm up tremendously. As a point of comparison, in the current climate, ocean areas that are ice covered at the end of winter do not achieve temperatures of more than about 2◦C in summer. Thus the problems posed by abrupt change in the North Atlantic region are: 1. How could sea ice extend so far south in winter during the stadials? 2. How, during the spring and summer of stadials, can there be such an enormous influx of heat as to melt the ice and warm the water below by close to 10◦ C. If 50 m of water needs to be warmed up by this much in four months, it would take an average net surface heat flux of 150 Wm−2, more than twice the current average between early spring and midsummer and more than can be accounted for by any increase in summer solar irradiance (as during the Younger Dryas). 3. How can this stadial state of drastic seasonality abruptly shift into one similar to that of today with a highly maritime climate in western Europe? Remember that both states can exist in the presence of large ice sheets over North America and Scandinavia.

12.9.2. Required Changes of Atmospheric Circulation Regimes and Heat Transports To solve the first two problems we must imagine a stadial climate in which the heat transport has almost the opposite seasonal cycle to that of today. Whereas now winter atmospheric and ocean heat transport holds back the sea ice in the North Atlantic, during stadials weak heat transport in the mid to high latitudes must have allowed the sea ice to advance south. In contrast, during the summers of stadials, there must have been strong transport to melt back the sea ice and establish mild summers. In thinking of ways to reduce the winter convergence of heat into the midand high-latitude North Atlantic, we might begin with the storm tracks and mean atmosphere circulation. The Atlantic storm track and jet stream have a clear southwestto-northeast trajectory, whereas the Pacific ones are more zonal over most of their longitudinal reach (Hoskins and Valdes 1990). If the Atlantic storm track and jet could be induced to take a more zonal track, akin to its Pacific cousin, the North Atlantic would cool. The cooling over western Europe and the North Atlantic Ocean would be driven by less southerly flow and advective warming in the winter stationary waves—the real reason for the mildness of west European winters (Seager et al. 2002). Transient eddy heat transports, which act diffusively on temperature, would oppose the cooling as the storm track—defined as the statistical mean of the routes taken by eddies or, for this

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purpose, as the location of the maximum eddy heat transport—relocated toward the Mediterranean, and Europe was placed on the eddy-heat-flux-convergence side of the storm track.

12.9.3. Impacts of a Hypothesized Shift to Zonal Circulation during Winter in the North Atlantic Sector An essentially zonal wind flow across the North Atlantic Ocean could set in motion a chain of events that establish a cold North Atlantic climate. First of all, the removal of warm southwesterly advection into the North Atlantic will directly cool the ocean and coastal regions there. In the current climate, the flow of water in the North Atlantic Drift from the Gulf Stream and into the Nordic Seas is controlled by the wind stress pattern and reflects the northward tilt of the Atlantic jet stream and storm track. This water is salty and, as it cools on its northward track, becomes dense enough to sink to the bottom. The northward flow of water was the ultimate explanation of Warren (1983) for why deep water is formed in the North Atlantic and not the North Pacific (see also Emile-Geay et al. [2003]). If the wind stress pattern becomes zonal, the North Atlantic Drift will flow directly across the Atlantic toward France and Spain instead of toward the Nordic Seas, reducing the salt flux into the subpolar Atlantic and causing the sinking branch of the THC to shift southward and reduce ocean heat flux convergence in the Nordic Seas. In the current climate, the sea ice edge is very much controlled by the pattern of the winds through the influence they have on both the ice drift and the atmosphere heat transport. For example, positive phases of the North Atlantic Oscillation (NAO) go along with less ice in the Greenland and Barents Seas, where the anomalous winds are from the south, and more ice in the Labrador Sea where the anomalous winds are from the north (Deser et al. 2000). On longer time scales, changes in wind will impact the ocean heat transport, and this will also influence the sea ice cover. A shift to more zonal winds across the North Atlantic will, by all processes, allow sea ice to extend further south than it currently does, cooling the North Atlantic regional climate. Currently much of the Gulf Stream water continues northward and becomes part of the deep overturning in the North Atlantic. As shown by Talley (2003) using data and by Boccaletti et al. (2005) using a model, in the North Atlantic Ocean, unlike other basins, the deep overturning heat transport is equal in magnitude to that by the horizontal gyres and intermediate overturning. The northward flow of the North Atlantic Drift helps sustain winter heat release from the ocean, especially north of Scotland. The heat release and diabatic heating of the atmosphere above helps maintain the Icelandic Low. This was shown in the GCM experiments of Seager et al. (2002) and is implicit in the modeling of Hoskins and Valdes (1990). The Icelandic Low maintains the northward deflection of the storm track, jet, stream, and the North Atlantic Drift. Thus the interaction between the atmosphere and ocean circulations over the North Atlantic

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appears to be self-reinforcing. Perhaps forcing of this system from outside, maybe in the Tropics, can set this reinforcing mechanism running in reverse and establish a stable zonal jet and storm track, a zonal North Atlantic Drift, a collapsed THC, and a cold North Atlantic. Although a shift to a zonal circulation across the North Atlantic would cause extensive cooling of the European region on its own, it would be assisted by the induced THC shutdown. To date, explorations of THC shutdowns have all focused on the impact of freshwater discharges from melting ice sheets or ice-dammed lakes. To our knowledge, the impact on the THC of a sudden shift in the Atlantic to a more Pacific-like wind stress regime has never been investigated. It would decrease the warm, salty water flow toward the Nordic Seas that currently, as it cools, becomes dense enough to sink to the abyss. With that inflow shut off by the altered wind stress pattern, it at least appears possible that the THC would shut down too.

12.9.4. Summer Climate and Abrupt Shifts in Seasonality During the Bølling-Allerød warm period and the period after the Younger Dryas, the climate of western Europe had a seasonality similar to that of today, but during the cold spells, while winters were extremely cold, summer temperatures remained close to 10◦C (Atkinson et al. 1987; Denton et al. 2005). Not surprisingly, such summer temperatures can be reproduced for these periods in an atmosphere GCM when the North Atlantic SSTs are specified to also be that warm (Renssen and Isarin 2001). During the cold periods, the SSTs need to warm from freezing to about 10◦C. Currently the only regions of the world ocean that have a seasonal cycle that large are in the western boundary currents east of North America and Asia. Here, warm moist advection around the summer subtropical anticyclones increases the SSTs, and cold dry advection of the continents, as well as transient eddies, decrease the SSTs in winter (Seager et al. 2003a). The summer warmth and increased seasonality require a much larger heat import into the North Atlantic region by the atmosphere and ocean during summer than currently occurs. One possible cause of this is the presence of extensive ice sheets over North America and Scandinavia. During winters, ice sheets do not significantly perturb the surface or planetary radiation budget, but in summer, they are a vast radiative sink as they reflect solar radiation that otherwise would be absorbed at the surface. This radiative sink, and the associated cold temperatures, will induce an anomalous convergence of atmospheric heat transport over the ice. This could cause a much larger export of heat from the Tropics into high latitudes during summer than is currently the case. Few GCM studies have examined the impact of ice sheets on seasonal energy transports. Hall et al. (1996) did find greatly increased summer atmosphere heat transport in an LGM simulation, although the fixed SSTs in the study make the results questionable. Summer ice sheets will also impact the stationary waves. Currently

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during summers, poleward warm advection around the Icelandic Low (which, unlike the Aleutian Low, is still present in summer, though weak) helps warm the coast of Scandinavia, while further south cold advection around the North Atlantic subtropical high keeps the coasts of Portugal and North Africa cool (Seager et al. 2003a). If, during the summers of cold periods such as the Younger Dryas, the Icelandic Low was strong and south of its current location, then it could cause advective warming of western Europe. It will be well worth examining how the Laurentide and Scandinavian ice sheets impacted the summer stationary wave climate, via their orography and albedo, and how this impacted summer temperatures and seasonality. 12.9.5. The Tropical Circulation and Abrupt Climate Change 12.9.5.1. Tropical Forcing of Global Climate Variations There are many reasons to think that the Tropics play an active role, and maybe an organizing one, in abrupt climate change and are not a backwater responding passively to climate changes organized by the North Atlantic Ocean. For one thing, the paleoclimate record shows that abrupt climate changes are relatively as large in the Tropics (e.g., the Asian monsoon and Andean ice cores) as in the North Atlantic region (Denton et al. 2005). While models do show that weakening and strengthening of the Atlantic THC can lead to significant tropical climate change (Zhang and Delworth 2005; Vellinga and Wood 2002), the changes seem to fall short of those that actually did occur. It is equally plausible that past abrupt changes occurred as part of a global climate reorganization instigated within the Tropics. The other reason for looking to the Tropics is that in the current climate, many climate changes around the globe throughout the twentieth century have been forced from the Tropics through varying SSTs and patterns of tropical convection. This includes the major droughts and pluvials over the Americas (Schubert et al. 2004; Seager et al. 2005; Huang et al. 2005), the drying of the Sahel (Giannini et al. 2003) and, quite possibly, the trends in the Northern Annular Mode (Hurrell et al. 2004; Hoerling et al. 2004; Schneider et al. 2003). The tropical Pacific may also be able to exert a control on the Atlantic THC (Latif et al. 2000) through its influence on rainfall over the tropical Americas and the Atlantic Ocean and the vapor flux across Central America. 12.9.5.2. Limitations of the ENSO Blueprint Although all this is true, so far it has not been demonstrated that the Tropics in any way control abrupt climate changes. The dominant mode of year-to-year and decadeto-decade climate variability in the Tropics relates to the El Niño-Southern Oscillation (ENSO). Numerous attempts to explain past tropical SST changes have invoked changes with an ENSO-like spatial pattern (e.g., Stott et al. 2002; Koutavas et al. 2002), and the global impacts of ENSO have been appealed to as a cause of climate change on glacial time scales (Cane 1998). Certainly ENSO responds to orbital and other external forcing

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and may even respond abruptly to gradual forcing (Clement et al. 1999, 2000, 2001; Mann et al. 2005). Clement et al. (2001) suggested that the peculiar orbital forcing of the Younger Dryas interval stabilized the tropical Pacific atmosphere-ocean system, resulting in long periods without interannual variability and a persistent change in the mean state with a La Niña-like pattern. While this result is compelling, the global climate changes during the Younger Dryas do not fit the typical La Niña pattern. Of the available paleo-records, only tropical cooling, as implied by Andean ice core records for the Younger Dryas, is consistent with a La Niña-like state. During stadials, reduced precipitation over northern South America, the weak Asian and west African monsoons, and reduced salinity in the tropical west Pacific are all more typical of an El Niño-like state. El Niño events also cool the midlatitudes of each hemisphere (Seager et al. 2003b), which clearly happened during the Younger Dryas and stadials. However, an El Niño-like state would also reduce precipitation in the part of northeast Brazil where speleothems indicate wet conditions during stadials. This, in combination with the dry Caribbean coast of South America (as inferred from the Cariaco Basin record), is more consistent with a weakened THC and a southward shift of the ITCZ (see Fig. 12.5). Finally, neither a THC shutdown nor an El Niño-like state can explain the cooling off Namibia during the Younger Dryas reported by Farmer et al. (2005). Despite some evidence for an El Niño-like state during stadials, currently the impacts of El Niño over the North Atlantic are far too weak to account for the dramatic climate changes that occurred there during stadials and interstadials. It is true that the climate response associated with persistent El Niño-like, or La Niña-like, anomalies need not be the same as the one that goes along with interannual variability. However, experiments with coupled models in which persistent El Niño or La Niña states were induced (Hazeleger et al. [2005] and unpublished results conducted by the first author with the GISS and CCM3 climate models) produced a response akin to the interannual one: during persistent El Niños, the Tropics warm, the midlatitudes cool, the poles warm, and there is a rather small global mean temperature change (Seager et al. 2003b). At least in these experiments, persistent changes in ENSO did not excite positive feedbacks involving water vapor or clouds that significantly amplified climate change. The inadequacy of the ENSO blueprint alone in explaining the spatial pattern of abrupt change indicates that in pursuing tropical involvement in abrupt climate change we need to think beyond ENSO. This need is highlighted by two problems of the model studies mentioned above. First, in interannual variability, an El Niño warming of the eastern tropical Pacific is caused by a transient adjustment of the ocean circulation and upper-ocean heat content. On longer time scales relevant to different climate regimes, the changes in ocean circulation and heat transport must be in equilibrium with the atmospheric circulation. In this equilibrated case, changes in SST must be sustained by different processes than for interannual changes in SST (Hazeleger et al. 2004). They will probably have different spatial patterns and different climate consequences.

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Second, the models that were used to examine the global response to persistent El Niños and La Niñas (Hazeleger et al. 2005) did not allow for ocean circulation adjustment. It is an open question how persistent tropical climate changes impact the global ocean circulation, thermocline structure, and heat transports. In both cases the challenge is to consider the full range of possible tropical and global atmosphere-ocean coupling and to think outside of the “ENSO box.” 12.9.5.3. Tropical Heating and Extratropical Jets and Storm Tracks Clearly, tropical climate changes have the potential to cause significant extratropical climate change. In an intriguing paper, Lee and Kim (2003) have argued that the Northern Hemisphere contains two dynamically distinct jet streams. One is a subtropical jet (STJ) on the poleward flank of the Hadley cell and the other is a polar front jet (PFJ) further poleward. As they point out, an STJ can be created in the absence of eddies by conservation of angular momentum in the poleward upper-level flow of the Hadley cell (Schneider 1977; Held and Hou 1980). In the real world, while this process is relevant, the STJ is also shaped by eddies. In contrast, eddies can create a jet within a zone of uniform baroclinicity but without a prior existing jet (Panetta 1993; Lee 1997). Lee and Kim argue that the PFJ has the characteristics of a jet generated in this manner. Such jets are sometimes referred to as “eddy-driven,” although this term is potentially misleading as both the STJ and PFJ are in thermal wind balance and affected by eddies. Nonetheless, the distinction between a STJ with maximum westerlies on the poleward edge of the Hadley cell and overlying a region of zero mean meridional surface wind, and an eddy-driven jet further poleward, coincident with maximum eddy momentum flux convergence and lying between zero lines of the mean meridional surface flow, is valid (Son and Lee, 2005; see also chapter 5 in this volume for further discussion of the work of Son, Lee, and Kim). The Asian sector has only a single jet with the character of an STJ. Over the Atlantic sector during winter, both jets exist with an STJ that begins west of Africa and extends across Africa and Asia, and a PFJ that begins over the southern United States and extends up to the British Isles and Scandinavia. This picture is broadly consistent with the observations of Palmen and Newton (1969). Lee and Kim argue that the STJ and PFJ compete for the attention of transient eddies. When the STJ is strong enough, the meridional temperature gradient with which it must coexist is potent enough to organize transient eddy activity resulting in a relatively southern storm track, as over the Pacific Ocean. Where the STJ is weaker, the self-reinforcing interaction between eddies that feed off temperature gradients and the momentum fluxes off those eddies that drive jets—and associated temperature gradients—allow the establishment of a relatively northern jet, as over the Atlantic Ocean. Lee and Kim show that the competition between the two jets is modulated by the strength of the tropical heating (Fig. 12.6). When this is strong, the STJ is strong enough to “capture” the transient eddies and merge the subtropical and eddy-driven

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Figure 12.6. Latitudes of the subtropical jet (line with open circles), the primary (line with closed circles), and secondary (dashed line with open squares) eddy-driven jets as a function of the strength of an imposed tropical heating (horizontal axis). Results were generated from simulations with simplified models with the eddy-driven jet latitudes predicted with the unstable normal modes of an axisymmetric flow. From Lee and Kim (2003). (Reproduced with c 2003.) permission from the American Meteorological Society


jets. Their results are based on a collection of idealized and theoretical calculations. Son and Lee (2005) reproduced similar results with the statistically steady states of a primitive-equation model subjected to different strengths of tropical heating and highlatitude cooling. Both studies suggest that there are distributions of tropical heating and high-latitude cooling that would cause the eddy-driven Atlantic jet to be captured by the STJ west of Africa. Were this to occur, the Atlantic storm track would become reoriented to extend directly eastward from the southern United States toward the Mediterranean region. In certain regimes, according to these studies, a modest change in the tropical heating could then allow reestablishment of the double-jet structure and set the same chain reaction off in the reverse direction. These interactions between jets and thermal driving, when coupled to the ocean, may provide a means for creating climate transitions such as those seen in the paleoclimate records. As stated in chapter 5 in this volume, the abrupt transitions of jet regimes here are governed by the dynamics of extratropical eddies and could also “lurk in our future as the climate warms.” According to Yin (2002), southward movement of tropical convection during northern winter strengthens the Northern Hemisphere STJ, which is consistent with the results of Lindzen and Hou (1988). A southward shift of the ITCZ in the longitudes of the Americas could lead to the Atlantic STJ “capturing” the transient eddies. This would cause a weakening of the eddy-driven jet over the North Atlantic, analogous to the mid-winter suppression of the North Pacific storm track (Yin 2002, but see chapter 4 in this volume for a purely midlatitude explanation of this phenomenon). This would

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reduce the atmospheric flux of heat into the northeast Atlantic sector. Further, the zonal elongation of the Atlantic jet would cause changes in the surface wind stress and curl and reduce, or eliminate, the flux of warm, salty water from the subtropical North Atlantic into the subpolar Atlantic, reducing the heat loss to the atmosphere there (and, hence, reinforcing zonal atmosphere flow), allowing sea ice to expand, and also reducing, or shutting down, the THC. These processes would also operate in the classical paradigm for abrupt climate change, which features abrupt shifts in the THC as the causal agent. For example, if there was a sudden resumption of the THC (for whatever reason) and the sea ice moved northward, the subtropical North Atlantic atmosphere and ocean would warm and the ITCZ would shift farther north (Chiang et al. 2003). A shift in the ITCZ northward across the equator would weaken the STJ and cause the storm track in the North Atlantic to strengthen and shift northward into the Nordic Seas. This would reestablish the transport of warm, salty water into the subpolar North Atlantic, deflecting the jet and storm track northwards and further encouraging a strong THC. It is also possible that the changes in the ITCZ could involve alterations in the latitudinal concentration of the heating such that, during stadials, as the ITCZ convection moves south, it also becomes more longitudinally confined. According to Hou and Lindzen (1992), this is another process that can intensify the Hadley circulation and potentially set in motion the same set of atmosphere and ocean processes and feedbacks described above. In this case, the Southern Hemisphere would be impacted too. A stronger STJ and weaker PFJ in the Southern Hemisphere could, if it led to an equatorward movement of the surface westerlies, reduce the flux of salty water from the Indian Ocean into the south Atlantic. This salt flux helps sustain the North Atlantic branch of the THC (Gordon et al. 1992), so this is another means whereby tropical heating distributions could have the potential to alter the THC and create global climate changes. Hence, interactions involving both the atmosphere and ocean in the North Atlantic-Americas sector could act as an amplifier of climate change linking the high latitudes and the Tropics. We have already discussed evidence that the ITCZ over the Atlantic and Americas does move south during Greenland stadials (Peterson et al. 2000; Wang et al. 2004). To date, this has been viewed as little more than a response to cooling of the North Atlantic Ocean, but it would be immediately fruitful to examine the complete nature of two-way coupling between the ITCZ and the atmosphere and ocean circulation in the mid- and high-latitude North Atlantic region, both in modern and glacial climates. Support for these ideas of reorganization of atmospheric circulation comes from simulations with coupled GCMs of the climate of the LGM such as the one conducted with NCAR’s Climate System Model (CSM) by Shin et al. (2003) and further analyzed by Camille Li at the University of Washington. Figure 12.7 shows the upper-tropospheric zonal wind and the transient eddy heat transport in the lower troposphere for the

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Figure 12.7. The Northern Hemisphere wintertime zonal wind in the upper troposphere (contour interval of 10 m s−1, contouring beginning at 30 m s−1) and the transient eddy heat transport in the lower troposphere (shading, m K s−1) from coupled GCM simulations of the modern climate (above) and the Last Glacial Maximum (below). Analysis conducted and figure provided by Camille Li of the University of Washington.

winters of a simulation of the modern climate and for the winters of the LGM. Not surprisingly, in the presence of ice sheets, and with lower levels of carbon dioxide, the meridional temperature gradient was stronger at the LGM and, consistently, the jet stream was also. However, in a situation analogous to the modern day midwinter suppression in the Pacific, the transient eddy heat flux was reduced throughout the

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Figure 12.8. Schematic of the meridional cross-section of the proposed atmosphere and ocean circulation for interstadial (above) and stadial (below) states of the glacial period. The stadial states have stronger overturning in the Tropics, a stronger subtropical jet and weaker polar front jet, weaker eddy heat transports (indicated by wavy arrows), and as a consequence of the changes in North Atlantic wind stress, a reduced THC.

winter. This could be a consequence of the orographic forcing of the Laurentide ice sheet generating a barotropic stationary wave that influences the structure of the Atlantic jet (see chapter 4 in this volume for a simple model demonstration of this). This atmospheric state, with reduced eddy heat transport, is suggestive of the stadial state during glacials. It will be interesting to see if some change can induce this state to flip over to one akin to the modern (or interstadial) state because if it can, then this is a viable means for explaining the observed abrupt changes of the glacial period. Figure 12.8 provides a highly idealized schematic of the circulations proposed for the stadial and interstadial states of the glacial period.

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An intriguing aspect of this idea is that during summer the stationary wave forced by the Laurentide ice sheet will be weaker but, because of the albedo effect, the overall baroclinicity in the North Atlantic region will remain. This could allow for a strong summer storm track stretching into the subpolar North Atlantic, much as we see in winter today, that would effectively warm European summers in agreement with the proxy evidence. If it is true that the ice sheet orography is important for allowing shifts in jet regimes, then it might be expected that at its most southward extent and largest volume, the orographic forcing of the flow may become sufficiently dominant to allow only one jet and eddy regime, that of the stadial state. Such stability would lead to a cessation of abrupt changes as actually did happen, broadly speaking, between Dansgaard-Oeschger Event 3 and the warm transition at the start of the Bølling-Allerød (see Fig. 12.1). If transitions during the glacial period between these two coupled atmosphereocean states are responsible for abrupt climate changes, then changes in the global ocean are likely to be important in explaining the long-term nature (centuries to a millennium or so) of each regime, a subject to which we now turn. 12.9.5.4. Near-Global Atmosphere-Ocean Coupling and Tropical Climate Reorganization ENSO and decadal ENSO are oscillations of the Tropical Pacific atmosphere and ocean. The climatological winds over the tropical Pacific Ocean can quite closely explain the spatial variation of tropical Pacific thermocline depths according to Sverdrup dynamics (Veronis 1973) and, with the inclusions of Ekman dynamics, the creation of a warm pool and a cold tongue (Clement et al. 2005). The same dynamics can explain the transient adjustment of the thermocline to varying winds (Cane 1984). A central element is an adjustment to balance at the equator between the thermocline (or sea-level height) tilt and the zonal wind stress. What this dynamics cannot explain is the mean thermocline depth. Boccaletti et al. (2004) point out that the mean thermocline depth has to be related to the global surface heat budget and ocean heat transport. This is because a shallow equatorial thermocline allows upwelling to expose cold water, which causes weak latent heat loss and, consequently, a net downward surface heat flux. The wind-driven overturning circulation exports this heat poleward. A deeper thermocline would allow less poleward ocean heat transport. While this is true, the details of the adjustment between the thermocline, winds, currents, and the ocean and atmosphere heat transports are opaque. Huang et al. (2000), Johnson and Marshall (2004), Cessi et al. (2004), and Timmermann et al. (2005) have pointed out that by reducing the transfer of mass from the upper layer of the ocean to deeper layers, cessation of North Atlantic Deep Water formation would cause the tropical Pacific thermocline to migrate downward. The adjustment occurs by coastal and equatorial Kelvin waves and begins within a

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decade or two, although the splitting of the equatorial Kelvin wave in the Atlantic into coastal Kelvin waves that propagate both north and south limits the strength of the signal outside of the North Atlantic. Nonetheless, a deeper tropical Pacific thermocline should cause a reduction of poleward tropical ocean heat transport. The tropical thermocline depth and structure must also be impacted by surface buoyancy fluxes, water mass transformation, and subduction within the extratropics where water that eventually upwells at the equator leaves the surface. Relatively little work has been done on this beyond theoretical explorations, but Hazeleger et al. (2001, 2004) have shown that variations in the midlatitude storm tracks and circulation can have a significant impact on the tropical Pacific thermocline. It remains to be seen how, for example, plausible shifts in the the storm tracks and jet streams in past climates impacted the tropical oceans. 12.9.5.5. Ocean and Atmosphere Heat Transports and the Global Climate If it is really true that global atmosphere-ocean coupling involving changes in storm tracks, the midlatitude westerlies, and the THC can impact the depth and structure of the tropical Pacific thermocline, then the potential exists to shift the partitioning between the tropical atmosphere and ocean heat transports. Held (2001) argued that this partitioning within the Tropics cannot change because of the dynamical coupling, by Ekman transports, between the Hadley cell and the meridional ocean overturning. However, the constraint is not nearly so tight once the tropical oceanic gyre transport, which moves heat equatorward, is considered (Hazeleger et al. 2004, 2005). It is the total transport by atmosphere and ocean that is most tightly constrained, apparently by the radiation budget at the top of the atmosphere (Clement and Seager 1999). Consequently a reduction of wind-driven tropical ocean heat transport causes a compensating increase in atmosphere heat transport, with the total heat transport varying by little, a universal result with all manner of models in all types of experiments including removal of the ocean component, removal of mountains, and removal of continents (Manabe et al. 1975; Cohen-Solal and Le Treut 1997; Clement and Seager 1999; Winton 2003; Czaja and Marshall 2006). It is plausible that a deeper tropical thermocline, by reducing the poleward ocean heat transport, could cause the atmosphere to carry a larger share of the total transport. It is also a common model result that, as the tropical ocean heat transport is reduced, the subtropical SSTs decrease and the marine low-level cloud cover and planetary albedo increase. In addition, tropical deep convection becomes more confined toward the equator, resulting in reduced atmospheric water vapor and reduced greenhouse trapping in the subtropics.4 Both effects cool the planet (Clement and Seager 1999; Winton 2003; Herweijer et al. 2005). Thus an ocean adjustment to a deeper equatorial thermocline, as could be induced by a THC shutdown or some unknown change of the extratropical atmosphere circulation, would be expected to cause a cooling of the global climate.

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12.9.6. Discussion The global atmosphere-ocean coupling idea of abrupt climate change, with an active or organizing role for the Tropics, is pure speculation. To date, no one has shown that the tropical-extratropical atmosphere-ocean circulation has different modes of operation with radically different climates, but then again, no one has tried. The tropical equivalent of a North Atlantic “hosing” experiment capable of causing a model climate to flip into an alternate state has yet to be devised. There also remains the tricky problem of what causes the climate to remain stuck in one regime or the other for centuries before rapidly switching back to the alternate state. Here the THC idea has little advantage over the global coupling idea. Although simplified climate models can simulate regime-like climate changes with abrupt transitions in between (Ganopolski and Rahmstorf 2001), this kind of behavior has not been found in coupled GCMs. Instead, after being forced to shut down by a very large forcing, the THC in current coupled GCMs tends to dribble back to full strength over the following few centuries. In contrast, in the paleoclimate record, the resumptions of deep water formation appear more abrupt than the shutdowns. For the global coupling idea, the long residence in one climate state or the other and then a switch would, presumably, have to involve the deep ocean circulation. At some point deep ocean climate change reaches a point whereby its influence on the coupled climate of the atmosphere and upper tropical oceans causes a switch between the different tropical climate states. At this point, the global coupling idea is largely based on intuition. However, there are interesting recent studies on (1) how the Tropics both respond to and organize global circulation and climate change, (2) new theories on the role of tropical heat transports in global climate and (3) new ideas on the global controls on the tropical thermocline, all ensuring that this idea will be actively pursued in the future.

12.10. Conclusions The abrupt climate changes that occurred during the last glaciation and deglaciation are mind boggling both in terms of rapidity and magnitude. That winters in the British Isles could switch between mild, wet ones very similar to today and ones in which winter temperatures dropped to as much as 20◦C below freezing, and do so in years to decades, is simply astounding. No state-of-the-art climate model, of the kind used to project future climate change within the Intergovernmental Panel on Climate Change process, has ever produced a climate change like this. The normal explanation of how such changes occurred is that deepwater formation in the Nordic Seas abruptly ceased or resumed forcing a change in ocean heat flux convergence and changes in sea ice. However, coupled GCMs only produce such rapid cessations in response to unrealistically large freshwater forcing and have not so far produced a rapid resumption.

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Even when they do produce cessations of deepwater formation, the climate change around the North Atlantic region is much less than the proxy reconstructions indicate, even though the sea ice cover in the models increases. According to coupled GCMs, cessations and resumptions of deep water formation do cause climate changes around the world qualitatively akin to those reconstructed within the paleoclimate record. When the deep sinking branch of the THC in the North Atlantic is forced to shut down, the Atlantic ITCZ moves south and the Asian monsoon weakens, both of which agree with reconstructions. However, the modeled change in the Asian monsoon is weaker than that reconstructed, while the North Atlantic Ocean circulation changes do not seem capable of causing—for the most recent such abrupt change (the Younger Dryas)—the South American cooling seen in Andean ice cores and the cooling in the southeast Atlantic. Here we have argued that the abrupt changes must involve more than changes in the North Atlantic Ocean circulation. In particular it is argued that the degree of winter cooling around the North Atlantic must be caused by a substantial change in the atmospheric circulation involving a great reduction of atmospheric heat transport into the region. Such a change could, possibly, be due to a switch to a regime of nearly zonal wind flow across the Atlantic, denying western Europe the warm advection within stationary waves that is the fundamental reason for why Europe’s winters are currently so mild. Such a change in wind regime would, presumably, also cause a change in the North Atlantic Ocean circulation as the poleward flow of warm, salty waters from the Tropics into the Nordic Seas is diverted south by the change in wind stress curl. This would impact the location and strength of deep water formation and allow sea ice to expand south. Changes in the distribution and strength of tropical convection are capable of causing such changes in the midlatitude wind regime, according to idealized GCM experiments. The tropical-forcing route is appealing because it could help explain the large abrupt changes in the monsoons and tropical climate that are known to have occurred, as well as force changes in midlatitude atmosphere and ocean circulation and climate. That the period of instrumental records has been free of dramatic abrupt changes, and that even the Holocene was quiet compared to the glacial period, argues that climate instability arises when there are continental ice sheets in North America and Eurasia and/or when the climate is colder. During glacial periods the climate can be described as fitting with Lorenz’s (1968, 1970) concept of “almost-intransitivity” in which the climate possesses successive very long periods with remarkably different states and abrupt transitions in between. It must be that either the presence of ice sheets, with their albedo and orographic forcing, or the colder mean state allows the atmosphere and ocean circulations to adopt almost-intransitivity. It is possible that long integrations of coupled GCMs with glacial boundary conditions will reveal these states. Climate modelers should hesitate before discarding a model simulation that produces a

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climate that is distinctly warm in many parts of the world even in the presence of glacial boundary conditions! Currently our knowledge of the general circulations of the atmosphere and ocean does not provide a means whereby we can imagine alternative states of the tropical and global climate and the ability to move rapidly between them. However, there has been enough recent work on the relationships between tropical atmosphere and ocean heat transports and the global controls on heat transports and the tropical thermocline to provide some hints that rearrangement may be possible. The situation is poised for a “Manabe and Stouffer” moment when with some clever, or fortuitous, experiment, alternative states are demonstrated in a coupled model. When, if ever, this will occur is unclear. The problem for dynamicists working in this area is that the period of instrumental observations, and model simulations of that period, do not provide even a hint that drastic climate reorganizations can occur. Our understanding of the general circulation is based fundamentally on this period or, more correctly, on the last 50 years of it, a time of gradual climate change or, at best, more rapid changes of modest amplitude. So it is not surprising that our encyclopedia of knowledge of the general circulations contains many ideas of negative feedbacks between circulation features that may help explain climate variability but also stabilize the climate (Bjerknes 1964; Hazeleger et al. 2005; Shaffrey and Sutton 2004). The modern period has not been propitious for studying how the climate can run away to a new state. Because of this, our understanding has to be limited. Possibly the extent of our understanding will be brought into question by climate change itself as the Earth’s climate changes more rapidly than we can extend our understanding of it. But we do not have to wait for that unfortunate event, as the past is already full of events that simply cannot be placed within our current understanding of the general circulation but are there, waiting to be explained.

Acknowledgments We wish to thank Amy Clement, Mark Cane, Peter deMenocal, Gavin Schmidt, and George Denton for many useful conversations; and Thomas Blunier, Julian Sachs, Jerry McManus, Rong Zhang, Michael Vellinga, Sukyoung Lee, and Camille Li for kindly providing figures. We also thank Tapio Schneider and Walter Robinson for excellent critical reviews of the manuscript. This work was supported by NOAA grant NA030AR4320179 PO7 (RS) and and NSF grant ATM-0502204 (DSB).

Notes 1. Alkenones are chemical compounds within the marine organisms that are resistant to decomposition, and laboratory studies have shown the unsaturation ratio to vary linearly with the temperature of the water that the organism is living in.

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Abrupt Climate Change | 365 2. The southward shift of the ITCZ during the Younger Dryas is not easy to reconcile with alkenone based SST reconstructions that show warming of the South American coast at 10◦N (Kim and Schneider 2003) and in the Caribbean Sea (Ruhlemann et al. 1999). Those SST reconstruction also disagree with the Mg/Ca-based ones of Lea et al. (2003), which show dramatic cooling north of Venezuela during the Younger Dryas. Curry and Oppo (1997) show cooling during Greenland stadials at 5◦N in the western tropical Atlantic based on the δ 18O in foraminifera which would also be consistent with a southward shifted ITCZ. 3. The sudden discharge of ice from continental ice sheets directly into the ocean is another source of freshwater. These are called Heinrich events and can be traced in ocean sediments by the debris carried within the ice. It has been suggested that they occur at the end of a cooling cycle (Clarke et al. 1999), but in general it is not well understood how they fit into climate changes and they are not dealt with here. See the review by Hemming (2004). 4. It is also possible that changes in the fluxes of moisture by transient eddies and the eddydriven mean meridional circulation cause drying in the subtropics, but this possibility was not addressed in these studies.

References Adegbie, A. T., Schneider, R. R., Rohl, U. and Wefer, G. (2003). Glacial millennial-scale fluctuations in central African precipitation recorded in terrigenous sediment supply and freshwater signals offshore Cameroon, Palaeogeog. Palaeoclim. Palaeoecol., 197, 323–333. Altabet, M. A., Higginson, M. J. and Murray, D. W. (2002). The effect of millennial-scale changes in Arabian Sea denitrification on atmospheric CO2, Nature, 415, 159–162. Atkinson, T. C., Briffa, K. R. and Coope, G. R. (1987). Seasonal temperatures in Britain during the past 22,000 years, reconstructed using beetle remains, Nature, 365, 587–592. Behl, R. J. and Kennett, J. P. (1996). Brief interstadial events in the Santa Barbara basin, NE Pacific, during the past 60 kyr, Nature, 379, 243–246. Bjerknes, J. (1964). Atlantic air-sea interaction, Adv. Geophys., 10, Academic Press, 1–82. Blunier, T. and Brook, E. J. (2001). Timing of millennial-scale climate change in Antarctica and Greenland during the last glacial period, Science, 291, 109–112. Boccaletti, G., Ferrari, R., Adcroft, A., Ferreira, D. and Marshall, J. (2005). The vertical structure of ocean heat transport, Geophys. Res. Lett., 32, doi:10.1029/2005GL022474. Boccaletti, G., Pacanowski, R. C., Philander, S. G. H. and Fedorov, A. V. (2004). The thermal structure of the upper ocean. J. Phys. Ocean, 34, 888–902. Bond, G., Broecker, W., Johnsen, S., McManus, J., Labeyrie, L., Jouzel, J. and Bonani, G. (1993). Correlations between climate records from North Atlantic sediments and Greenland ice, Nature, 365, 143–147. Bond, G., Showers, W., Cheseby, M., Lotti, R., Almasi, P., deMenocal, P., Priore, P., Cullen, H., Hajdas, I. and Bonani, G. (1997). A pervasive millennial-scale cycle in North Atlantic Holocene and glacial climates, Science, 278, 1257–1266. Broecker, W. S., Peteet, D. M. and Rind, D. (1985). Does the ocean-atmosphere system have more than one stable mode of operation? Nature, 315, 21–26. Broecker, W. S. (2003). Does the trigger for abrupt climate change reside in the ocean or in the atmosphere? Science, 300, 1519–1522. Broecker, W. S. (2006). Was the Younger Dryas triggered by a flood? Science, 312, 1146–1148.

June 7, 2007

Time:

02:33pm

chapter12.tex

366 | Seager and Battisti Brook, E. J., Harder, S., Severinghaus, J. and Bender, M. (1999). Atmospheric methane and millennial-scale climate change. In Mechanisms of global climate change at millennial time scales, Clark, P. U., Webb, R. S. and Keigwin, L. D., Eds., American Geophysical Union, Washington D.C., 165–175. Cane, M. A. (1984). Modeling sea level during El Niño, J. Phys. Oceanogr., 14, 1864–1874. Cane, M. A. (1998). A role for the tropical Pacific, Science, 282, 59–61. Cessi, P., Bryan, K. and Zhang, R. (2004). Global seiching of thermocline waters between the Atlantic and the Indian-Pacific Ocean basins, Geophys. Res. Lett., 31, L06201, doi:10.1029/2003GL019091. Chiang, J. C. H., Biasutti, M. and Battisti, D. S. (2003). Sensitivity of the Atlantic Intertropical Convergence Zone to Last Glacial Maximum boundary conditions, Paleoceanogr., 18, doi:10.1029/2003PA000916. Clark, P. U., Marshall, S. J., Clarke, G. K. C., Hostetler, S. W., Licciardi, J. M. and Teller, J. T. (2001). Freshwater forcing of abrupt climate change during the last glaciation, Science, 293, 283–287. Clarke, G. K. C., Marshall, S. J., Hillaire-Marcel, C., Bilodeau, G. and Veiga-Pires, C. (1999). A glaciological perspective on Heinrich events. In Mechanisms of global climate change at millennial time scales, Clark, P. U., Webb, R. S. and Keigwin, L. D., Eds., American Geophysical Union, Washington D.C., 243–262. Clement A. C., Cane M. A. and Seager R. (2001). An orbitally driven tropical source for abrupt climate change, J. Climate, 14, 2369–2375. Clement A. C. and Seager R. (1999). Climate and the tropical oceans, J. Climate, 12, 3383–3401. Clement A. C., Seager R. and Cane M. A. (1999). Orbital controls on the El Niño/Southern Oscillation and the tropical climate, Paleoceanogr., 15, 731–737. Clement, A. C., Seager, R. and Cane, M. A. (2000). Suppression of El Niño during the midHolocene by changes in the Earth’s orbit, Paleoceanogr., 14, 441–456. Clement, A. C., Seager, R. and Murtugudde, R. (2005). Why are there tropical warm pools? J. Climate, 18, 5294–5311. Cohen-Solal, E. and Le Treut, H. (1997). Role of the oceanic heat transport in climate dynamics: A sensitivity study with an atmospheric general circulation model, Tellus, 49A, 371–387. Cuffey, K. M., Clow, G. D., Alley, R. B., Stuiver, M., Waddington, E. D. and Saltus, R. W. (1995). Large arctic temperature change at the Wisconsin-Holocene glacial transition, Nature, 270, 455–458. Curry, W. B. and Oppo, D. W. (1997). Synchronous, high frequency oscillations in tropical sea surface temperatures and North Atlantic Deep Water production during the last glacial cycle, Paleoceanogr., 12, 1–14. Czaja, A. and Marshall, J. C. (2006). The partitioning of poleward heat transport between the atmosphere and ocean, J. Atmos. Sci., in press. Dahl, S. O., and Nesje, A. (1992). Paleoclimatic implications based on equilibrium-line altitude depressions of reconstructed Younger Dryas and Holocene cirque glaciers in inner Nordfjord, western Norway, Palaeogeorg. Palaeoclimatol. Palaeocecol., 94, 87–97. deMenocal, P., Ortiz, J., Guilderson, T., Adkins, J., Sarnthein, M., Baker, L. and Yarusinsky, M. (2000). Abrupt onset and termination of the African Humid Period: Rapid climate responses to gradual insolation forcing, Quat. Sci. Rev., 19, 347–361. Denton, G. H., Alley, R. B., Comer, G. C. and Broecker, W. S. (2005). The role of seasonality in abrupt climate change, Quat. Sci. Rev., 24, 1159–1182. Denton, G. H., and Hendy, C. H. (1994). Younger Dryas age advance of Franz Joseph glacier in the southern Alps of New Zealand, Science, 264, 1434–1437.

June 7, 2007

Time:

02:33pm

chapter12.tex

Abrupt Climate Change | 367 Deser, C., Walsh, J. E., and Timlin, M. S. (2000). Arctic sea ice variability in the context of recent atmospheric circulation trends, J. Climate, 13, 617–633. Emile-Geay, J., Cane, M. A., Naik, N., Seager, R., Clement, A. and van Geen, A. (2003). Warren revisited: Atmospheric freshwater fluxes and “why is no deep water formed in the North Pacific?” J. Geophys. Res. Oceans, 108, No. C6, Art. No. 3178, 10.1029/2001JC001058. Farmer, E. C., deMenocal, P. B. and Marchitto, T. M. (2005). Holocene and deglacial ocean temperature variability in the Benguela upwelling region: Implications for low latitude atmospheric circulation, Paleoceanogr., 20, PA2018, doi:10.1029/2004PA001049. Ganapolski, A. and Rahmstorf, S. (2001). Rapid changes of glacial climate simulated in a coupled climate model, Nature, 409, 153–158. Gasse, F. (2000). Hydrological changes in the African Tropics since the Last Glacial Maximum, Quat. Sci. Rev., 19, 189–211. Giannini, A., Saravanan, R. and Chang P. (2003). Oceanic forcing of Sahel rainfall on interannual and interdecadal time scales, Science, 302, 1027–1030. Gordon, A. L., Weiss, R. F., Smethie, W. M. and Warner, M. J. (1992). Thermocline and intermediate water communication between the South Atlantic and Indian Oceans, J. Geophys. Res., 97, 7223–7240. Hall, N. M. J., Dong, B. and Valdes, P. J. (1996). Atmospheric equilibrium, instability and energy transport at the last glacial maximum, Clim. Dyn., 12, 497–511. Hays, J. D., Imbrie, J. and Shackleton, N. J. (1976). Variations in the Earth’s orbit, pacemaker of the ice ages, Science, 194, 1121–1132. Hazeleger, W., Seager, R., Cane, M. A. and Naik, N. H. (2004). How can tropical Pacific ocean heat transport vary, J. Phys. Oceanogr., 24, 320–333. Hazeleger, W., Seager, R., Visbeck, M., Naik, N. H. and Rodgers, K. (2001). On the impact of the mid-latitude stormtrack on the upper Pacific, J. Phys. Oceanogr., 31, 616–636. Hazeleger, W., Severijns, C., Seager, R. and Molteni, F. (2005). Tropical Pacific-driven decadal energy transport variability, J. Climate, 18, 2037–2051. Held, I. M. (2001). The partitioning of the poleward energy transport between the tropical ocean and atmosphere, J. Atmos. Sci., 58, 943–948. Held, I. M. and Hou, A. Y. (1980). Nonlinear axially symmetric circulations in a nearly inviscid atmosphere, J. Atmos. Sci., 37, 515–533. Hemming, S. R. (2004). Heinrich events: Massive late Pleistocene detritus layers of the North Atlantic and their global imprint, Rev. Geophys., 42, RG1005, doi:10.1029/2003RG000128. Herweijer, C., Seager, R., Winton, M. and Clement, A. C. (2005). Why ocean heat transport warms the global mean climate, Tellus, 57, 662–675. Hewitt, C. D., Stouffer, R. J., Brocolli, A. J., Mitchell, J. B. and Valdes, P. J. (2003). The effect of ocean dynamics in a coupled GCM simulation of the Last Glacial Maximum, Clim. Dyn., 20, 203–218. Hoerling, M. P., Whitaker, J. S., Kumar, A. and Wang, W. (2004). Twentieth century climate change. Part II: Understanding the effect of Indian Ocean warming, Clim. Dyn., 23, 391–405. Hoskins, B. J. and Valdes, P. J. (1990). On the existence of storm tracks, J. Atmos. Sci., 47, 1854– 1864. Hou, A. Y. and Lindzen, R. S. (1992). The influence of concentrated heating on the Hadley circulation, J. Atmos. Sci., 49, 1233–1241. Huang, R.-X., Cane, M. A., Naik, N. and Goodman, P. (2000). Global adjustment of the thermocline in response to deepwater formation, Geophys. Res. Let., 27, 759–762.

June 7, 2007

Time:

02:33pm

chapter12.tex

368 | Seager and Battisti Huang, H.-P., Seager, R. and Kushnir, Y. (2005). The 1976/77 transition in precipitation over the Americas and the influence of tropical SST, Climate Dyn., 24, 721–740. Hughen, K. A., Overpeck, J. T., Lehman, S. J., Kashgarian, M., Southon, J., Peterson, L. C., Alley, R. and Sigman, D. M. (1998). Deglacial changes in ocean circulation from an extended radiocarbon calibration, Nature, 391, 65–68. Hughen, K. A., Overpeck, J. T., Peterson, L. C. and Trumbore, S. (1996). Rapid climate changes in the tropical Atlantic region during the last deglaciation, Nature, 380, 51–54. Hughen, K. A., Southon, J., Lehman, S. J. and Overpeck, J. T. (2000). Synchronous radiocarbon and climate shifts during the last deglaciation, Science, 290, 1951–1954. Hurrell, J. W., Hoerling, M. P., Phillips, A. S. and Xu, T. (2004). Twentieth century North Atlantic climate change. Part I: Assessing determinism, Clim. Dyn., 23, 371–389. Johnson, H. L. and Marshall, D. P. (2004). Global teleconnections of meridional overturning circulation anomalies, J. Phys. Oceanogr., 34, 1702–1722. Kim, J.-H. and Schneider, R. R. (2003). Low-latitude control of interhemispheric sea-surface temperature contract in the tropical Atlantic over the past 21 k years: The possible role of SE trade winds, Clim. Dyn., 21, 337–347. Koutavas, A., Lynch-Stieglitz, J., Marchitto Jr., T. M. and Sachs, J. P. (2002). El Niño-like pattern in ice age tropical Pacific sea surface temperature, Science, 297, 226–230. Latif, M., Roeckner, E., Mikolajewicz, U. and Voss, R. (2000). Tropical stabilization of the thermohaline circulation in a greenhouse warming simulation, J. Climate, 13, 1809–1813. Lau, N.-C. (1979). The observed structure of tropospheric stationary waves and the local balances of vorticity and heat, J. Atmos. Sci., 36, 996–1016. Lea, D. W., Pak, D. P., Peterson, L. C. and Hughen, K. A. (2003). Synchroneity of tropical and high-latitude Atlantic temperatures over the last glacial termination, Science, 301, 1361–1364. Lee, S. (1997). Maintenance of multiple jets in a baroclinic flow, J. Atmos. Sci., 54, 1726–1738. Lee, S. and Kim, H.-K. (2003). The dynamical relationship between subtropical and eddy-driven jets, J. Atmos. Sci., 60, 1490–1503. Lezine, A.-M., Duplessey, J. C. and Cazet, J. P. (2005). West African monsoon variability during the last deglaciation and the Holocene: Evidence from fresh water algae, pollen and isotope data from core KW31, Gulf of Guinea, Palaeogeog. Plaeoclim. Palaeoecol., 219, 225–237. Licciardi, J. M., Clark, P. U., Brook, E. J., Elmore, D. and Sharma, P. (2004). Variable responses of western U.S. glaciers during the last deglaciation, Geol. Soc. of America, 32-1, 81–84. Licciardi, J. M., Teller, J. T. and Clark, P. U. (1999). Freshwater routing by the Laurentide ice sheet during the last deglaciation. In Mechanisms of global climate change at millennial time scales, 794, 177–201. Lindzen, R. S. and Hou, A. Y. (1988). Hadley circulation for zonally averaged heating centered off the equator, J. Atmos. Sci., 45, 2416–2427. Lorenz, E. N. (1968). Climatic determinism, Meteor. Monogr., 8, 1–3. Lorenz, E. N. (1970). Climatic change as a mathematical problem, J. Appl. Meteor., 9, 325–329. Manabe, S., Bryan, K., and Spelman, M. J. (1975). A global ocean-atmosphere climate model. Part I. The atmospheric circulation, J. Phys. Oceanogr., 5, 3–29. Manabe, S. and Stouffer, R. J. (1988). Two stable equilibria of a coupled ocean-atmosphere model, J. Climate, 1, 841–866. Manabe, S. and Stouffer, R. J. (1997). Coupled ocean-atmosphere model response to freshwater input: Comparison to Younger Dryas event, Paleoceanogr., 12, 321–336. Mann, M. E., Cane, M. A., Zebiak, S. E. and Clement, A. (2005). Volcanic and solar forcing of the tropical Pacific over the past 1000 years, J. Climate, 18, 447–456.

June 7, 2007

Time:

02:33pm

chapter12.tex

Abrupt Climate Change | 369 Mantua, N. J., Hare, S. R., Zhang, Y., Wallace, J. M. and Francis, R. C. (1997). A Pacific interdecadal climate oscillation with impacts on salmon production, Bull. Amer. Meteor. Soc., 78, 1069–1079. McManus, J. F., Francois, R., Gherardi, J.-M., Kelgwin, L. D. and Brown-Leger, S. (2004). Collapse and rapid resumption of Atlantic meridional circulation linked to deglacial climate changes, Nature, 428, 834–837. Ohkouchi, N., Eglington, T. L., Keigwin, L. D. and Hayes, J. M. (2002). Spatial and temporal offsets between proxy records in a sediment drift, Science, 298, 1224–1227. Palmen, E. and Newton, C. W. (1969). Atmospheric circulation systems: Their structure and physical interpretation, Academic Press, New York and London, 603 pp. Panetta, R. L. (1993). Zonal jets in wide baroclinically unstable regions: Persistence and scale selection, J. Atmos. Sci., 50, 2073–2106. Peterson, L. C., Haug, G. H., Hughen, K. A. and Rohl, U. (2000). Rapid changes in the hydrologic cycle of the tropical Atlantic during the last glacial, Science, 290, 1947–1951. Renssen, H. and Isarin, R. F. B. (2001). The two major warming phases of the last deglaciation at ∼ 14.7 and ∼ 11.5 ka cal BP in Europe: Climate reconstructions and AGCM experiments, Global and Planetary Change, 30, 117–154. Renssen, H., Isarin, R. F. B., Jacob, D., Podzun, R. and Vandenberghe, J. (2001). Simulation of the Younger Dryas climate in Europe using a regional climate model nested in an AGCM: Preliminary results, Global and Planetary Change, 30, 41–57. Rind, D., Russell, G., Schmidt, G., Sheth, S., Collins, D., deMenocal, P. and Teller, J. (2001). Effects of glacial meltwater in the GISS coupled atmosphere-ocean model. 1. North Atlantic Deep Water response, J. Geophys. Res., 106, 27335–27353. Roe, G. H. and Steig, E. J. (2004). Characterization of millennial-scale climate variability, J. Climate, 17, 1929–1944. Ruddiman, W. F. and McIntyreu, A. (1981). Oceanic mechanisms for amplifying the 23,000 year ice-volume cycle, Science, 212, 617–627. Rühlemann, C., Mulitza, S., Müller, P. J., Wefer, G. and Zhan, R. (1999). Warming of the tropical Atlantic Ocean and slowdown of thermohaline circulation during the last deglaciation, Nature, 402, 511–514. Sachs, J. P. and Lehman, S. J. (1999). Subtropical North Atlantic temperatures 60,000 to 30,000 years ago, Science, 286, 756–759. Schmidt, M. W., Spero, H. J. and Lea, D. W. (2004). Links between salinity variation in the Caribbean and North Atlantic thermohaline circulation, Nature, 428, 160–163. Schneider, E. K. (1977). Axially symmetric steady-state models of the basic state for instability and climate studies. Part II: Nonlinear calculations, J. Atmos. Sci., 34, 280–297. Schneider, E. K., Bengtsson, L., and Hu, Z. Z. (2003). Forcing of Northern Hemisphere climate trends, J. Atmos. Sci., 60, 1504–1521. Schubert, S. D., Suarez, M. J., Region, P. J., Koster, R. D. and Bacmeister, J. T. (2004). Causes of long-term drought in the United States Great Plains, J. Climate, 17, 485–503. Schultz, H., von Rad, U. and Erlenkeuser, H. (1998). Correlation between Arabian Sea and Greenland climate oscillations of the past 110,000 years, Nature, 393, 54–57. Seager, R., Battisti, D. S., Yin, J., Gordon, N., Naik, N. H., Clement, A. C. and Cane, M. A. (2002). Is the Gulf Stream responsible for Europe’s mild winters, Quart. J. Roy. Meteor. Soc., 128, 2563–2586. Seager, R., Harnik, N., Kushnir, Y., Robinson, W. and Miller, J. (2003b). Mechanisms of hemispherically symmetric climate variability, J. Climate, 16, 2960–2978.

June 7, 2007

Time:

02:33pm

chapter12.tex

370 | Seager and Battisti Seager, R., Karspeck, A. R., Cane, M. A., Kushnir, Y., Giannini, A., Kaplan, A., Kerman, B. and Velez, J. (2004). Predicting Pacific decadal variability. In Earth’s climate: The oceanatmosphere interaction, Wang, C., Xie, S. P. and Carton, J. A., Eds., American Geophysical Union, Washington D.C., 105–120. Seager, R., Kushnir, Y., Herweijer, C., Naik, N. and Velez, J. (2005). Modeling of tropical forcing of persistent droughts and pluvials over western North America: 1856–2000, J. Climate, 18, 4068–4091. Seager, R., Murtugudde, R., Naik, N., Clement, A., Gordon, N. and Miller, J. (2003a). Air-sea interaction and the seasonal cycle of the subtropical anticyclones, J. Climate, 16, 1948–1966. Severinghaus, J. P. and Brook, E. J. (1999). Abrupt climate change at the end of the last glacial period inferred from trapped air in polar ice, Science, 286, 930–934. Severinghaus, J. P., Sowers, T., Brook, E. J., Alley, R. A. and Bender, M. L. (1998). Timing of abrupt climate change at the end of the Younger Dryas interval from thermally fractionated gases in polar ice, Nature, 391, 141–146. Shaffrey, L. and Sutton, R. (2004). The interannual variability of energy transports within and over the Atlantic Ocean in a coupled climate model, J. Climate, 17, 1433–1448. Shin, S. I., Liu, Z., Otto-Bliesner, B., Brady, E. C., Kutzbach, J. E. and Harrison, S. P. (2003). A simulation of the Last Glacial Maximum climate using the NCAR-CCSM, Clim. Dyn., 20, 127–151. Shindell, D. T., Miller, R. L., Schmidt, G. A., and Pandolfo, L. (1999). Simulation of recent northern winter climate trends by greenhouse-gas forcing, Nature, 399, 452–455. Shuman, B., Webb III, T., Bartlein, P. and Williams, J. W. (2002). The anatomy of a climatic oscillation: Vegetation change in eastern North America during the Younger Dryas chronozone, Quat. Sci. Rev., 20, 1777–1791. Son, S.-W. and Lee, S. (2005). The response of westerly jets to thermal driving in a primitive equation model, J. Atmos. Sci., 62, 3741–3757. Stephenson, D. B., Hannachi, A. and O’Neill, A. (2004). On the existence of multiple climate regimes, Quart. J. Roy. Meteor. Soc., 130, 583–605. Stott, L., Poulsen, C., Lund, S. and Thunell, R. (2002). Super ENSO and global climate oscillations at millennial time scales, Science, 297, 222–226. Talley, L. D. (2003). Shallow, intermediate and deep overturning components of the global heat budget, J. Phys. Oceanogr., 33, 530–560. Thompson, L. G., Davis, M. E., Mosley-Thompson, E., Sowers, T. A., Henderson, K. A., Zagorodnov, V. S., Lin, P.-N., Mikhalenko, V. N., Campen, R. K., Bolzan, J. F., Cole-Dai, J., and Francou, B. (1998). A 25,000-year tropical climate history from Bolivian ice cores, Science, 282, 1858–1864. Timmermann, A., An, S.-I., Krebs, U. and Goosse, H. (2005). ENSO suppression due to weakening of the North Atlantic thermohaline circulation, J. Climate, 18, 3122–3139. Trenberth, K. and Caron, J. M. (2001). Estimates of meridional atmosphere and ocean heat transports, J. Climate, 14, 3433–3443. Vellinga, M. and Wood, R. A. (2002). Global climatic impacts of a collapse of the Atlantic thermohaline circulation, Climatic Change, 54, 251–267. Vellinga, M., Wood, R. A. and Gregory, J. M. (2002). Processes governing the recovery of a perturbed thermohaline circulation in HadCM3, J. Climate, 15, 764–780. Veronis, G. (1973). Model of world ocean circulation: 1. Wind-driven, two layer, J. Mar. Res., 31, 228–288.

June 7, 2007

Time:

02:33pm

chapter12.tex

Abrupt Climate Change | 371 Wang, X., Auler, A. S., Edwards, R. L., Cheng, H., Cristalli, P. S., Smart, P. L., Richards, D. A. and Shen, C.-C. (2004). Wet periods in northeastern Brazil over the past 210 kyr linked to distant climate anomalies, Nature, 432, 740–743. Warren, B. A. (1983). Why is no deep water formed in the North Pacific? J. Mar. Res., 41, 21–26. Winton, M. (2003). On the climatic impact of ocean circulation, J. Climate, 16, 2875–2889. Wunsch, C. (2003). Greenland-Antarctic phase relations and millennial time-scale climate fluctuations in the Greenland ice-cores, Quat. Sci. Rev., 22, 1631–1646. Yin, J. (2002). The peculiar behavior of baroclinic waves during the midwinter suppression of the Pacific storm track. Ph.D. thesis, University of Washington, 121 pp. Yuan, D., Cheng, H., Edwards, R. L., Dykoski, C. A., Kelly, M. J., Zhang, M., Qing, J., Lin, Y., Wang, Y., Wu, J., Dorale, J. A., An, Z. and Cai, Y. (2004). Timing, duration, and transitions of the last interglacial Asian monsoon, Science, 304, 575–578. Zhang, R. and Delworth, T. L. (2005). Simulated tropical response to a substantial weakening of the Atlantic thermohaline circulation, J. Climate, 18, 1853–1860. Zhang, Y., Wallace, J. M. and Battisti, D. S. (1997). ENSO-like decade-to-century scale variability: 1900–93, J. Climate, 10, 1004–1020.

June 7, 2007

Time:

02:33pm

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Time: 12:59pm

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Contributors

David S. Battisti, Professor, Department of Atmospheric Sciences, and Associate Vice Provost and Director, Earth Initiative, University of Washington, Seattle. Christopher S. Bretherton, Professor, Departments of Atmospheric Sciences and Applied Mathematics, University of Washington, Seattle. Hélène Brogniez, Research Associate, Climate Systems Center, Department of Geophysical Sciences, University of Chicago. Kerry Emanuel, Professor, Department of Earth, Atmospheric, and Planetary Sciences, Massachusetts Institute of Technology. Isaac M. Held, Senior Research Scientist, Geophysical Fluid Dynamics Laboratory, National Oceanic and Atmospheric Administration, and Lecturer with Rank of Professor, Department of Geosciences, Princeton University. Richard S. Lindzen, Alfred Sloan Professor of Meteorology, Department of Earth, Atmospheric, and Planetary Sciences, Massachusetts Institute of Technology. Edward N. Lorenz, Professor Emeritus, Department of Earth, Atmospheric, and Planetary Sciences, Massachusetts Institute of Technology. J. David Neelin, Professor, Department of Atmospheric Sciences and Institute of Geophysics and Planetary Physics, University of California, Los Angeles. Raymond T. Pierrehumbert, Louis Block Professor in Geophysical Sciences and The College, Department of the Geophysical Sciences, University of Chicago. R. Alan Plumb, Professor of Meteorology and Director, Program in Atmospheres, Oceans, and Climate, Department of Earth, Atmospheric, and Planetary Sciences, Massachusetts Institute of Technology.

June 12, 2007

Time: 12:59pm

contributors.tex

374 | Contributors

Walter A. Robinson, Professor, Department of Atmospheric Sciences, University of Illinois at Urbana-Champaign. Rémy Roca, Researcher, Laboratoire de Météorologie Dynamique, Centre National de la Recherche Scientifique, Paris, France. Tapio Schneider, Assistant Professor of Environmental Science and Engineering, California Institute of Technology. Richard Seager, Doherty Senior Research Scientist, Lamont-Doherty Earth Observatory, Columbia University. Adam H. Sobel, Associate Professor, Department of Applied Physics and Applied Mathematics and Department of Earth and Environmental Sciences, Columbia University. Kyle L. Swanson, Associate Professor of Atmospheric Sciences, Department of Mathematical Sciences, University of Wisconsin-Milwaukee. Pablo Zurita-Gotor, Ramon y Cajal Research Scientist, Departamento de Ciencias de la Tierra, Geofísica y Meteorología Facultad de Ciencias Físicas, Universidad Complutense de Madrid.

June 12, 2007

Time:

03:43pm

index.tex

Index

Abrupt climate change atmospheric circulation regimes, heat transport and, 350–51 Caribbean, Tropical Atlantic, Africa and, 336–37, 342 deep ocean circulation and, 339–40 ENSO and, 353–55 global mean climate and, 339 lack of modern analogues to, 342–43 northern extratropics and, 337–38 ocean, atmosphere heat transport and, 361 overview of, 331–33, 362–64 polar ice cores and, 333–35 seasonality around North Atlantic Ocean, 340–41, 349–50 Southern Hemisphere and, 338–39 spatial character of, 343 summer climate and, 352–53 surface Atlantic Ocean and, 335–36 thermohaline circulation and, 343–49 tropical climate reorganization and, 360–61 tropical forcing of, 353 tropical heating, extratropical jets, storm tracks and, 355–60 winter shift to zonal circulation and, 351–52 Adiabatic lapse rate, 53, 70, 189, 191, 195–97, 199–200 Advection-condensation model, 154 African Humid Period, 332–33 Ageostrophic mass flux, 61–62 Albedo abrupt climate change and, 339

ice feedback, 147, 150–51 northern African summer monsoon and, 282 thermal stratification and, 50 Alkenones, 335–36, 364–65 Almost intransitive systems, 335 Angular momentum, 54–55, 220, 252, 255 Annular modes, 110, 134–37 Anomalous gross moist stability mechanism, 294, 295 Antarctica, 332, 333–35 Arakawa-Schubert convection scheme, 190–91 Atlantic Ocean, 335–37, 342. See also North Atlantic Ocean Atmospheric circulation regimes, abrupt climate change and, 350–51 Atmospheric general circulation models (AGCMs) cloud biases and, 308–11 Madden-Julian Oscillation (MJO) bias and, 307 superparameterization and, 315–16 warm eastern ocean/double ITCZ bias and, 303–5 Available potential energy, moist eddies and, 12–13 Back pressure effect, 224–25 Back trajectories, 156–58, 164, 175–78 Background flow, storm track dynamics and, 89–90 Baroclinic adjustment

June 12, 2007

Time:

03:43pm

index.tex

376 | Index boundary layers and, 33–37 Charney problem and, 33 eddy lifecycles and, 37–40 forcing, dissipation and, 40–42 overview of, 25–28 quasigeostrophic constraints on, 70–71, 74 temperature gradients and, 22, 23–24 two-layer QG model and, 4–5, 28–33 Baroclinic eddy fluxes circulation background state and, 110 Eliassen-Palm fluxes and, 138 equatorial superrotation and, 131 jet and storm track shifts and, 125–27 potential vorticity and, 111–19 thermal stratification and, 49, 55–59, 72 tropical warming and, 120, 128 vertical extent of, 65–67 Baroclinic zones, storm tracks and, 79, 85–86, 88–89 Barotropic governor mechanism, 9, 18 Barotropic jets, 17–18 Bermuda Rise, 335–36 Beta-plume, 257 Bølling-Allerød warm period, 331, 335–37, 339–40, 349, 360 Boundary layer. See also Planetary boundary layer (PBL) baroclinic adjustment and, 26, 29–30, 33–37, 40–41 potential vorticity gradients and, 34–35 quasi-equilibrium and, 231 troposphere as, 48 Boussinesq approximation, 12, 38 British Isles, 337, 340, 349 Brunt-Väisälä frequency, 79 Bulk moist stability. See Gross moist stability Buoyancy flux, 13, 188 Capacity of barotropic jets, 17–18 Carbon dioxide, 147, 148–49, 181 Cariaco Basin, 336–37, 339, 345 Caribbean, 336–37, 340 Charney problem, 33, 38 Charney-Stern condition for instability, 29–31 Circulation constraint, 253–57, 257–61 Cirrostratus/cirrocumulus (CsCc), 285–87

Clausius-Clapeyron relation, 145, 153, 159, 180 Climate change. See Abrupt climate change; Global warming Cloud radiative forcing, defined, 308 Cloud-resolving cumulus parameterization. See Superparameterization Cloud-resolving models (CRMs) AGCM response to climate warming and, 318 cloud-climate feedback uncertainties and, 317–18 DARE climate sensitivity results and, 324–26 feedbacks in CRM simulations and, 318–24 global climate dynamics and, 314–17 modeling and, 302–3, 326 tropical biases and, 311–14 Clouds CISK and, 221 convection, precipitation and, 285–87 effect of biases in on modeling, 308–11 effective static stability and, 285–87 radiative effects of, 182, 205–6, 214, 291 surface flux mechanisms and, 293 thermal stratification and, 50 trajectory approach and, 158 water vapor-radiation interactions and, 205–6 Clouds and the Earth’s Radiant Energy System (CERES), 308, 311 Cold trap, 162–163, 171–72 Column radiation model, 49 Condensation, 145, 154, 159–63 Conditional instability of the second kind (CISK), 221, 228 Convection, 53–55, 71, 74, 187–90, 207–10. See also Deep ocean circulation Convective adjustment, 24–25 Convective available potential energy (CAPE) Madden-Julian Oscillation (MJO) bias and, 307 PBL convergence and, 245 precipitation and, 230–31 quasi-equilibrium and, 191 tropical biases and, 311

June 12, 2007

Time:

03:43pm

index.tex

Index | 377 Convective inhibition (CIN), 231, 245 Convective quasi-equilibrium. See also Quasi-equilibrium tropical circulation model (QTCM) gross moist stability and, 278–79 moist static energy budget and, 280–81 moist teleconnection and, 289 overview of, 267–68 Convergence. See also Intertropical convergence zone (ITCZ); Moisture convergence CISK arguments and, 228 convective available potential energy (CAPE) and, 245–46 East Pacific Investigation of the Climate (EPIC) and, 239 Lindzen-Nigam model and, 224 momentum fluxes and, 117, 229 precipitation and, 220 vertical shear and, 39 Coupled ocean-atmosphere general circulation models (CGCMs), 303–5, 306, 326 Coupling abrupt climate change and, 360–61 atmospheric circulation regimes, heat transports and, 350–51 global climate dynamics and, 361–62 monsoons, deserts and, 333 North Atlantic climate change and, 349–50 oceans and, 206–7 summer climate and, 352–53 tropical circulation and, 353–60 tropical climate reorganization and, 360–61 warm eastern ocean/double ITCZ bias and, 303–5 winter shift to zonal circulation and, 351–52 Critical slope of isentropes, 3 Cyclones, 55, 204, 210–13, 230. See also Storm tracks Dansgaard-Oeschger Event, 360 Deep ocean circulation abrupt climate change and, 333, 339–40, 361–62 convective available potential energy (CAPE) and, 230

North Atlantic Deep Water and, 360–61, 362–63 thermodynamic control of, 227 Density temperature, defined, 215 Diabatic Acceleration and REscaling (DARE), 314, 316–17, 324–26 Diabatic processes, 41, 60, 270 Diurnal cycle biases, 307–8, 312–13 Double ITCZ bias, 229, 244, 303–5 Downdrafts, stratification and, 199 Dynamical constraints, thermal stratification and, 49–53 Dynamical time scale, baroclinic adjustment and, 27 Eady growth-rate, 79 Earth Radiation Budget Experiment (ERBE), 280, 308 East Pacific Investigation of the Climate (EPIC), 226–27, 239 Eddy diffusivity, structure of, 67 Eddy lifecycle analysis, 37–40 Eddy momentum fluxes forcing, dissipation and, 40–42 Hadley circulation and, 255 inviscid life cycles and, 37–40 thermal stratification and, 62–65 tropospheric dynamics and, 17 two-layer QG model and, 4 upper trophsphere and, 17 Eddy-mediated responses abrupt climate change and, 355–56 circulation constraint in upper atmosphere and, 257–61 dynamical mechanisms of, 125–27 low latitude-extratropical interactions and, 104–11, 137–39 mean flow state and, 22–23 structural circulation changes and, 127–33 theory of, 111–19 tropical forcing of annular modes and, 134–37 tropical warming and, 119–25 Effective static stability, 12, 276, 285–87 Ekman damping, 61, 85, 86–87, 93 El Niño-Southern Oscillation (ENSO) abrupt climate change and, 353–55

June 12, 2007

Time:

03:43pm

index.tex

378 | Index biases in, 302–3, 306 changes following, 105, 106 climate change and, 342, 353–55 equatorial warming and, 106–8 mechanisms for warming and, 290 modeling of, 219–20, 270–72 precipitation anomalies and, 110 radiative cooling and, 292 Southern Annular Mode (SAM) and, 134–37 storm track dynamics and, 88 wave dynamics and, 287, 289 Eliassen-Palm flux, 81, 96, 115, 138 Emission temperature, tropopause height and, 73–74 Energy fluxes, FHZ idealized moist GCM and, 15–16 Enthalpy flux, 188, 198 Entropy flux, 10–11, 65–69 Eocene, abrupt climate change and, 332 Equatorial forcing, Rossby waves and, 104 Equatorial superrotation, 128–33, 138–39 Equivalent potential temperature, 231 Evaporation fallacy, 153 Feedbacks cloud radiative, 285–87, 317–18 cloud-resolving models (CRMs) and, 317–24 eddy, 125–127 ice albedo, 147, 150–51 lapse-rate, 321 moisture-convection, 206 surface flux mechanisms and, 293 tropical perturbations and, 198–99 water vapor and, 52, 143–44 WISHE and, 204–5, 207, 214 Ferrel cell, 61, 125 Forcings abrupt climate change and, 353 of annular modes, 134–37 baroclinic adjustment and, 27 cloud radiative, 285–87 eddy-mediated responses and, 40–42, 111 of equatorial waveguides, 204 planetary boundary layer (PBL) and, 244–45

potential vorticity gradients and, 42 radiative-convective equilibrium and, 189 storm track dynamics and, 98–101 Free tropospheric humidity (FTH), 152–54 General circulation models (GCMs). See also Atmospheric general circulation models (AGCMs); Coupled ocean-atmosphere general circulation models (CGCMs) abrupt climate change and, 357–58 annular modes and, 134–36 biases in, 222, 312 eddies and, 23 ENSO and, 270–71 moisture convergence and, 222 orbital changes and, 332 relative humidity and, 143–44, 149–50, 177–79 storm track dynamics and, 91–92 temperature gradients and, 22 THC-driving theory and, 344–46 tropical warming and, 132 tropical-extratropical teleconnections and, 110, 123–24 two-layer QG dynamics and, 3 ventilation mechanism and, 283 Geostrophic mass flux, 59–60, 61, 62–65 Gill model, 202, 223–27 Global warming. See also Abrupt climate change AGCM response to, 318 changes in storm tracks and, 80 equatorial superrotation and, 138–39 midlatitude eddy fluxes and, 12 poleward displacement of surface westerlies and, 19 QTCM and, 279–80 relative humidity and, 143, 174–80 Southern Annular Mode (SAM) and, 136–37 tropical precipitation changes and, 272, 293–95 uncertainty in cloud feedback and, 317–24 water vapor and, 146 GMS (gross moist stability) multiplier effect, 290, 291, 292

June 12, 2007

Time:

03:43pm

index.tex

Index | 379 Greenhouse gases, 146–47, 150–51 Greenland, 333–35, 340 GRIP, 334 Gross moist stability. See also GMS (gross moist stability) multiplier effect fluctuations in, 56–58 intertropical convergence zones (ITCZs) and, 278, 296 overview of, 276–78 rich-get-richer mechanism and, 294 Group propagation dynamics, storm tracks and, 82–84 Gulf Stream, THC-driving theory and, 343–44 Hadley cells abrupt climate change and, 355 ageostrophic mass flux and, 61 equatorial superrotation and, 131 extratropical temperature changes and, 124–25 jets and, 132–33 lack of symmetry in, 110 monsoon circulations and, 252 redistribution of heat by, 111 storm tracks, extratropical jet and, 41 temperature and, 122 tropical warming and, 125 zonal asymmetry and, 255 Heat fluxes abrupt climate change and, 350–51, 361–62 Gill model and, 223 Hadley cells and, 111 potential vorticity and, 112–13 storm track dynamics and, 95–96 surface winds and, 225 WISHE and, 204–5, 207, 214 zonal-mean circulation anomalies and, 108 Heinrich events, 365 Held-Hou theory of Hadley circulation, 252–53 Hoskins-Rodwell effect, 263, 282, 283 Humidity, 206. See also Relative humidity Hurricanes, 204, 212–13, 230. See also Cyclones Hysteresis, 100, 130

Ice albedo feedback, 147, 150–51 Ice cores, 333–35, 340–41 Ice microphysics, 310 Ice sheets, 331, 332, 352–53 Icelandic Low, 351, 353 Idealized global climate models dry, 19, 27–28, 69–71 moist, 14–16 relative humidity and, 143–44 simulations using, 318–24 thermal stratification and, 22, 69–71 Interactive Rodwell-Hoskins (IRH) mechanism, 282, 283 Interior atmosphere, mean mass fluxes and, 60–61 Intermediate models, abrupt climate change and, 348 Intertropical convergence zone (ITCZ) abrupt climate change and, 337, 356, 357–58 double ITCZ bias and, 229, 244, 303–5 factors controlling, 220 gross moist stability and, 278, 296 moist static energy and, 279 precipitation and, 228–29 sea surface temperatures and, 240–43 surface winds and, 226–27 THC-driving theory and, 345 Younger Dryas and, 365 Intransitive systems, 335, 363–64 Inverse energy cascade, 6–7, 23 Isentropic coordinates, 58 Isentropic slope, 26, 29, 58–59. See also Static stability Jets abrupt climate change and, 355–60 eddy forcing and, 23 Hadley circulation and, 132–33 lower tropospheric eddies and, 261–64 meridional shift of, 108 momentum flux, convergence and, 39 tropical-extratropical teleconnections and, 127–33 Kapingamarangi, 192 Kelvinoid solution, 287–89, 292–93, 306

June 12, 2007

Time:

03:43pm

index.tex

380 | Index La Niña, abrupt climate change and, 354 Land convective zones, 280 Land-ocean contrast, 280–84, 286–87 Lapse-rate feedback, 321 Last Glacial Maximum, 332, 338, 340, 347 Latent heating, 12, 14, 16, 276 Laurentide ice sheet, 352–53, 359–60 Levy flights, 174 Life-cycle studies, 22, 37–40, 133 Lindzen-Nigam model, 224–28, 245–46 Local modes, 5, 91–92 Longwave cloud forcing, 308 Macroturbulence, 1, 67 Madden-Julian Oscillation (MJO), 186, 205, 212, 306–7, 315 Markov processes, 174 Mass fluxes, balance condition on, 59–65 Matsuno model, 202, 223–27 Maxwell’s relations, 196 Mean field theories, 89 Mean meridional circulation (MMC), 38, 124, 137 Mediation, convective quasi-equilibrium, 290, 291 Meridional mass fluxes, 60, 61 Methane records, 333 Midwinter minimum in storm track activity, 80 Miles-Howard condition, 28 Mixing drying, transport and, 158–63 interior vs. boundary, 33–37 thermal stratification and, 67 trajectory approach and, 154–55, 157–58 transport, drying and, 158–63 Mixing length theory, 22–23, 35 Mixing slope, 11 Moist adiabat, defined, 192 Moist adiabatic lapse rate, 53–54, 189, 191, 195–97, 199–200 Moist available potential energy, 12, 13–14 Moist convection. See also Global warming; Monsoons baroclinic adjustment and, 27–28 challenges in modeling of, 206, 302–3 damping of, 203

precipitation anomalies and, 270–73, 279–80, 285, 289–93 static stability of midlatitudes and, 16 thermal stratification and, 49, 55–58 Tropics and, 52–53 Moist criticality, 192 Moist entropy, 230–31 Moist static energy budget land convective zones and, 280 land-sea contrast and, 280–81 large-scale advection and, 198 overview of, 273–79 precipitation and, 231–38, 245–46, 247 QTCM and, 279–80 surface flux mechanisms and, 293 as tracer, 312 ventilation mechanism and, 282 Moist wave dynamics, 287–89, 291–92 Moisture convergence, 206, 221, 222, 227, 276 Moisture-convection feedback, 206 Momentum fluxes baroclinic adjustment and, 38 convergence and, 117, 229 entropy budget and, 10–11 equatorial superrotation and, 129 forcing and, 40–42 Hadley circulation and, 252, 255 jets and, 39 Kelvinoid solution and, 288 life-cycle studies and, 37–40, 133 limitation of surface winds and, 220 Lindzen-Nigam model and, 225 PBL control and, 228–29 precipitation and, 227–28 radiative-convective equilibrium and, 189 storm track dynamics and, 96–97 temperature and, 41, 62–65, 108 transient eddy heat fluxes and, 108 two-layer QG model and, 4 upper trophsphere and, 17 vertical redistribution of, 38 vertical shear and, 39 zonal-mean circulation anomalies and, 108 Monsoons abrupt climate change and, 338, 345, 346 circulation constraint in upper atmosphere and, 253–56, 257–61

June 12, 2007

Time:

03:43pm

index.tex

Index | 381 dynamical constraints on, 252–53, 264–65 lower tropospheric eddies and, 261–64 nonlocal coupling of with deserts, 333 overview of, 268–70 setting of poleward boundaries for, 281–84 sustained divergent circulations and, 256–57 Montgomery streamfunction/potential, 59, 255–56 Multiplier effect, 290, 291, 293 Multiscale modeling, 314–16 Newtonian relaxation, 69, 93–94, 98 Nigam-Lindzen model, 224–28, 245–46 Nino 3.4 SST Index, 107, 135 Non-locality problem, baroclinic adjustment and, 25 Normal modes, 90–93 North Atlantic Deep Water, 360–61, 362–63 North Atlantic Ocean abrupt climate change and, 340–41, 349–50 Greenland sea surface temperatures and, 335 impact of shift to zonal circulation in winter and, 351–52 THC-driving theory and, 343–44, 345–46, 347 winter shift to zonal circulation and, 351–52 Northern Annular Mode (NAM), 135–36, 342, 353 Northern hemisphere, abrupt climate change and, 332 Oceanic storm tracks, inviscid life cycle calculations and, 40 Oceans. See also Deep ocean circulation; Specific oceans abrupt climate change and, 361 convective quasi-equilibrium and, 280–81, 284–85 coupling to, 206–7 radiative damping over, 293 tropical circulations and, 214 Orbital changes, 331, 332, 354 Outgoing longwave radiation (OLR) baroclinic adjustment and, 25 elevation and, 152 factors determining, 147–51

Madden-Julian Oscillation (MJO) bias and, 307 upper-level potential vorticity and, 258–59 water vapor and, 146–47 Oxygen isotope analysis, 333–35, 338–39 Pacific Ocean, THC-driving theory and, 345 Parallel shear flow instability, 28 Phase speed, storm structure and, 82 Planetary boundary layer (PBL) East Pacific Investigation of the Climate (EPIC) studies and, 239 ENSO biases and, 306 mechanisms for destabilization of, 244–45 precipitation and, 227–29 surface winds and, 224–25 Polar amplification, 147 Polar front jet (PFJ), abrupt climate change and, 355–56 Polar ice cores, 333–35, 340–41 Potential vorticity baroclinic eddies and, 111–19 critical shear and, 29 divergent tropical flow and, 255–56 equatorial superrotation and, 130 forcing and, 42 mixing of interior, 34–37 quasigeostrophic conservation of, 112 static stability and, 33–34 storm track dynamics and, 86–87, 93–94 tropospheric dynamics and, 17 two-layer model and, 31–33 zonal asymmetry and, 253 Prandtl problem, 188–89 Precipitation cloud radiative feedbacks and, 285–87 diurnal cycle biases and, 307–8, 312–13 El Niño and, 110 at given sea surface temperatures, 219–20, 244–47 Lindzen-Nigam model and, 225 Madden-Julian Oscillation (MJO) bias and, 306–7 mechanisms for anomalies in with global warming, 293–95 moist convection and, 268 moist static energy and, 272–73, 284

June 12, 2007

Time:

03:43pm

index.tex

382 | Index moist teleconnection and, 290 monsoons and, 252, 268–70, 281–84 PBL momentum control of, 228–29 radiative-convective equilibrium and, 188–89 theories for, 220–23, 227–28, 238–44 thermodynamic control of, 230–38 Probability density functions (PDF), 156–57, 164–66, 174–78 Pseudo-adiabatic processes, 215 Pseudomomentum, 96–97 Quasi-equilibrium, 231, 289, 291. See also Convective quasi-equilibrium Quasi-equilibrium tropical circulation model (QTCM) defined, 220 ENSO and, 271–72 examples of application of, 210–11 formulation of hypothesis of, 190–91 overview of, 279–80 schematic representation of, 272 surface winds and, 223 theoretical and empirical bases for, 192–95 ventilation mechanism and, 283 Radiation budget equilibrium in, 187–90, 207–10 static stability and, 9–10, 25, 49–53 tropospheric humidity and, 152–53 water vapor and, 146, 147–51 Radiative constraints, 25, 49–53 Radiative equilibrium, 3, 52 Radiative-convective equilibrium baroclinic adjustment and, 27 departures from, 235 effect of departures from, 203 examples of closure and, 210–11 feedback of air motion on temperature and, 198–99 finite-amplitude perturbations and, 207–10 fixed sea surface temperature and, 202–3 humidity fluctuations and, 206 moist-adiabatic lapse rate, tropical disturbances and, 195–97 oceans and, 206–7 overview of, 50, 187–90

precipitation, local surface evaporation and, 220–21 quasi-linear equatorial system for first baroclinic mode and, 199–201 sea surface temperature anomalies and, 201–2 WISHE and, 204–5 Random walk model, 169–70 Reflection principle, 164 Relative humidity advection-condensation model and, 154 atmospheric saturation and, 151–58 climate change and, 174–79 diffusion-condensation model and, 159–63 fluctuations in, 206 forced equilibria and, 169–74 overview of, 143–45, 179–83 radiation budget and, 147–51 stochastic drying and, 163–69 thermal stratification and, 50 water vapor properties and, 145–47 Relaxed equilibrium, defined, 191 Rhines scale, 14, 15, 18 Rich-get-richer mechanism, 294, 295 Rodwell-Hoskins effect, 263, 282, 283 Rossby radius, 23 Rossby waves baroclinic eddies as linear, 126–27 ENSO and, 136, 137, 306 Equatorial forcing and, 104 equatorial superrotation and, 129 interactive Rodwell-Hoskins (IRH) mechanism and, 282 potential vorticity gradients and, 17 storm track dynamics and, 81, 95 Sahara, greening of, 332 Sajama, 338 Salinity, 340, 357 Saturation deficit, moist eddies and, 13 Saturation fallacy, 153 Saturation moist entropy, 54, 195 Saturation specific humidity, 145, 158, 180 Saturation vapor pressure, 145 Sea ice, abrupt climate change and, 349–50 Sea surface temperatures

June 12, 2007

Time:

03:43pm

index.tex

Index | 383 abrupt climate change and, 335–36, 339, 352 anomalies in, equilibrium and, 201–2 El Niño and, 105, 106, 107 fixed, equilibrium and, 202–3 idealized Walker circulation CRM simulations and, 320–24 moist static energy budget and, 284–85 precipitation and surface wind modeling and, 219–20, 244–47 Southern Annular Mode (SAM) and, 134 surface flux mechanisms and, 293 teleconnection and, 291, 293 tropical circulations and, 210, 214 tropical warming and, 119–25 warm eastern ocean/double ITCZ bias and, 304–5 Shear, 28–33, 189, 190 Shear flow instability, 28 Shortwave cloud forcing, 308 Shortwave cutoff, 40 Slantwise convection, 53–55, 71, 74 Snowball Earth, 149, 150 Source of wave activity, 96–97 Southern Annular Mode (SAM), 110, 134–37 Southern Hemisphere, abrupt climate change and, 332, 338–39 Specific humidity, moist static energy and, 272–73 Speleothems, 337, 338 Stability, 230–31. See also Gross moist stability; Static stability Stalactites and stalagmites, 337, 338 Static energy. See Moist static energy budget Static stability baroclinic adjustment and, 33–34, 39 effective, 12, 276, 285–87 isentropic slope and, 11 of midlatitudes, latent heating and, 16 moist convection and, 56 thermal stratification and, 68 two-layer QG model and, 4, 9–10 Stationary waves, 85, 136 Steering level, potential vorticity gradients and, 37 Stochastic forcing, 23, 163–69, 169–74

Stochastic models, storm track dynamics and, 93–95 Storm tracks abrupt climate change and, 355–60 heuristic models for, 95–98 linear theories for, 89–90 nonlinear behavior of, 98–101 normal modes and, 90–93 overview of, 78–84, 101–2 perturbations in, 88–89 stochastic models for, 93–95 tropical-extratropical teleconnections and, 127–33 variability example for, 84–88 Stratosphere, 52, 137 Strict equilibrium, 191, 203 Subtropical jet (STJ), 355–56 Supercooling, 192 Supercriticality insensitivity of to changes in forcing, 31–32 overview of, 28–29 storm track dynamics and, 98–100 thermal stratification and, 65–70, 72–73 two-layer QG model and, 3, 42 Superparameterization, 314–16 Superrotation, 128–33, 138–39 Surface processes, baroclinic adjustment and, 40 Surface temperature, tropopause height and, 50–53 Surface winds ENSO biases and, 306 Gill model for, 223–24, 225–26 at given sea surface temperatures, 219–20, 244–47 Lindzen-Nigam model and, 224–26 moist static energy and, 234 precipitation and, 220 theories for, 220–23 warm eastern ocean/double ITCZ bias and, 304–5 Synoptic development, 26–27 Synoptic eddy life cycles, 81 Teleconnections, 270–73, 279–80, 285, 289–93

June 12, 2007

Time:

03:43pm

index.tex

384 | Index Temperature convective quasi-equilibrium constraint and, 278–79 entropy budget and, 10–11 evaporation and, 153 feedback of air motion, equilibrium and, 198–99 of free troposphere, 201 gross moist stability and, 276 moist convection and, 268 moist static energy and, 272–73 oxygen isotopes as proxy for, 333–35 relative humidity and, 149 saturation vapor pressure and, 145 transient eddy heat and momentum fluxes and, 108 tropical soundings of, 192–93 water vapor, climate change and, 147 Thermal stratification baroclinic eddies and, 58–65 baroclinic entropy flux, supercriticality and, 65–69 eddy scale reduction and, 34 factors determining, 22–24, 49–73 moist convection, baroclinic eddies and, 55–58 radiative and dynamical constraints on, 49–53 simulations with idealized GCM and, 69–71 slantwise convection and, 53–55 two-layer QG model and, 5 Thermocline depth, 360 Thermodynamic control, 230–38, 244 Thermodynamic disequilibrium, 204 Thermohaline circulation (THC) abrupt climate change and, 332, 353, 354, 357–58 deep water formation and, 339–40 impact of shift to zonal circulation in winter and, 352 overview of, 343–44 sea ice and, 349–50 spatial pattern of, 344–47 temporal behavior of, 347–48 Thermostat theory, 26 Tibetan anticyclone, 253, 255–56, 258–60 Time of last saturation model, 154

Topography, storm track dynamics and, 98 Trajectory approach, 154–58, 174–77, 182 Transformed Eulerian mean (TEM) circulation, 115, 117–19, 124, 138 Transient eddies abrupt climate change and, 350–51, 355–56 mean meridional circulation (MMC) and, 124 storm track dynamics and, 81–84, 86–87, 92, 95–96 thermal stratification and, 55–69 zonal-mean circulation anomalies and, 108 Transport, mixing, drying and, 158–63 Tropical-extratropical teleconnections dynamical mechanisms of, 125–27 eddy-mediated, 104–11, 137–39 structural circulation changes and, 127–33 theory of eddy-mediated, 111–19 tropical forcing of annular modes and, 134–37 tropical warming and, 119–25 Tropics. See also Radiative-convective equilibrium abrupt climate change and, 332, 336–37, 338–39, 353, 355–60 cyclones of, 210, 211–13 dry air and, 151 greenhouse effect and, 148–49 Hadley cell responses in, 111, 122 limitations on data collection in, 186 moist convection and, 52–53 moist teleconnection mechanisms in, 289–93 moist-adiabatic lapse rate, disturbances and, 195–97 overview of quasi-equilibrium dynamics of, 186–87, 213–15 red spectrum of, 203, 204 responses of idealized models to warming in, 119–25 Rossby waves and, 104 tropopause height and, 74 Tropopause boundary layer theories and, 48 moist adiabatic lapse rate and, 200 as rigid lid, 199, 204, 212

June 12, 2007

Time:

03:43pm

index.tex

Index | 385 static stability and, 10 surface temperature, tropospheric lapse rate and, 50–53 Troposphere/SST disequilibrium mechanisms, 291, 293 Turbulence, baroclinic forcing and, 98 Two-box model, 207–10 Two-layer QG model critical shear of, 28–33 eddy closure in, 4–5 entropy budget and, 10–11 homogeneous limit and, 5–9 overview of, 2–4 static stability and, 9–10 storm tracks and, 84–93 Upped-ante mechanism, 290–91, 292, 294, 295 Upper tropospheric humidity, 152–54 Variability, storm tracks and, 79–80 Venezuela, 336–37, 339, 345 Ventilation mechanism, 282–84, 295 Vertical shear, 25–26, 39, 41, 79 Walker circulation model, 207–10, 319–22 Water vapor. See also Relative humidity

basic properties of, 145–47 cloud-radiation interactions and, 205–6 feedback modeling and, 143–44 radiation budget and, 147–51 tropopause height and, 52 two-layer QG model and, 12 Wave dynamics, moist teleconnection and, 289, 291 Waveguides, 89, 95–97, 127, 204, 211 Weak temperature gradient (WTG) approximation, 230 Webster model, 202, 223–27 Wind-evaporation feedback, 204–5, 207, 214 Wind-induced surface heat exchange (WISHE), 204–5, 207, 214 Winter, storm tracks and, 80, 85 Younger Dryas. See Abrupt climate change Zonal asymmetry Hadley circulation and, 252–53, 255 Lindzen-Nigam model and, 224 Southern Annular Mode (SAM) and, 136–37 storm track dynamics and, 85–86, 90–93, 97–98